There are no particles, there are only fields

14 Pages·2013·1.32 MB·English

by  Art Hobson

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There are no particles, there are only fields
ArtHobsona)
DepartmentofPhysics,UniversityofArkansas,Fayetteville,Arkansas72701
(Received18April2012;accepted15January2013)
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 that unbounded fields, not
bounded particles, are fundamental. This is especially clear for relativistic systems, implying that
it’s also true of nonrelativistic systems. Particles are epiphenomena arising from fields. Thus, the
Schr€odingerfieldisaspace-fillingphysicalfieldwhosevalueatanyspatialpointistheprobability
amplitude for an interaction to occur at that point. The field for an electron is the electron; each
electron extendsoverbothslitsinthetwo-slitexperimentandspreadsovertheentirepattern;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 two-slit experiment, rigorous theorems showing particles are inconsistent with relativistic
quantum theory, and several phenomena showing particles are incompatible with quantum field
theories.VC 2013AmericanAssociationofPhysicsTeachers.
[http://dx.doi.org/10.1119/1.4789885]
I. INTRODUCTION Here, I’ll discuss just one fundamental quantum issue:
field-particle (or wave-particle) duality. More precisely, this
Physicistsarestillunabletoreachconsensusontheprinci- paper answers the following question: Based on standard
plesormeaningofscience’smostfundamentalandaccurate nonrelativistic and relativistic quantum physics, do experi-
theory, quantum physics. An embarrassment of enigmas mentandtheoryleadustoconcludethattheuniverseisulti-
abounds concerning wave-particle duality, measurement, matelymadeoffields,orparticles,orboth,orneither?There
nonlocality, superpositions, uncertainty, and the meaning of are other embarrassing quantum enigmas, especially the
quantum states.1 After more than a century of quantum his- measurementproblem,aswellastheultimateontology(i.e.,
tory,thisisscandalous.2,3 reality)impliedbyquantumphysics.Thispaperstudiesonly
It’s not only an academic matter. This confusion has field-particle duality. In particular, it is neutral on the inter-
hugereal-lifeimplications.Inaworldthatcriesoutforgen- pretations (e.g., many worlds) and modifications (e.g., hid-
eral scientific literacy,4 quantum-inspired pseudoscience has den variables, objective collapse theories) designed to
become dangerous to science and society. What the Bleep resolvethemeasurementproblem.
DoWeKnow,apopular 2004film,wonseveral filmawards Many textbooks and physicists apparently don’t realize
and grossed $10 million; its central tenet is that we create that a strong case, supported by leading quantum field theo-
our own reality through consciousness and quantum rists, for a pure fields view has developed during the past
mechanics. It features physicists saying things like “The three decades.10–17 Three popular books are arguments for
materialworldaroundusisnothingbutpossiblemovements an all-fields perspective.18–20 I have argued the advantages
of consciousness,” it purports to show how thoughts change of teaching nonrelativistic quantum physics (NRQP, or
thestructureoficecrystals,anditinterviewsa35,000year- “quantum mechanics”) from an all-fields perspective;21 my
old spirit “channeled” by a psychic.5 “Quantum mysticism” conceptualphysicstextbookfornon-sciencecollegestudents
ostensibly provides a basis for mind-over-matter claims assumesthisviewpoint.22
from ESP to alternative medicine, and provides intellectual Thereisplentyofevidencetodayforphysiciststocometo
support for the postmodern assertion that science has no a consensus supporting an all-fields view. Such a consensus
claim on objective reality.6 According to the popular televi- would make it easier to resolve other quantum issues. But
sion physician Deepak Chopra, “quantum healing” can cure fields-versus-particles is still alive and kicking, as you can
all our ills by the application of mental power.7 Chopra’s see by noting that “quantum field theory” (QFT) and
book Ageless Body, Timeless Mind, a New York Times “particle physics” are interchangeable names for the same
Bestseller that sold over two million copies worldwide, is discipline! And there’s a huge gap between the views of
subtitled The Quantum Alternative to Growing Old.8 Quan- leading quantum physicists on the one hand; (Refs. 10–18)
tum Enigma, a highly advertised book from Oxford Univer- and virtually every quantum physics textbook on the other
sity Press that’s used as a textbook in liberal arts physics hand.
courses at the University of California and elsewhere, bears Physicists are schizophrenic about fields and particles. At
the sub-title Physics Encounters Consciousness.9 It is the high-energy end, most quantum field theorists agree for
indeed scandalous when librarians and book store managers good reasons (Secs. III, V, and VI) that relativistic quantum
wonder whether to shelve a book under “quantum physics,” physicsisaboutfieldsandthatelectrons,photons,andsoforth
“religion,” or “new age.” For further documentation of this aremerelyexcitations(waves)inthefundamentalfields.Butat
point, see the Wikipedia article “Quantum mysticism” and thelow-energyend,mostNRQPeducationandpopulartalkis
referencestherein. about particles. Working physicists, teachers, and NRQP
211 Am.J.Phys.81(3),March2013 http://aapt.org/ajp VC 2013AmericanAssociationofPhysicsTeachers 211
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textbookstreatelectrons,photons,protons,atoms,etc.aspar- Hegerfeldt showing that, even if we assume a very broad
ticlesthatexhibitparadoxicalbehavior.YetNRQPisthenon- definitionof“particle”(namely,thataparticleshouldextend
relativisticlimitofthebroaderrelativistictheory,namelyQFT, over only a finite, not infinite, region), particles contradict
which for all the world appears to be about fields. If QFT is relativistic quantum physics. Section VI argues that quan-
aboutfields,howcanitsrestrictiontononrelativisticphenom- tizedfieldsimplyaquantumvacuumthatcontradictsanall-
enabeaboutparticles?Doinfinitelyextendedfieldsturninto particlesviewwhileconfirmingthefieldview.Furthermore,
boundedparticlesastheenergydrops? two vacuum effects—the Unruh effect and single-quantum
Asan exampleofthefield/particle confusion, thetwo-slit nonlocality—implyafieldview.Thus,manylinesofreason-
experimentis often considered paradoxical, and itis a para- ing contradict the particles view and confirm the field view.
doxifoneassumesthattheuniverseismadeofparticles.For SectionVIIsummarizestheconclusions.
