# Spacetime

Posted by jackdoan on September 19, 2013

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The original spacetime accidentally was lost  during a save See 4 parts of it above.

10-24-15 Update : For the larger parts of the following collection of Prior Art pertaining to my discloser was coped unchanged from its source.

3 -23-15 Continued from page 2012 and beyond.   Clarifying J_Cosmic String  compared to current science cosmic string. The inflation right after the  big bang some stringy matter goes away from the high density and temperature to lower density and cooler volume. Current science at this fraction of a second matter as quarks. J_Cosmic String at this fraction of a second is composed of the tiny strings from "the  string theory" create in taking the path of least resistance J_Cosmic String J_Torus with outside surface formed like a Sine wave and the inside surface a cylinder. Inside J_Torus J_Wave Length like a Sine Wave length is a J_Rod the same length and volume as J_Wave. J_Mass of these tiny strings is contained in both of these volumes. The physics fact that for every action there is an equal and opposite reaction applies to J_Tours J_Wave with J_Mass moving near the speed of light outward on a J_Spiral the J_Rod reacts with its = mass inward along same segment at near the speed of light. The force and momentum is caused by the pressure coming from big bang.

3-25-15 I will also be entering some Cosmic String Physics on 2012 and beyond Page & below :

10-18-15 Update: For the larger parts of the following collection of Prior Art pertaining to my discloser was coped unchanged from its source.

## Cosmic String

We've blown through black holes and wormholes, but there's yet another possible means of time traveling via theoretic cosmic phenomena. For this scheme, we turn to physicist J. Richard Gott, who introduced the idea of cosmic string back in 1991. As the name suggests, these are stringlike objects that some scientists believe were formed in the early universe.

These strings may weave throughout the entire universe, thinner than an atom and under immense pressure. Naturally, this means they'd pack quite a gravitational pull on anything that passes near them, enabling objects attached to a cosmic string to travel at incredible speeds and benefit from time dilation. By pulling two cosmic strings close together or stretching one string close to a black hole, it might be possible to warp space-time enough to create what's called a closed timelike curve.

What is a 'cosmic string'?

Super strings are what physicists consider the 'end of the line' in describing the structure of both particles and fields in nature. There typical sizes are believed to be near 10^-33 centimeters, although some recent proposals suggest that these strings can be a lot bigger. String theory is the current theory which looks like it might be able to describe, in one consistent language, how all of the particles and fields we know about, work, and blend together to form the physical world and all of its laws and 'symmetries'. It will explain why he have gravity, why the electron has the mass it does, and why the electron/proton mass ratio is 1/1864, among many other things.

Cosmic strings are another thing entirely.

When the universe was inflating after the Big Bang, the field that caused this rapid inflation 'crystalized' so that space-itself became a patina of stringy discontinuities and other patterns. Its hard to describe this, but when you look at the surface of a pond, you see plates of ice grow until they collide with thtringeir neighbors, and you get these patterns on the ice that look like lines. When 3-d space cooled, some cosmologists think that space also took-on a texture which consists of lines ( strings) and other kinds of 'textures'. When matter cooled to form galaxies, it tended to fall into these stringy boundaries too. They are called Cosmic Strings, but they have nothing to do with Super Strings.

3-26-15 J_Texture is created by the J_Spiralling of the J_Strings Disclosed here in respect to "( strings) and other kinds of 'textures'. " Just 3-4 lines up in the the above article that I agree with.

3-29-15 J_Paradigm new and unique (J_Cosmic J_String J_Wave) correlates loosely with the above article "physics Richard Gott". Next at " gravitational pull " J_Paradigm including J_Aether that is  also a (J_Cosmic J_String J_Wave) And the magnitude of J_Gravity is the push of tremendous number of J_Waves pushing due to J_Spiraling from J_Masses . J_Paradigm J_Particles are made up from J_String formation of elementary J_Particles. Like Einstein Mass push concept of a stretched rubber sheet the masses are pushed toward each other in 3 dimensional space time, J_Mass carried in the J_Torus cress  of J_Wave an moving  the same fraction of J_Mass J_Rod at nearly C in along the same section of  J_Spiral in the opposite direction toward The origin J_Mass of the J_Spiral creating the equal and opposite reaction  in J_Mass. This continuing Disclosure of J_Paradigm is using a lot of specifitivity for thorough disclosure:

3-30-15

J_Paradigm J_BlackHole J_CosmicString circulation of J_wave disassembling in the extremely hot center of J_BlackHole, into anew J_CosmicString and Jeted back through event horizon in a new J_Inflation  cycle. The J_Bigbang is the same J_Process which is as follows:

Paralleling "the tiny strings of THE STRING THEORY"@current physicists WEBSITES  sciences languages. J_Quarks are like the particle physics debris that look like strings in a supper colider cloud chamber. Using their re'engineering  J_Re'Engineering J_String @ ( 10^-33 Centimeters size J_Black-Hole J_Big-bang extremely density Hottest possibly tiny volume with a highest number of other J_Strings my J_Process  of them taking the path of least resistance form J_Wave 4 dimensional dynamic  J_Elementary units of closed loop J_Closed-loop-String, Open loop J_Open-loop-String, J_Cosmic_J_Strings are formed. J_Aether is created by this same J_Process including J_Spacetime is reset. My J_Paradigm in respect to current particle physist re'engineering quarks and their spins established insist they are madeup of J_Wave J_Strings. The new and unique J_Wave J_Mass 1/2 in the J_Tours outer crest wave like a sign wave and iner surface a cylinder and with the other 1/2 J_Mass in J_Rod both J_Wavelength moving at near the speed of light in opposite direction along the same segmrent of a J_Spiral creating J_Aether, J_SpaceTime, J_Steady State J_Recycled J_Universe. This J_Paradigm incompasses new and unique J_Disclosures: The J_Spiral and its relation toJ_Gravity I have been thinking about all my life  since highschool  physics and aljabra.

3-31-15 The J_Paradigm has evolved over 78 years of thought about since in highschool when Einstein  was talking to a boy as steam locomotive was passing on the RR tracks. He told the boy that it could be an illusion it coulld be created from waves only. When my school teachers couldn't explain my questions adequately J_Paradign evolved further. I had trouble with my Highscool Aljbra and my teacher Mr. Saddler taugh me to in my mind the real phenomenon istead of the symbol X or Y. This why my skill in math language is limited, J_Paradigm Skill benefited.

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3-31-15 As I continue J_Paradigm I  have copied the Article below from XFER ext hard drive on MacPro and an identical copy above which I will not alter. The one below I will alter to simplified and note alterations here just above copy below: The misspelling underlined in red on my Dell Desktop Windows 7 PC I delete from below copy. I deleted these up to Simple NCIBI Directory & then deleted more 0n 4-3-15 Then on 4-15-15 I deleted all of that copy above.

Disclosing J-INERTIA J_GRAVITY J_ Mass & J_Momentum My perception of J_Aether are all due to J_Wave Phenomenon. First I Disclose what is inside of J_Wave and J_Rod That J_Mass consist of. The tiny strings of "The String Theory are similar to J_Base-string but new and unique J_Base-string form in taking the initial path of least resistance as the cooling from the hot center of J_Black whole or the big bang the cooling mass adapts to string shape of J_Mass. The J_Process continues as J_Open and J_Closed J_Base-strings form J_Wave-string created in taking the next path of least resistance. As J_Base-string J_Wave takes the third path of  least resistance. The Speed of J_Wave-crest is near <C (speed of light) a very high vibration along J_Wave-crest of this tiny 10^-33 CM J_Torus diameter with [

1 Planck length =
SI units
16.162×10−36 m 16.162×10−27 nm
Natural units
11.706 S 305.42×10−27 a0
US customary units (Imperial units)
53.025×10−36 ft 636.30×10−36 in

In physics, the Planck length, denoted P, is a unit of length, equal to 1.616199(97)×10−35 metres. It is a base unit in the system of Planck units, developed by physicist Max Planck. The Planck length can be defined from three fundamental physical constants: the speed of light in a vacuum, the Planck constant, and the gravitational constant.]   a Planck (h length) J-Wavelength. time = distance h /<C. The Time Frequency = h/<C J_Wave-crest J_Vibration. J_Mass in The J_Wave-crest is where the J_Wave J_Gravity J_Inertia J_Phonon Starts.

4-3-15 J_Mass-dark-mater and J-Mass-visible-matter possess J_Higgs quality at the third  path to least resistance J-String. J-Spirals' J-Aether and J_Inertia are both part of J_Unified-J_Field. Our entire J_Universe in no mater how infinitely small by its inversely square of the distances has mutual effect on all J_Waves.

# Planck time

In physics, the Planck time (tP) is the unit of time in the system of natural units known as Planck units. It is the time required for light to travel, in a vacuum, a distance of 1 Planck length.[1] The unit is named after Max Planck, who was the first to propose it.

The Planck time is defined as:[2]

$t_\mathrm{P} \equiv \sqrt{\frac{\hbar G}{c^5}}\approx 5.39106 (32) \times 10^{-44}\ \mathrm{s}$

where:

ħ = h/π is the reduced Planck constant (sometimes h is used instead of ħ in the definition[1])
G = gravitational constant
c = speed of light in a vacuum
s is the SI unit of time, the second.

The two digits between parentheses denote the standard error of the estimated value.

## Physical significance

The Planck time is the unique combination of the gravitational constant G, the special-relativistic constant c, and thequantum constant ħ, to produce a constant with units of time. Because the Planck time comes from dimensional analysis, which ignores constant factors, there is no reason to believe that exactly one unit of Planck time has any special physical significance. Rather, the Planck time represents a rough time scale at which quantum gravitational effects are likely to become important. The nature of those effects, and the exact time scale at which they would occur, would need to be derived from an actual theory of quantum gravity. All scientific experiments and human experiences happen over billions of billions of billions of Planck times,[3] making any events happening at the Planck scale hard to detect. As of May 2010, the smallest time interval uncertainty in direct measurements is on the order of 12 attoseconds (1.2 × 10−17 seconds), about 3.7 × 1026 Planck times.[4]

4-4-15 The J_Cosmic-strings both in J_Aether and J_Gravity&J_Inertia J_Strings with J_Mass. The J_Field-J_Mass is created in this third path of least resistance inside J_Wave J_Toroidal J_Spin of J_Closed-loop and J_Open-loop tiny J_Strings rolling J_Toroidal in  J_Wave-tours. This J_Mass-field is J_Paramount in all J_Forces. J_Unified-force-field like Albert Einstein was looking for in the end of his endeavours. Just below I inserted a rough TurboCad 6 drawing that I can refer to in this blog space-time page. Delineation of sub-microscopic J_Strings disclosure of this new and unique invention of Jack L Doan 4-4-15

4-6-15 I isert the Heading LXLE Eee PC, got its TCW J_Waves DWG : [

J_Cosmic-String containing tiny J_Wave open and closed J_Strings in torus J_Wave-Crest J_Spin creating J_Mass in donut shaped moving at near speed of light out from high density and heat along J_Spiral. An equal volume of J_Mass J_Rod moving at near the speed of light in the opposite direction:

Scan_Doc0001.pdf

]

4-8-15 J_Waves' J_Force-field is the J_Push between two or more bodies of J_Mass trying to  occupy the same J_Space-time at the same time. This J_Action and J_Reaction  is the same as J_Magnetic J_Force-feild. J_Unified-force-feild is a new and unique Disclosure.

4-9-15 I reviewed all pages of my blog Inserting the title ABOUT on the About Page. I haven't found where I Disclosed electromagnetics, I will review some more but at the top of this page where I stated the original Spacetime page was lost during a save may have been where that Disclosure was on that part of my Unified Force Field. I never got in touch with Concrete5. I found some of that Spacetime printed 3/25/14 5 double sided paper record. I will scan it and put at the top of this page latter.

J_Unified-Force-Field Magnetism J_Strings of a body of J_Magnetic material's J_Atom J_Electron-Shells contain to create this field and force.

4-10-15 Though there isn't an absolute reference plane in the J_Universe there is instantaneous reference J_Plane localized in a J_Wave-String where the J_Wave-Crest is moving out along theJ_Spiral and the J_Rod is moving in along that same J_Spiral. This point could be either in part of an atom or J_Aether. J_Electromagnetism or J_Magnetism. Both depend on J_Aether to function. J_Magnetic material a small body or the size of the earth J_Magnetic-Feild associated with this material  exists in J_Aether. This is also the case for J_Electromagnetic radiation including light. J_Electro-mechanical machinery and a electric charge work through J_Aether. APM magnetic and electrical J_Charges exists in J_Aether. I was puzzled in high school physics about the field lines of a magnet when a plane like a sheet pf paper was sit on top or under it with iron filings was sprinkled on it they would align themselves to that field. That J_Field from the magnetic material are J_Stringes from the atoms of that material which the the iron filings line up with in J_Aether-fabric. This like .n the case of J_Gravity the J_Magnetic-material (iron) is pushed into alignment as they are magnetized. Even if this material becomes sub-microscopic into the J_Quatum domain where quantum gravity is in effect because the extremely small bit of J_Magnetic-Material is a J_Quantum. The J_Referance-plane can be where simple J_Math in J_Wave domain computations. j_Aether-Fabric exerts less J_Push J_Force than J_Magnetic-Field J_Push J_Unified-Field of J_Gravity and J_Magnetisim in J_APM is Disclosed.

## Notes and references

1. a b "Big Bang models back to Planck time"Georgia State University. 19 June 2005.
2. ^ CODATA Value: Planck Time – The NIST Reference on Constants, Units, and Uncertainty.
3. ^ "First Second of the Big Bang". How The Universe Works 3. 2014. Discovery Science.
4. ^ "12 attoseconds is the world record for shortest controllable time". 2010-05-12. Retrieved 2012-04-19.

4-13-15 Copied from My E-mail Quora Digest:

# There are no particles, there are only fields

Quantum foundations are still unsettled, with mixed effects on science and society. By now it should be possible to obtain consensus on at least one issue: Are the fundamental constituents fields or particles? As this paper shows, experiment and theory imply unbounded fields, not bounded particles, are fundamental. This is especially clear for relativistic systems, implying it's also true of non-relativistic systems. Particles are epiphenomena arising from fields. Thus the Schroedinger field is a space-filling physical field whose value at any spatial point is the probability amplitude for an interaction to occur at that point. The field for an electron is the electron; each electron extends over both slits in the 2-slit experiment and spreads over the entire pattern; and quantum physics is about interactions of microscopic systems with the macroscopic world rather than just about measurements. It's important to clarify this issue because textbooks still teach a particles- and measurement-oriented interpretation that contributes to bewilderment among students and pseudoscience among the public. This article reviews classical and quantum fields, the 2-slit experiment, rigorous theorems showing particles are inconsistent with relativistic quantum theory, and several phenomena showing particles are incompatible with quantum field theories.
Art Hobson There are no particles, there are only fields 1 TO BE PUBLISHED IN AMERICAN JOURNAL OF PHYSICS There are no particles, there are only fields Art Hobson a Department of Physics, University of Arkansas, Fayetteville, AR, ahobson@uark.edu Quantum foundations are still unsettled, with mixed effects on science and society. By now it should be possible to obtain consensus on at least one issue: Are the fundamental constituents fields or particles? As this paper shows, experiment and theory imply unbounded fields, not bounded particles, are fundamental. This is especially clear for relativistic systems, implying it's also true of non-relativistic systems. Particles are epiphenomena arising from fields. Thus the Schroedinger field is a space-filling physical field whose value at any spatial point is the probability amplitude for an interaction to occur at that point. The field for an electron is the electron; each electron extends over both slits in the 2-slit experiment and spreads over the entire pattern; and quantum physics is about interactions of microscopic systems with the macroscopic world rather than just about measurements. It's important to clarify this issue because textbooks still teach a particles- and measurement-oriented interpretation that contributes to bewilderment among students and pseudoscience among the public. This article reviews classical and quantum fields, the 2-slit experiment, rigorous theorems showing particles are inconsistent with relativistic quantum theory, and several phenomena showing particles are incompatible with quantum field theories. I. INTRODUCTION Physicists are still unable to reach consensus on the principles or meaning of science's most fundamental and accurate theory, namely quantum physics. An embarrassment of enigmas abounds concerning wave-particle duality, measurement, nonlocality, superpositions, uncertainty, and the meaning of quantum states.1 After over a century of quantum history, this is scandalous. 2, 3 It's not only an academic matter. This confusion has huge real-life implications. In a world that cries out for general scientific literacy, 4 quantuminspired pseudoscience has become dangerous to science and society. What the Bleep Do We Know, a popular 2004 film, won several film awards and grossed $10 million; it's central tenet is that we create our own reality through consciousness and quantum mechanics. It features physicists saying things like "The material world around us is nothing but possible movements of consciousness," it purports Art Hobson There are no particles, there are only fields 2 to show how thoughts change the structure of ice crystals, and it interviews a 35,000 year-old spirit "channeled" by a psychic.5 "Quantum mysticism" ostensibly provides a basis for mind-over-matter claims from ESP to alternative medicine, and provides intellectual support for the postmodern assertion that science has no claim on objective reality.6 According to the popular television physician Deepak Chopra, "quantum healing" can cure all our ills by the application of mental power. 7 Chopra's book Ageless Body, Timeless Mind, a New York Times Bestseller that sold over two million copies worldwide, is subtitled The Quantum Alternative to Growing Old. 8 Quantum Enigma, a highly advertised book from Oxford University Press that's used as a textbook in liberal arts physics courses at the University of California and elsewhere, bears the sub-title Physics Encounters Consciousness. 9 It's indeed scandalous when librarians and book store managers wonder whether to shelve a book under "quantum physics," "religion," or "new age." For further documentation of this point, see the Wikipedia article "Quantum mysticism" and references therein. Here, I'll discuss just one fundamental quantum issue: field-particle (or wave-particle) duality. More precisely, this paper answers the following question: Based on standard non-relativistic and relativistic quantum physics, do experiment and theory lead us to conclude that the universe is ultimately made of fields, or particles, or both, or neither? There are other embarrassing quantum enigmas, especially the measurement problem, as well as the ultimate ontology (i.e. reality) implied by quantum physics. This paper studies only field-particle duality. In particular, it's neutral on the interpretations (e.g. many worlds) and modifications (e.g. hidden variables, objective collapse theories) designed to resolve the measurement problem. Many textbooks and physicists apparently don't realize that a strong case, supported by leading quantum field theorists, 10, 11 12, 13, 14, 15, 16, 17 for a pure fields view has developed during the past three decades. Three popular books are arguments for an all-fields perspective. 18, 19, 20 I've argued the advantages of teaching non-relativistic quantum physics (NRQP, or "quantum mechanics") from an all-fields perspective; 21 my conceptual physics textbook for non-science college students assumes this viewpoint.22 There is plenty of evidence today for physicists to come to a consensus supporting an all-fields view. Such a consensus would make it easier to resolve other quantum issues. But fields-versus-particles is still alive and kicking, as you can see by noting that "quantum field theory" (QFT) and "particle physics" are interchangeable names for the same discipline! And there's a huge gap between the views of leading quantum physicists (Refs. 10-18) and virtually every quantum physics textbook. Art Hobson There are no particles, there are only fields 3 Physicists are schizophrenic about fields and particles. At the high-energy end, most quantum field theorists agree for good reasons (Secs. III, V, VI) that relativistic quantum physics is about fields and that electrons, photons, and so forth are merely excitations (waves) in the fundamental fields. But at the low-energy end, most NRQP education and popular talk is about particles. Working physicists, teachers, and NRQP textbooks treat electrons, photons, protons, atoms, etc. as particles that exhibit paradoxical behavior. Yet NRQP is the non-relativistic limit of the broader relativistic theory, namely QFTs that for all the world appear to be about fields. If QFT is about fields, how can its restriction to non-relativistic phenomena be about particles? Do infinitely extended fields turn into bounded particles as the energy drops? As an example of the field/particle confusion, the 2-slit experiment is often considered paradoxical, and it is a paradox if one assumes that the universe is made of particles. For Richard Feynman, this paradox was unavoidable. Feynman was a particles guy. As Frank Wilczek puts it, "uniquely (so far as I know) among physicists of high stature, Feynman hoped to remove field-particle dualism by getting rid of the fields " (Ref. 16). As a preface to his lecture about this experiment, Feynman advised his students, Do not take the lecture too seriously, feeling that you really have to understand in terms of some model what I am going to describe, but just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, "But how can it be like that?" because you will get "down the drain," into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.23 There are many interpretational difficulties with the 2-slit experiment, and I'm certainly not going to solve all of them here. But the puzzle of wave-particle duality in this experiment can be resolved by switching to an all-fields perspective (Sect. IV). Physics education is affected directly, and scientific literacy indirectly, by what textbooks say about wave-particle duality and related topics. To find out what textbooks say, I perused the 36 textbooks in my university's library having the words "quantum mechanics" in their title and published after 1989. 30 implied a universe made of particles that sometimes act like fields, 6 implied the fundamental constituents behaved sometimes like particles and sometimes like fields, and none viewed the universe as made of fields that sometimes appear to be particles. Yet the leading quantum field theorists argue explicitly for the latter view (Refs. 10-18). Something's amiss here. Art Hobson There are no particles, there are only fields 4 The purpose of this paper is to assemble the strands of the fields-versusparticles discussion in order to hasten a consensus that will resolve the waveparticle paradoxes while bringing the conceptual structure of quantum physics into agreement with the requirements of special relativity and the views of leading quantum field theorists. Section II argues that Faraday, Maxwell, and Einstein viewed classical electromagnetism as a field phenomenon. Section III argues that quantum field theory developed from classical electrodynamics and then extended the quantized field notion to matter. Quantization introduced certain particle-like characteristics, namely individual quanta that could be counted, but the theory describes these quanta as extended disturbances in space-filling fields. Section IV analyzes the 2-slit experiment to illustrate the necessity for an all-fields view of NRQP. The phenomena and the theory lead to paradoxes if interpreted in terms of particles, but are comprehensible in terms of fields. Section V presents a rigorous theorem due to Hegerfeldt showing that, even if we assume a very broad definition of "particle" (namely that a particle should extend over only a finite, not infinite, region), particles contradict both relativity and quantum physics. Section VI argues that quantized fields imply a quantum vacuum that contradicts an allparticles view while confirming the field view. Furthermore, two vacuum effects-- the Unruh effect and single-quantum nonlocality--imply a field view. Thus, many lines of reasoning contradict the particles view and confirm the field view. Section VII summarizes the conclusions. II. A HISTORY OF CLASSICAL FIELDS Fields are one of physics' most plausible notions, arguably more intuitively credible than pointlike particles drifting in empty space. It's perhaps surprising that, despite the complete absence of fields from Isaac Newton's Principia (1687), Newton's intuition told him the universe is filled with fields. In an exchange of letters with Reverend Richard Bentley explaining the Principia in non-scientists' language, Newton wrote: It is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact… That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.24 Art Hobson There are no particles, there are only fields 5 But Newton couldn't find empirical evidence to support a causal explanation of gravity, and any explanation remained purely hypothetical. When writing or speaking of a possible underlying mechanism for gravity, he chose to remain silent, firmly maintaining "I do not feign hypotheses" (Ref. 18, p. 138). Thus it was generally accepted by the beginning of the 19th century that a fundamental physical theory would contain equations for direct forces-at-a-distance between tiny indestructible atoms moving through empty space. Before long, however, electromagnetism and relativity would shift the emphasis from action-at-a-distance to fields. The shift was largely due to Michael Faraday (1791-1867). Working about 160 years after Newton, he introduced the modern concept of fields as properties of space having physical effects. 25 Faraday argued against action-at-a-distance, proposing instead that interactions occur via space-filling "lines of force" and that atoms are mere convergences of these lines of force. He recognized that a demonstration that non-instantaneous electromagnetic (EM) interactions would be fatal to action-at-a-distance because interactions would then proceed gradually from one body to the next, suggesting that some physical process occurred in the intervening space. He saw lines of force as space-filling physical entities that could move, expand, and contract. He concluded that magnetic lines of force, in particular, are physical conditions of "mere space" (i.e. space containing no material substance). Today this description of fields as "conditions of space" is standard.26 James Clerk Maxwell (1831-1879) was less visionary, more Newtonian, and more mathematical than Faraday. By invoking a mechanical ether that obeyed Newton's laws, he brought Faraday's conception of continuous transmission of forces rather than instantaneous action-at-a-distance into the philosophical framework of Newtonian mechanics. Thus Faraday's lines of force became the state of a material medium, "the ether," much as a velocity field is a state of a material fluid. He found the correct dynamical field equations for EM phenomena, consistent with all known experimental results. His analysis led to the predictions of (1) a finite transmission time for EM actions, and (2) light as an EM field phenomenon. Both were later spectacularly confirmed. Despite the success of his equations, and despite the non-appearance of ether in the actual equations, Maxwell insisted throughout his life that Newtonian mechanical forces in the ether produce all electric and magnetic phenomena, a view that differed crucially from Faraday's view of the EM field as a state of "mere space." Experimental confirmations of the field nature of light, and of a time delay for EM actions, were strong confirmations of the field view. After all, light certainly seems real. And a time delay demands the presence of energy in the intervening space in order to conserve energy. That is, if energy is emitted here Art Hobson There are no particles, there are only fields 6 and now, and received there and later, then where is it in the meantime? Clearly, it's in the field.27 Faraday and Maxwell created one of history's most telling changes in our physical worldview: the change from particles to fields. As Albert Einstein put it, “Before Maxwell, Physical Reality …was thought of as consisting in material particles…. Since Maxwell's time, Physical Reality has been thought of as represented by continuous fields, ...and not capable of any mechanical interpretation. This change in the conception of Reality is the most profound and the most fruitful that physics has experienced since the time of Newton.”28 As the preceding quotation shows, Einstein supported a "fields are all there is" view of classical (but not necessarily quantum) physics. He put the final logical touch on classical fields in his 1905 paper proposing the special theory of relativity, where he wrote "The introduction of a 'luminiferous' ether will prove to be superfluous."29 For Einstein, there was no material ether to support light waves. Instead, the "medium" for light was space itself. That is, for Einstein, fields are states or conditions of space. This is the modern view. The implication of special relativity (SR) that energy has inertia further reinforces both Einstein's rejection of the ether and the significance of fields. Since fields have energy, they have inertia and should be considered "substance like" themselves rather than simply states of some substance such as ether. The general theory of relativity (1916) resolves Newton's dilemma concerning the "absurdity" of gravitational action-at-a-distance. According to general relativity, the universe is full of gravitational fields, and physical processes associated with this field occur even in space that is free from matter and EM fields. Einstein's field equations of general relativity are Rµν(x) - (1/2)gµν(x)R(x) = Tµν(x) , (1) where x represents space-time points, µ and ν run over the 4 dimensions, gµν(x) is the metric tensor field, Rµν(x) and R(x) are defined in terms of gµν(x), and Tµν(x) is the energy-momentum tensor of matter. These field (because they hold at every x) equations relate the geometry of space-time (left-hand side) to the energy and momentum of matter (right-hand side). The gravitational field is described solely by the metric tensor gµν(x). Einstein referred to the left-hand side of Eq. (1) as "a palace of gold" because it represents a condition of space-time and to the righthand side as "a hovel of wood" because it represents a condition of matter. Art Hobson There are no particles, there are only fields 7 Thus by 1915 classical physics described all known forces in terms of fields- -conditions of space--and Einstein expressed dissatisfaction that matter couldn't be described in the same way. III. A HISTORY AND DESCRIPTION OF QUANTUM FIELDS From the early Greek and Roman atomists to Newton to scientists such as Dalton, Robert Brown, and Rutherford, the microscopic view of matter was always dominated by particles. Thus the non-relativistic quantum physics of matter that developed in the mid-1920s was couched in particle language, and quantum physics was called "quantum mechanics" in analogy with the Newtonian mechanics of indestructible particles in empty space.30 But ironically, the central equation of the quantum physics of matter, the Schroedinger equation, is a field equation. Rather than an obvious recipe for particle motion, it appears to describe a time-dependent field Ψ(x,t) throughout a spatial region. Nevertheless, this field picked up a particle interpretation when Max Born proposed that Ψ(x0,t) is the probability amplitude that, upon measurement at time t, the presumed particle "will be found" at the point x0. Another suggestion, still in accord with the Copenhagen interpretation but less confining, would be that Ψ(x0,t) is the probability amplitude for an interaction to occur at x0. This preserves the Born rule while allowing either a field or particle interpretation. In the late 1920s, physicists sought a relativistic theory that incorporated quantum principles. EM fields were not described by the nonrelativistic Schroedinger equation; EM fields spread at the speed of light, so any quantum theory of them must be relativistic. Such a theory must also describe emission (creation) and absorption (destruction) of radiation. Furthermore, NRQP says energy spontaneously fluctuates, and SR (E=mc 2 ) says matter can be created from non-material forms of energy, so a relativistic quantum theory must describe creation and destruction of matter. Schroedinger's equation needed to be generalized to include such phenomena. QFTs, described in the remainder of this Section, arose from these efforts. A. Quantized radiation fields "How can any physicist look at radio or microwave antennas and believe they were meant to capture particles?"31 It's implausible that EM signals transmit from antenna to antenna by emitting and absorbing particles; how do antennas "launch" or "catch" particles? In fact, how do signals transmit? Instantaneous Art Hobson There are no particles, there are only fields 8 transmission is ruled out by the evidence. Delayed transmission by direct actionat-a-distance without an intervening medium has been tried in theory and found wanting. 32 The 19-century answer was that transmission occurs via the EM field. Quantum physics preserves this notion, while "quantizing" the field. The field itself remains continuous, filling all space. The first task in developing a relativistic quantum theory was to describe EM radiation--an inherently relativistic phenomenon--in a quantum fashion. So it's not surprising that QFT began with a quantum theory of radiation. 33 34 35 This problem was greatly simplified by the Lorentz covariance of Maxwell's equations-- they satisfy SR by taking the same form in every inertial reference frame. Maxwell was lucky, or brilliant, in this regard. A straightforward approach to the quantization of the "free" (no source charges or currents) EM field begins with the classical vector potential field A(x,t) from which we can derive E(x,t) and B(x,t). 36 Expanding this field in the set of spatial fields exp(±ik•x) (orthonormalized in the delta-function sense for large spatial volumes) for each vector k having positive components, A(x,t) = ∑k [a(k,t) exp(ik•x) + a* (k,t) exp(-ik•x)]. (2) The field equation for A(x,t) then implies that each coefficient a(k,t) satisfies a classical harmonic oscillator equation. One regards these equations as the equations of motion for a mechanical system having an infinite number of degrees of freedom, and quantizes this classical mechanical system by assuming the a(k,t) are operators aop(k,t) satisfying appropriate commutation relations and the a*(k,t) are their adjoints. The result is that Eq. (2) becomes a vector operator-valued field Aop(x,t) = ∑k [aop(k,t) exp(ik•x)+a* op(k,t) exp(-ik•x)], (3) in which the amplitudes a*op(k,t) and aop(k,t) of the kth "field mode" satisfy the Heisenberg equations of motion (in which the time-dependence resides in the operators while the system's quantum state |Ψ> remains fixed) for a set of quantum harmonic oscillators. Assuming a* op and aop, commute for bosons, one can show that these operators are the familiar raising and lowering operators from the harmonic oscillator problem in NRQP. Aop(x,t) is now an operator-valued field whose dynamics obey quantum physics. Since the classical field obeyed SR, the quantized field satisfies quantum physics and SR. Thus, as in the harmonic oscillator problem, the kth mode has an infinite discrete energy spectrum hfk(Nk+1/2) with Nk = 0, 1, 2, ... where fk = c|k|/2π is the mode's frequency. 