384 CHAPTER25. AGLIMPSEOFQUANTUMFIELDTHEORY
Ofcourse,aphononisn’t”really”aparticle,inthesensethatonecouldtrapsuch
anobject,extractitfromthesolid,and(say)lookatitunderamicroscope. What
is”really”inthesolidaretheatomswhichcomposeit. Phonons,eventhoughthey
carryenergyandmomentum,andeventhoughtheyscatter(likeparticles)offimpu-
ritiesinthesolidandoffoneanother,aremerelyconvenientlabelsforthequantized
vibrationalstatesoftheunderlyingsolid.
Whattosay,then,aboutphotons?Arethey,oraretheynot,elementaryparticles?
25.2 The Quantization of Light
QuantummechanicsbeganwithPlanck’sdiscoverythattheenergyofanelectromag-
neticfieldmustbedivisibleinunits ofhf. ThencameEinstein’s identificationof
thoseunitsasparticles,”photons.”ThencametheComptoneffect,theBohratom,
DeBrogliewaves,theSchrodingerequation...andthencamealot,someofwhichwe
havetouchedoninthiscourse. Nowits timeto returntothebeginning. Whyis
theenergyofaradiationfieldquantizedinunitsofhf? Whathappensifweapply
theprinciplesofquantummechanics,whichwasdevelopedtodescribethemotionof
electronsandotherparticles,tothedynamicsoftheelectromagneticfield?
Firstofall,whatarethedegreesoffreedomofthesystemwearequantizing?An
electrononlyhasafew:thepositiondegreesoffreedom(x,y,z),andthespinstate.
Theelectromagneticfield,ontheotherhand,hasanelectricfieldE%andamagnetic
fieldB% defined ateverypointinspace; thenumberof degreesof freedommustbe
infinite!Buthowmanyindependentdegreesoffreedomarethere,exactly,perpoint?
Afirstguessmightbethattherearesixdegreesoffreedomperpoint,namelythe
threecomponentsoftheelectricfield,andthethreecomponentsofthemagneticfield.
Butthiscan’tberight.Firstofall,notallofthecomponentsofE%andB%areindepen-
dent(thinkoftheno-monopoleequation∇·B=0).Secondly,Maxwell’sequations
arefirst-orderdifferentialequations,whichmeansthat theymustbeHamiltonian,
ratherthanEuler-Lagrange,equationsofmotion. ThisimpliesthatsomeoftheE,B
fieldsareactuallycanonical”momenta”,ratherthancanonical”coordinates.”
Thekeytoobtainingthedegreesoffreedomcorrespondingtocanonicalcoordi-
natesistheintroductionofthe4-vectorpotentialAμ(x)={A 0 ,A 1 ,A 2 ,A 3 }.OftenA 0
iscalledthe”scalarpotential,”andtheremainingthreecomponentsA%={A 1 ,A 2 ,A 3 }
arereferredtoasthe”3-vectorpotential,”orjust”vectorpotential.”Intermsofthe
4-vectorpotential,theelectricandmagneticfieldstrengthsareexpressedas
E% = −∇A 0 −∂tA%
B% = ∇×A% (25.31)Writteninthisform,theE,Bfieldsautomaticallysatisythetheno-monopoleequa-
tion,andalsoFaraday’sLaw.Allthatisnecessaryistowriteanactionasafunctional