Richard Feynman, this paradox was unavoidable. Feynman
was a particles guy. As Frank Wilczek put it, “uniquely (so
II. AHISTORYOFCLASSICALFIELDS
far as I know) among physicists of high stature, Feynman
hoped to remove field-particle dualism by getting rid of the Fieldsareoneofphysics’mostplausiblenotions,arguably
fields.”16 As a preface to his lecture about this experiment, more intuitively credible than point-like particles drifting in
Feynmanadvisedhisstudents, empty space. It’s perhaps surprising that, despite the com-
plete absence of fields from Isaac Newton’s Principia
(1687), Newton’s intuition told him the universe is filled
Do not take the lecture too seriously, feeling that
withfields.InanexchangeofletterswithReverendRichard
you really have to understand in terms of some
BentleyexplainingthePrincipiainnon-scientists’language,
model what I am going to describe, but just relax
Newtonwrote:
and enjoy it. I am going to tell you what nature
behaves like. If you will simply admit that maybe It is inconceivable that inanimate brute matter
she does behave like this, you will find her a should, without the mediation of something else
delightful, entrancing thing. Do not keep saying to whichisnotmaterial,operateuponandaffectother
yourself,ifyoucanpossiblyavoidit,“Buthowcan matterwithoutmutualcontact…Thatgravityshould
it be like that?” because you will get “down the be innate, inherent, and essential to matter, so that
drain,”intoablindalleyfromwhichnobodyhasyet onebodymayactuponanotheratadistancethrough
escaped.Nobodyknowshowitcanbelikethat.23 avacuum,withoutthemediationofanythingelse,by
and through which their action and force may be
There are many interpretational difficulties with the two-
conveyedfrom onetoanother, is tomesogreatan
slitexperiment,andI’mnotgoingtosolveallofthemhere.
absurdity that I believe no man who has in
But the puzzle of wave-particle duality in this experiment
philosophicalmattersacompetentfacultyofthinking
can be resolved by switching to an all-fields perspective
caneverfallintoit.24
(Sec.IV).
Physics education is affected directly, and scientific liter- ButNewton couldn’tfindempiricalevidence tosupporta
acy indirectly, by what textbooks say about wave-particle causalexplanationofgravity,andanyexplanationremained
duality and related topics. To find out what textbooks say, I purely hypothetical. When writing or speaking of a possible
perused the 36 textbooks in my university’s library having underlyingmechanismforgravity,hechosetoremainsilent,
the words “quantum mechanics” in their title and published firmly maintaining “I do not feign hypotheses” (Ref. 18,
after1989.Ofthese,30impliedauniversemadeofparticles p. 138). Thus, it was generally accepted by the beginning of
that sometimes act like fields, 6 implied the fundamental the 19th century that a fundamental physical theory would
constituentsbehavedsometimeslikeparticlesandsometimes containequationsfordirectforces-at-a-distancebetweentiny
like fields, and none viewed the universe as made of fields indestructible atoms moving through empty space. Before
thatsometimesappear tobe particles.Yettheleadingquan- long, however, electromagnetism and relativity would shift
tum field theorists argue explicitly for the latter view (Refs. theemphasisfromaction-at-a-distancetofields.
10–18).Something’samisshere. TheshiftwaslargelyduetoMichaelFaraday(1791–1867).
Thepurposeofthispaperistoassemblethestrandsofthe Working about 160 years after Newton, he introduced the
fields-versus-particlesdiscussioninordertohastenaconsen- modernconceptoffieldsaspropertiesofspacehavingphysi-
sus that will resolve the wave-particle paradoxes while caleffects.25Faradayarguedagainstaction-at-a-distance,pro-
bringing the conceptual structure of quantum physics into posing instead that interactions occur via space-filling “lines
agreementwiththerequirementsofspecialrelativityandthe offorce”andthatatomsaremereconvergencesoftheselines
views of leading quantum field theorists. Section II argues of force. He recognized that a demonstration of non-
that Faraday, Maxwell, and Einstein viewed classical elec- instantaneous electromagnetic (EM) interactions would be
tromagnetismasafieldphenomenon.SectionIIIarguesthat fatal to action-at-a-distance because interactions would then
quantumfieldtheorydevelopedfromclassicalelectrodynam- proceedgraduallyfromonebodytothenext,suggestingthat
ics and then extended the quantized field notion to matter. some physical process occurred in the intervening space. He
Quantization introduced certain particle-like characteristics, sawlinesofforceasspace-fillingphysicalentitiesthatcould
namely, individual quanta that could be counted, but the move,expand,andcontract.Heconcludedthatmagneticlines
theory describes these quanta as extended disturbances in offorce,inparticular,arephysicalconditionsof“merespace”
space-filling fields. Section IV analyzes the two-slit experi- (i.e., space containing no material substance). Today this
ment to illustrate the necessity for an all-fields view of descriptionoffieldsas“conditionsofspace”isstandard.26
NRQP. The phenomena and the theory lead to paradoxes if James Clerk Maxwell (1831–1879) was less visionary,
interpreted in terms of particles but are comprehensible in more Newtonian, and more mathematical than Faraday. By
termsoffields.SectionVpresentsarigoroustheoremdueto invoking a mechanical ether that obeyed Newton’s laws, he
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brought Faraday’s conception of continuous transmission of ofspace-timeandtotheright-handsideas“ahovelofwood”
forces rather thaninstantaneousaction-at-a-distance intothe becauseitrepresentsaconditionofmatter.
philosophical framework of Newtonian mechanics. Thus Thusby1915classicalphysicsdescribedallknownforces
Faraday’s lines of force became the state of a material me- in terms of fields—conditions of space—and Einstein
dium,“theether,”muchasavelocityfieldisastateofama- expresseddissatisfactionthatmattercouldn’tbedescribedin
terial fluid. He found the correct dynamical field equations thesameway.
for EM phenomena, consistent with all known experimental
results. His analysis led to the predictions of (1) a finite III. AHISTORYANDDESCRIPTIONOFQUANTUM
transmission time for EM actions, and (2) light as an EM
FIELDS
field phenomenon. Both were later spectacularly confirmed.