37 As Max Planck had hypothesized, the energy of a single mode has an infinite spectrum of discrete possible values separated by ΔE=hfk. Nk is the Art Hobson There are no particles, there are only fields 9 number of Planck's energy bundles or quanta in the kth mode. Each quantum is called an "excitation" of the field, because its energy hfk represents additional field energy. EM field quanta are called "photons," from the Greek word for light. A distinctly quantum aspect is that, even in the vacuum state where Nk = 0, each mode has energy hfk/2. This is because the a(k, t) act like quantum harmonic oscillators, and these must have energy even in the ground state because of the uncertainty principle. Another quantum aspect is that EM radiation is "digitized" into discrete quanta of energy hf. You can't have a fraction of a quantum. Because it defines an operator for every point x throughout space, the operator-valued field Eq. (3) is properly called a "field." Note that, unlike the NRQP case, x is not an operator but rather a parameter, putting x on an equal footing with t as befits a relativistic theory. E.g. we can speak of the expectation value of the field Aop at x and t, but we cannot speak of the expectation value of x because x is not an observable. This is because fields are inherently extended in space and don't have specific positions. But what does the operator field Eq. (3) operate on? Just as in NRQP, operators operate on the system's quantum state |Ψ>. But the Hilbert space for such states cannot have the same structure as for the single-body Schroedinger equation, or even its N-body analog, because N must be allowed to vary in order to describe creation and destruction of quanta. So the radiation field's quantum states exist in a Hilbert space of variable N called "Fock space." Fock space is the (direct) sum of N-body Hilbert spaces for N = 0, 1, 2, 3, .... Each component Nbody Hilbert space is the properly symmetrized (for bosons or fermions) product of N single-body Hilbert spaces. Each normalized component has its own complex amplitude, and the full state |Ψ> is (in general) a superposition of states having different numbers of quanta. An important feature of QFT is the existence of a vacuum state |0>, a unit vector that must not be confused with the zero vector (having "length" zero in Fock space), having no quanta (Nk=0 for all k). Each mode's vacuum state has energy hfk/2. The vacuum state manifests itself experimentally in many ways, which would be curious if particles were really fundamental because there are no particles (quanta) in this state. We'll expand on this particular argument in Sect. VI. The operator field Eq. (3) (and other observables such as energy) operates on |Ψ>, creating and destroying photons. For example, the expected value of the vector potential is a vector-valued relativistic field A(x,t) = = <Ψ| Aop(x,t) |Ψ>, an expression in which Aop(x,t) operates on |Ψ>. We see again that Aop(x,t) is actually a physically meaningful field because it has a physically measurable expectation value at every point x throughout a region of space. So a classical field that is quantized does not cease to be a field. Art Hobson There are no particles, there are only fields 10 Some authors conclude, incorrectly, that the countability of quanta implies a particle interpretation of the quantized system.38 Discreteness is a necessary but not sufficient condition for particles. Quanta are countable, but they are spatially extended and certainly not particles. Eq. (3) implies that a single mode's spatial dependence is sinusoidal and fills all space, so that adding a monochromatic quantum to a field uniformly increases the entire field's energy (uniformly distributed throughout all space!) by hf. This is nothing like adding a particle. Quanta that are superpositions of different frequencies can be more spatially bunched and in this sense more localized, but they are always of infinite extent. So it's hard to see how photons could be particles. Phenomena such as "particle" tracks in bubble chambers, and the small spot appearing on a viewing screen when a single quantum interacts with the screen, are often cited as evidence that quanta are particles, but these are insufficient evidence of particles 39, 40 (Sec. IV). In the case of radiation, it's especially difficult to argue that the small interaction points are evidence that a particle impacted at that position because photons never have positions--position is not an observable and photons cannot be said to be "at" or "found at" any particular point.41, 42, 43, 44, 45 Instead, the spatially extended radiation field interacts with the screen in the vicinity of the spot, transferring one quantum of energy to the screen. B. Quantized matter fields. QFT puts matter on the same all-fields footing as radiation. This is a big step toward unification. In fact, it's a general principal of all QFTs that fields are all there is (Refs. 10-21). For example the Standard Model, perhaps the most successful scientific theory of all time, is a QFT. But if fields are all there is, where do electrons and atoms come from? QFT's answer is that they are field quanta, but quanta of matter fields rather than quanta of force fields.46 "Fields are all there is" suggests beginning the quantum theory of matter from Schroedinger's equation, which mathematically is a field equation similar to Maxwell's field equations, and quantizing it. But you can't create a relativistic theory (the main purpose of QFT) this way because Schroedinger's equation is not Lorentz covariant. Dirac invented, for just this purpose, a covariant generalization of Schroedinger’s equation for the field Ψ(x,t) associated with a single electron.47 It incorporates the electron's spin, accounts for the electron's magnetic moment, and is more accurate than Schroedinger's equation in predicting the hydrogen atom's spectrum. It however has undesirable features such as the existence of nonphysical negative-energy states. These can be overcome by treating Dirac's equation as a classical field equation for matter analogous to Maxwell's equations Art Hobson There are no particles, there are only fields 11 for radiation, and quantizing it in the manner outlined in Sec. III A. The resulting quantized matter field Ψop(x,t) is called the "electron-positron field." It's an operator-valued field operating in the anti-symmetric Fock space. Thus the nonquantized Dirac equation describes a matter field occupying an analogous role in the QFT of matter to the role of Maxwell's equations in the QFT of radiation (Refs. 12, 45). The quantized theory of electrons comes out looking similar to the preceding QFT of the EM field, but with material quanta and with field operators that now create or destroy these quanta in quantum-antiquantum pairs (Ref. 36). It's not difficult to show that standard NRQP is a special case, for nonrelativistic material quanta, of relativistic QFT (Ref. 36). Thus the Schroedinger field is the non-relativistic version of the Dirac equation's relativistic field. It follows that the Schroedinger matter field, the analogue of the classical EM field, is a physical, space-filling field. Just like the Dirac field, this field is the electron. C. Further properties of quantum fields Thus the quantum theory of electromagnetic radiation is a re-formulation of classical electromagnetic theory to account for quantization--the "bundling" of radiation into discrete quanta. It remains, like the classical theory, a field theory. The quantum theory of matter introduces the electron-positron field and a new field equation, the Dirac equation, the analog for matter of the classical Maxwell field equations for radiation. Quantization of the Dirac equation is analogous to quantization of Maxwell's equations, and the result is the quantized electronpositron field. The Schroedinger equation, the non-relativistic version of the Dirac equation, is thus a field equation. There are no particles in any of this, there are only field quanta--excitations in spatially extended continuous fields. For over three decades, the Standard Model--a QFT--has been our best theory of the microscopic world. It's clear from the structure of QFTs (Secs. III A and III B) that they actually are field theories, not particle theories in disguise. Nevertheless, I'll offer further evidence for their field nature here and in Secs. V and VI. Quantum fields have one particle-like property that classical fields don't have: They are made of countable quanta. Thus quanta cannot partly vanish but must (like particles) be entirely and instantly created or destroyed. Quanta carry energy and momenta and can thus "hit like a particle." Following three centuries of particle-oriented Newtonian physics, it's no wonder that it took most of the 20th century to come to grips with the field nature of quantum physics. Were it not for Newtonian preconceptions, quantum physics might have been recognized as a field theory by 1926 (Schroedinger's equation) or 1927 Art Hobson There are no particles, there are only fields 12 (QFT). The superposition principle should have been a dead giveaway: A sum of quantum states is a quantum state. Such superposition is characteristic of all linear wave theories and at odds with the generally non-linear nature of Newtonian particle physics. A benefit of QFTs is that quanta of a given field must be identical because they are all excitations of the same field, somewhat as two ripples on the same pond are in many ways identical. Because a single field explains the existence and nature of gazillions of quanta, QFTs represent an enormous unification. The universal electron-positron field, for example, explains the existence and nature of all electrons and all positrons. When a field changes its energy by a single quantum, it must do so instantaneously, because a non-instantaneous change would imply that, partway through the change, the field had gained or lost only a fraction of a quantum. Such fractions are not allowed because energy is quantized. Field quanta have an all-ornothing quality. The QFT language of creation and annihilation of quanta expresses this nicely. A quantum is a unified entity even though its energy might be spread out over light years--a feature that raises issues of nonlocality intrinsic to the quantum puzzle. "Fields are all there is" should be understood literally. For example, it's a common misconception to imagine a tiny particle imbedded somewhere in the Schroedinger field. There is no particle. An electron is its field. As is well known, Einstein never fully accepted quantum physics, and spent the last few decades of his life trying to explain all phenomena, including quantum phenomena, in terms of a classical field theory. Nevertheless, and although Einstein would not have agreed, it seems to me that QFT achieves Einstein's dream to regard nature as fields. QFT promotes the right-hand side of Eq. (1) to field status. But it is not yet a "palace of gold" because Einstein's goal of explaining all fields entirely in terms of zero-rest-mass fields such as the gravitational field has not yet been achieved, although the QFT of the strong force comes close to this goal of "mass without mass" (Refs. 13, 16, 17). IV. THE 2-SLIT EXPERIMENT A. Phenomena Field-particle duality appears most clearly in the context of the time-honored 2-slit experiment, which Feynman claimed "contains the only mystery." 48, 49 Figures 1 and 2 show the outcome of the 2-slit experiment using a dim light beam (Fig. 1) and a "dim" electron beam (Fig. 2) as sources, with time-lapse photography. The set-up is a source emitting monochromatic light (Fig. 1) or Art Hobson There are no particles, there are only fields 13 mono-energetic electrons (Fig. 2), an opaque screen with two parallel slits, and a detection screen with which the beam collides. In both figures, particle-like impacts build up on the detection screen to form interference patterns. The figures show both field aspects (the extended patterns) and particle aspects (the localized impacts). The similarity between the two figures is striking and indicates a fundamental similarity between photons and electrons. It's intuitively hard to believe that one figure was made by waves and the other by particles. Fig. 1: The 2-slit experiment outcome using dim light with time-lapse photography. In successive images, an interference pattern builds up from particlelike impacts. 50 Fig 2: The 2-slit experiment outcome using a "dim" electron beam with time-lapse photography. An interference pattern builds up from particle-like impacts. 51 Art Hobson There are no particles, there are only fields 14 Consider, first, the extended pattern. It's easy to explain if each quantum (photon or electron) is an extended field that comes through both slits. But could the pattern arise from particles? The experiments can be performed using an ensemble of separately emitted individual quanta, implying the results cannot arise from interactions between different quanta.52 Preparation is identical for all the quanta in the ensemble. Thus, given this particular experimental context (namely the 2-slit experiment with both slits open, no detector at the slits, and a "downstream" screen that detects interactions of each ensemble member), each quantum must carry information about the entire pattern that appears on the screen (in order, e.g., to avoid all the nodes). In this sense, each quantum can be said to be spread out over the pattern. If we close one slit, the pattern shifts to the single-slit pattern behind the open slit, showing no interference. Thus each quantum carries different information depending on whether two or one slits are open. How does one quantum get information as to how many slits are open? If a quantum is a field that is extended over both slits, there's no problem. But could a particle coming through just one slit obtain this information by detecting physical forces from the other, relatively distant, slit? The effect is the same for photons and electrons, and the experiment has been done with neutrons, atoms, and many molecular types, making it difficult to imagine gravitational, EM, or nuclear forces causing such a long-distance force effect. What more direct evidence could there be that a quantum is an extended field? Thus we cannot explain the extended patterns by assuming each quantum is a particle, but we can explain the patterns by assuming each quantum is a field. 53 Now consider the particle-like small impact points. We can obviously explain these if quanta are particles, but can we explain them with fields? The flashes seen in both figures are multi-atom events initiated by interactions of a single quantum with the screen. In Fig. 2, for example, each electron interacts with a portion of a fluorescent film, creating some 500 photons; these photons excite a photo cathode, producing photo electrons that are then focused into a point image that is displayed on a TV monitor (Ref. 51). This shows that a quantum can interact locally with atoms, but it doesn't show that quanta are point particles. A large object (a big balloon, say) can interact quite locally with another object (a tiny needle, say). The localization seen in the two figures is characteristic of the detector, which is made of localized atoms, rather than of the detected quanta. The detection, however, localizes ("collapses"--Secs. IV B and IV C) the quantum. Similar arguments apply to the observation of thin particle tracks in bubble chambers and other apparent particle detections. Localization is characteristic of the detection process, not of the quantum that is being detected. Thus the interference patterns in Figs. 1 and 2 confirm field behavior and Art Hobson There are no particles, there are only fields 15 rule out particle behavior, while the small interaction points neither confirm particle behavior nor rule out field behavior. The experiment thus confirms field behavior. As Dirac famously put it in connection with experiments of the 2-slit type, "The new theory [namely quantum mechanics], which connects the wave function with probabilities for one photon, gets over the difficulty [of explaining the interference] by making each photon go partly into each of the two components. Each photon then interferes only with itself." 54 The phrases in square brackets are mine, not Dirac's. Given the extended field nature of each electron, Fig. 2 also confirms von Neumann's famous collapse postulate:55 Each electron carries information about the entire pattern and collapses to a much smaller region upon interaction. Most textbooks set up a paradox by explicitly or implicitly assuming each quantum to come through one or the other slit, and then struggle to resolve the paradox. But if each quantum comes through both slits, there's no paradox. B. Theory, at the slits 56 , 57 Now assume detectors are at each slit so that a quantum passing through slit 1 (with slit 2 closed) triggers detector 1, and similarly for slit 2. Let |1 > and |2 >, which we assume form an orthonormal basis for the quantum's Hilbert space, denote the states of a quantum passing through slit 1 with slit 2 closed, and through slit 2 with slit 1 closed, respectively. We assume, with von Neumann, that the detector also obeys quantum physics, with |ready> denoting the "ready" state of the detectors, and |1> and |2> denoting the "clicked" states of each detector. Then the evolution of the composite quantum + detector system, when the quantum passes through slit i alone (with the other slit closed), is of the form |ψi> |ready> → |ψ i> |i> (i=1,2) (assuming, with von Neumann, that these are "ideal" processes that don't disturb the state of the quantum). With both slits open, the single quantum approaching the slits is described by a superposition that's extended over both slits: (|1 > + |2 >)/√2 ≡ |ψ>. (4) Linearity of the time evolution implies that the composite system's evolution during detection at the slits is |ψ> |ready> → (|1 > |1> + |2 > |2>)/√2 ≡ |Ψslits>. (5) Art Hobson There are no particles, there are only fields 16 The "measurement state" |Ψslits> involves both spatially distinguishable detector states |j>. It is a "Bell state" of nonlocal entanglement between the quantum and the detector (Ref. 57, pp. 29, 32). If the detectors are reliable, there must be zero probability of finding detector i in the state |i> when detector j≠i is in its clicked state |j>, so |1> and |2> are orthogonal and we assume they are normalized. It's mathematically convenient to form the pure state density operator ρslits ≡ |Ψslits><Ψslits|, (6) and to form the reduced density operator for the quantum alone by tracing over the detector: ρq slits = Trdetector (ρslits) = (|1 >< 1 | + | 2 >< 2 |)/2 . (7) Eq. (7) has a simple interpretation: Even though the quantum is in the entangled superposition Eq. (5), the result of any experiment involving the quantum alone will come out precisely as though the quantum were in one of the pure states |1 > or |2 > with probabilities of 1/2 for each state (Ref. 57). In particular, Eq. (7) predicts that the quantum does not interfere with itself, i.e. there are no interferences between | 1 > and |2 >. This of course agrees with observation: When detectors provide "which path" information, the interference pattern (i.e. the evidence that the quantum came through both slits) vanishes. The quantum is said to "decohere" (Ref. 57). To clearly see the field nature of the measurement, suppose there is a "which slit" detector only at slit 1 with no detector at slit 2. Then |ψi> |ready> → |ψi> |i> holds only for i=1, while for i=2 we have |ψ2> |ready> → |ψ2> |ready>. The previous analysis still holds, provided the "clicked" state |1> is orthogonal to the unclicked state |ready> (i.e. if the two states are distinguishable with probability 1). The superposition Eq. (4) evolves just as before, and Eq. (7) still describes the quantum alone just after measurement. So the experiment is unchanged by removal of one slit detector. Even though there is no detector at slit 2, when the quantum comes through slit 2 it still encodes the presence of a detector at slit 1. This is nonlocal, and it tells us that the quantum extends over both slits, i.e. the quantum is a field, not a particle. Thus the experiment (Sec. IV A) and the theory both imply each quantum comes through both slits when both slits are open with no detectors, but through Art Hobson There are no particles, there are only fields 17 one slit when there is a detector at either slit, just as we expect a field (but not a particle) to do. C. Theory, at the detecting screen We'll see that the above analysis at the slits carries over at the detecting screen, with the screen acting as detector. The screen is an array of small but macroscopic detectors such as single photographic grains. Suppose one quantum described by Eq. (4) passes through the slits and approaches the screen. Expanding in position eigenstates, just before interacting with the screen the quantum's state is |ψ> = ∫ |x> dx = ∫ |x> ψ(x) dx (8) where the integral is over the 2-dimensional screen, and ψ(x) is the Schroedinger field. Eq. (8) is a (continuous) superposition over position eigenstates, just as Eq. (4) is a (discrete) superposition over slit eigenstates. Both superpositions are extended fields. Rewriting Eq. (8) in a form that displays the quantum's superposition over the non-overlapping detection regions, |ψ> = Σι ∫i |x> ψ(x) dx ≡ Σi Ai |ψi> (9) where |ψi> ≡ (1/Ai)∫i |x> ψ(x) dx and Ai ≡ [∫i |ψ(x)|2 dx] 1/2. The detection regions are labeled by "i" and the |ψi> form an orthonormal set. Eq. (9) is analogous to Eq. (4). The detection process at the screen is represented by the analog of Eq. (5): |ψ> |ready> → Σi Ai |ψi> |i> ≡ |Ψscreen> (10) where |i> represents the "clicked" state of the ith detecting region whose output can be either "detection" or "no detection" of the quantum. Localization occurs at the time of this click. Each region i responds by interacting or not interacting, with just one region registering an interaction because a quantum must give up all, or none, of it's energy. As we'll see in Sec. VI C, these other sections of the screen actually register the vacuum--a physical state that can entangle nonlocally with the registered quantum. The non-locality inherent in the entangled superposition state |Ψscreen> has been verified by Bell-type measurements (Sec. VI C). As was the Art Hobson There are no particles, there are only fields 18 case for detection at the slits (Eq. (5)), Eq. (10) represents the mechanism by which the macro world registers the quantum's impact on the screen. The argument from Eq. (10) goes through precisely like the argument from Eq. (5) to Eq. (7). The result is that, assuming the states |i> are reliable detectors, the reduced density operator for the quantum alone is ρq screen = Σi |ψi> Ai 2 <ψi|. (11) Eq. (11) tells us that the quantum is registered either in region 1 or region 2 or .... It's this "all or nothing" nature of quantum interactions, rather than any presumed particle nature of quanta, that produces the particle-like interaction regions in Figures 1 and 2. In summary, "only spatial fields must be postulated to form the fundamental objects to be quantized, ...while apparent 'particles' are a mere consequence of decoherence" [i.e. of localization by the detection process]. 58 V. RELATIVISTIC QUANTUM PHYSICS NRQP (Sec. IV) is not the best basis for analyzing field-particle duality. The spontaneous energy fluctuations of quantum physics, plus SR's principle of mass-energy equivalence, imply that quanta, be they fields or particles, can be created or destroyed. Since relativistic quantum physics was invented largely to deal with such creation and destruction, one might expect relativistic quantum physics to offer the deepest insights into fields and particles. Quantum physics doesn't fit easily into a special-relativistic framework. As one example, we saw in Sec. III A that photons (relativistic phenomena for sure) cannot be quantum point particles because they don't have position eigenstates. A more striking example is nonlocality, a phenomenon shown by Einstein, Podolsky, and Rosen,59 and more quantitatively by John Bell,60 to inhere in the quantum foundations. Using Bell's inequality, Aspect, Clauser and others showed rather convincingly that nature herself is nonlocal and that this would be true even if quantum physics were not true. 61 The implication is that, by altering the way she measures one of the quanta in an experiment involving two entangled quanta, Alice in New York City can instantly (i.e. in a time too short to allow for signal propagation) change the outcomes observed when Bob measures the other quantum in Paris. This sounds like it violates the special-relativistic prohibition on super-luminal signaling, but quantum physics manages to avoid a contradiction by camouflaging the signal so that Alice's measurement choice is "averaged out" in the statistics of Bob's observations in such a way that Bob detects no change in the Art Hobson There are no particles, there are only fields 19 statistics of his experiment. 62 Thus Bob receives no signal, even though nonlocality changes his observed results. Quantum physics' particular mixture of uncertainty and non-locality preserves consistency with SR. It's only when Alice and Bob later compare their data that they can spot correlations showing that Alice's change of measurement procedure altered Bob's outcomes. Quantum physics must thread a fine needle, being "weakly local" in order to prevent superluminal signaling but, in order to allow quantum non-locality, not "strongly local" (Ref. 62). Quantum field spreading can transmit information and is limited by the speed of light, while non-local effects are related to superluminal field collapse and cannot transmit information lest they violate SR. When generalizing NRQP to include such relativistic quantum phenomena as creation and destruction, conflicts with SR can arise unless one generalizes carefully. Hegerfeldt63 and Malament64 have each presented rigorous "no-go theorems" demonstrating that, if one assumes a universe containing particles, then the requirements of SR and quantum physics lead to contradictions. This supports the "widespread (within the physics community) belief that the only relativistic quantum theory is a theory of fields." 65 Neither theorem assumes QFT. They assume only SR and the general principles of quantum physics, plus broadly inclusive definitions of what one means by a "particle." Each then derives a contradiction, showing that there can be no particles in any theory obeying both SR and quantum physics. I will present only Hegerfeldt's theorem here, because it is the more intuitive of the two, and because Malement's theorem is more subject to difficulties of interpretation. Hegerfeldt shows that any free (i.e. not constrained by boundary conditions or forces to remain for all time within some finite region) relativistic quantum "particle" must, if it's localized to a finite region to begin with, instantly have a positive probability of being found an arbitrarily large distance away. But this turns out to violate Einstein causality (no superluminal signaling). The conclusion is then that an individual free quantum can never--not even for a single instant--be localized to any finite region. More specifically, a presumed particle is said to be "localized" at to if it is prepared in such a way as to ensure that it will upon measurement be found, with probability 1, to be within some arbitrarily large but finite region Vo at to. Hegerfeldt then assumes two conditions: First, the presumed particle has quantum states that can be represented in a Hilbert space with unitary time-development operator Ut = exp(-iHt), where H is the energy operator. Second, the particle's energy spectrum has a lower bound. The first condition says that the particle obeys standard quantum dynamics. The second says that the Hamiltonian that drives the dynamics cannot provide infinite energy by itself dropping to lower and lower energies. Hegerfeldt then proves that a particle that is localized at to is not Art Hobson There are no particles, there are only fields 20 localized at any t >to. See Ref. 63 for the proof. It's remarkable that even localizability in an arbitrarily large finite region can be so difficult for a relativistic quantum particle--its probability amplitude spreads instantly to infinity. Now here is the contradiction: Consider a particle that is localized within Vo at t0. At any t > t0, there is then a nonzero probability that it will be found at any arbitrarily large distance away from Vo. This is not a problem for a non-relativistic theory, and in fact such instantaneous spreading of wavefunctions is easy to show in NRQP.66 But in a relativistic theory, such instantaneous spreading contradicts relativity's prohibition on superluminal transport and communication, because it implies that a particle localized on Earth at t0 could, with nonzero probability, be found on the moon an arbitrarily short time later. We conclude that "particles" cannot ever be localized. To call a thing a "particle" when it cannot ever be localized is surely a gross misuse of that word. Because QFTs reject the notion of position observables in favor of parameterized field observables (Sec. III), QFTs have no problem with Hegerfeldt's theorem. In QFT interactions, including creation and destruction, occur at specific locations x, but the fundamental objects of the theory, namely the fields, do not have positions because they are infinitely extended. Summarizing: even under a broadly inclusive definition of "particle," quantum particles conflict with Einstein causality. VI. THE QUANTUM VACUUM The Standard Model, a QFT, is today the favored way of looking at relativistic quantum phenomena. In fact, QFT is "the only known version of relativistic quantum theory." 