582-chapter9 - 9 Quantization of Gauge Fields We will now turn to the problem of the quantization of gauge theories We will begin with the simplest

582-chapter9 - 9 Quantization of Gauge Fields We will now...

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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-trivial dynamics. 9.1 Canonical Quantization of the Free Electromagnetic Field Maxwell’s theory was the first field theory to be quantized. However, the quantization procedure 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 configurations of vector potentials represent physically distinct states. It could be argued that one should abandon the picture based on the vector potential and go back to a picture based on electric and magnetic fields instead. However, there is no local Lagrangian that can describe the time evolution of the system now. 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 momenta. For that reason, one sticks to the Lagrangian formulation with the vector potential A μ as its independent coordinate-like variable. The Lagrangian for Maxwell’s theory L = 1 4 F μ ν F μ ν (9.1)
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9.1 Canonical Quantization of the Free Electromagnetic Field 261 where F μ ν = μ A ν ν A μ , can be written in the form L = 1 2 ( E 2 B 2 ) (9.2) where E j = 0 A j j A 0 B j = ϵ jk k A (9.3) The electric field E j and the space components of the vector potential A j form a canonical pair since, by definition, the momentum Π j conjugate to A j is Π j ( x ) = L δ∂ 0 A j ( x ) = 0 A j + j A 0 = E j (9.4) Notice that since L does not contain any terms which include 0 A 0 , the momentum Π 0 , conjugate to A 0 , vanishes Π 0 = δ L δ∂ 0 A 0 = 0 (9.5) A consequence of this result is that A 0 is essentially arbitrary and it plays the role of a Lagrange multiplier. Indeed it is always possible to find a gauge transformation φ A 0 = A 0 + 0 φ A j = A j j φ (9.6) such that A 0 = 0. The solution is 0 φ = A 0 (9.7) which is consistent provided that A 0 vanishes both in the remote part and in the remote future, x 0 ± . The canonical formalism can be applied to Maxwell’s electrodynamics if we notice that the fields A j ( x ) and Π j ( x ) obey the equal-time Poisson Brackets { A j ( x ) , Π j ( x ) } P B = δ jj δ 3 ( x x ) (9.8) or, in terms of the electric field E , { A j ( x ) , E j ( x ) } P B = δ jj δ 3 ( x x ) (9.9) The classical Hamiltonian density is defined in the usual manner H = Π j 0 A j L (9.10)
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262 Quantization of Gauge Fields We find H ( x ) = 1 2 ( E 2 + B 2 ) A 0 ( x ) · E ( x ) (9.11) Except for the last term, this is the usual answer. It is easy to see that the
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