Despite the success of his equations, and despite the non- FromtheearlyGreekandRomanatomiststoNewtontosci-
appearanceofetherintheactualequations,Maxwellinsisted entistssuchasDalton,RobertBrown,andRutherford,themi-
throughout his life that Newtonian mechanical forces in the croscopicviewofmatterwasalwaysdominatedbyparticles.
ether produce all electric and magnetic phenomena, a view Thus,thenonrelativisticquantumphysicsofmatterthatdevel-
that differed crucially from Faraday’s view of the EM field opedinthemid-1920swascouchedinparticlelanguage,and
asastateof“merespace.” quantumphysicswascalled“quantummechanics”inanalogy
Experimental confirmations of the field nature of light, with the Newtonian mechanics of indestructible particles in
and of a time delay for EM actions, were strong confirma- emptyspace.30Butironically,thecentralequationofthequan-
tions of the field view. After all, light certainly seems real. tum physics of matter, the Schr€odinger equation, is a field
And a time delay demands the presence of energy in the equation.Ratherthananobviousrecipeforparticlemotion,it
intervening space in order to conserve energy. That is, if appearstodescribeatime-dependentfieldW(x,t)throughouta
energyisemittedhereandnow,andreceivedthereandlater, spatial region. Nevertheless, this field picked up a particle
thenwhereisitinthemeantime?Clearly,it’sinthefield.27 interpretation when Max Born proposed that W(x ,t) is the
0
FaradayandMaxwellcreatedoneofhistory’smosttelling probability amplitude that, upon measurement at time t, the
changesinourphysicalworldview:thechangefromparticles presumedparticle“willbefound”atthepointx .Anothersug-
0
tofields.AsAlbertEinsteinputit,“BeforeMaxwell,Physical gestion,stillinaccordwiththeCopenhageninterpretationbut
Reality…wasthoughtofasconsistinginmaterialparticles…. lessconfining,wouldbethatW(x ,t)istheprobabilityampli-
0
Since Maxwell’s time, Physical Reality has been thought of tudeforaninteractiontooccuratx .ThispreservestheBorn
0
asrepresentedbycontinuousfields,…andnotcapableofany rulewhileallowingeitherafieldorparticleinterpretation.
mechanical interpretation. This change in the conception of Inthelate1920s,physicistssoughtarelativistictheorythat
Realityisthemostprofoundandthemostfruitfulthatphysics incorporated quantum principles. EM fields were not
hasexperiencedsincethetimeofNewton.”28 described by the nonrelativistic Schr€odinger equation; they
As the preceding quotation shows, Einstein supported a spread at the speed of light, so any quantum theory of them
“fields are all there is” view of classical (but not necessarily mustberelativistic.Suchatheorymustalsodescribeemission
quantum)physics.He putthefinal logical touch onclassical (creation) and absorption (destruction) of radiation. Further-
fieldsinhis1905paperproposingthespecialtheoryofrelativ- more, NRQP says energy spontaneously fluctuates, and SR
ity, where he wrote “The introduction of a “luminiferous” (E mc2)saysmattercanbecreatedfromnon-materialforms
etherwillprovetobesuperfluous.”29ForEinstein,therewas ofe¼nergy,soarelativisticquantumtheorymustdescribecrea-
no material ether to support light waves. Instead, the tionanddestructionofmatter.Schr€odinger’sequationneeded
“medium”forlightwasspaceitself.Thatis,forEinstein,fields tobegeneralizedtoincludesuchphenomena.QFTs,described
arestatesorconditionsofspace.Thisisthemodernview.The intheremainderofthissection,arosefromtheseefforts.
implication of special relativity (SR) that energy has inertia
furtherreinforcesbothEinstein’srejectionoftheetherandthe
A. Quantizedradiationfields
significanceoffields.Sincefieldshaveenergy,theyhaveiner-
tia and should be considered “substance like” themselves “Howcananyphysicistlookatradioormicrowaveanten-
ratherthansimplystatesofsomesubstancesuchasether. nasandbelievetheyweremeanttocaptureparticles?”31It’s
The general theory of relativity (1916) resolves Newton’s implausible that EM signals transmit from antenna to
dilemma concerning the “absurdity” of gravitational action- antenna by emitting and absorbing particles; how do anten-
at-a-distance. According to general relativity, the universe is nas “launch” or “catch” particles? In fact, how do signals
full of gravitational fields, and physical processes associated propagate? Instantaneous transmission is ruled out by the
withthisfieldoccureveninspacethatisfreefrommatterand evidence. Delayed transmission by direct action-at-a-dis-
EMfields.Einstein’sfieldequationsofgeneralrelativityare tancewithoutaninterveningmediumhasbeentriedintheory
and found wanting.32 The 19th-century answer was that
R x 1=2 g x R x T x ; (1) transmission occurs via the EM field. Quantum physics pre-
lvð Þ#ð Þ lvð Þ ð Þ ¼ lvð Þ
serves this notion, while “quantizing” the field. The field
where x represents space-time points, l and ! run over the itselfremainscontinuous,fillingallspace.
fourdimensions,g (x)isthemetrictensorfield,R (x)and The first task in developing a relativistic quantum theory
l! l!
R(x)aredefinedintermsofg (x),andT (x)istheenergy- wastodescribeEMradiation—aninherentlyrelativisticphe-
l! l!
momentum tensor of matter. These field (because they hold nomenon—in a quantum fashion. So it’s not surprising that
ateveryx)equationsrelatethegeometryofspace-time(left- QFT began with a quantum theory of radiation.33–35 This
hand side) to the energy and momentum of matter (right- problemwasgreatlysimplifiedbytheLorentzcovarianceof
hand side). The gravitational field is described solely by the Maxwell’s equations: they satisfy SR by taking the same
metrictensorg (x).Einsteinreferredtotheleft-handsideof form in every inertial reference frame. Maxwell was lucky,
l!
Eq.(1)as“apalaceofgold”becauseitrepresentsacondition orbrilliant,inthisregard.
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A straightforward approach to the quantization of the W . But the Hilbert space for such states cannot have the
j i
“free” (no source charges or currents) EM field begins with same structure as for the single-body Schr€odinger equation,
theclassical vectorpotential fieldA(x,t)fromwhichwecan or even its N-body analog, because N must be allowed to
calculate E(x,t) and B(x,t).36 Expanding this field in the set vary in order to describe creation and destruction of quanta.
of spatial fields exp(6ik x) (orthonormalized in the delta- So the radiation field’s quantum states exist in a Hilbert
!
function sense for large spatial volumes) for each vector k space of variable N, called “Fock space.” Fock space is the
havingpositivecomponents,wewrite (direct) sum of N-body Hilbert spaces for N 0, 1, 2, 3,…
¼
Each component N-body Hilbert space is the properly sym-
A x;t a k;t exp ik x a’ k;t exp ik x : metrized (for bosons or fermions) product of N single-body
ð Þ¼ ½ ð Þ ð ! Þ þ ð Þ ð( ! Þ)
k Hilbert spaces. Each normalized component has its own
X
(2) complex amplitude, and the full state W is (in general) a
j i
superpositionofstateshavingdifferentnumbersofquanta.