67 Since NRQP can also be expressed as a QFT, 68 all of quantum physics can be expressed consistently as QFTs. We've seen (Sec. V) that quantum particles conflict with SR. This suggests (but doesn't prove) that QFTs are the only logically consistent version of relativistic quantum physics. 69 Thus it appears that QFTs are the natural language of quantum physics. Because it has energy and non-vanishing expectation values, the QFT vacuum is embarrassing for particle interpretations. If one believes particles to be the basic reality, then what is it that has this energy and these values in the state that has no particles? 70 Because it is the source of empirically verified phenomena such as the Lamb shift, the Casimir effect, and the electron's anomalous magnetic moment, this "state that has no particles" is hard to ignore. This Section discusses QFT vacuum phenomena that are difficult to reconcile with particles. Sec. VI A discusses the quantum vacuum itself. The remaining parts are implications of the quantum vacuum. The Unruh effect (Sec. VI B), related to Hawking radiation, has Art Hobson There are no particles, there are only fields 21 not yet been observed, while single-quantum nonlocality (Sec. VI C) is experimentally confirmed. On the other hand, we do not yet really understand the quantum vacuum. The most telling demonstration of this is that the most plausible theoretical QFT estimate of the energy density of the vacuum implies a value of the cosmological constant that is some 120 orders of magnitude larger than the upper bound placed on this parameter by astronomical observations. Possible solutions, such as the anthropic principle, have been suggested, but these remain speculative.71 A. The necessity for the quantum vacuum72 Both theory and experiment demonstrate that the quantized EM field can never be sharply (with probability one) zero, but rather that there must exist, at every spatial point, at least a randomly fluctuating "vacuum field" having no quanta. Concerning the theory, recall (Sec. III) that a quantized field is equivalent to a set of oscillators. An actual mechanical oscillator cannot be at rest in its ground state because this would violate the uncertainty principle; its ground state energy is instead hf/2. Likewise, each field oscillator must have a ground state where it has energy but no excitations. In the "vacuum state," where the number of excitations Nk is zero for every mode k, the expectation values of E and B are zero yet the expectation values of E2 and B2 are not zero. Thus the vacuum energy arises from random "vacuum fluctuations" of E and B around zero. As a second more direct argument for the necessity of EM vacuum energy, consider a charge e of mass m bound by an elastic restoring force to a large mass of opposite charge. The equation of motion for the Heisenberg-picture position operator x(t) has the same form as the corresponding classical equation, namely d2 x/dt2 + ωo 2 x = (e/m)[Err(t) + Eo(t)]. (12) Here, ωo is the oscillator's natural frequency, Err(t) is the "radiation reaction" field produced by the charged oscillator itself, Eo(t) is the external field, and it's assumed that the spatial dependence of Eo(t) can be neglected. It can be shown that the radiation reaction has the same form as the classical radiation reaction field for an accelerating charged particle, Err(t) = (2e/3c3 ) d3 x/dt3 , so Eq. (12) becomes d2 x/dt2 + ωo 2 x - (2e2 /3mc3 )d3 x/dt3 = (e/m)Eo(t). (13) If the term Eo(t) were absent, Eq. (13) would become a dissipative equation with x(t) exponentially damped, and commutators like [z(t), pz(t)] would approach zero Art Hobson There are no particles, there are only fields 22 for large t, in contradiction with the uncertainty principle and in contradiction with the unitary time development of quantum physics according to which commutators like [z(t), pz(t)] are time-independent. Thus Eo(t) cannot be absent for quantum systems. Furthermore, if Eo(t) is the vacuum field then commutators like [z(t), pz(t)] turn out to be time-independent. B. The Unruh effect QFT predicts that an accelerating observer in vacuum sees quanta that are not there for an inertial observer of the same vacuum. More concretely, consider Mort who moves at constant velocity in Minkowski space-time, and Velma who is uniformly accelerating (i.e. her acceleration is unchanging relative to her instantaneous inertial rest frame). If Mort finds himself in the quantum vacuum, Velma finds herself bathed in quanta--her “particle” detector clicks. Quantitatively, she observes a thermal bath of photons having the Planck radiation spectrum with kT = ha/4π 2 c where a is her acceleration. 73 This prediction might be testable in high energy hadronic collisions, and for electrons in storage rings.74 In fact it appears to have been verified years ago in the Sokolov-Ternov effect.75 The Unruh effect lies at the intersection of QFT, SR, and general relativity. Combined with the equivalence principle of general relativity, it entails that strong gravitational fields create thermal radiation. This is most pronounced near the event horizon of a black hole, where a stationary (relative to the event horizon) Velma sees a thermal bath of particles that then fall into the black hole, but some of which can, under the right circumstances, escape as Hawking radiation.76 The Unruh effect is counterintuitive for a particle ontology, as it seems to show that the particle concept is observer-dependent. If particles form the basic reality, how can they be present for the accelerating Velma but absent for the nonaccelerating Mort who observes the same space-time region? But if fields are basic, things fall into place: Both experience the same field, but Velma's acceleration promotes Mort's vacuum fluctuations to the level of thermal fluctuations. The same field is present for both observers, but an accelerated observer views it differently. C. Single-quantum nonlocality Nonlocality is pervasive, arguably the characteristic quantum phenomenon. It would be surprising, then, if it were merely an "emergent" property possessed by two or more quanta but not by a single quantum. Art Hobson There are no particles, there are only fields 23 During the 1927 Solvay Conference, Einstein noted that "a peculiar actionat-a-distance must be assumed to take place" when the Schroedinger field for a single quantum passes through a single slit, diffracts in a spherical wave, and strikes a detection screen. Theoretically, when the interaction localizes as a small flash on the screen, the field instantly vanishes over the rest of the screen. Supporting de Broglie's theory that supplemented the Schroedinger field with particles, Einstein commented "if one works only with Schroedinger waves, the interpretation of psi, I think, contradicts the postulate of relativity."77 Since that time, however, the peaceful coexistence of quantum nonlocality and SR has been demonstrated (Refs. 62, 67). It's striking that Einstein's 1927 remark anticipated single-quantum nonlocality in much the same way that Einstein's EPR paper (Ref. 59) anticipated nonlocality of two entangled quanta. Today, single-quantum nonlocality has a 20- year history that further demonstrates nonlocality as well as the importance of fields in understanding it. Single-photon nonlocality was first described in detail by Tan et. al. in 1991. 78 In this suggested experiment, a single photon passed through a 50-50 beam-splitting mirror (the "source"), with reflected and transmitted beams (the "outputs") going respectively to "Alice" and "Bob." They could be any distance apart and were equipped with beam splitters with phase-sensitive photon detectors attached to these detectors' outputs. But nonlocality normally involves two entangled quantum entities. With just one photon, what was there to entangle with? If photons are field mode excitations, the answer is natural: the entanglement was between two quantized field modes, with one of the modes happening to be in the vacuum state. Like all fields, each mode fills space, making nonlocality between modes more intuitive than nonlocality between particles: If a space-filling mode were to instantly change states, the process would obviously be non-local. This highlights the importance of thinking of quantum phenomena in terms of fields. 79 In Tan et. al.'s suggested experiment, Alice's and Bob's wave vectors were the two entangled modes. According to QFT, an output "beam" with no photon is an actual physical state, namely the vacuum state |0>. Alice's mode having wave vector kA was then in a superposition |1>A+|0>A of having a single excitation and having no excitation, Bob's mode kB was in an analogous superposition |1>B+|0>B, and the two superpositions were entangled by the source beam splitter to create a 2-mode composite system in the nonlocal Bell state |ψ> = |1>A|0>B + |0>A|1>B (14) Art Hobson There are no particles, there are only fields 24 (omitting normalization). Note the analogy with Eq. (5): In Eq. (14), Alice and Bob act as detectors for each others' superposed quanta, collapsing (decohering) both quanta. This entangled superposition state emerged from the source; Alice then detected only mode kA and Bob detected only mode kB. Quantum theory predicted that coincidence experiments would show correlations that violated Bell's inequality, implying nonlocality that cannot be explained classically. Analogously to Eq. (7), Alice's and Bob's reduced density operators are ρA = TrB(|ψ><ψ|) = |1>A A<1| + |0>A A<0| ⎫ ⎬ (15) ρB = TrA(|ψ><ψ|) = |1>B B<1| + |0>B B<0|. ⎭ Each observer has a perfectly random 50-50 chance of receiving 0 or 1, a "signal" containing no information. All coherence and non-locality are contained in the composite state Eq. (14). This returns us to Einstein's concerns: In the single-photon diffraction experiment (Sec. IV), interaction of the photon with the screen creates a non-local entangled superposition (Eq. (10)) that is analogous (but with N terms) to Eq. (14). As Einstein suspected, this state is odd, nonlocal. Violation of Bell's inequality shows that the analogous state Eq. (14) is, indeed, nonlocal in a way that cannot be interpreted classically. In 1994, another single-photon experiment was proposed to demonstrate nonlocality without Bell inequalities. 80 The 1991 and 1994 proposals triggered extended debate about whether such experiments really demonstrate nonlocality involving only one photon.81 The discussion generated three papers describing proposed new experiments to test single-photon nonlocality. 82 One of these proposals was implemented in 2002, when a single-photon Bell state was teleported to demonstrate (by the nonlocal teleportation) the single-photon nonlocality. In this experiment, "The role of the two entangled quantum systems which form the nonlocal channel is played by the EM fields of Alice and Bob. In other words, the field modes rather than the photons associated with them should be properly taken as the information and entanglement carriers" (italics in the original).83 There was also an experimental implementation of a single-photon Bell test based on the 1991 and 1994 proposals.84 It was then suggested that the state Eq. (14) can transfer its entanglement to two atoms in different locations, both initially in their ground states |g>, by using the state Eq. (14) to generate the joint atomic state |e>A|g>B + |g>A|e>B (note that the vacuum won't excite the atom).85 Here, |e> represents an excited state of an atom, while A and B now refer to different modes kA and kB of a matter field (different beam directions for atoms A and B). Thus the atoms (i.e. modes kA and kB) are placed in a nonlocal entangled superposition of being excited and not Art Hobson There are no particles, there are only fields 25 excited. Since this nonlocal entanglement arises from the single-photon nonlocal state by purely local operations, it's clear that the single-photon state must have been nonlocal too. Nevertheless, there was controversy about whether this proposal really represents single-quantum nonlocality. 86 Another experiment, applicable to photons or atoms, was proposed to remove all doubt as to whether these experiments demonstrated single-quantum nonlocality. The proposal concluded by stating, "This strengthens our belief that the world described by quantum field theory, where fields are fundamental and particles have only a secondary importance, is closer to reality than might be expected from a naive application of quantum mechanical principles." 87 VII. CONCLUSION There are overwhelming grounds to conclude that all the fundamental constituents of quantum physics are fields rather than particles. Rigorous analysis shows that, even under the broadest definition of "particle," particles are inconsistent with the combined principles of relativity and quantum physics (Sec. V). And photons, in particular, cannot be point particles because relativistic and quantum principles imply that a photon cannot "be found" at a specific location even in principle (Sec. III A). Many relativistic quantum phenomena are paradoxical in terms of particles but natural in terms of fields: the necessity for the quantum vacuum (Sec. VI A), the Unruh effect where an accelerated observer detects quanta while an inertial observer detects none (Sec. VI B), and single-quantum nonlocality where two field modes are put into entangled superpositions of a singly-excited state and a vacuum state (Sec. VI C). Classical field theory and experiment imply fields are fundamental, and indeed Faraday, Maxwell, and Einstein concluded as much (Sec. II). Merely quantizing these fields doesn't change their field nature. Beginning in 1900, quantum effects implied that Maxwell's field equations needed modification, but the quantized equations were still based on fields (Maxwell's fields, in fact, but quantized), not particles (Sec. III A). On the other hand, Newton's particle equations were replaced by a radically different concept, namely Schroedinger's field equation, whose field solution Ψ(x,t) was however inconsistently interpreted as the probability amplitude for finding, upon measurement, a particle at the point x. The result has been confusion about particles and measurements, including mentally-collapsed wave packets, students going "down the drain into a blind alley," textbooks filled almost exclusively with "particles talk," and pseudoscientific fantasies (Sec. I). The relativistic generalization of Schroedinger's equation, namely Dirac's equation, is clearly a field equation that is quantized to Art Hobson There are no particles, there are only fields 26 obtain the electron-positron field, in perfect analogy to the way Maxwell's equations are quantized (Sec. III B). It makes no sense, then, to insist that the nonrelativistic version of Dirac's equation, namely the Schroedinger equation, be interpreted in terms of particles. After all, the electron-positron field, which fills all space, surely doesn't shrink back to tiny particles when the electrons slow down. Thus Schroedinger's Ψ(x,t) is a spatially extended field representing the amplitude for an electron (i.e. the electron-positron field) to interact at x rather than an amplitude for finding, upon measurement, a particle. In fact, the field Ψ(x,t) is the so-called "particle." Fields are all there is. Analysis of the 2-slit experiment (Sec. IV) shows why, from a particle viewpoint, "nobody knows how it [i.e. the experiment] can be like that": The 2-slit experiment is in fact logically inconsistent with a particle viewpoint. But everything becomes consistent, and students don't get down the drain, if the experiment is viewed in terms of fields. Textbooks need to reflect that fields, not particles, form our most fundamental description of nature. This can be done easily, not by trying to teach the formalism of QFT in introductory courses, but rather by talking about fields, explaining that there are no particles but only particle-like phenomena caused by field quantization (Ref. 21). In the 2-slit experiment, for example, the quantized field for each electron or photon comes simultaneously through both slits, spreads over the entire interference pattern, and collapses non-locally, upon interacting with the screen, into a small (but still spread out) region of the detecting screen. Field-particle duality exists only in the sense that quantized fields have certain particle-like appearances: quanta are unified bundles of field that carry energy and momentum and thus "hit like particles;" quanta are discrete and thus countable. But quanta are not particles; they are excitations of spatially unbounded fields. Photons and electrons, along with atoms, molecules, and apples, are ultimately disturbances in a few universal fields. ACKNOWLEDGEMENTS My University of Arkansas colleagues Julio Gea-Banacloche, Daniel Kennefick, Michael Lieber, Surendra Singh, and Reeta Vyas discussed my incessant questions and commented on the manuscript. Rodney Brooks and Peter Milonni read and commented on the manuscript. I also received helpful comments from Stephen Adler, Nathan Argaman, Casey Blood, Edward Gerjuoy, Daniel Greenberger, Nick Herbert, David Mermin, Michael Nauenberg, Roland Omnes, Marc Sher, and Wojciech Zurek. I especially thank the referees for their careful attention and helpful comments. Art Hobson There are no particles, there are only fields 27 a Email ahobson@uark.edu 1 M. Schlosshauer, Elegance and Enigma: The Quantum Interviews (Springer-Verlag, Berlin, 2011). 2 N. G. van Kampen, "The scandal of quantum mechanics," Am. J. Phys. 76 (11), 989- 990 (2008); A. Hobson, "Response to 'The scandal of quantum mechanics,' by N. G. Van Kampen," Am. J. Phys. 77 (4), 293 (2009). 3 W. Zurek, "Decoherence and the transition from quantum to classical," Phys. Today 44 (10), 36-44 (1991): "Quantum mechanics works exceedingly well in all practical applications. ...Yet well over half a century after its inception, the debate about the relation of quantum mechanics to the familiar physical world continues. How can a theory that can account with precision for everything we can measure still be deemed lacking?" 4 C. Sagan, The Demon-Haunted World: Science as a Candle in the Dark (Random House, New York, 1995), p. 26: "We've arranged a civilization in which most crucial elements profoundly depend on science and technology. We have also arranged things so that almost no one understands science and technology. This is a prescription for disaster. ...Sooner or later this combustible mixture of ignorance and power is going to blow up in our faces." 5 M. Shermer, "Quantum Quackery," Scientific American 292 (1), 34 (2005). 6 V. Stenger "Quantum Quackery," Sceptical Inquirer 21 (1), 37-42 (1997). It's striking that this article has, by coincidence, the same title as Ref. 5. 7 D. Chopra, Quantum Healing: Exploring the Frontiers of Mind/Body Medicine (Bantam, New York, 1989). 8 D. Chopra, Ageless Body, Timeless Mind: The Quantum Alternative to Growing Old (Harmony Books, New York, 1993). 9 B. Rosenblum and F. Kuttner, Quantum Enigma: Physics Encounters Consciousness (Oxford University Press, New York, 2006). 10 S. Weinberg, Dreams of a Final Theory: The Search for the Fundamental Laws of Nature (Random House, Inc., New York, 1992): "Furthermore, all these particles are bundles of the energy, or quanta, of various sorts of fields. A field like an electric or magnetic field is a sort of stress in space... The equations of a field theory like the Standard Model deal not with particles but with fields; the particles appear as manifestations of those fields" (p. 25). 11 S. Weinberg, Facing Up: Science and its Cultural Adversaries (Harvard University Press, Cambridge, MA, 2001): "Just as there is an electromagnetic field whose energy and momentum come in tiny bundles called photons, so there is an electron field whose energy and momentum and electric charge are found in the bundles we call electrons, and likewise for every species of elementary particles. The basic ingredients of nature are fields; particles are derivative phenomena." 12 R. Mills, Space, Time, and Quanta: An Introduction to Modern Physics (W. H. Freeman, New York, 1994), Chp 16: "The only way to have a consistent relativistic theory is to treat all the particles of nature as the quanta of fields, like photons. Electrons and positrons are to be treated as the quanta of the electron-positron field, whose 'classical' field equation, the analog of Maxwell's equations for the EM field, turns out to be the Dirac equation, which started life as a relativistic version of the single-particle Art Hobson There are no particles, there are only fields 28 Schroedinger equation. …This approach now gives a unified picture, known as quantum field theory, of all of nature." 13 F. Wilczek, "Mass Without Mass I: Most of Matter," Physics Today 52 (11), 11-13 (1999): "In quantum field theory, the primary elements of reality are not individual particles, but underlying fields. Thus, e.g., all electrons are but excitations of an underlying field, ..the electron field, which fills all space and time. 14 M. Redhead, "More ado about nothing," Foundations of Physics 25 (1), 123-137 (1995): "Particle states are never observable--they are an idealization which leads to a plethora of misunderstandings about what is going on in quantum field theory. The theory is about fields and their local excitations. That is all there is to it." 15A. Zee, Quantum Field Theory in a Nutshell (Princeton University Press, Princeton, NJ, 2003), p 24: "We thus interpret the physics contained in our simple field theory as follows: In region 1 in spacetime there exists a source that sends out a 'disturbance in the field,' which is later absorbed by a sink in region 2 in spacetime. Experimentalists choose to call this disturbance in the field a particle of mass m." 16 F. Wilczek, "The persistence of ether," Physics Today 52 (1), 11-13 (1999). 17 F. Wilczek, "Mass Without Mass II: The Medium Is the Massage," Physics Today 53 (1), 13-14 (2000). 18 F. Wilczek, The Lightness of Being: Mass, Ether, and the Unification of Forces (Basic Books, New York, 2008). 19 R. Brooks, Fields of Color: The Theory That Escaped Einstein (Rodney A. Brooks, Prescott, AZ, 2nd ed. 2011). This is a lively history of classical and quantum fields, with many quotations from leading physicists, organized to teach quantum field theory to the general public. 20 P. R. Wallace, Paradox Lost: Images of the Quantum (Springer-Verlag, New York, 1996). 21 A. Hobson, "Electrons As Field Quanta: A Better Way to Teach Quantum Physics in Introductory General Physics Courses," Am. J. Phys. 73, 630-634 (2005); "Teaching quantum physics without paradoxes," The Physics Teacher 45, 96-99 (Feb. 2007); "Teaching quantum uncertainty," The Physics Teacher 49, 434-437 (2011); "Teaching quantum nonlocality," The Physics Teacher 50, 270-273 (2012). 22 A. Hobson, Physics: Concepts & Connections (Addison-Wesley/Pearson, San Francisco, 2010). 23 R. Feynman, The Character of Physical Law (The MIT Press, Cambridge, MA, 1965), p. 129. Feynman also says, in the same lecture, "I think I can safely say that nobody understands quantum mechanics." 24 A. Janiak, Newton: Philosophical Writings (Cambridge University Press, Cambridge, 2004), p. 102. 25 N. J. Nersessian, Faraday to Einstein: Constructing Meaning in Scientific Theories, (Martinus Nijhoff Publishers, Boston, 1984), p. 37. The remainder of Sec. III relies strongly on this book. 26 S. Weinberg, Facing Up: Science and its Cultural Adversaries (Harvard University Press, Cambridge, MA, 2001), p. 167: "Fields are conditions of space itself, considered apart from any matter that may be in it." Art Hobson There are no particles, there are only fields 29 27 This argument was Maxwell's and Einstein's justification for the reality of the EM field. R. H. Stuewer, Ed., Historical and Philosophical Perspectives of Science, (Gordon and Breach, New York, 1989), p. 299. 28 A. Einstein, "Maxwell's influence on the development of the conception of physical reality," in James Clerk Maxwell: A Commemorative Volume 1831-1931 (The Macmillan Company, New York, 1931), pp. 66-73. 29 A. Einstein, "Zur Elektrodynamik bewegter Koerper," Annalen der Physik 17, 891-921 (1905). 30 I. Newton, Optiks (4th edition, 1730): "It seems probable to me that God in the beginning formed matter in solid, massy, hard, impenetrable, movable particles ...and that these primitive particles being solids are incomparably harder than any porous bodies compounded of them, even so hard as never to wear or break in pieces...." 31 R. Brooks, author of Ref. 19, private communication. 32 J. A. Wheeler and R. P. Feynman, "Interaction With the Absorber as the Mechanism of Radiation," Revs. Mod. Phys. 17, 157-181 (1945). 33 P. A. M. Dirac, "The quantum theory of the emission and absorption of radiation," Proceedings of the Royal Society A114, 243-267 (1927). 34 M. Kuhlmann, The Ultimate Constituents Of The Material World: In Search Of An Ontology For Fundamental Physics (Ontos Verlag, Heusenstamm, Germany, 2010), Chp. 4; a brief but detailed history of QFT. 35 The first comprehensive account of a general theory of quantum fields, in particular the method of canonical quantization, was presented in W. Heisenberg and W. Pauli, "Zur quantendynamik der Wellenfelder, Zeitschrift fuer Physik 56, 1-61 (1929). 36 E. G. Harris, A Pedestrian Approach to Quantum Field Theory (Wiley-Interscience, New York, 1972). 37 More precisely, there are two vector modes for each non-zero k, one for each possible field polarization direction, both perpendicular to k. See Ref. 36 for other details. 38 For example, L. H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1996), p 131: "This completes the justification for interpreting N(k) as the number operator and hence for the particle interpretation of the quantized theory." 39 H. D. Zeh, "There are no quantum jumps, nor are there particles!" Phys. Lett. A 172, 189-195 (1993): "All particle aspects observed in measurements of quantum fields (like spots on a plate, tracks in a bubble chamber, or clicks of a counter) can be understood by taking into account this decoherence of the relevant local (i.e. subsystem) density matrix." 