Table Of ContentSchriftenderMathematisch-naturwissenschaftlichenKlasse
derHeidelbergerAkademiederWissenschaften
Nr.17(2005)
Hans Günter Dosch
Volkhard F. Müller
Norman Sieroka
Quantum FieldTheory
in a SemioticPerspective
HansGu¨nterDosch
Institutfu¨rTheoretischePhysik
Universita¨tHeidelberg
Philosophenweg16,69120Heidelberg,Germany
[email protected]
VolkhardF.Mu¨ller
FachbereichPhysik
derTechnischenUniversita¨tKaiserslautern
Postfach3049,67653Kaiserslautern,Germany
[email protected]
NormanSieroka
ETHZu¨rich
Professurfu¨rPhilosophie
ETHZentrumRACG16,8092Zu¨rich,Switzerland
[email protected]
LibraryofCongressControlNumber:2005931712
ISBN3-540-28211-4 SpringerBerlinHeidelbergNewYork
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Contents
1 Introduction............................................... 1
2 RelativisticQuantumFieldTheories
ViewedasPhysicalTheories................................. 5
2.1 TheEmpiricalDomain ................................... 5
2.2 TheNotionofaRelativisticQuantumField.................. 6
2.3 LocalGaugeSymmetry .................................. 7
2.4 ParticlesversusQuantumFields ........................... 8
2.5 TheFacetsofQuantumFieldTheory ....................... 9
2.5.1 GeneralTheoryofQuantumFields................... 10
2.5.2 PerturbativeRenormalisationTheory ................. 11
2.5.3 ConstructiveQuantumFieldTheory.................. 14
2.5.4 LatticeGaugeTheory .............................. 15
2.5.5 LocalQuantumPhysics ............................ 16
2.6 RenormalisabilityversusEffectiveFieldTheories ............ 17
3 ParticlesandFields......................................... 21
4 TheoriesofSignsandSymbols,andStructuralRealism ......... 27
5 ATheoryofSymbolsforQuantumFieldTheory................ 41
5.1 SymbolsandRelations ................................... 41
5.2 ASemioticViewofQuantumFieldTheory .................. 44
6 Summary ................................................. 51
References................................................... 53
NameIndex.................................................. 57
SubjectIndex ................................................ 59
1
Introduction
Inthisessaywediscussepistemologicalimplicationsofrelativisticquantum
fieldtheory.Theempiricaldomainofsuchatheoryisformedbyphenomena
ascribedtosubnuclearparticles,sometimesstillcalledelementaryparticles.
Thislattermoretraditionaldesignationreflectsthelastingdesireofphysicists
toeventuallyfindandisolateirreducibleconstituentsofmatter.Goingdown
totheatomiclevel,electronsappeartoplaysucharoˆle,whereasthenucleiof
atomscanbeconsideredascompoundsystemsofprotonsandneutrons,i.e.of
twospeciesofparticles.Thisviewmakessense,sincetherespectivenumber
of these two types of constituents essentially identifies an atomic nucleus.
Extractedfromanucleus,however,the‘free’neutronisanunstableparticle:
it decays spontaneously into a proton, an electron and an anti-neutrino. In
thepastfiftyyearsorsobasicallythebombardmentofmatterbyprotonsor
byelectronsinspeciallydevisedexperimentshasrevealedalargevarietyof
furthersubnuclearobjects.Successivegenerationsofacceleratorsandrefined
collisiondevicesprovidedhigherandhighercollisionenergies.Allthesesub-
nuclearobjectsaretermed‘particles’inthephysicscommunity,nearlyallof
these objects are unstable and decay spontaneously into other ones.The re-
spectivelifetimesofthedistincttypes,however,differwidely,rangingfrom
relativelylong(103sec)toextremelyshort(10−25sec).Becauseofthishuge
disparity in lifetime the notion of a particle deserves particular attention, a
pointlaidstressoninourconsideration.Thestudyofthephysicalbehaviour
ofthesesubnuclearparticlesledtodistinguishthreetypesofinteractions:the
strong, the electromagnetic and the weak interaction.As the names suggest
these interactions differ in their respective strength. Furthermore, each type
shows characteristic conservation laws obeyed in the observed reactions of
the subnuclear particles. On the theoretical side the Standard Model of par-
ticlephysicshasemergedinthecourseoftime.Thisstrikingachievementis
supposedtoaccountforthefullhierarchyofthestrong,theelectromagnetic
andtheweakinteraction.1
1Averylucid,non-technicaloverviewofthesubnuclearworldandofbasicelements
ofitstheoreticalrepresentationcanbefoundinVeltman(2003).