ThefieldequationforA(x,t)thenimpliesthateachcoefficient AnimportantfeatureofQFTistheexistenceofavacuum
a(k,t) satisfies a classical harmonic oscillator equation. One state 0 ,aunitvectorinFockspace(whichmustnotbecon-
regards these equations as the equations of motion fora me- fusedjwiith the zero vector, whose length is zero), having no
chanicalsystemhavinganinfinitenumberofdegreesoffree- quanta (N 0 for all k). Each mode’s vacuum state has
dom, and quantizes this classical mechanical system by energyhf /k2¼.Thevacuumstatemanifestsitselfexperimentally
k
assumingthea(k,t)areoperatorsa (k,t)satisfyingappropriate inmanyways,whichwouldbecuriousifparticleswerereally
op
commutation relations and the a*(k,t) are their adjoints. The fundamental because there are no particles (quanta) in this
resultisthatEq.(2)becomesavectoroperator-valuedfield, state.We’llexpandonthisparticularargumentinSect.VI.
TheoperatorfieldofEq.(3),likeotherobservablessuchas
Aopðx;tÞ¼ ½aopðk;tÞexpðik!xÞþa’opðk;tÞexpðik!xÞ); energy,operatesonjWi,creatinganddestroyingphotons.For
k example,theexpectedvalueofthevectorpotentialisavector-
X
(3) valued relativistic field A(x,t) A (x,t) WA (x,t) W ,
¼h op i¼h j op j i
anexpressioninwhichA (x,t)operateson W .Weseeagain
op j i
in which the amplitudes a* (k,t) and a (k,t) of the kth thatA (x,t)isactuallyaphysicallymeaningfulfieldbecause
op op op
“field mode” satisfy the Heisenberg equations of motion (in ithasaphysicallymeasurableexpectationvalueateverypoint
whichthetimedependenceresidesintheoperatorswhilethe xthroughoutaregionofspace.Soaclassicalfieldthatisquan-
system’squantumstate W remainsfixed)forasetofquan- tizeddoesnotceasetobeafield.
j i
tum harmonic oscillators. For bosons, one can show that Some authors conclude, incorrectly, that the countability
a* and a are the familiar raising and lowering operators of quanta implies a particle interpretation of the quantized
op op
from the harmonic oscillator problem in NRQP, satisfying system.38Discretenessisanecessarybutnotsufficientcondi-
the same commutation relation. Hence, A (x,t) is now an tionforparticles.Quantaarecountable,buttheyarespatially
op
operator-valued field whose dynamics obey quantum extended and certainly not particles. Equation (3) implies
physics. Since the classical field obeyed SR, the quantized thatasinglemode’sspatialdependenceissinusoidalandfills
fieldsatisfiesquantumphysicsandSR. allspace,sothataddingamonochromaticquantumtoafield
Thus,asintheharmonicoscillatorproblem,thekthmode uniformly increases the entire field’s energy (uniformly dis-
has an infinite discrete energy spectrum hf (N 1/2) with tributed throughout all space!) by hf. This is nothing like
N 0,1,2,…,wheref ck/2pisthemokde’skfþrequency.37 addingaparticle.Quantathataresuperpositionsofdifferent
k¼ k¼ j j
As Max Planck had hypothesized, the energy of a single frequencies can be more spatially bunched and in this sense
mode has an infinite spectrum of discrete possible values morelocalized,buttheyarealwaysofinfiniteextent.Soit’s
separated by DE hf . The integer N is the number of hardtoseehowphotonscouldbeparticles.
¼ k k
Planck’s energy bundles or quanta in the kth mode. Each Phenomena such as “particle” tracks in bubble chambers,
quantum is called an “excitation” of the field, because its andthesmallspotappearingonaviewingscreenwhenasin-
energy hf represents additional field energy. EM field glequantuminteractswiththescreen,areoftencitedasevi-
k
quanta are called “photons,” from the Greek word for light. dence that quanta are particles, but these are insufficient
Adistinctlyquantumaspectisthat,eveninthevacuumstate evidence of particles39,40(see Sec. IV).In the case of radia-
where N 0, each mode has energy hf /2. This is because tion, it’s especially difficult to argue that the small interac-
k¼ k
the individual modes act like quantum harmonic oscillators, tion points are evidence that a particle impacted at that
andthesemusthaveenergyeveninthegroundstatebecause position because photons never have positions–position is
of the uncertainty principle. Another quantum aspect is that not an observable and photons cannot be said to be “at” or
EMradiationis“digitized”intodiscretequantaofenergyhf. “found at” any particular point.41–45 Instead, the spatially
Youcan’thaveafractionofaquantum. extendedradiationfieldinteractswiththescreeninthevicin-
Becauseitdefinesanoperatorforeverypointxthroughout ity of the spot, transferring one quantum of energy to the
space, the operator-valued field Eq. (3) is properly called a screen.
“field.”Notethat,unliketheNRQPcase,xisnotanoperator
but rather a parameter, putting x on an equal footing with t
B. Quantizedmatterfields
as befits a relativistic theory. For example, we can speak of
theexpectationvalueofthefieldA atxandt,butwecan- QFT puts matter on the same all-fields footing as radia-
op
not speak of the expectation value of x because x is not an tion.Thisisabigsteptowardunification.Infact,it’sagen-
observable. Thisisbecause fieldsareinherently extendedin eralprincipleofallQFTsthatfieldsareallthereis.10–21For
spaceanddon’thavespecificpositions. example the Standard Model, perhaps the most successful
But what does the operator field Eq. (3) operate on? Just scientific theory of all time, is a QFT. But if fields are all
asinNRQP,operatorsoperateonthesystem’squantumstate there is, where do electrons and atoms come from? QFT’s
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answer is that they are field quanta, but quanta of matter physics,it’snowonderthatittookmostofthe20thcentury
fieldsratherthanquantaofforcefields.46 tocometogripswiththefieldnatureofquantumphysics.
“Fields are all there is” suggests beginning the quantum Were it not for Newtonian preconceptions, quantum
theoryofmatterfromSchr€odinger’sequation,whichmathe- physics might have been recognized as a field theory by
maticallyisafieldequationsimilartoMaxwell’sfieldequa- 1926(Schr€odinger’sequation)or1927(QFT).Thesuperpo-
tions, and quantizing it. But you can’t create a relativistic sitionprincipleshouldhavebeenadeadgiveaway:Asumof
theory (the main purpose of QFT) this way because quantum states is a quantum state. Such superposition is
Schr€odinger’s equation is not Lorentz covariant. Dirac characteristicofalllinearwavetheoriesandatoddswiththe
invented, for just this purpose, a covariant generalization of generallynonlinearnatureofNewtonianparticlephysics.