40 C. Blood, "No evidence for particles," http://arxiv.org/pdf/0807.3930.pdf: "There are a number of experiments and observations that appear to argue for the existence of particles, including the photoelectric and Compton effects, exposure of only one film grain by a spread-out photon wave function, and particle-like trajectories in bubble chambers. It can be shown, however, that all the particle-like phenomena can be explained by using properties of the wave functions/state vectors alone. Thus there is no evidence for particles. Wave-particle duality arises because the wave functions alone have both wave-like and particle-like properties." 41 T.D. Newton and E.P. Wigner, "Localized states for elementary systems," Revs. Mod. Phys. 21 (3), 400-406 (1949). Art Hobson There are no particles, there are only fields 30 42 I. Bialynicki-Birula and Z. Bialynicki-Birula, "Why photons cannot be sharply localized," Phys. Rev. A 79, 032112 (2009). 43 L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (New York, Cambridge UP, 1995). 44 R. E. Peierls, Surprises in Theoretical Physics (Princeton UP, 1979), pp 12-14. 45 M.G. Raymer and B.J. Smith, "The Maxwell wave function of the photon," SPIE Conference on Optics and Photonics, San Diego, Aug 2005, Conf #5866: The Nature of Light. 46 Molecules, atoms, and protons are "composite fields" made of the presumably fundamental standard model fields. 47 The Dirac field is a 4-component relativistic "spinor" field Ψi(x, t) (i = 1, 2, 3, 4). 48 R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. I (Addison-Wesley Publishing Co. Reading, MA, 1963), p. 37-2: "[The 2-slit experiment is] a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot explain the mystery in the sense of 'explaining' how it works. We will tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics." The italics are in the original. 49 Nick Herbert, Quantum Reality: Beyond the new physics (Doubleday, New York, 1985), pp. 60-67; conceptual discussion of the wave-particle duality of electrons. 50 Wolfgang Rueckner and Paul Titcomb, "A lecture demonstration of single photon interference," Am J. Phys. 64 (2), 184-188 (1996). Images courtesy of Wolfgang Rueckner, Harvard University Science Center. 51 A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Exawa, "Demonstration of single-electron buildup of an interference pattern," Am. J. Phys. 57 (2), 117-120 (1989). 52 Michler et al, "A quantum dot single-photon turnstile device," Science 290, 2282-2285. 53 For a more formal argument, see A. J. Leggett, "Testing the limits of quantum mechanics: motivation, state of play, prospects," J. Phys: Condensed Matter 14, R415- R451 (2002). 54 P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford at the Clarendon Press, Oxford, 3rd edition 1947), p. 9. The quoted statement appears in the 2nd, 3rd, and 4th editions, published respectively in 1935, 1947, and 1958. 55 J. von Neumann, The Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955), p. 351. 56 The central feature of this analysis, namely how decoherence localizes the quantum, was first discussed in W. K. Wootters and W. H. Zurek, "Complementarity in the doubleslit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle," Phys. Rev. D 19, 473-484 (1979). 57 M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition (Springer Verlag, Berlin, 2007), pp. 63-65. 58 H. D. Zeh, "There is no 'first' quantization," Phys. Lett. A 309, 329-334 (2003). 59 A. Einstein, B. Podolsky, and N. Rosen, "Can quantum-mechanical description of physical reality be considered complete?", Phys. Rev. 47 (10), 777-780 (1935). 60 J. Bell, "On the Einstein Podolsky Rosen Paradox," Physics 1 (3), 195-200 (1964). Art Hobson There are no particles, there are only fields 31 61 A. Aspect, "To be or not to be local," Nature 446, 866-867 (2007); S. Groblacher, A. Zeilinger, et al, "An experimental test of nonlocal realism," Nature 446, 871-875 (2007); G. C. Ghirardi, "The interpretation of quantum mechanics: where do we stand?" Fourth International Workshop DICE 2008, Journal of Physics: Conf Series 174 012013 (2009); T. Norsen, "Against 'realism'," Found. Phys. 37, 311-340 (2007); D. V. Tansk, "A criticism of the article, 'An experimental test of nonlocal realism'," arXiv 0809.4000 (2008); 62 L. E. Ballentine and J. P. Jarrett, "Bell's theorem: Does quantum mechanics contradict relativity?" Am. J. Phys. 55, 696-701 (Aug 1987). 63 Gerhard C. Hegerfeldt, "Particle localization and the notion of Einstein causality," in Extensions of Quantum Theory 3, edited by A. Horzela and E. Kapuscik (Apeiron, Montreal, 2001), pp. 9-16; "Instantaneous spreading and Einstein causality in quantum theory," Annalen der Physik 7, 716-725 (1998); "Remark on causality and particle localization," PR D 10 (1974), 3320-3321. 64 D. B. Malament, "In defense of dogma: why there cannot be a relativistic QM of localizable particles." Perspectives on quantum reality (Kluwer Academic Publishers, 1996, Netherlands), pp. 1-10. See also Refs. 34 and 65. 65 H. Halvorson and R. Clifton, "No place for particles in relativistic quantum theories?" Philosophy of Science 69, 1-28 (2002). 66 Rafael de la Madrid, "Localization of non-relativistic particles," International Journal of Theoretical Physics, 46, 1986-1997 (2007). Hegerfeldt's result for relativistic particles generalizes Madrid's result. 67 P. H. Eberhard and R. R. Ross, "Quantum field theory cannot provide faster-then-light communication," Found. Phys. Letts. 2, 127-148 (1989). 68 In other words, the Schroedinger equation can be quantized, just like the Dirac equation. But the quantized version implies nothing that isn't already in the nonquantized version. See Ref. 36. 69 S. Weinberg, Elementary Particles and the Laws of Physics, The 1986 Dirac Memorial Lectures (Cambridge University Press, Cambridge, 1987), pp. 78-79: "Although it is not a theorem, it is widely believed that it is impossible to reconcile quantum mechanics and relativity, except in the context of a quantum field theory." 70 Michael Redhead, "A philosopher looks at quantum field theory," in Philosophical Foundations of Quantum Field Theory, ed by Harvey R. Brown and Rom Harre (Oxford UP, 1988), pp. 9-23: "What is the nature of the QFT vacuum? In the vacuum state ... there is still plenty going on, as evidenced by the zero-point energy ...[which] reflects vacuum fluctuations in the field amplitude. These produce observable effects ....I am now inclined to say that vacuum fluctuation phenomena show that the particle picture is not adequate to QFT. QFT is best understood in terms of quantized excitations of a field and that is all there is to it." 71 S. Weinberg, "The cosmological constant problem," Revs. Mod. Phys. 61, 1-23 (1989). 72 My main source for Secs. VI A and B is Peter W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics (Academic Press Limited, London, 1994). 73 W. G. Unruh, "Notes on black hole evaporation," Phys. Rev. D 14, 870-892 (1976); P.C.W. Davies. “Scalar production in Schwarzschild and Rindler metrics,” Journal of Physics A 8, 609 (1975). Art Hobson There are no particles, there are only fields 32 74 S. Barshay and W. Troost, "A possible origin for temperature in strong interactions," Phys. Lett. 73B, 4 (1978); S. Barshay, H. Braun, J. P. Gerber, and G. Maurer, "Possible evidence for fluctuations in the hadronic temperature," Phys. Rev. D 21, 7 (1980); J. S. Bell and J. M. Leinaas, "Electrons as accelerated thermometers," Nucl. Phys. B212, 131 (1983). 75 Emil T. Akhmedov and Douglas Singleton, "On the physical meaning of the Unruh effect," arXiv:0705.2525v3 [hep-th], 19 Oct 2007. 76 B. S. DeWitt, "Quantum gravity: the new synthesis," in S. Hawking and W. Israel, Editors, General Relativity: An Einstein Centenary (Cambridge University Press, Cambridge, 1979), pp. 680-745. 77 M. Jammer The Philosophy of Quantum Mechanics (Wiley-Interscience, New York, 1974), pp 115-118. 78 S. M. Tan, D. F. Walls, and M. J. Collett, "Nonlocality of a single photon," Phys. Rev. Lett. 66, 252-255 (1991). 79 M.O. Terra Cunha, J.A. Dunningham, V. Vedral, "Entanglement in single-particle systems," Proc. Royal Soc. A 463, 2277-2286 (2007): "If we want to increase the breadth of applicability of entanglement, we should think in terms of fields which are a fundamental description of nature. Particles are only a manifestation of certain special configurations of quantum fields. If entanglement is to be considered a fundamental property of nature, and even a resource to be understood and applied, one would like to understand entangled fields." 80 L Hardy, “Nonlocality of a single photon revisited,” Phys. Rev. Lett. 73, 2277-2286 (1994). 81 L. Hardy, "N-measurement Bell inequalities, N-atom entangled, states, and the nonlocality of one photon," Phys. Lett. A 160, 1-8 (1991); E. Santos, "Comment on 'Nonlocality of a single photon'," Phys. Rev. Lett. 68, 894 (1992); D. M. Greenberger, M. A. Horne, and A. Zeilinger, "Nonlocality of a single photon?" Phys. Rev. Lett. 75, 2064 (1995); Lev Vaidman, "Nonlocality of a single photon revisited again," Phys. Rev. Lett 75, 2063 (1995); A. Peres, "Nonlocal effects in Fock space," Phys. Rev. Lett. 74, 4571 (1995); L. Hardy, "Hardy replies," Phys. Rev. Lett. 75, 2065-2066 (1995); R. J. C. Spreeuw, "A classical analogy of entanglement," Founds. Phys. 28, 361-374 (1998); 82 C. C. Gerry, "Nonlocality of a single photon in cavity QED," Phys. Rev. A 53, 4583- 4586 (1996); H. W. Lee and J. Kim, "Quantum teleportation and Bell's inequality using single-particle entanglement," Phys. Rev. A 63, 012305 (2000); G. Bjork, P. Jonsson, L. L. Sanchez-Soto, "Single-particle nonlocality and entanglement with the vacuum," Phys. Rev. A 64, 042106 (2001). 83 E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini, "Teleportation of a vacuum-- one-photon qubit," Phys. Rev. Lett. 88, 070402 (2002). 84 B. Hessmo, P. Usachev, H. Heydari, and G. Bjork, “Experimental demonstration of single photon nonlocality,” Phys. Rev. Lett. 92, 2004. 85 S. J. van Enk, “Single-particle entanglement,” Phys. Rev. A 72, 064306 (2005). 86 A. Drezet, "Comment on single-particle entanglement," Phys. Rev. 74, 026301 (2006). 87 J. Dunningham and V. Vedral, “Nonlocality of a single particle,” Phys. Rev. Lett. 99, 180404 (2007). No place for particles in relativistic quantum theories? Hans Halvorson and Rob Clifton Department of Philosophy, University of Pittsburgh hphst1@pitt.edu, rclifton@pitt.edu Abstract Several recent arguments purport to show that there can be no relativistic, quantum-mechanical theory of localizable particles and, thus, that relativity and quantum mechanics can be reconciled only in the context of quantum field theory. We point out some loopholes in the existing arguments, and we provide two no-go theorems to close these loopholes. However, even with these loopholes closed, it does not yet follow that relativity plus quantummechanics exclusively requires a field ontology, since relativistic quantum field theory itself might permit an ontology of localizable particles supervenient on the fundamental fields. Thus, we provide another no-go theorem to rule out this possibility. Finally, we allay potential worries about this conclusion by arguing that relativistic quantum field theory can nevertheless explain the possibility of “particle detections”, as well as the pragmatic utility of “particle talk.” 1 Introduction It is a widespread belief, at least within the physics community, that there is no particle mechanics that is simultaneously relativistic and quantum-theoretic; and, thus, that the only relativistic quantum theory is a field theory. This belief has received much support in recent years in the form of rigorous “no-go theorems” by Malament (1996) and Hegerfeldt (1998a, 1998b). In particular, Hegerfeldt shows that in a generic quantum theory (relativistic or non-relativistic), if there are states with localized particles, and if there is a lower bound on the system’s energy, then superluminal spreading of the wavefunction must occur. Similarly, Malament shows the inconsistency of a few intuitive desiderata for a relativistic, quantum-mechanical theory of (localizable) particles. Thus, it appears that there is a fundamental conflict between the demands of relativistic causality and the requirements of a theory of localizable particles. 1 What is the philosophical lesson of this apparent conflict between relativistic causality and localizability? One the one hand, if we believe that the assumptions of Malament’s theorem must hold for any theory that is descriptive of our world, then it follows that our world cannot be correctly described by a particle theory. On the other hand, if we believe that our world can be correctly described by a particle theory, then one (or more) of the Malament’s assumptions must be false. Malament clearly endorses the first response; that is, he argues that his theorem entails that there is no relativistic quantum mechanics of localizable particles (insofar as any relativistic theory precludes act-outcome correlations at spacelike separation). Others, however, have argued that the assumptions of Malament’s theorem need not hold for any relativistic, quantum-mechanical theory (cf. Fleming and Butterfield 1999), or that we cannot judge the truth of the assumptions until we resolve the interpretive issues of elementary quantum mechanics (cf. Barrett 2000). Although we do not think that these arguments against Malament’s assumptions succeed, there are other reasons to doubt thatMalament’s theorem is sufficient to support a sound argument against the possibility of a relativistic quantum mechanics of localizable particles. First, Malament’s theorem depends on a specific assumption about the structure ofMinkowski spacetime—a “no preferred reference frame” assumption—that could be seen as having less than full empirical warrant. Second, Malament’s theorem establishes only that there is no relativistic quantum mechanics in which particles can be completely localized in spatial regions with sharp boundaries; it leaves open the possibility that there might be a relativistic quantum mechanics of “unsharply” localized particles. In this paper, we present two new no-go theorems which, together, suffice to close these loopholes in the argument against relativistic quantum mechanics. First, we present a strengthened no-go theorem that subsumes the results of Malament and Hegerfeldt, and which does not depend on the “no preferred frame” assumption (Theorem 1). Second, we derive a generalized version of Malament’s theorem that shows that there is no relativistic quantum mechanics of “unsharply” localized particles (Theorem 2). However, it would be a mistake to think that these result show—or, are intended to show—that a field ontology, rather than a particle ontology, is appropriate for relativistic quantum theories. While these results show that there are no position observables that satisfy certain relativistic constraints, quantum field theories—both relativistic and non-relativistic—already reject the notion of position observables in favor of “localized” field observables. Thus, no-go results against relativistic position operators have nothing to say about the possibility that relativistic quantum field theory might permit a “particle interpretation,” in which localized particles are supervenient on the underlying localized field observables. To exclude this latter possibility, we formulate (in Section 6) a necessary condition for a generic quantum theory to permit a particle interpretation, and we then show 2 that this condition fails in any relativistic theory (Theorem 3). Since our world is presumably both relativistic and quantum-theoretic, these results show that there are no localizable particles. However, in Section 7 we shall argue that relativistic quantum field theory itself warrants an approximate use of “particle talk” that is sufficient to save the phenomena. 2 Malament’sTheorem Malament’s theorem shows the inconsistency of a few intuitive desiderata for a relativistic quantum mechanics of (localizable) particles. It strengthens previous results (e.g., Schlieder 1971) by showing that the assumption of “no superluminal wavepacket spreading” can be replaced by the weaker assumption of “microcausality,” and by making it clear that Lorentz invariance is not needed to derive a conflict between relativistic causality and localizability. In order to present Malament’s result, we assume that our background spacetime M is an affine space, with a foliation S into spatial hyperplanes. (For ease, we can think of an affine space as a vector space, so long as we do not assign any physical significance to the origin.) This will permit us to consider a wide range of relativistic (e.g., Minkowski) as well as non-relativistic (e.g., Galilean) spacetimes. The pure states of our quantum-mechanical system are given by rays in some Hilbert space H. We assume that there is amapping ! !" E! of bounded subsets of hyperplanes in M into projections on H. We think of E! as representing the proposition that the particle is localized in !; or, from a more operational point of view, E! represents the proposition that a position measurement is certain to find the particle within !. We also assume that there is a strongly continuous representation a !" U(a) of the translation group of M in the unitary operators on H. Here strong continuity means that for any unit vector ! #H,$!,U(a)!% " 1
as a " 0; and it is equivalent (via Stone’s theorem) to the assumption that there are
energy and momentum observables for the particle. If all of the preceding conditions
hold, we say that the triple (H,! !" E!, a !" U(a)) is a localization system
over M.
The following conditions should hold for any localization system—either relativistic
or non-relativistic—that describes a single particle.
Localizability: If ! and !! are disjoint subsets of a single hyperplane, then
E!E!! = 0.
Translation covariance: For any ! and for any translation a of M,
U(a)E!U(a)" = E!+a.
3
Energy bounded below: For any timelike translation a of M, the generator H(a)
of the one-parameter group {U(ta) : t # R} has a spectrum bounded from
below.
We recall briefly the motivation for each of these conditions. “Localizability” says
that the particle cannot be detected in two disjoint spatial sets at a given time.
“Translation covariance” gives us a connection between the symmetries of the
spacetime M and the symmetries of the quantum-mechanical system. In particular,
if we displace the particle by a spatial translation a, then the original wavefunction
! will transform to some wavefunction !a. Since the statistics for the
displaced detection experiment should be identical to the original statistics, we
have $!,E!!% =$!a,E!+a!a%. ByWigner’s theorem, however, the symmetry
is implemented by some unitary operator U(a). Thus, U(a)! = !a, and
U(a)E!U(a)" = E!+a. In the case of time translations, the covariance condition
entails that the particle has unitary dynamics. (This might seem to beg the
question against a collapse interpretation of quantum mechanics; we dispell this
worry at the end of this section.) Finally, the “energy bounded below” condition
asserts that, relative to any free-falling observer, the particle has a lowest possible
energy state. If it were to fail, we could extract an arbitrarily large amount of
energy from the particle as it drops down through lower and lower states of energy.
We now turn to the “specifically relativistic” assumptions needed for Malament’s
theorem. The special theory of relativity entails that there is a finite upper
bound on the speed at which (detectable) physical disturbances can propagate
through space. Thus, if ! and !! are distant regions of space, then there is a
positive lower bound on the amount of time it should take for a particle localized
in ! to travel to !!. We can formulate this requirement precisely by saying that
for any timelike translation a, there is an " > 0 such that, for every state !, if
$!,E!!% = 1 then$!,E!!+ta!% = 0 whenever 0 & t < ". This is equivalent to
the following assumption.
Strong causality: If ! and !! are disjoint subsets of a single hyperplane, and if
the distance between ! and !! is nonzero, then for any timelike translation
a, there is an " > 0 such that E!E!!+ta = 0 whenever 0 & t < ".
(Note that strong causality entails localizability.) Although strong causality is a
reasonable condition for relativistic theories, Malament’s theorem requires only
the following weaker assumption (which he himself calls “locality”).
Microcausality: If ! and !! are disjoint subsets of a single hyperplane, and if the
distance between ! and !! is nonzero, then for any timelike translation a,
there is an " > 0 such that [E!,E!!+ta] = 0 whenever 0 & t < ".
4
If E! can be measured within !, microcausality is equivalent to the assumption
that a measurement within ! cannot influence the statistics of measurements performed
in regions that are spacelike to ! (see Malament 1996, 5). Conversely, a
failure of microcausality would entail the possibility of act-outcome correlations at
spacelike separation. Note that both strong and weak causality make sense for nonrelativistic
spacetimes (as well as for relativistic spacetimes); though, of course, we
should not expect either causality condition to hold in the non-relativistic case.
Theorem (Malament). Let (H,! !" E!, a !" U(a)) be a localization system
over Minkowski spacetime that satisfies:
1. Localizability
2. Translation covariance
3. Energy bounded below
4. Microcausality
Then E! = 0 for all !.
Thus, in every state, there is no chance that the particle will be detected in any local
region of space. AsMalament claims, this serves as a reductio ad absurdum of any
relativistic quantum mechanics of a single (localizable) particle.
Several authors have claimed that Malament’s theorem is not sufficient to rule
out a relativistic quantum mechanics of localizable particles. In particular, these authors
argue that it is not reasonable to expect the conditions ofMalament’s theorem
to hold for any relativistic, quantum-mechanical theory of particles. For example,
Dickson (1997) argues that a ‘quantum’ theory does not need a position operator
(equivalently, a system of localizing projections) in order to treat position as
a physical quantity; Barrett (2000) argues that time-translation covariance is suspect;
and Fleming and Butterfield (1999) argue that the microcausality assumption
is not warranted by special relativity. We now show, however, that none of these
arguments is decisive against the assumptions of Malament’s theorem.
Dickson (1997, 214) cites the Bohmian interpretation of the Dirac equation as
a counterexample to the claim that any ‘quantum’ theory must represent position
by an operator. In order to see what Dickson might mean by this, recall that the
Dirac equation admits both positive and negative energy solutions. If H denotes
the Hilbert space of all (both positive and negative energy) solutions, then we may
define the ‘standard position operator’ Q by setting Q!(x) = x · !(x) (Thaller
1992, 7). If, however, we restrict to the Hilbert space Hpos ' H of positive energy
solutions, then the probability density given by the Dirac wavefunction does
5
not correspond to a self-adjoint position operator (Thaller 1992, 32). According to
Holland (1993, 502), this lack of a position operator on Hpos precludes a Bohmian
interpretation of !(x) as a probability amplitude for finding the particle in an elementary
volume d3x around x.
Since the Bohmian interpretation of the Dirac equation uses all states (both
positive and negative energy), and the corresponding position observable Q, it is
not clear what Dickson means by saying that the Bohmian interpretation of the
Dirac equation dispenses with a position observable. Moreover, since the energy
is not bounded below in H, this would not in any case give us a counterexample
to Malament’s theorem. However, Dickson could have developed his argument by
appealing to the positive energy subspace Hpos. In this case, we can talk about
positions despite the fact that we do not have a position observable in the usual
sense. In particular, we shall show in Section 5 that, for talk about positions, it
suffices to have a family of “unsharp” localization observables. (And, yet, we shall
show that relativistic quantum theories do not permit even this attenuated notion of
localization.)
Barrett (2000) argues that the significance of Malament’s theorem cannot be
assessed until we have solved the measurement problem:
If we might have to violate the apparently weak and obvious assumptions
that go into proving Malament’s theorem in order to get a satisfactory
solution to the measurement problem, then all bets are off
concerning the applicability of the theorem to the detectible entities
that inhabit our world. (Barrett 2000, 16)
In particular, a solution to the measurement problem may require that we abandon
unitary dynamics. But if we abandon unitary dynamics, then the translation covariance
condition does not hold, and we need not accept the conclusion that there is
no relativistic quantum mechanics of (localizable) particles.
Unfortunately, it is not clear that we could avoid the upshot of Malament’s theorem
by moving to a collapse theory. Existing (non-relativistic) collapse theories
take the empirical predictions of quantum theory seriously. That is, the “statistical
algorithm” of quantum mechanics is assumed to be at least approximately correct;
and collapse is introduced only to ensure that we obtain determinate properties
at the end of a measurement. However, in the present case, Malament’s theorem
shows that the statistical algorithm of any quantum theory predicts that if there are
local particle detections, then act-outcome correlations are possible at spacelike
separation. Thus, if a collapse theory is to stay close to these predictions, it too
would face a conflict between localizability and relativistic causality.
Perhaps, then, Barrett is suggesting that the price of accomodating localizable
particles might be a complete abandonment of unitary dynamics, even at the level
6
of a single particle. In other words, wemay be forced to adopt a collapse theory
without having any underlying (unitary) quantum theory. But even if this is correct,
it wouldn’t count against Malament’s theorem, which was intended to show that
there is no relativistic quantum theory of localizable particles. Furthermore, noting
that Malament’s theorem requires unitary dynamics is one thing; it would be quite
another thing to provide a model in which there are localizable particles—at the