2 1 Introduction
Sincewearesolelyinterestedinfirmlybasedconclusions,weconfineour-
selvestomathematicallycoherentandexperimentallyverywellcorroborated
aspects of quantum field theory. Therefore, we focus on various aspects re-
latedtotheStandardModelofparticlephysicsandleaveoutallspeculations
presently in vogue, as e.g. string theory – interesting as they might be. The
StandardModelisconsideredtoessentiallydescribetherealmofsubnuclear
particlesuptothecurrentexperimentallimitenergy,probingdistancesdown
to10−16cm.Weareawareofsomeindications–bothexperimentalandthe-
oretical–thattheStandardModelshouldbemodified.Webelieve,however,
that future developments which might crystallise from today’s more specu-
lativeinvestigations,willfitneatlyintheepistemicschemewepropose,too.
This holds also for the variety of partial models, motivated by the Standard
Modelbutaugmentedbycrucialadditionalassumptionsorapproximations;
thesemodelsarenotconsideredhereeither.
Theprincipalaimofthisessayistoshowthatvariousfacetsofquantum
fieldtheorydoenterintothetheoryofsubnuclearphenomenaanddealwith
complementing domains. In our opinion these various facets are not always
adequately considered in the existing philosophical literature, where most
emphasisisputonperturbationtheory.Inparticular,thenotionofaparticle
ishighlyintricate,aswillbeexposed;forphysicalreasonsthereisnothinglike
a unique definition. Even more, the relation between particles and quantum
fieldsisanimplicitone,determinedbytheinherentdynamicalcontentofthe
theoryenvisaged.
Our essay is organised as follows: chapter 2 presents a concise general
descriptionofrelativisticquantumfieldtheoriesviewedasphysicaltheories,
avoiding technical formulations as far as possible.Various facets are eluci-
datedanddistinguishedaccordingtodifferingaimspursued.Inchapter3we
returningreaterdetailtothecentralquestion:inwhichwaydoparticles,i.e.
theobjectsobservedexperimentally,emergeinthetheoryfromthequantised
fields,i.e.fromthetheoreticalbuildingblocks?Theresultingrathercomplex
answermanifests,towhichextendthetheorycopeswiththebasicphenom-
enonofrelativisticphysics,thatmasscanbetransmutedintoenergyandvice
versa.Itisimportanttonoticethatourconsiderationsenvisagefromthevery
beginning quantum theories with their probabilistic physical interpretation,
determined by expectation values; hence, we do not discuss the so-called
particle-wavedualism.Inchapter4wetracebacktheepistemicdiscussionto
thetime,whenthenotionoftheclassicalelectromagneticfieldgainedanau-
tonomousstatus,freedfromtheattemptsofamechanicalfoundation.There,
1 Introduction 3
we encounter the birth of the symbolic interpretation of physical theories –
notably in the work of Heinrich Hertz. In chapter 5 we look at the various
facets of quantum field theory from a semiotic perspective. In chapter 6 we
giveashortre´sume´.
