9 Quantization of Gauge Fields We will now turn to the problem of the quantization of gauge theories. We will begin with the simplest gauge theory, the free electromagnetic field. This is an abelian gauge theory. After that we will discuss at length the quantization of non-abelian gauge fields. Unlike abelian theories, such as the free electromagnetic field, even in the absence of matter fields non-abelian gauge theories are not free fields and have highly non-trivialdynamics. 9.1 Canonical quantization of the free electromagnetic field The Maxwell theory was the first field theory to be quantized. The quanti- zation procedure of a gauge theory, even for a free field, involves a number of subtleties not shared by the other problems that we have considered so far. The issue is the fact that this theory has a local gauge invariance. Unlike systems which only have global symmetries, not all the classical configu- rations of vector potentials represent physically distinctstates.Itcouldbe argued that one should abandon the picture based on the vectorpotential and go back to a picture based on electric and magnetic fields instead. How- ever, there is no local Lagrangian that can describe the time evolution of the system in that representation. Furthermore, is not clear which fields, E or B (or some other field) plays the role of coordinates and which can play the role of momentum. For that reason, and others, one sticks to the Lagrangian formulation with the vector potential Aµ as its independent coordinate-like variable. The Lagrangian for the Maxwell theory 1 µν L = − F F (9.1) 4 µν 262 Quantization of Gauge Fields where Fµν = ∂µAν − ∂νAµ,canbewrittenintheform 1 L = E2 − B2 (9.2) 2 where ( ) Ej = −∂0Aj − ∂jA0,Bj = −"jk"∂kA" (9.3) The electric field Ej and the space components of the vector potential Aj form a canonical pair since, by definition, the momentum Πj conjugate to Aj is δL Πj x = = ∂0Aj + ∂jA0 = −Ej (9.4) δ∂0Aj x Notice that since L(does) not contain any terms which include ∂0A0,the ( ) momentum Π0,conjugatetoA0,vanishes δL Π0 = = 0(9.5) δ∂0A0 AconsequenceofthisresultisthatA0 is essentially arbitrary and it plays the role of a Lagrange multiplier. Indeed, it is always possible to find a gauge transformation φ ′ ′ A0 = A0 + ∂0φAj = Aj − ∂jφ (9.6) ′ such that A0 = 0. The solution is ∂0φ = −A0 (9.7) which is consistent provided that A0 vanishes both in the remote part and in the remote future, x0 → ±∞. The canonical formalism can be applied to Maxwell electrodynamics if ′ we notice that the fields Aj x and Πj′ x obey the equal-time Poisson Brackets ′ 3 ′ Aj x , Π(j′ )x PB =(δjj)′ δ x − x (9.8) or, in terms of the electric field E, { ( ) ( )} ( ) ′ 3 ′ Aj x ,Ej′ x PB = −δjj′ δ x − x (9.9) Thus, the spatial components of the vector potential and the components { ( ) ( )} ( ) of the electric field are canonical pairs. However, the time component of the vector field, A0,doesnothaveacanonicalpair.Thus,thequantization procedure treats it separately, as a Lagrange multiplier field that is imposing aconstraint,thatwewillseeistheGaussLaw.However,attheoperator level, the condition Π0 = 0mustthenbeimposedasaconstraint.Thisfact 9.1 Canonical quantization of the free electromagnetic field 263 led Dirac to formulate the theory of quantization of systems with constraints (Dirac, 1966). There is, however, another approach, also initiated by Dirac, consisting in setting A0 = 0andtoimposetheGaussLawasaconstraint on the space of quantum states. As we will see, this amounts to fixing the gauge first (at the price of manifest Lorentz invariance). The classical Hamiltonian density is defined in the usual manner H = Πj∂0Aj − L (9.10) We find 1 H x = E2 + B2 − A x ! ⋅ E x (9.11) 2 0 Except for the last term, this is the usual answer. It is easy toseethatthe ( ) ( ) ( ) ( ) last term is a constant of motion.Indeedtheequal-timePoissonBracket between the Hamiltonian density H x and ! ⋅ E y is zero. By explicit calculation, we get ( ) ( ) 3 δH x δ! ⋅ E y δH x δ! ⋅ E y H x , ! ⋅ E y PB = d z − + ! δAj z δEj z δEj z δAj z ( ) ( ) ( ) (9.12)( ) { ( ) ( )} ! " But ( ) ( ) ( ) ( ) δH x 3 δH x δBk w 3 w = d w = d wBk w δ x − w "k"j %" δ w − z δAj z ! δBk w δAj z ! ( ) ( ) ( ) = −" %z( )d3(wB w) δ x − w (δ w − z) ( ) ( ) ( ) k"j " ! k (9.13) ( ) ( ) ( ) Hence δH x z x = "j"k %" Bk x δ x − z = "j"kBk x %" δ x − z (9.14) δAj z ( ) Similarly, we get ( ( ) ( )) ( ) ( ) ( ) δ! ⋅ E y δ! ⋅ E y = y = %j δ y − z , 0(9.15) δEj z δAj z ( ) ( ) Thus, the Poisson Bracket is ( ) ( ) ( ) H x , ! ⋅ E y = d3z −" B x %x δ x − z %y δ y − z PB ! j"k k " j = x y −"j"kBk x %" %j δ x − y { ( ) ( )} [ x ( )x ( ) ( )] = "j"kBk x %" %j δ x − y = 0 ( ) ( ) (9.16) ( ) ( ) 264 Quantization of Gauge Fields provided that B x is non-singular. Thus, !