price of non-unitary dynamics—but which is also capable of reproducing the wellconfirmed
quantum interference effects at the micro-level. Until we have such a
model, pinning our hopes for localizable particles on a failure of unitary dynamics
is little more than wishful thinking.
Like Barrett, Fleming (Fleming and Butterfield 1999, 158ff) disagrees with
the reasonableness of Malament’s assumptions. Unlike Barrett, however, Fleming
provides a concrete model in which there are localizable particles (viz., using the
Newton-Wigner position operator as a localizing observable) and in which Malament’s
microcausality assumption fails. Nonetheless, Fleming argues that this failure
of microcausality is perfectly consistent with relativistic causality.
According to Fleming, the property “localized in !” (represented by E!) need
not be detectable within !. As a result, [E!,E!! ] (= 0 does not entail that it is
possible to send a signal from ! to !!. However, by claiming that local beables
need not be local observables, Fleming undercuts the primary utility of the notion
of localization, which is to indicate those physical quantities that are operationally
accessible in a given region of spacetime. Indeed, it is not clear what motivation
there could be—aside from indicating what is locally measurable—for assigning
observables to spatial regions. If E! is not measurable in !, then why should we
say that “E! is localized in !”? Why not say instead that “E! is localized in !!”
(where !! (= !)? Does either statement have any empirical consequences and,
if so, how do their empirical consequences differ? Until these questions are answered,
we maintain that local beables are always local observables; and a failure
of microcausality would entail the possibility of act-outcome correlations at spacelike
separation. Therefore, the microcausality assumption is an essential feature of
any relativistic quantum theory with “localized” observables. (For a more detailed
argument along these lines, see Halvorson 2001, Section 6.)
Thus, the arguments against the four (explicit) assumptions ofMalament’s theorem
are unsuccessful; these assumptions are perfectly reasonable, and we should
expect them to hold for any relativistic, quantum-mechanical theory. However,
there is another difficulty with the argument against any relativistic quantum mechanics
of (localizable) particles: Malament’s theorem makes tacit use of specific
features of Minkowski spacetime which—some might claim—have less than perfect
empirical support. First, the following example shows that Malament’s theorem
fails if there is a preferred reference frame.
7
Example 1. Let M = R1 ) R3 be full Newtonian spacetime (with a distinguished
timelike direction a). To any set of the form {(t, x) : x # !}, with t # R, and !
a bounded open subset of R3, we assign the spectral projection E! of the position
operator for a particle in three dimensions. Let H(a) = 0 so that U(ta) = eit0 = I
for all t # R. Since the energy inevery state is zero, the energy condition is trivially
satisfied.
Note, however, that if the background spacetime is not regarded as having a
distinguished timelike direction, then this example violates the energy condition.
Indeed, the generator of an arbitrary timelike translation has the form
H(b) = b · P = b00 + b1P1 + b2P2 + b3P3 = b1P1 + b2P2 + b3P3, (1)
where b = (b0, b1, b2, b3) # R4 is a timelike vector, and Pi are the three orthogonal
components of the total momentum. But since each Pi has spectrum R, the
spectrum of H(b) is not bounded from below when b (= a. !
Malament’s theorem does not require the full structure of Minkowski spacetime
(e.g., the Lorentz group). Rather, it suffices to assume that the affine space M
satisfies the following condition.
No absolute velocity: Let a be a spacelike translation of M. Then there is a pair
(b, c) of timelike translations of M such that a = b − c.
Despite the fact that “no absolute velocity” is a feature of all post-Galilean spacetimes,
there are some who claim that the existence of a (undetectable) preferred
reference frame is perfectly consistent with the empirical evidence on which relativistic
theories are based (cf. Bell 1987, Chap. 9). What is more, the existence of
a preferred frame is an absolutely essential feature of a number of “realistic” interpretations
of quantum theory (cf. Maudlin 1994, Chap. 7). Thus, this tacit assumption
of Malament’s theorem has the potential to be a major source of contention
for those wishing to maintain that there can be a relativistic quantum mechanics of
localizable particles.
There is a further worry about the generality of Malament’s theorem: It is not
clear whether the result can be expected to hold for arbitrary relativistic spacetimes,
or whether it is an artifact of peculiar features of Minkowski spacetime
(e.g., that space is infinite). To see this, suppose that M is an arbitrary globally
hyperbolic manifold. (That is, M is a manifold that permits at least one foliation
S into spacelike hypersurfaces). Although M will not typically have a translation
group, we suppose that M has a transitive Lie group G of diffeomorphisms. (Just
as a manifold is locally isomorphic to Rn, a Lie group is locally isomorphic to a
group of translations.) We require that G has a representation g !" U(g) in the
8
unitary operators on H; and, the translation covariance condition now says that
Eg(!) = U(g)E!U(g)" for all g # G.
The following example shows that Malament’s theorem fails even for the very
simple case where M is a two-dimensional cylinder.
Example 2. Let M = R ) S1, where S1 is the one-dimensional unit circle, and
let G denote the Lie group of timelike translations and rotations of M. It is not
difficult to construct a unitary representation of G that satisfies the energy bounded
below condition. (We can use the Hilbert space of square-integrable functions
from S1 into C, and the procedure for constructing the unitary representation is
directly analogous to the case of a single particle moving on a line.) Fix a spacelike
hypersurface ", and let μ denote the normalized rotation-invariant measure on ".
For each open subset ! of ", let E! = I if μ(!) + 2/3, and let E! = 0 if
μ(!) < 2/3. Then localizability holds, since for any pair (!,!!) of disjoint open
subsets of ", either μ(!) < 2/3 or μ(!!) < 2/3. !
Nonetheless, Examples 1 and 2 hardly serve as physically interesting counterexamples
to a strengthened version of Malament’s theorem. In particular, in
Example 1 the energy is identically zero, and therefore the probability for finding
the particle in a given region of space remains constant over time. In Example 2,
the particle is localized in every region of space with volume greater than 2/3, and
the particle is never localized in a region of space with volume less than 2/3. In
the following two sections, then, we will formulate explicit conditions to rule out
such pathologies, and we will use these conditions to derive a strengthened version
of Malament’s theorem that applies to generic spacetimes.
3 Hegerfeldt’sTheorem
Hegerfeldt’s (1998a, 1998b) recent results on localization apply to arbitrary (globally
hyperbolic) spacetimes, and they do not make us of the “no absolute velocity”
condition. Thus, we will suppose henceforth thatM is a globally hyperbolic spacetime,
and we will fix a foliation S of M, aswell as a unique isomorphism between
any two hypersurfaces in this foliation. If " # S, wewillwrite "+t for the hypersurface
that results from “moving " forward in time by t units”; and if! is a subset
of ", wewill use!+t to denote the corresponding subset of "+t. We assume that
there is a representation t !" Ut of the time-translation group R in the unitary operators
on H, and we will say that the localization system (H,! !" E!, t !" Ut)
satisfies time-translation covariance just in case UtE!U−t = E!+t for all ! and
all t # R.
Hegerfeldt’s result is based on the following root lemma.
9
Lemma 1 (Hegerfeldt). Suppose that Ut = eitH, where H is a self-adjoint operator
with spectrum bounded from below. Let A be a positive operator (e.g., a
projection operator). Then for any state !, either
$Ut!,AUt!% (= 0, for almost all t # R, or$Ut!,AUt!% = 0, for all t # R.
Hegerfeldt claims that this lemma has the following consequence for localization:
If there exist particle states which are strictly localized in some finite
region at t = 0and later move towards infinity, then finite propagation
speed cannot hold for localization of particles. (Hegerfeldt 1998a,
243)
Hegerfeldt’s argument for this conclusion is as follows:
Now, if the particle or system is strictly localized in ! at t = 0 it
is, a fortiori, also strictly localized in any larger region !! containing
!. If the boundaries of !! and ! have a finite distance and if finite
propagation speed holds then the probability to find the system in !!
must also be 1 for sufficiently small times, e.g. 0 & t < ". But then
[Lemma 1], with A , I−E!!, states that the systemstays in!! for all
times. Now, we can make !! smaller and let it approach !. Thuswe
conclude that if a particle or system is at time t = 0 strictly localized
in a region !, then finite propagation speed implies that it stays in
! for all times and therefore prohibits motion to infinity. (Hegerfeldt
1998a, 242–243; notation adapted, but italics in original)
Let us attempt now to put this argument into a more precise form.
First, Hegerfeldt claims that the following is a consequence of “finite propagation
speed”: If ! - !!, and if the boundaries of ! and !! have a finite distance,
then a state initially localized in ! will continue to be localized in !! for some
finite amount of time. We can capture this precisely by means of the following
condition.
No instantaneous wavepacket spreading (NIWS): If ! - !!, and the boundaries
of ! and !! have a finite distance, then there is an " > 0 such that E! &
E!!+t whenever 0 & t < ".
10
(Note that NIWS plus localizability entails strong causality.) In the argument,
Hegerfeldt also assumes that if a particle is localized in every one of a family of
sets that “approaches” !, then it is localized in !. We can capture this assumption
in the following condition.
Monotonicity: If {!n : n # N} is a downward nested family of subsets of " such
that !n !n =!, then "n E!n = E!.
Using this assumption, Hegerfeldt argues that if NIWS holds, and if a particle is
initially localized in some finite region !, then it will remain in ! for all subsequent
times. In other words, if E!! = !, then E!Ut! = Ut! for all t + 0. We
can now translate this into the following rigorous no-go theorem.
Theorem (Hegerfeldt). Suppose that the localization system (H,! !" E!, t !"
Ut) satisfies:
1. Monotonicity
2. Time-translation covariance
3. Energy bounded below
Then UtE!U−t = E! for all ! ' " and all t # R.
(For the proof of this theorem, see Appendix A.)
Thus, conditions 1–4 can be satisfied only if the particle has trivial dynamics. If
M is an affine space, and if we add “no absolute velocity” as a fifth condition in this
theorem, then we get the stronger conclusion that E! = 0 for all bounded ! (see
Lemma 2, appendix). Thus, there is an obvious similarity between Hegerfeldt’s
and Malament’s theorems. However, NIWS is a stronger causality assumption than
microcausality. In fact, while NIWS plus localizability entails strong causality (and
hence microcausality), the following example shows that NIWS is not entailed by
the conjunction of strong causality, monotonicity, time-translation covariance, and
energy bounded below.
Example 3. Let Q, P denote the standard position and momentum operators on
H = L2(R), and let H = P2/2m for some m > 0. Let ! !" EQ
! denote
the spectral measure for Q. Fix some bounded subset !0 of R, and let E! =
EQ
! . EQ
!0
(a projection operator on H.H) for all Borel subsets ! of R. Thus,
! !" E! is a (non-normalized) projection-valued measure. Let Ut = I.eitH, and
let E!+t = UtE!U−t for all t # R. It is clear thatmonotonicity, time-translation
11
covariance, and energy bounded below hold. To see that strong causality holds, let
! and !! be disjoint subsets of a single hyperplane ". Then,
E!UtE!!U−t = EQ
!EQ
!! . EQ
!0EQ
!0+t = 0. EQ
!EQ
!0+t = 0, (2)
for all t # R. On the other hand, UtE!U−t (= E! for any nonempty ! and for
any t (= 0. Thus, it follows fromHegerfeldt’s theorem that NIWS fails. !
Thus, we could not recapture the full strength of Malament’s theorem simply
by adding “no absolute velocity” to the conditions of Hegerfeldt’s theorem.
4 AStrengthenedHegerfeldt-MalamentTheorem
Example 3 shows that Hegerfeldt’s theorem fails if NIWS is replaced by strong
causality (or by microcausality). On the other hand, Example 3 is hardly a physically
interesting counterexample to a strengthened version ofHegerfeldt’s theorem.
In particular, if " is a fixed spatial hypersurface, and if {!n : n # N} is a covering
of " by bounded sets (i.e., #n !n = "), then $n E!n = I .E!0 (= I .I. Thus, it is not certain that the particle will be detected somewhere or other in space. In fact, if {!n : n # N} is a covering of " and {#n : n # N} is a covering of "+t, then %n$N
E!n = I . E!0 (= I . E!0+t = %n$N E"n. (3) Thus, the total probability for finding the particle somewhere or other in space can change over time. It would be completely reasonable to require that$n E!n = I whenever {!n :
n # N} is a covering of ". This would be the case, for example, if the mapping
! !" E! (restricted to subsets of ") were the spectral measure of some position
operator. However, we propose that—at the very least—any physically interesting
model should satisfy the following weaker condition.
Probability conservation: If {!n : n # N} is a covering of ", and {#n : n # N}
is a covering of "+t, then $n E!n =$n E"n.
Probability conservation guarantees that there is a well-defined total probability
for finding the particle somewhere or other in space, and this probability remains
constant over time. In particular, if both {!n : n # N} and {#n : n # N} consist
of pairwise disjoint sets, then the localizability condition entails that $n E!n = &n E!n and$n E"n = &n E"n. In this case, probability conservation is equivalent
to
'n$N Prob!(E!n) = 'n$N
Prob!(E"n) , (4)
12
for any state !. Note, finally, that probability conservation is neutral with respect
to relativistic and non-relativistic models.1
Theorem 1 (Strengthened Hegerfeldt-Malament Theorem). Suppose that the localization
system (H,! !" E!, t !" Ut) satisfies:
1. Localizability
2. Probability conservation
3. Time-translation covariance
4. Energy bounded below
5. Microcausality
Then UtE!U−t = E! for all ! and all t # R.
(For the proof of this theorem, see Appendix A.)
IfM is an affine space, and if we add “no absolute velocity” as a sixth condition
in this theorem, then it follows that E! = 0for all!(see Lemma 2). Thus, modulo
the probability conservation condition, Theorem 1 recaptures the full strength of
Malament’s theorem. Moreover, we can now trace the difficulties with localization
to microcausality alone: there are localizable particles only if it is possible to have
act-outcome correlations at spacelike separation.
We now give examples to show that each condition in Theorem 1 is indispensable;
that is, no four of the conditions suffices to entail the conclusion. (Example 1
shows that conditions 1–5 can be simultaneously satisfied.) Suppose for simplicity
that M is two-dimensional. (All examples work in the four-dimensional case as
well.) Let Q, P be the standard position and momentum operators on L2(R), and
let H = P2/2m. Let " be a spatial hypersurface in M, and suppose that a coordinatization
of " has been fixed, so that there is a natural association between each
bounded open subset ! of " and a corresponding spectral projection E! of Q.
(1+2+3+4) (a) Consider the standard localization system for a single non-relativistic
particle. That is, let " be a fixed spatial hyperplane, and let ! !" E! (with
domain the Borel subsets of ") be the spectral measure for Q. For " + t,
set E!+t = UtE!U−t, where Ut = eitH. (b) TheNewton-Wigner approach
to relativistic QM uses the standard localization system for a non-relativistic
1Probability conservation would fail if a particle could escape to infinity in a finite amount of time
(cf. Earman 1986, 33). However, a particle can escape to infinity only if there is an infinite potential
well, and this would violate the energy condition. Thus, given the energy condition, probability
conservation should also hold for non-relativistic particle theories.
13
particle, only replacing the non-relativistic Hamiltonian P2/2m with the relativistic
Hamiltonian (P2 +m2I)1/2, whose spectrum is also bounded from
below.
(1+2+3+5) (a) For a mathematically simple (but physically uninteresting) example,
take the first example above and replace the Hamiltonian P2/2m with
P. In this case, microcausality trivially holds, since UtE!U−t is just a
shifted spectral projection of Q. (b) For a physically interesting example,
consider the relativistic quantum theory of a single spin-1/2 electron (see
Section 2). Due to the negative energy solutions of the Dirac equation, the
spectrum of the Hamiltonian is not bounded from below.
(1+2+4+5) Consider the the standard localization system for a non-relativistic particle,
but set E!+t = E! for all t # R. Thus, we escape the conclusion of
trivial dynamics, but only by disconnecting the (nontrivial) unitary dynamics
from the (trivial) association of projections with spatial regions.
(1+3+4+5) (a) Let !0 be some bounded open subset of ", and let E!0 be the
corresponding spectral projection of Q. When ! (= !0 , let E! = 0. Let
Ut = eitH, and let E!+t = UtE!U−t for all !. This example is physically
uninteresting, since the particle cannot be localized in any region besides
!0, including proper supersets of !0. (b) See Example 3.
(2+3+4+5) Let !0 be some bounded open subset of ", and let E!0 be the corresponding
spectral projection of Q. When ! (= !0 , let E! = I. Let
Ut = eitH, and let E!+t = UtE!U−t for all !. Thus, the particle is always
localized in every region other than !0, and is sometimes localized in !0 as
well.
5 ArethereUnsharplyLocalizableParticles?
We have argued that attempts to undermine the four explicit assumptions of Malament’s
theorem are unsuccessful. We have also now shown that the tacit assumption
of “no absolute velocity” is not necessary to derive Malament’s conclusion.
And, yet, there is one more loophole in the argument against a relativistic quantum
mechanics of localizable particles. In particular, the basic assumption of a family
{E!} of localizing projections is unnecessary; it is possible to have a quantummechanical
particle theory in the absence of localizing projections. What is more,
one might object to the use of localizing projections on the grounds that they represent
an unphysical idealization—viz., that a “particle” can be completely contained
in a finite region of space with a sharp boundary, when in fact it would require an
14
infinite amount of energy to prepare a particle in such a state. Thus, there remains
a possibility that relativistic causality can be reconciled with “unsharp” localizability.
To see how we can define “particle talk” without having projection operators,
consider the relativistic theory of a single spin-1/2 electron (where we now restrict
to the subspace Hpos of positive energy solutions of the Dirac equation). In order
to treat the ‘x’ of the Dirac wavefunction as an observable, we need only to define
a probability amplitude and density for the particle to be found at x; and these can
be obtained from the Dirac wavefunction itself. That is, for a subset ! of ", we set
Prob!(x # !) = (!
|!(x)|2dx . (5)
Now let ! !" E! be the spectral measure for the standard position operator on
the Hilbert space H (which includes both positive and negative energy solutions).
That is, E! multiplies a wavefunction by the characteristic function of !. Let F
denote the orthogonal projection of H onto Hpos. Then,
(!
|!(x)|2dx = $!,E!!% =$!,FE!!%, (6)
for any ! # Hpos. Thus, we can apply the standard recipe to the operator FE!
(defined on Hpos) to compute the probability that the particle will be found within
!. However, FE! does not define a projection operator on Hpos. (In fact, it can
be shown that FE! does not have any eigenvectors with eigenvalue 1.) Thus, we
do not need a family of projection operators in order to define probabilities for
localization.
Now, in general, to define the probability that a particle will be found in !, we
need only assume that there is an operator A! such that $!,A!!% # [0, 1] for any unit vector !. Suchoperators are called effects, and include the projection operators as a proper subclass. Thus, we say that the triple (H,! !" A!, a !" U(a)) is an unsharp localization system over M just in case ! !" A! is a mapping from subsets of hyperplanes in M to effects on H, and a !" U(a) is a continuous representation of the translation group ofM in unitary operators onH. (We assume for the present that M is again an affine space.) Most of the conditions from the previous sections can be applied, with minor changes, to unsharp localization systems. In particular, since the energy bounded below condition refers only to the unitary representation, it can be carried over intact; and translation covariance also generalizes straightforwardly. However, we will need to take more care with microcausality and with localizability. If E and F are projection operators, [E,F] = 0 just in case for any state, the statistics of a measurement of F are not affected by a non-selective measurement 15 of E and vice versa (cf. Malament 1996, 5). This fact, along with the assumption that E! is measurable in !, motivates themicrocausality assumption. For the case of an association of arbitrary effects with spatial regions, Busch (1999, Proposition 2) has shown that [A!,A!!] = 0 just in case for any state, the statistics for a measurement of A! are not affected by a non-selective measurement of A!! and vice versa. Thus, we may carry over the microcausality assumption intact, again seen as enforcing a prohibition against act-outcome correlations at spacelike separation. The localizability condition is motivated by the idea that a particle cannot be simultaneously localized (with certainty) in two disjoint regions of space. In other words, if ! and !! are disjoint subsets of a single hyperplane, then$!,E!!% = 1
entails that $!,E!!!% = 0. It is not difficult to see that this last condition is equivalent to the assumption that E! + E!! & I. That is,$!, (E! + E!!)!% & $!, I!% , (7) for any state !. Now, it is an accidental feature of projection operators (as opposed to arbitrary effects) that E! + E!! & I is equivalent to E!E!! = 0. Thus, the apropriate generalization of localizability to unsharp localization systems is the following condition. Localizability: If ! and !! are disjoint subsets of a single hyperplane, then A! + A!! & I. That is, the probability for finding the particle in !, plus the probability for finding the particle in some disjoint region !!, never totals more than 1. It would, in fact, be reasonable to require a slightly stronger condition, viz., the probability of finding a particle in ! plus the probability of finding a particle in !! equals the probability of finding a particle in ! / !!. If this is true for all states !, we have: Additivity: If ! and !! are disjoint subsets of a single hyperplane, then A! + A!! = A!%!! . With just these mild constraints, Busch (1999) was able to derive the following no-go result. Theorem (Busch). Suppose that the unsharp localization system (H,! !" A!, a !" U(a)) satisfies localizability, translation covariance, energy bounded below, microcausality, and no absolute velocity. Then, for all !, A! has no eigenvector with eigenvalue 1. Thus, it is not possible for a particle to be localized with certainty in any bounded region !. Busch’s theorem, however, leaves it open question whether there are (nontrivial) “strongly unsharp” localization systems that satisfy microcausality. The following result shows that there are not. 16 Theorem 2. Suppose that the unsharp localization system (H,! !" A!, a !" U(a)) satisfies: 1. Additivity 2. Translation covariance 3. Energy bounded below 4. Microcausality 5. No absolute velocity Then A! = 0 for all !. (For the proof of this theorem, see Appendix B.) Theorem 2 shows that invoking the notion of unsharp localization does nothing to resolve the tension between relativistic causality and localizability. For example, we can now show that the (positive energy) Dirac theory—in which there are localizable particles—violates relativistic causality. Indeed, it is clear that the conclusion of Theorem 2 fails.2 On the other hand, additivity, translation covariance, energy bounded below, and no absolute velocity hold. Thus, microcausality fails, and the (positive energy) Dirac theory permits superluminal signalling. Unfortunately, Theorem 2 does not generalize to arbitrary globally hyperbolic spacetimes, as the following example shows. Example 4. Let M be the cylinder spacetime from Example 2. Let G denote the group of timelike translations and rotations of M, and let g !" U(g) be a positive energy representation of G in the unitary operators on a Hilbert space H. For any " # S, let μ denote the normalized rotation-invariant measure on ", and let A! = μ(!)I. Then, conditions 1–5 of Theorem2 are satisfied, but the conclusion of the theorem is false. ! The previous counterexample can be excluded if we require there to be a fixed positive constant # such that, for each !, there is a state ! with$!,A!!% + #.
In fact, with this condition added, Theorem 2 holds for any globally hyperbolic
spacetime. (The proof is an easy modification of the proof we give in Appendix B.)
However, it is not clear what physical motivation there could be for requiring this
further condition. Note also that Example 4 has trivial dynamics; i.e., UtA!U−t =
A! for all !. We conjecture that every counterexample to a generalized version of
Theorem 2 will have trivial dynamics.
2For any unit vector ! ! Hpos, there is a bounded set ! such that )! |!|2dx "= 0. Thus,
A! "= 0.
17
Theorem 2 strongly supports the conclusion that there is no relativistic quantum
mechanics of a single (localizable) particle; and that the only consistent combination