2
Relativistic Quantum FieldTheories
Viewed as PhysicalTheories
2.1 TheEmpiricalDomain
Phenomena ascribed to subnuclear particles and their interaction form the
physicaldomainofrelativisticquantumfieldtheory.Suchasubnuclearpar-
ticleisidentifiedbyitsmassandspin,whichdetermineitsbehaviourunder
space-time transformations, and by its electric charge and further charge-
likeinnerquantumnumbers.Whenastableparticleisisolatedfromexternal
perturbations it moves freely with constant velocity: its energy E and its
momentumvectorp satisfytherelativistickinematicalrelation
E2 = m2c4+c2p2, (2.1)
where m is the rest mass of the particle and c the vacuum velocity of light.
Eventually,theparticleisrecognisedwhenittriggersanappropriate(macro-
scopic)detectiondevice.Theserecordingsconstitutetheempiricaldatatobe
metbyarelativisticquantumfieldtheory.
Thesalientfeaturethatcharacterisestheinteractionofsubnuclearparticles
isthepossibletransmutationofmatterintoenergyandviceversa:particlescan
be created or annihilated, provided certain conservation laws are respected
– as for instance charge conservation – when a mass m is converted into its
energyequivalent(cid:1)E = mc2orviceversa.Hence,definiteconfigurationsof
particlesexistonlyasymptoticallyintimebeforeandafteraprocessofcolli-
sion,whentheparticlesarewellseparatedandmutuallynon-interacting,due
totheshortrangeoftheirinteraction.Becauseoftheseproperties,scattering
experiments, where a certain initial state of particles is carefully prepared
and the resulting final states are analysed by measurements, play such an
importantroˆleinparticlephysics.
Many of the objects, however, which are conveniently called subnuclear
particles,arenotstablebutdecayspontaneouslyintolighterparticles.Thus,
strictlyspeaking,theycannotappearinanasymptoticstate.Nevertheless,it
is theoretically appealing and proves to be empirically justified to treat an
6 2 RelativisticQuantumFieldTheoriesViewedasPhysicalTheories
unstable particle also as forming asymptotic states, provided its lifetime is
largecomparedwiththereactiontimeinascatteringprocess.Thisindicates
already,thatthenotionofasubnuclearparticleisstronglybasedontherelated
theoreticalperspective.Weshallbeconfrontedwiththisproblemseveraltimes
inthisessay.
2.2 TheNotionofaRelativisticQuantumField
The basic concept of the theory are quantum fields defined on space-time,
not particles. Space-time is assumed to be a four-dimensional real vector
space with given metrical properties and Einstein causality, such that the
Poincare´ group (constituted by translations and Lorentz-transformations) is
implied as symmetry group. This space-time structure fixed in advance –
called Minkowski space – forms the register for recording physical events.
Thepredictionsofarelativisticquantumfieldtheoryontheoutcomeofscat-
tering processes are of probabilistic nature, in this respect similar to those
of(non-relativistic)quantummechanics.However,anovelfeatureoccurs:in
theseprocessesparticlescanbecreatedandannihilated.Thequantumfields,
in terms of which the theory is constructed, are operators that depend on
space-time and act on the space of physical state vectors. This dependence
on space-time, however, shows the behaviour of a generalised function or
distribution. Phrased technically, a relativistic quantum field is an operator-
valueddistribution.Therefore,awell-definedoperatorcannotberelatedtoa
definitespace-timepointx,butonlytoaspace-timedomainoffiniteextent:
aproperoperatoremergesfromthequantumfieldφ(x)by‘smearing’witha
correspondingtestfunctionf(x):1
(cid:1)
φ(f) := d4xf(x)φ(x). (2.2)
Theveryfact,thataquantumfieldbehaveslikeadistribution,hasfar-reaching
consequences:ingeneral,pointwiseproductsofdistributionsarenotdefined
mathematically. Hence, the nth power φn(x),n = 2,3,4,··· of a quantum
fieldφ(x)atagivenspace-timepointxisapriorinotdefined.Incontradistinc-
tion, however, such products are well defined in the realm of classical field
theory, since a classical field ϕ(x) is a (generally differentiable) function.
Converting indiscriminately such products of classical fields into the cor-
responding products of quantum fields produces mathematically ill-defined
1Therighthandsideoftheequationisonlyaformalrepresentation.