Schr€odinger’sequationforthefieldW(x,t)associatedwitha A benefit of QFTs is that quanta of a given field must be
singleelectron.47Itincorporatestheelectron’sspin,accounts identical because they are all excitations of the same field,
for the electron’s magnetic moment, and is more accurate somewhatastworipplesonthesamepondareinmanyways
than Schr€odinger’s equation in predicting the hydrogen identical. Because a single field explains the existence and
atom’s spectrum. It, however, has undesirable features such nature of gazillions of quanta, QFTs represent an enormous
as the existence of non-physical negative-energy states. unification. The universal electron-positron field, for exam-
These can be overcome by treating Dirac’s equation as a ple,explainstheexistenceandnatureofallelectronsandall
classical field equation for matter analogous to Maxwell’s positrons.
equations for radiation, and quantizing it in the manner out- When a field changes its energy by a single quantum, it
lined in Sec. IIIA. The resulting quantized matter field must do so instantaneously, because a non-instantaneous
W (x,t) is called the “electron-positron field.” It’s an change would imply that, partway through the change, the
op
operator-valued field operating in the anti-symmetric Fock field had gained or lost only a fraction of a quantum. Such
space. Thus the non-quantized Dirac equation describes a fractions are not allowed because energy is quantized. Field
matterfieldoccupyingananalogousroleintheQFTofmat- quanta have an all-or-nothing quality. The QFT language of
ter to the role of Maxwell’s equations in the QFT of radia- creation and annihilation of quanta expresses this nicely. A
tion.12,45 The quantized theory of electrons comes out quantum is a unified entity even though its energy might be
looking similar to the preceding QFT of the EM field, but spread out over light years—a feature that raises issues of
withmaterialquantaandwithfieldoperatorsthatnowcreate nonlocalityintrinsictothequantumpuzzle.
ordestroythesequantainquantum-antiquantumpairs.36 “Fieldsareallthereis”shouldbeunderstoodliterally.For
It’s not difficult to show that standard NRQP is a special example,it’sacommonmisconceptiontoimagineatinypar-
case, for nonrelativistic material quanta, of relativistic ticleimbeddedsomewhereintheSchr€odingerfield.Thereis
QFT.36Thus,theSchr€odingerfieldisthenonrelativisticver- noparticle.Anelectronisitsfield.
sion of the Dirac equation’s relativistic field. It follows that As is well known, Einstein never fully accepted quantum
the Schr€odinger matter field, the analog of the classical EM physics, and spent the last few decades of his life trying to
field, is a physical, space-filling field. Just like the Dirac explain all phenomena, including quantum phenomena, in
field,thisfieldistheelectron. 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
promotestheright-handsideofEq.(1)tofieldstatus.Butit
C. Furtherpropertiesofquantumfields
is not yet a “palace of gold” because Einstein’s goal of
Thus the quantum theory of electromagnetic radiation is a explainingallfieldsentirelyintermsofzero-rest-massfields
re-formulation of classical electromagnetic theory to account such as the gravitational field has not yet been achieved,
for quantization—the “bundling” of radiation into discrete although the QFT of the strong force comes close to this
quanta.Itremains,liketheclassicaltheory,afieldtheory.The goalof“masswithoutmass.”13,16,17
quantumtheoryofmatterintroducestheelectron-positronfield
and a new field equation, the Dirac equation, the analog for
matter of the classical Maxwell field equations for radiation. IV. THETWO-SLITEXPERIMENT
Quantization of the Dirac equation is analogous to quantiza-
A. Phenomena
tion of Maxwell’s equations, and the result is the quantized
electron-positronfield.TheSchr€odingerequation,thenonrela- Field-particle duality appears most clearly in the context
tivisticversionofthe Diracequation,isthusa fieldequation. of the time-honored two-slit experiment, which Feynman
There are no particles in any of this; there are only field claimed “contains the only mystery.”48,49 Figures 1 and 2
quanta—excitationsofspatiallyextendedcontinuousfields. show the outcome of the two-slit experiment using a dim
Foroverthreedecades,theStandardModel—aQFT—has light beam (Fig. 1) and a “dim” electron beam (Fig. 2) as
been our best theory of the microscopic world. It’s clear sources,withtime-lapsephotography.Theset-upisasource
from the structure of QFTs (Secs. IIIA and IIIB) that they emitting monochromatic light (Fig. 1)50 or mono-energetic
actually are field theories, not particle theories in disguise. electrons(Fig.2),51anopaquescreenwithtwoparallelslits,
Nevertheless,I’llofferfurtherevidencefortheirfieldnature andadetectionscreenwithwhichthebeamcollides.Inboth
hereandinSecs.VandVI. figures,particle-likeimpactsbuilduponthedetectionscreen
Quantumfieldshaveoneparticle-likepropertythatclassi- to form interference patterns. The figures show both field
cal fields don’t have: They are made of countable quanta. aspects (the extended patterns) and particle aspects (the
Thusquantacannotpartlyvanishbutmust(likeparticles)be localizedimpacts).Thesimilaritybetweenthetwofiguresis
entirely and instantly created or destroyed. Quanta carry strikingandindicatesafundamentalsimilaritybetweenpho-
energy and momenta and can thus “hit like a particle.” Fol- tons and electrons. It’s intuitively hard to believe that one
lowing three centuries of particle-oriented Newtonian figurewasmadebywavesandtheotherbyparticles.
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Fig.1.Thetwo-slitexperimentoutcomeusingdimlightwithtime-lapsephotography.Insuccessiveimages,aninterferencepatternbuildsupfromparticle-
likeimpacts.(ImagescourtesyofWolfgangRueckner,HarvardUniversityScienceCenter.SeeRef.50.)
Consider,first,theextendedpattern.It’seasytoexplainif teristic of the detector, which is made of localized atoms,
each quantum (photon or electron) is an extended field that rather than of the detected quanta. The detection, however,
comes through both slits. But could the pattern arise from localizes(“collapses”—Secs.IVBandIVC)thequantum.
particles? The experiments can be performed using an en- Similarargumentsapplytotheobservationofthinparticle
semble of separately emitted individual quanta, implying tracksinbubblechambersandotherapparentparticledetec-
thattheresultscannotarisefrominteractionsbetweendiffer- tions. Localization is characteristic of the detection process,
entquanta.52Preparationisidenticalforallthequantainthe notofthequantumthatisbeingdetected.
ensemble. Thus, given this particular experimental context ThustheinterferencepatternsinFigs.1and2confirmfield
(namely,thetwo-slitexperimentwithbothslitsopen,node- behaviorandruleoutparticlebehavior,whilethesmallinter-
tector at the slits, and a “downstream” screen that detects action points neither confirm particle behavior nor rule out
each ensemble member), each quantum mustcarryinforma- field behavior. The experiment thus confirms field behavior.
tion about the entire pattern that appears on the screen (in As Dirac famously put it in connection with experiments of
order, e.g., to avoid all the nodes). In this sense, each quan- thetwo-slittype,“Thenewtheory[namelyquantummechan-
tumcanbesaidtobespreadoutoverthepattern. ics], which connects the wave function with probabilities for
Ifwecloseoneslit,thepatternshiftstothesingle-slitpat- onephoton,getsoverthedifficulty[ofexplainingtheinterfer-
ternbehindtheopenslit,showingnointerference.Thuseach ence] by making each photon go partly into each of the two
quantumcarriesdifferentinformationdependingonwhether components. Each photon then interferes only with itself.”54
twoslitsareopenorjustone. (Thephrasesinsquarebracketsaremine,notDirac’s.)