of special relativity and quantum mechanics is in the context of quantum
field theory. However, neither Theorem 1 nor Theorem 2 says anything about the
ontology of relativistic quantum field theory itself; they leave open the possibility
that relativistic quantum field theory might permit an ontology of localizable particles.
To eliminate this latter possibility, we will now proceed to present a more
general result which shows that there are no localizable particles in any relativistic
quantum theory.
6 ArethereLocalizableParticlesinRQFT?
The localizability assumption is motivated by the idea that a “particle” cannot be
detected in two disjoint spatial regions at once. However, in the case of a manyparticle
system, it is certainly possible for there to be particles in disjoint spatial
regions. Thus, the localizability condition does not apply to many-particle systems;
and Theorems 1 and 2 cannot be used to rule out a relativistic quantum mechanics
of n > 1 localizable particles.
Still, one might argue that we could use E! to represent the proposition that
a measurement is certain to find that all n particles lie within !, in which case
localizability should hold. Note, however, that when we alter the interpretation of
the localization operators {E!}, wemust alter our interpretation of the conclusion.
In particular, the conclusion now shows only that it is not possible for all n particles
to be localized in a bounded region of space. This leaves open the possibility that
there are localizable particles, but that they are governed by some sort of “exclusion
principle” that prohibits them all from clustering in a bounded spacetime region.
Furthermore, Theorems 1 and 2 only show that it is impossible to define position
operators that obey appropriate relativistic constraints. But it does not immediately
follow from this that we lack any notion of localization in relativistic
quantum theories. Indeed,
...a position operator is inconsistent with relativity. This compels us to
find another way of modeling localization of events. In field theory, we
model localization by making the observables dependent on position
in spacetime. (Ticiatti 1999, 11)
However, it is not a peculiar feature of relativistic quantum field theory that it lacks
a position operator: Any quantumfield theory (either relativistic or non-relativistic)
will model localization by making the observables dependent on position in spacetime.
Moreover, in the case of non-relativistic QFT, these “localized” observables
18
suffice to provide us with a concept of localizable particles. In particular, for each
spatial region !, there is a “number operator” N! whose eigenvalues give the
number of particles within the region !. Thus, we have no difficultly in talking
about the particle content in a given region of space despite the absence of any
position operator.
Abstractly, a number operator N on H is any operator with eigenvalues contained
in {0, 1, 2, . . .}. In order to describe the number of particles locally, we
require an association ! !" N! of subsets of spatial hyperplanes in M to number
operators on H, where N! represents the number of particles in the spatial region
!. If a !" U(a) is a unitary representation of the translation group, we say that
the triple (H,! !" N!, a !" U(a)) is a system of local number operators over
M. Note that a localization system(H,! !" E!, a !" U(a)) is a special case of a
system of local number operators where the eigenvalues of each N! are restricted
to {0, 1}. Furthermore, ifwe loosen our assumption that number operators have
a discrete spectrum, and instead require only that they have spectrum contained in
[0,0), then we can also include unsharp localization systems within the general
category of systems of local number operators. Thus, a system of local number
operators is the minimal requirement for a concept of localizable particles in any
quantum theory.
In addition to the natural analogues of the energy bounded below condition,
translation covariance, and microcausality, we will be interested in the following
two requirements on a system of local number operators:3
Additivity: If ! and !! are disjoint subsets of a single hyperplane, then
N! + N!! = N!%!! .
Number conservation: If {!n : n # N} is a disjoint covering of ", then the
sum &n N!n converges to a densely defined, self-adjoint operator N on
H (independent of the chosen covering), and U(a)NU(a)" = N for any
timelike translation a of M.
Additivity asserts that, when ! and !! are disjoint, the expectation value (in any
state !) for the number of particles in ! / !! is the sum of the expectations for
the number of particles in ! and the number of particles in !!. In the pure case, it
asserts that the number of particles in !/!! is the sum of the number of particles
3Due to the unboundedness of number operators, we would need to take some care in giving
technically correct versions of the following conditions. In particular, the additivity condition should
technically include the clause that N! and N!! have a common dense domain, and the operator
N!"!! should be thought of as the self-adjoint closure of N! + N!!. In the number conservation
condition, the sum N = &n N!n can be made rigorous by exploiting the correspondence between
self-adjoint operators and “quadratic forms” on H. In particular,we can think of N as deriving from
the upper bound of quadratic forms corresponding to finite sums.
19
in ! and the number of particles in !!. The “number conservation” condition tells
us that there is a well-defined total number of particles (at a given time), and that
the total number of particles does not change over time. This condition holds for
any non-interacting model of QFT.
It is a well-known consequence of the Reeh-Schlieder theorem that relativistic
quantum field theories do not admit systems of local number operators (cf. Redhead
1995). We will now derive the same conclusion from strictly weaker assumptions.
In particular, we show that microcausality is the only specifically relativistic
assumption needed for this result. The relativistic spectrum condition—which requires
that the spectrum of the four-momentum lie in the forward light cone, and
which is used in the proof of the Reeh-Schlieder theorem—plays no role in our
proof.4
Theorem 3. Suppose that the system (H,! !" N!, a !" U(a)) of local number
operators satisfies:
2. Translation covariance
3. Energy bounded below
4. Number conservation
5. Microcausality
6. No absolute velocity
Then N! = 0 for all !.
(For the proof of the theorem, see Appendix C.)
Thus, in every state, there are no particles in any local region. This serves
as a reductio ad absurdum for any notion of localizable particles in a relativistic
quantum theory.
Unfortunately, Theorem 3 is not the strongest result we could hope for, since
“number conservation” can only be expected to hold in the (trivial) case of noninteracting
fields. However, we would need a more general approach in order
to deal with interacting relativistic quantum fields, because (due to Haag’s theorem;
cf. Streater and Wightman, 2000, 163) their dynamics are not unitarily implementable
on a fixed Hilbert space. On the other hand it would be wrong to think of
4Microcausality is not only sufficient, but also necessary for the proof that there are no local
number operators. The Reeh-Schlieder theorem entails the cyclicity of the vacuum state. But the
cyclicity of the vacuum state alone does not entail that there are no local number operators; we must
also assume microcausality (cf. Halvorson 2001, Requardt 1982).
20
this as indicating a limitation on the generality of our conclusion: Haag’s theorem
also entails that interacting models of RQFT have no number operators—either
global or local.5 Still, it would be interesting to recover this conclusion (perhaps
working in a more general algebraic setting) without using the full strength of
Haag’s assumptions.
7 ParticleTalkwithoutParticleOntology
The results of the previous sections show that, insofar as we can expect any relativistic
quantum theory theory to satisfy a few basic conditions, these theories
do not admit (localizable) particles into their ontology. We also considered and
rejected several arguments which attempt to show that one (or more) of these conditions
can be jettisoned without doing violence to the theory of relativity or to
quantum mechanics. Thus, we have yet to find a good reason to reject one of the
premises on which our argument against localizable particles is based. However,
Segal (1964) and Barrett (2000) claim that we have independent grounds for rejecting
the conclusion; that is, we have good reasons for believing that there are
localizable particles.
The argument for localizable particles appears to be very simple: Our experience
shows us that objects (particles) occupy finite regions of space. But the reply
to this argument is just as simple: These experiences are illusory! Although no object
is strictly localized in a bounded region of space, an object can be well-enough
localized to give the appearance to us (finite observers) that it is strictly localized.
In fact, relativistic quantum field theory itself shows how the “illusion” of localizable
particles can arise, and how talk about localizable particles can be a useful
fiction.
In order to assess the possibility of “approximately localized” objects in relativistic
quantum field theory, we shall now pursue the investigation in the framework
of algebraic quantum field theory.6 Here, one assumes that there is a correspondence
O !" R(O) between bounded open subsets of M and subalgebras
of observables on some Hilbert space H. Observables in R(O) are considered to
be “localized” (i.e., measurable) in O. Thus, if O and O! are spacelike separated,
we require that [A,B] = 0 for any A # R(O) and B # R(O!). Furthermore,
5If a total number operator exists in a representation of the canonical commutation relations, then
that representation is quasiequivalent to a free-field (Fock) representation (Chaiken 1968). However,
Haag’s theorem entails that in relativistic theories, representations with nontrivial interactions are not
quasiequivalent to a free-field representation.
6For general information on algebraic quantum field theory, see (Haag 1992) and (Buchholz
2000). For specific information on particle detectors and “almost local” observables, see Chapter 6
of (Haag 1992) and Section 4 of (Buchholz 2000).
21
we assume that there is a continuous representation a !" U(a) of the translation
group of M in unitary operators on H, and that there is a unique “vacuum” state
$#Hsuch that U(a)$ = $for all a. This latter condition entails that the vacuum appears the same to all observers, and that it is the unique state of lowest energy. In this context, a particle detector can be represented by an effect C such that$$,C$% = 0. That is, C should register no particles in the vacuum state. However,
the Reeh-Schlieder theorem entails that no positive local observable can have zero
expectation value in the vacuum state. Thus, we again see that (strictly speaking) it
is impossible to detect particles by means of local measurements; instead, we will
have to think of particle detections as “approximately local” measurements.
If we think of an observable as representing a measurement procedure (or, more
precisely, an equivalence class of measurement procedures), then the norm distance
1C − C!1 between two observables gives a quantitative measure of the physical
similarity between the corresponding procedures. (In particular, if 1C − C!1 < #,
then the expectation values of C and C! never differ by more than #.)7 Moreover,
in the case of real-world measurements, the existence of measurement errors and
environmental noise make it impossible for us to determine precisely which measurement
procedure we have performed. Thus, practically speaking, we can at best
determine a neighborhood of observables corresponding to a concrete measurement
procedure.
In the case of present interest, what we actually measure is always a local
observable—i.e., an element of R(O), where O is bounded. However, given a
fixed error bound #, if an observable C is within norm distance # from some local
observable C! # R(O), then ameasurement of C! will be practically indistinguishable
from a measurement of C. Thus, ifwe let
R"(O) = {C : 2C! # R(O) such that 1C − C!1 < #}, (8)
denote the family of observables “almost localized” in O, then ‘FAPP’ (i.e., ‘for
all practical purposes’) we can locally measure any observable from R"(O). That
is, measurement of an element from R"(O) can be simulated to a high degree of
accuracy by local measurement of an element from R(O). However, for any local
region O, and for any # > 0, R"(O) does contain (nontrivial) effects that annihilate
the vacuum.8 Thus, particle detections can always be simulated by purely
local measurements; and the appearance of (fairly-well) localized objects can be
7Recall that #C − C!# is defined as the supremum of #(C − C!)!# as ! runs through the
unit vectors in H. It follows, then, fromthe Cauchy-Schwarz inequality that |%!, (C − C!)!&| '
#C − C!# for any unit vector !.
8Suppose that A ! R(O), and let A(x) = U(x)AU(x)#. If f is a test function on M whose
Fourier transform is supported in the complement of the forward light cone, then L = ) f(x)A(x)dx
is almost localized in O and %", L"& = 0 (cf. Buchholz 2000, 7).
22
explained without the supposition that there are localizable particles in the strict
sense.
However, it may not be easy to pacify Segal and Barrett with a FAPP solution
to the problem of localization. Both appear to think that the absence of localizable
particles (in the strict sense) is not simply contrary to our manifest experience, but
would undermine the very possiblity of objective empirical science. For example,
Segal claims that,
...it is an elementary fact, without which experimentation of the usual
sort would not be possible, that particles are indeed localized in space
at a given time. (Segal 1965, 145; our italics)
Furthermore, “particles would not be observable without their localization in space
at a particular time” (1964, 139). In other words, experimentation involves observations
of particles, and these observations can occur only if particles are localized
in space. Unfortunately, Segal does not give any argument for these claims. It
seems to us, however, that the moral we should draw from the no-go theorems is
that Segal’s account of observation is false. In particular, it is not (strictly speaking)
true that we observe particles. Rather, there are ‘observation events’, and these observation
events are consistent (to a good degree of accuracy) with the supposition
that they are brought about by (localizable) particles.
Like Segal, Barrett (2000) claims that we will have trouble explaining how empirical
science can work if there are no localizable particles. In particular, Barrett
claims that empirical science requires that we be able to keep an account of our
measurement results so that we can compare these results with the predictions of
our theories. Furthermore, we identify measurement records bymeans of their location
in space. Thus, if there were no localized objects, then there would be no
identifiable measurement records, and “...it would be difficult to account for the
possibility of empirical science at all” (Barrett 2000, 3).
However, it’s not clear what the difficulty here is supposed to be. On the
one hand, we have seen that relativistic quantum field theory does predict that
the appearances will be FAPP consistent with the supposition that there are localized
objects. So, for example, we could distinguish two record tokens at a given
time if there were two disjoint regions O and O! and particle detector observables
C # R"(O) and C! # R"(O!) (approximated by observables strictly localized in
O and O respectively) such that $!,C!% 3 1 and$!,C!!% 3 1. Now, itmay
be that Barrett is also worried about how, given a field ontology, we could assign
any sort of trans-temporal identity to our record tokens. But this problem, however
important philosophically, is distinct from the problem of localization. Indeed, it
also arises in the context of non-relativistic quantum field theory, where there is
no problem with describing localizable particles. Finally, Barrett might object that
23
once we supply a quantum-theoretical model of a particle detector itself, then the
superposition principle will prevent the field and detector from getting into a state
where there is a fact of the matter as to whether, “a particle has been detected in
the region O.” But this is simply a restatement of the standard quantum measurement
problem that infects all quantum theories—and we have made no pretense of
solving that here.
8 Conclusion
Malament claims that his theorem justifies the belief that,
...in the attempt to reconcile quantum mechanics with relativity theory...
one is driven to a field theory; all talk about “particles” has to be
understood, at least in principle, as talk about the properties of, and
interactions among, quantized fields. (Malament 1996, 1)
We have argued that the first claim is correct—quantum mechanics and relativity
can be reconciled only in the context of quantum field theory. In order, however, to
close a couple of loopholes in Malament’s argument for this conclusion, we provided
two further results (Theorems 1 and 2) which show that the conclusion continues
to hold for generic spacetimes, as well as for “unsharp” localization observables.
We then went on to show that relativistic quantum field theory also does not
permit an ontology of localizable particles; and so, strictly speaking, our talk about
localizable particles is a fiction. Nonetheless, relativistic quantum field theory does
permit talk about particles—albeit, if we understand this talk as really being about
the properties of, and interactions among, quantized fields. Indeed, modulo the
standard quantum measurement problem, relativistic quantum field theory has no
trouble explaining the appearance of macroscopically well-localized objects, and
shows that our talk of particles, though a façon de parler, has a legitimate role to
play in empirically testing the theory.
Acknowledgments: We would like to thank Jeff Barrett and David Malament
A Appendix
Theorem (Hegerfeldt). Suppose that the localization system (H,! !" E!, t !"
Ut) satisfies monotonicity, time-translation covariance, energy bounded below, and
NIWS. Then UtE!U−t = E! for all ! ' " and all t # R.
24
Proof. The formal proof corresponds directly to Hegerfeldt’s informal proof. Thus,
let ! be a subset of some spatial hypersurface ". If E! = 0 then obviously
UtE!U−t = E! for all t # R. So, suppose that E! (= 0, and let ! be a unit
vector such that E!! = !. Since " is a manifold, and since ! (= ", there is
a family {!n : n # N} of subsets of " such that, for each n # N, the distance
between the boundaries of !n and ! is nonzero, and such that !n !n = !. Fix
n # N. By NIWS and time-translation covariance, there is an "n > 0 such that
E!nUt! = Ut! whenever 0 & t < "n. That is, $Ut!,E!nUt!% = 1 whenever 0 & t < "n. Since energy is bounded from below, wemay apply Lemma 1with A = I − E!n to conclude that$Ut!,E!nUt!% = 1 for all t # R. That is,
E!nUt! = Ut! for all t # R. Since this holds for all n # N, and since (by
monotonicity) E! = "n E!n, it follows that E!Ut! = Ut! for all t # R. Thus,
UtE!U−t = E! for all t # R.
Lemma 2. Suppose that the localization system (H,! !" E!, a !" U(a)) satisfies
localizability, time-translation covariance, and no absolute velocity. Let ! be
a bounded spatial set. If U(a)E!U(a)" = E! for all timelike translations a of
M, then E! = 0.
Proof. By no absolute velocity, there is a pair (a, b) of timelike translations such
that !+(a − b) is in " and is disjoint from !. By time-translation covariance,
we have,
E!+(a−b) = U(a)U(b)"E!U(b)U(a)" = E!. (9)
Thus, localizability entails that E! is orthogonal to itself, and so E! = 0.
Lemma 3. Let {!n : n = 0, 1, 2, . . .} be a covering of ", and let E = $& n=0 E!n. If probability conservation and time-translation covariance hold, then UtEU−t = E for all t # R. Proof. Since {!n + t : n # N} is a covering of "+ t, probability conservation entails that$n E!n+t = E. Thus,
UtEU−t = Ut* &
%n=0
E!n +U−t =
&
%n=0
*UtE!nU−t+ (10)
=
&
%n=0
E!n+t = E, (11)
where the third equality follows from time-translation covariance.
25
In order to prove the next result, we will need to invoke the following lemma
from Borchers (1967).
Lemma (Borchers). Let Ut = eitH, where H is a self-adjoint operator with spectrum
bounded from below. Let E and F be projection operators such that EF = 0.
If there is an " > 0 such that
[E,UtFU−t] = 0, 0 & t < ",
then EUtFU−t = 0 for all t # R.
Lemma 4. Let Ut = eitH, where H is a self-adjoint operator with spectrum
bounded from below. Let {En : n = 0, 1, 2, . . .} be a family of projection operators
such that E0En = 0 for all n + 1, and let E = $& n=0 En. If UtEU−t = E for all t # R, and if for each n + 1 there is an "n > 0 such that [E0, UtEnU−t] = 0, 0 & t < "n, (12) then UtE0U−t = E0 for all t # R. Proof. If E0 = 0 then the conclusion obviously holds. Suppose then that E0 (= 0, and let ! be a unit vector in the range of E0. Fix n + 1. Using (12) and Borchers’ lemma, it follows that E0UtEnU−t = 0 for all t # R. Then, 1EnU−t!12 =$U−t!,EnU−t!% = $!,UtEnU−t!% (13) =$E0!,UtEnU−t!% = $!,E0UtEnU−t!% = 0, (14) for all t # R. Thus, EnU−t! = 0for all n + 1, and consequently, [$n'1 En]U−t! =
0. Since E0 = E − [$n'1 En], and since (by assumption) EU−t = U−tE, it follows that E0U−t! = EU−t! = U−tE! = U−t!, (15) for all t # R. Theorem 1. Suppose that the localization system (H,! !" E!, t !" Ut) satisfies localizability, probability conservation, time-translation covariance, energy bounded below, and microcausality. Then UtE!U−t = E! for all!and all t # R. Proof. Let ! be an open subset of ". If! = "then probability conservation and time-translation covariance entail that E! = E!+t = UtE!U−t for all t # R. If ! (= " then, since " is a manifold, there is a covering {!n : n # N} of "\! such that the distance between !n and ! is nonzero for all n. Let E0 = E!, and 26 let En = E!n for n + 1. Then 1 entails that E0En = 0 when n + 1. If we let E =$&
n=0 En then probability conservation entails that UtEU−t = E for all
t # R (see Lemma 3). By time-translation covariance and microcausality, for each
n + 1 there is an "n > 0 such that
[E0, UtEnU−t] = 0, 0 & t < "n. (16)
Since the energy is bounded from below, Lemma 4 entails that UtE0U−t = E0 for
all t # R. That is, UtE!U−t = E! for all t # R.
B Appendix
Theorem 2. Suppose that the unsharp localization system (H,! !" A!, a !"
U(a)) satisfies additivity, translation covariance, energy bounded below, microcausality,
and no absolute velocity. Then A! = 0 for all !.
Proof. We prove by induction that 1A!1 & (2/3)m, for each m # N, and for each
bounded !. For this, let F! denote the spectral measure for A!.
(Base case: m = 1) Let E! = F!(2/3, 1). We verify that (H,! !" E!, a !"
U(a)) satisfies the conditions of Malament’s theorem. Clearly, no absolute velocity
and energy bounded below hold. Moreover, since unitary transformations
preserve spectral decompositions, translation covariance holds; and since spectral
projections of compatible operators are also compatible, microcausality holds. To
see that localizability holds, let ! and !! be disjoint bounded subsets of a single
hyperplane. Then microcausality entails that [A!,A!!] = 0, and therefore E!E!!
is a projection operator. Suppose for reductio ad absurdum that ! is a unit vector in
the range of E!E!!. By additivity, A!%!! = A! + A!!, and we therefore obtain
1 + $!,A!%!!!% =$!,A!!% + $!,A!!!% + 2/3 + 2/3 . (17) Thus, E!E!! = 0, andMalament’s theorem entails that E! = 0 for all !. Therefore, A! = A!F!(0, 2/3) has spectrum lying in [0, 2/3], and 1A!1 & 2/3 for all bounded !. (Inductive step) Suppose that 1A!1 & (2/3)m−1 for all bounded !. Let E! = F!((2/3)m, (2/3)m−1). In order to see thatMalament’s theorem applies to (H,! !" E!, a !" U(a)), we need only check that localizability holds. For this, suppose that ! and !! are disjoint subsets of a single hyperplane. By microcausality, [A!,A!!] = 0, and therefore E!E!! is a projection operator. Suppose for reductio ad absurdum that ! is a unit vector in the range of E!E!!. Since 27 ! / !! is bounded, the induction hypothesis entails that 1A!%!!1 & (2/3)m−1. By additivity, A!%!! = A! + A!!, and therefore we obtain the contradiction: (2/3)m−1 +$!,A!%!!!% = $!,A!!%+$!,A!!!% + (2/3)m+(2/3)m . (18)
Thus, E!E!! = 0, andMalament’s theorem entails that E! = 0 for all !. Therefore,
1A!1 & (2/3)m for all bounded !.
C Appendix
Theorem 3. Suppose that the system (H,! !" N!, a !" U(a)) of local number
operators satisfies additivity, translation covariance, energy bounded below, number
conservation, microcausality, and no absolute velocity. Then, N! = 0 for all
bounded !.
Proof. Let N be the unique total number operator obtained from taking the sum
&n N!n where {!n : n # N} is a disjoint covering of ". Note that for any
! - ", we can choose a covering containing !, and hence, N = N! + A,
where A is a positive operator. By microcausality, [N!,A] = 0, and therefore
[N!,N] = [N!,N! + A] = 0. Furthermore, for any vector ! in the domain of
N, $!,N!!% &$!,N!%.
Let E be the spectral measure for N, and let En = E(0, n). Then, NEn is a
bounded operator with norm at most n. Since [En,N!] = 0, it follows that
$!,N!En!% =$En!,N!En!% & \$En!,NEn!% & n , (19)
for any unit vector !. Thus, 1N!En1 & n. Since #&
n=1 En(H) is dense in H, and
since En(H) is in the domain of N! (for all n), it follows that if N!En = 0, for
all n, then N! = 0. We nowconcentrate on proving the antecedent.
For each !, let A! = (1/n)N!En. We show that the structure (H,! !"
A!, a !" U(a)) satisfies the conditions of Theorem 2. Clearly, energy bounded
below and no absolute velocity hold. It is also straightforward to verify that additivity
and microcausality hold. To check translation covariance, we compute:
U(a)A!U(a)" = U(a)N!EnU(a)" = U(a)N!U(a)"U(a)EnU(a)" (20)
= U(a)N!U(a)"En = N!+aEn = A!+a. (21)
The third equality follows from number conservation, and the fourth equality follows
from translation covariance. Thus, N!En = A! = 0 for all !. Since this
holds for all n # N, N! = 0 for all !.
28
REFERENCES
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30