⋅E x is a constant of motion. It is easy to check that !⋅E generates infinitesimal gauge transformations. We will prove this( ) statement directly in the quantum( ) theory. Since ! ⋅ E x is a constant of motion, if we pick a value for it at some initial time x0 = t0,itwillremainconstantintime.Thuswecanwrite ( ) ! ⋅ E x = ρ x (9.17) which we recognize to be Gauss’s Law. Naturally, an external charge distri- ( ) ( ) bution may be explicitly time dependent and then d ! ∂ ! ∂ ⋅ E = ⋅ E = ρext x,x0 (9.18) dx0 ∂x0 ∂x0 Before turning to the( quantization) ( of this) theory, we( must) notice that A0 plays the role of a Lagrange multiplier field whose variation yields the Gauss Law, ! ⋅ E = 0. Hence, the Gauss Law should be regarded as a constraint rather than an equation of motion. This issue becomes very important in the quantum theory. Indeed, without the constraint ! ⋅ E = 0, the theory is absolutely trivial, and wrong. Constraints impose very severe restrictions on the allowed states of a quantum theory. Consider for instance a particle of mass m moving freely in three dimensional space. Its stationary states have plane wave wave functions 2 r,x E p = p Ψp 0 ,withanenergy 2m .Ifweconstraintheparticletomove only on the surface of a sphere of radius R,itbecomesequivalenttoarigid 2 2 ( ) I =( mR) " = h̵ & & + rotor of moment of inertia and energy eigenvalues "m 2I 1 where & = 0, 1, 2,...,and m ≤ &.Thus,eventhesimpleconstraintr 2 = R2, does have non-trivial effects. ( ) Unlike the case of a particle∣ ∣ forced to move on the surface of a sphere, the constraints that we have to impose when quantizing Maxwell electrodynam- ics do not change the energy spectrum. This is so because we canreduce the number of degrees of freedom to be quantized by taking advantage of the gauge invariance of the classical theory. This procedureiscalledgauge fixing.Forexample,theclassicalequationofmotion 2 µ µ ν ∂ A − ∂ ∂νA = 0(9.19) in the Coulomb gauge, A = 0and! ⋅ A = 0, becomes 0 ( ) 2 ∂ Aj = 0(9.20) However the Coulomb gauge is not compatible with the Poisson Bracket ′ ′ A x , Π , x = δ ′ δ x − x (9.21) j j PB jj $ ( ) ( )% ( ) 9.1 Canonical quantization of the free electromagnetic field 265 since the spatial divergence of the delta function does not vanish. It will follow that the quantization of the theory in the Coulomb gauge is achieved at the price of a modification of the commutation relations. Since the classical theory is gauge-invariant, we can alwaysfixthegauge without any loss of physical content. The procedure of gauge fixing has the attractive that the number of independent variables is greatly reduced. A standard approach to the quantization of a gauge theory is to fix the gauge first, at the classical level, and to quantize later. However, a number of problems arise immediately. For instance, in most gauges, such as the Coulomb gauge, Lorentz invariance is lost, or at least it is manifestly so. Thus, although the Coulomb gauge, also known as the radiation or transverse gauge, spoils Lorentz invariance, it has the attractive feature that the nature of the physical states (the photons)isquitetrans- parent. We will see below that the quantization of the theory in this gauge has some peculiarities. Another standard choice is the Lorentz gauge µ ∂µA = 0(9.22) whose main appeal is its manifest covariance. The quantization of the system is this gauge follows the method developed by and Gupta and Bleuer. While highly successful, it requires the introduction of states with negative norm (known as ghosts) which cancel all the gauge-dependent contributions to physical quantities. This approach is described in detail inthebookby Itzykson and Zuber (Itzykson and Zuber, 1980). More general covariant gauges can also be defined. A general approach consists not on imposing a rigid restriction on the degrees offreedom,but to add new terms to the Lagrangian which eliminate the gauge freedom. For instance, the modified Lagrangian 1 1 µ L = − F 2 + ∂ A x 2 (9.23) 4 µν 2α µ is not gauge invariant because of the presence( ( )) of the last term. We can easily see that this term weighs gauge equivalent configurations differently and the parameter 1 α plays the role of a Lagrange multiplier field. In fact, in the limit α → 0werecovertheLorentzgaugecondition.Inthepath integral quantization/ of the Maxwell theory it is proven thatthisapproach is equivalent to an average over gauges of the physical quantities. If α = 1, 2 the equations of motion become very simple, i.e. ∂ Aµ = 0. This is the Feynman gauge. In this gauge the calculations are simplest although, here too, the quantization of the theory has subtleties (such as ghosts, etc.).