How does one quantum get information as to how many Given the extended field nature of each electron, Fig. 2
slits are open? If a quantum is a field that is extended over also confirms von Neumann’s famous collapse postulate:55
both slits, there’s no problem. But could a particle coming Eachelectroncarriesinformationabouttheentirepatternand
through just one slit obtain this information by detecting collapses to a much smaller region upon interaction. Most
physical forces from the other, relatively distant, slit? The textbookssetupaparadoxbyexplicitlyorimplicitlyassum-
effect is thesame for photonsandelectrons, andthe experi- ingeach quantum to come through oneor theother slit,and
menthasbeendonewithneutrons,atoms,andmanymolecu- then struggle to resolve the paradox. But if each quantum
lartypes,makingitdifficulttoimaginegravitational,EM,or comesthroughbothslits,there’snoparadox.
nuclear forces causing such a long-distance force effect.
What more direct evidence could therebe that aquantum is
B. Theory,attheslits
anextendedfield?Thuswecannotexplaintheextendedpat-
terns by assuming each quantum is a particle, but we can Now assume detectors are at each slit so that a quantum
explainthepatternsbyassumingeachquantumisafield.53 passingthroughslit1(withslit2closed)triggersdetector1,
Now consider the particle-like small impact points. We andsimilarly forslit2.Let w and w ,whichweassume
1 2
j i j i
can obviously explain these if quanta are particles, but can form an orthonormal basis for the quantum’s Hilbert space,
weexplainthemwithfields?Theflashesseeninbothfigures denotethestatesofaquantumpassingthroughslit1withslit
are multi-atom events initiated by interactions of a single 2 closed, and through slit 2 with slit 1 closed, respectively.
quantum with the screen. In Fig. 2, for example, each elec- Weassume,withvonNeumann,thatthedetectoralsoobeys
tron interacts with a portion of a fluorescent film, creating quantum physics, with ready denoting the “ready” state of
j i
some 500 photons; these photons excite a photo cathode, thedetectors,and 1 and 2 denotingthe“clicked”statesof
j i j i
producing photo-electrons that are then focused into a point each detector. Then the evolution of the composite quan-
imagethatisdisplayedonaTVmonitor.51Thisshowsthata tum detectorsystem,whenthequantumpassesthroughslit
þ
quantumcaninteractlocallywithatoms,butitdoesn’tshow ialone(withtheotherslitclosed),isoftheform w ready
i
j ij i
thatquantaarepointparticles.Alargeobject(abigballoon, w i (i 1,2)(assuming,withvonNeumann,thatthese
i
!j iji ¼
say)caninteractquitelocallywithanotherobject(atinynee- are “ideal” processes that don’t disturb the state of the
dle, say). The localization seen in the two figures is charac- quantum).56,57
216 Am.J.Phys.,Vol.81,No.3,March2013 ArtHobson 216
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With both slits open, the single quantum approaching the
slits is described by a superposition that’s extended over
bothslits:
w w =!2 w : (4)
ðj 1iþj 2iÞ $j i
Linearity of the time evolution implies that the composite
system’sevolutionduringdetectionattheslitsis
jw1ijreadyi!ðjw1ij1iþjw2ij2iÞ=!2$jWslitsi: (5)
The“measurementstate” W involvesbothspatiallydis-
j slitsi
tinguishabledetectorstates j .Itisa“Bellstate”ofnonlocal
ji
entanglement between the quantum and the detector (Ref.
57, pp. 29, 32). If the detectors are reliable, there must be
zeroprobabilityoffindingdetectoriinthestate i whende-
ji
tectorj iisinitsclickedstate j ,so 1 and 2 areorthog-
6¼ ji j i j i
onalandweassumetheyarenormalized.
It’smathematicallyconvenienttoformthepurestateden-
sityoperator
qslits $jWslitsihWslitsj; (6)
and to form the reduced density operator for the quantum
alonebytracingoverthedetector:
qqslits ¼TrdetectorðqslitsÞ¼ðjw1ihw1jþjw2ihw2jÞ=2:
(7)
Equation (7) has a simple interpretation: Even though the
quantum is in the entangled superposition of Eq. (5), the
result of any experiment involving the quantum alone will
comeoutpreciselyasthoughthequantumwereinoneofthe
pure states w or w with probabilities of 1/2 for each
1 2
state.57 In pajrticiular,jEqi. (7) predicts that the quantum does
not interfere with itself, i.e., there are no interferences
between w and w . This of course agrees with observa-
1 2
j i j i
tion: 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”57or“collapse”toasingleslit.
To clearly see the field nature of the measurement, sup-
posethereisa“whichslit”detectoronlyatslit1withnode-
tector at slit 2. Then w ready w i holds only for
j ii j i ! j ii ji
i 1, while for i 2 we have w ready w ready .
¼ ¼ j 2i j i ! j 2i j i
Thepreviousanalysisstillholds,providedthe“clicked”state
1 is orthogonal to the unclicked state ready (i.e., if the
j i j i
twostatesaredistinguishablewithprobability1).Thesuper-
position 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.
Eventhoughthereisnodetectoratslit2,whenthequantum
comesthroughslit2itstillencodesthepresenceofadetec-
Fig.2.Thetwo-slitexperimentoutcomeusinga“dim”electronbeamwith
toratslit1.Thisbehaviorisnonlocal,andittellsusthatthe
time-lapse photography. As in Fig. 1, an interference pattern builds up
fromparticle-likeimpacts(ReprintedwithpermissionfromA.Tonomura, quantum extends over both slits, i.e., the quantum is a field,
J. Endo, T. Matsuda, T. Kawasaki, and H. Exawa, Am. J. Phys. 57(2), notaparticle.
117–120 (1989).Copyright # 1989 American Association of Physics Thustheexperiment(Sec.IVA)andthetheorybothimply
Teachers) that each quantum comes through both slits when both slits
areopenwithnodetectors,butthroughoneslitwhenthereis
a detector at either slit, just as we expect a field (but not a
particle)todo.