__________________________________________________________________________

Mass Is All That Matters in the Size–Weight Illusion  Myrthe A. Plaisier1,2,* and Jeroen B. J. Smeets2

Friedemann Paul, Editor

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Abstract

An object in outer space is weightless due to the absence of gravity, but astronauts can still judge whether one object is heavier than another one by accelerating the object. How heavy an object feels depends on the exploration mode: an object is perceived as heavier when holding it against the pull of gravity than when accelerating it. At the same time, perceiving an object’s size influences the percept: small objects feel heavier than large objects with the same mass (size–weight illusion). Does this effect depend on perception of the pull of gravity? To answer this question, objects were suspended from a long wire and participants were asked to push an object and rate its heaviness. This way the contribution of gravitational forces on the percept was minimised. Our results show that weight is not at all necessary for the illusion because the size–weight illusion occurred without perception of weight. The magnitude of the illusion was independent of whether inertial or gravitational forces were perceived. We conclude that the size–weight illusion does not depend on prior knowledge about weights of object, but instead on a more general knowledge about the mass of objects, independent of the contribution of gravity. Consequently, the size–weight illusion will have the same magnitude on Earth as it should have on the Moon or even under conditions of weightlessness.

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Introduction

The size-weight illusion is the well-known effect that large objects are perceived to be lighter than small objects of the same weight [1]. Although this illusion originally was discovered as a multi–sensory phenomenon in which the visually perceived size of an object influences its perceived weight, the illusion also occurs in the absence of vision if haptic size cues are available [2]. This shows that the illusion is not a multi-sensory one per se. It is based on a general effect of perceived size on perceived weight. In fact, showing an object before lifting, but not during lifting already triggers the illusion [3]. Similarly, there exists also a material–weight illusion [4], [5]: objects that are perceived to be made out of a denser material are perceived to be lighter than objects with the same size and weight that appear to be made out of a less dense material. Apparently, our percept of how heavy an object feels is biased by prior knowledge about the general relationship between object properties and the weight of an object. This suggests that the size–weight illusion occurs because we have learned that there is a correlation between size and weight. This idea is supported by a study in which it was shown that the illusion can be reversed: repeatedly lifting a set of objects manufactured such that the smaller objects had more mass than the larger objects for several days reduced the illusion and finally reversed it [6].

Combining information sources, such as size and weight, is common in human perception. It has been shown that when judging the size of an object through vision and touch simultaneously, the two estimates are integrated in a statistically optimal fashion [7]. This means that the combined percept is more precise than either of the two percepts independently. In fact, one can even learn to integrate two unrelated perceptual signals such as stiffness and luminance [8]. Sometimes, several information sources are combined with prior assumptions. This can be modelled using Bayesian statistics [9]. In the case of the size–weight illusion a perceptual estimate of size is combined with an estimate of the weight together with a prior for large objects being heavier. The way these information sources are combined in the size–weight illusion is fundamentally different from the previous examples, as it makes the percept less accurate and can be regarded as anti-Bayesian [10].

A prior for larger objects being heavier would suggest that a larger lifting force is applied for lifting large objects than for lifting small objects [11]. It has been shown that initially larger lifting forces are applied when lifting large objects, but the difference in the applied forces disappears within a few lifts while the perceived weight difference remains constant [12], [13]. This suggests that the illusion is not caused by applying more force when lifting a large object than when lifting a smaller object.

Since the illusion is usually referred to as size-weight illusion, one would expect it to be related to the weight of an object. Note that weight is another word for the gravitational force acting upon an object, which is proportional to the (gravitational) mass of an object. The mass of an object can also be experienced without weight through inertial forces proportional to the (inertial) mass acting during acceleration of an object [14]. This is why the mass of an object can be judged in the absence of gravity, such as in outer–space. Since gravitational and inertial masses of an object are the same (Einstein’s equivalence principle), one might expect that the two types of mass appear to be the same for the perceptual system. Surprisingly, an object is perceived to be almost twice as heavy through perception of gravitational pull than through perception of inertial forces [15], [16]. So, clearly the perceived heaviness of an object depends on whether inertial or gravitational forces are perceived, even though the underlying object property, mass, is the same.

In the present study we investigated whether the size-weight illusion depends on perceiving the pull of gravity, i.e. whether it is caused by a prior for weight or a by more general prior for the mass of an object. To this end we investigated whether the size–weight illusion occurs in the absence of weight through perception of inertial forces only. If the size–weight illusion occurs independent of gravitational forces, the illusion must be related to the mass of an object independent of the forces acting upon it. One reason for not expecting the illusion to occur in the absence of gravitational forces is that in daily life we rarely experience the heaviness of an object without perceiving gravitational forces. This means that the prior for larger objects being heavier may be limited to perception of gravitational forces, i.e. weight. Secondly, for perceiving inertial mass it is necessary to combine information about the acceleration of an object with efferent or afferent information about the applied amount of force. Gravitational forces (weight), in contrast, can be perceived purely tactual through the pressure of an object on the skin of the static hand. Therefore, fundamentally different sources of information are being used for perception of mass through inertial or gravitational forces.

To investigate whether the size–weight illusion occurs in the absence of weight, a set of objects differing in size but with the same mass was constructed. To let participants perceive inertial forces only, we suspended the objects from a long pair of wires and asked the participant to give the objects a short push after which they rated the perceived heaviness. This way perception of gravitational forces acting on the object was minimised. We let participants perform this task with and without visual feedback of the trajectory of the object after pushing to test whether participants used visual information about how far the object travelled. Finally, we also asked participants to rate heaviness after lifting the objects and placing them back as a control task. This allowed us to test whether the magnitude of the illusion as obtained through perception of inertial forces differed from the illusion obtained in the traditional way.

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Materials and Methods

Participants

Twenty self-reported right-handed participants volunteered in the experiment (age range 22 to 40 years). All participants were naive as to the purpose of the experiment. Half of the participants performed the experiment pushing the objects with full vision. The other half also pushed the objects, but without visual feedback of the object’s trajectory. Eight of the subjects that performed the pushing with full visual feedback task also performed the control task of lifting with full vision.

Ethics Statement

The experiment was conducted as part of a program that was approved by the ethical committee of the Faculty of Human Movement Sciences of VU University. All participants signed a statement of informed consent prior to participation in the experiments.

Stimuli

The stimuli consisted of a set of four objects constructed out of MDF (medium density fibre). Three test objects differing in size (small, medium, large) were weighed down such that the mass of each object was 250 g (Fig. 1A). The fourth object (reference) had the same dimensions as the medium sized test object, but had a mass of 200 g. The ratings for the two medium sized objects were used to convert the heaviness ratings of the other objects into grams.

 Figure 1 Description of the stimuli and set-up.

A small infrared Light Emitting Diode was attached in the centre of each object’s surface facing away from the subject. The position of the object was recorded at 500 Hz using an Optotrak position tracking system (Northern Digital). These data were used to calculate the objects’ velocities.

In the pushing task the objects were suspended just above table height from a pair of long wires in front of the participant. The participants were asked to give the object a short push such that it travelled over a distance of 50 cm (Fig. 1B) and rate how heavy the object felt using arbitrary numbers (i.e. method of absolute magnitude estimation [17]). These ratings were converted into z-scores by taking the difference between the individual ratings of a participant and his or her average rating, before dividing by the standard deviation of the ratings. For the task without visual feedback a screen was placed in front of the subject, such that the object was initially visual, but it disappeared behind the screen shortly after the push.

The procedure in the control task was the same as in the pushing task, but now participants were instructed to lift the objects between their thumb and index finger. They lifted the objects grasping them in the centre along the 6 cm axis, such that grip aperture was the same for all objects (Fig. 1C).

In all tasks the test objects were presented in 15 sets of three trials; in every set each object was presented once and the order of presentation within each set was randomised. After these 15 sets of three trials, another 10 trials were performed in which the reference object was presented 5 times randomly interleaved with 5 times one of the test objects. All trials were performed in one continuous run such that subjects were not aware of the introduction of the reference object.

Analysis

Repeated measures ANOVA was performed on the z-scores of the heaviness ratings with object size as a repeated factor and visual feedback as a between subjects factor. The effect of object size was also tested with a repeated measures ANOVA on the peak velocities of the objects. Finally, a repeated measures ANOVA with object size and task as repeated factors was performed to compare the lifting and pushing with visual feedback conditions.

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Results

Pushing

The heaviness ratings were converted into z-scores and are shown for the participants in the full vision tasks in Fig. 2A. Using the difference in the ratings for the two medium sized objects, the z-scores were converted into grams. The results show that the small 250 g object was consistently rated as being almost 100 g heavier than the large 250 g object. This was the case with full visual feedback and without vision of the object’s trajectory (Fig. 2B): a repeated measures ANOVA with object size as a within subjects factor and visual feedback as between subjects factor on the z-scores showed an effect of object size () but no interaction between visual feedback and object size (). Therefore, the size–weight illusion occurs without any haptic or visual information about gravitational forces acting on the object. Furthermore, visual feedback of the trajectory did not affect the illusion size.

4-14-15 Heard on PBS News Hour that Dark Matter, Dark Energy new map from Chile radio telescope camera that captures 100,000 Galaxies at a time. A map was made and 25% Dark Mater, 70% Dark Energy, 5% everything else like Visible Matter and ordinary Energy.

4-17-15 I looked at NATIONAL GEOGRAPHIC APRIL 2014 Page 85 COSMIC QUESTIONS. "WHAT IS OUR UNIVERSE MADE OF ?  Stars, dust, and gas--the stuff we can discern--make up less than 5 percent of the universe." The Universe 24% Dark matter, 4% Gas , 0.5% Planets and stars, 71% Planets and Stars, 71.5% Dark energy.

My New And Unique Paradigm Conceives that the complete Universe is created from J_Waves in J_Strings. This is my Disclosure of J_Theory of Everything ( J_TOE ).

4-18-15 Copied from my e-mail Quora Digest :