217 Am.J.Phys.,Vol.81,No.3,March2013 ArtHobson 217
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C. Theory,atthedetectingscreen V. RELATIVISTICQUANTUMPHYSICS
We’llseethattheaboveanalysisattheslitscarriesoverat NRQP (Sec. IV) is not the best basis for analyzing field-
thedetectingscreen,withthescreenactingasdetector. particle duality. The spontaneous energy fluctuations of
Thescreenisanarrayofsmallbutmacroscopicdetectors quantumphysics,plusSR’sprincipleofmass-energyequiva-
such as single photographic grains. Suppose one quantum lence, imply that quanta, be they fields or particles, can be
describedbyEq.(4)passesthroughtheslitsandapproaches createdordestroyed.Sincerelativisticquantumphysicswas
the screen. Expanding in position eigenstates, just before invented largely to deal with such creation and destruction,
interactingwiththescreenthequantum’sstateis one might expect relativistic quantum physics to offer the
deepestinsightsintofieldsandparticles.
Quantum physics doesn’t fit easily into a special-
w x dx xw x w x dx; (8)
j i¼ j i h j i¼ j i ð Þ relativisticframework.Asoneexample,wesawinSec.IIIA
ð ð that photons (relativistic phenomena for sure) cannot be
where the integral is over the two-dimensional screen, and quantum point particles because they don’t have position
w(x) is the Schr€odinger field. Equation (8) is a (continuous) eigenstates.
superposition over position eigenstates, just as Eq. (4) is a A more striking example is nonlocality, a phenomenon
(discrete)superpositionoversliteigenstates.Bothsuperposi- shownbyEinstein,Podolsky,andRosen,59andmorequanti-
tionsareextendedfields. tativelybyJohnBell,60toinhereinthequantumfoundations.
Rewriting Eq. (8) in a form that displays the quantum’s Using Bell’s inequality, Aspect, Clauser, and others showed
superpositionoverthenon-overlappingdetectionregions, experimentallythatnatureisnonlocalandthatthiswouldbe
trueevenifquantumphysicswerenottrue.61Theimplication
isthat,byalteringthewayshemeasuresoneofthequantain
w x w x dx A w ; (9)
j i¼ j i ð Þ $ i ij ii anexperimentinvolvingtwoentangledquanta,AliceinNew
i ði
X X York City can instantly (i.e.,in a time too shortto allowfor
where w (1/A) x w(x)dx and A [ w(x)2dx]1/2. signalpropagation)changetheoutcomesobservedwhenBob
j ii$ i i j i i $ i j j
The detection regions are labeled by i and the w form an measurestheotherquantuminParis.Thissoundslikeitvio-
orthonormalset.EqÐuation(9)isanalogoustoÐEqj.(4ii). latesthespecial-relativistic prohibitiononsuper-luminalsig-
The detection process at the screen is represented by the naling,butquantumphysicsmanagestoavoidacontradiction
analogofEq.(5): by camouflaging the signal so that Alice’s measurement
choiceis“averagedout”inthestatisticsofBob’sobservations
jwijreadyi! Aijwiijii$jWscreeni; (10) ihnissuexchpearimwaeynt.t6h2atTBhuosbBdeotbecrtesceniovecshannogesiginnatlh,eesvteantistthicosugohf
i
X nonlocality changes his observed results. Quantum physics’
where i represents the “clicked” state of the ith detecting particular mixture of uncertainty and nonlocality preserves
region,jiwhose output can be either “detection” or “no consistencywithSR.It’sonlywhenAliceandBoblatercom-
detection”ofthequantum.Localizationoccursatthetimeof pare their data that they can spot correlations showing that
thisclick.Eachregionirespondsbyinteractingornotinter- Alice’schangeofmeasurementprocedurealteredBob’sout-
acting, with just one region registering an interaction comes. Quantum physics must thread a fine needle, being
because a quantum must give up all, or none, of its energy. “weaklylocal”inordertopreventsuperluminalsignalingbut,
As we’ll see in Sec. VIC, these other sections of the screen inordertoallowquantumnonlocality,not“stronglylocal.”62
actuallyregisterthevacuum—aphysicalstatethatcanentan- Quantumfieldspreadingcantransmitinformationandislim-
gle nonlocally with the registered quantum. The nonlocality itedbythespeedoflight,whilenonlocaleffectsarerelatedto
inherent in the entangled superposition state W has superluminal field collapse and cannot transmit information
been verified by Bell-type measurements (Secj. sVcrIeeCni). As lesttheyviolateSR.
wasthecasefordetectionattheslits(Eq.(5)),Eq.(10)rep- When generalizing NRQP to include such relativistic
resents the mechanism by which the macro world registers quantum phenomena as creation and destruction, conflicts
thequantum’simpactonthescreen. with SR can arise unless one proceeds carefully. Heger-
The argument from Eq. (10) goes through precisely like feldt63andMalament64haveeachpresentedrigorous“no-go
the argument from Eq. (5) to Eq. (7). The result is that, theorems”demonstratingthat,ifoneassumesauniversecon-
assuming the states i are reliable detectors, the reduced taining particles, then the requirements of SR and quantum
ji
densityoperatorforthequantumaloneis physicsleadtocontradictions.Thissupportsthe“widespread
(withinthephysicscommunity)beliefthattheonlyrelativis-
q w A2 w : (11) tic quantum theory is a theory of fields.”65 Neither theorem
qscreen ¼ j ii ih ij assumesQFT.TheyassumeonlySRandthegeneralprinci-
i
X ples of quantum physics, plus broadly inclusive definitions
Equation(11)tellsusthatthequantumisregisteredeitherin ofwhatonemeansbya“particle.”Eachthenderivesacon-
region 1or region 2or…It’s this“allor nothing” natureof tradiction, showing that there can be no particles in any
quantum interactions, rather than any presumed particle na- theoryobeyingbothSRandquantumphysics.Iwilldescribe
ture of quanta, that produces the particle-like interaction onlyHegerfeldt’stheoremhere,becauseitisthemoreintui-
regionsinFigures1and2. tive of the two, and because Malement’s theorem is more
In summary, “only spatial fields must be postulated to subjecttodifficultiesofinterpretation.
formthefundamentalobjectstobequantized,.whileapparent Hegerfeldt shows that any free (i.e., not constrained by
“particles”are amere consequence of decoherence” (i.e.,of boundary conditions or forces to remain for all time within
localizationbythedetectionprocess).58 some finite region) relativistic quantum “particle” must, if
218 Am.J.Phys.,Vol.81,No.3,March2013 ArtHobson 218
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it’slocalizedtoafiniteregiontobeginwith,instantlyhavea verified phenomena such as the Lamb shift, the Casimir
positive probability of being found an arbitrarily large dis- effect, and the electron’s anomalous magnetic moment, this
tance away. Butthisresult turnsout toviolate Einsteincau- “statethathasnoparticles”ishardtoignore.Thissectiondis-
sality (no superluminal signaling). The conclusion is then cussesQFTvacuumphenomenathataredifficulttoreconcile
that an individual free quantum can never—not even for a with particles. Section VIA discusses the quantum vacuum
singleinstant—belocalizedtoanyfiniteregion. itself. The remaining parts are implications of the quantum
More specifically, a presumed particle is said to be vacuum. The Unruh effect (Sec. VIB), related to Hawking
“localized” at t if it is prepared in such a way as to ensure radiation, has not yet been observed, while single-quantum
o
that it will upon measurement be found, with probability 1, nonlocality(Sec.VIC)isexperimentallyconfirmed.
to be within some arbitrarily large but finite region V at t . On the other hand, we do not yet really understand the
o o
Hegerfeldtthenassumestwoconditions:First,thepresumed quantum vacuum. The most telling demonstration of this is
particle has quantum states that can be represented in a thatthemostplausibletheoreticalQFTestimateoftheenergy
Hilbert space with unitary time-development operator U density of the vacuum implies a value of the cosmological
t
exp( iHt), where H is the energy operator. Second, the constantthatissome120ordersofmagnitudelargerthanthe
¼ "
particle’senergyspectrumhasalowerbound.Thefirstcon- upperboundplacedonthisparameterbyastronomicalobser-
ditionsaysthattheparticleobeysstandardquantumdynam- vations. Possible solutions, such as the anthropic principle,
ics. The second says that the Hamiltonian that drives the havebeensuggested,buttheseremainspeculative.71
dynamicscannotprovideinfiniteenergybyitselfdroppingto
lowerandlowerenergies.Hegerfeldtthenprovesthatapar- A. Thenecessityforthequantumvacuum
ticle that is localized at t is not localized at any t>t . See
0 0
Ref.63fortheproof.It’sremarkablethatevenlocalizability Both theory and experiment demonstrate that the quan-
in an arbitrarily large finite region can be so difficult for tized EM field can never be sharply (with probability 1)
a relativistic quantum particle: its probability amplitude zero, but rather that there must exist, at every spatial point,
spreadsinstantlytoinfinity. at least a randomly fluctuating “vacuum field” having no
Now here is the contradiction: Consider a particle that is quanta.72Concerningthetheory,recall(Sec.III)thataquan-
localizedwithinV att .Atanyt>t ,thereisthenanonzero tizedfieldisequivalenttoasetofoscillators.Anactualme-
0 0 0
probability that it will be found at any arbitrarily large dis- chanical oscillator cannot be at rest in its ground state
tanceawayfromV .Thisisnotaproblemforanonrelativistic because this would violate the uncertainty principle; its
0
theory,andinfactsuchinstantaneousspreadingofwavefunc- ground state energy is instead hf/2. Likewise, each field os-
tionsiseasytoshowinNRQP.66Butinarelativistictheory, cillatormusthaveagroundstatewhereithasenergy butno
such instantaneous spreading contradicts relativity’s prohibi- excitations.Inthe“vacuumstate,”wherethenumberofexci-
tiononsuperluminaltransportandcommunication,becauseit tationsNkiszeroforeverymodek,theexpectationvaluesof
impliesthataparticlelocalizedonEarthatt could,withnon- EandBarezeroyettheexpectationvaluesofE2andB2are
0
zero probability, be found on the moon an arbitrarily short not zero. Thus the vacuum energy arises from random
timelater.Weconcludethat“particles”cannoteverbelocal- “vacuumfluctuations”ofEandBaroundzero.
ized.Tocallathinga“particle”whenitcannoteverbelocal- Asasecondandmoredirectargumentforthenecessityof
izedinanyfiniteregionissurelyagrossmisuseofthatword. EMvacuumenergy,considerachargeeofmassmboundby
Because QFTs reject the notion of position observables in anelasticrestoringforcetoalargemassofoppositecharge.
favorofparameterizedfieldobservables(Sec.III),QFTshave The equation of motion for the Heisenberg-picture position
no problem with Hegerfeldt’s theorem. In QFT the interac- operator x(t) has the same form as the corresponding classi-
tions, including creation and destruction, occur at specific calequation,namely
locationsx,butthefundamentalobjectsofthetheory,namely,
the fields, do not have positions because they are infinitely d2x=dt2þx2ox¼ðe=mÞ½ErrðtÞþEoðtÞ’: (12)
extended.
Summarizing:evenunderabroadlyinclusivedefinitionof Here, xo is the oscillator’s natural frequency, Err(t) is the
“particle,”quantumparticlesconflictwithEinsteincausality. “radiation reaction” fieldproduced bythe charged oscillator
itself, E (t) is the external field, and it’s assumed that the
o
spatial dependence of E (t) can be neglected. It can be
VI. THEQUANTUMVACUUM o
shown that the radiation reaction has the same form as the
The Standard Model, a QFT (more precisely two QFTs), classical radiation reaction field for an accelerating charged
is today the favored way of looking at relativistic quantum particle,Err(t)¼(2e/3c3)d3x/dt3,soEq.(12)becomes
phenomena.Infact,QFTis“theonlyknownversionofrela-
tivistic quantum theory.”67 Since NRQP can also be d2x=dt2þx2ox"ð2e2=3mc3Þd3x=dt3¼ðe=mÞEoðtÞ:
expressed as a QFT,68 all of quantum physics can be (13)
expressed consistently as QFTs. We’ve seen (Sec. V) that
quantum particles conflict with SR. This suggests (but IfthetermE (t)wereabsent,Eq.(13)wouldbecomeadissi-
o
doesn’t prove) that QFTs are the only logically consistent pativeequationwithx(t)exponentiallydamped,andcommu-
version of relativistic quantum physics.69 Thus, it appears tators like [z(t), p(t)] would approach zero for large t, in
z
thatQFTsarethenaturallanguageofquantumphysics. contradictionwiththeuncertaintyprincipleandincontradic-
Because it has energy and nonvanishing expectation val- tion with the unitary time development of quantum physics
ues, theQFT vacuum isembarrassing forparticle interpreta- according to which commutators like [z(t), p(t)] are time-
z
tions. If one believes particles to be the basic reality, then independent. Thus E (t) cannot be absent for quantum
o
whatisitthathasthisenergyandthesevaluesinthestatethat systems.Furthermore,ifE (t)isthevacuumfieldthencom-
o
has no particles?70 Because it is the source of empirically mutatorslike[z(t),p(t)]turnouttobetime-independent.
z
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