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Chapter 14 Path Integral for Gauge Fields

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Chapter 14 Path Integral for Gauge Fields Chapter 14 Path integral for gauge fields All fundamental theories in particle physics are gauge theories. These theories contain first class constraints which generate the (time-independent) gauge transformations and hence must be quantized along the lines outlined above. We shall first recall the classical canonical structure of pure Yang-Mills theories with particular emphasis on the constraints. At the end we specialize to the Abelian case and set some of the potentials and field strengths to zero to recover the path integral for the Schwinger model. 14.1 Classical Yang-Mills Theories In Minkowski spacetime the Lagrangian for a non-Abelian gauge theory reads 1 = tr F F µν , (14.1) L −4 µν where the (hermitian) field strength is F = ∂ A ∂ A i[A , A ]. The chromoelectric µν µ ν − ν µ − µ ν and chromomagnetic fields are the generalization of the electric and magnetic fields in electro- magnetism, F = E and F = ǫ B (14.2) 0i i ij − ijk k Expanding the potential and field strength as dim G dim G µ µ µν µν A = Aa Ta , F = Fa Ta, aX=1 aX=1 where the (hermitian) generators Ta of the Lie algebra obey the commutation relations [Ta, Tb]= ifabcTc, (14.3) 120 CHAPTER 14. GAUGE FIELDS 14.1. Classical Yang-Mills Theories 121 with totally anti-symmetric and real structure constants fabc, we find the following formulae for the components in group-space, d 1 E = A A0 + f A0A , B = A f A A . (14.4) a dt a −∇ a abc b c a −∇ × a − 2 abc b × c We have set A =(A1, A2, A3), E =(E1, E2, E3) and B =(B1, B2, B3). One would have the usual sign convention [51] if one would take A =(A1, A2, A3) that is replace A by A. In the − non-covariant notation the Lagrangian reads 1 E 2 B 2 = ( a a ). (14.5) L 2 a − X µν The non-covariant form of the Yang-Mills equations DνF are the generalized Gauss- and Ampere law D E =0 E + f A E =0 · ⇐⇒ ∇· a abc b · c D E =(D B) ∂ E + f A0E = B + f (A B ). (14.6) t × ⇐⇒ t a abc b c ∇ × a abc b × c The corresponding identities in two dimensions for F01 = E are obtained by setting E = (E, 0, 0), B =0 and A2 = A3 =0 in the above equations. 14.1.1 Hamiltonian structure Our task is to build a Hamiltonian scheme, which will give rise to these Yang-Mills equations. The first problem in passing to a Hamiltonian description arises from the fact that does not ˙ 0 0 L depend on Aa and thus there is no momentum conjugate to Aa. To remedy this we use the gauge 0 freedom to choose the temporal gauge Aa =0. In this gauge we have 1 = (A˙ 2 B 2) (14.7) L 2 a − a and the Gauss- and Ampere laws take the simple forms (D E) =0 and E˙ =(D B) . (14.8) · a a × a The momentum density conjugate to A is gotten by differentiating in (14.7) with respect to a L the ’velocity’ A˙ , δ πa(x)= L = A˙ a = Ea (14.9) δA˙ a(x) which then lead to the following Hamiltonian and Hamiltonian density, 3 1 E 2 B 2 H = d x , where = ( a + a ). (14.10) Z H H 2 ———————————— A. Wipf, Path Integrals CHAPTER 14. GAUGE FIELDS 14.1. Classical Yang-Mills Theories 122 The canonical equal time commutation relations read (here we do not distinguish between upper i and lower indices, in particular A = Ai) Ai (t, x), Ej(t, y) = δ δ δ3(x y), (14.11) { a b } ab ij − from which follows that i j k B (t, x), E (t, y) = ǫ δ ∂ k δ(x y) f A δ(x y) . (14.12) { a b } ijk ab x − − abc c − Now it is rather straightforward to calculate the time-derivative of the canonical fields. On obtains ˙ i i 3 i j j i Aa(x)= Aa(x),H = d y Aa(x), Eb (y) Eb (y)= Ea(x) (14.13) { } Z { } and similarly, using (14.11), E˙ i (x)= Ei (x),H = ǫ ∂ Bk + f AjBk (14.14) a { a } ijk j a abc b c and hence the Hamiltonian equations reproduce Ampere’s law (14.8) and the definition of Ea in terms of A˙ a. However, Gauss’s law has yet not emerged, since it is a fixed-time constraint between canonical variables. To understand the role of the Gauss constraints C (x)=(D E) = ∂ Ei + f Ai Ei (14.15) a · a i a abc b c more clearly let us calculate the commutator of these constraints with the canonical variables. One finds A (y),C (x) = δ δ(x y) f A δ(x y) { b a } ab∇x − − abc c − E (y),C (x) = f E δ(x y). (14.16) { b a } − abc c − Smearing the constraints with arbitrary test functions θa as 3 Cθ = d xθa(x)Ca(x), (14.17) Z these commutation relations become A (y),C = θ (y)+ f θ (y)A (y) { a θ} −∇ a abc b c E (y),C = f θ (y)E (y). (14.18) { a θ} abc b c From the first equation one may obtains B (y),C = f θ (y)B (y). (14.19) { a θ} abc b c ———————————— A. Wipf, Path Integrals CHAPTER 14. GAUGE FIELDS 14.1. Classical Yang-Mills Theories 123 Now we shall see, that the constraints generate the time-independent gauge transformations A e−iθAeiθ + ie−iθ eiθ, E e−iθEeiθ, B e−iθBeiθ. (14.20) −→ ∇ −→ −→ The corresponding small transformations of the gauge potential and field strengths are δ A = θ i[θ, A] δ E = i[θ, E] and δ B = i[θ, B], (14.21) θ −∇ − θ − θ − which, after expanding θ = θaTa read in component form δ A = θ + f θ A , δ E = f θ E and δ B = f θ B (14.22) θ a −∇ a abc b c θ a abc b c θ a abc b c which are identical with the corresponding commutation relations in (14.18,14.19) with the smeared constraint Cθ. Hence the Gauss-constraints generate the time-independent gauge trans- formations. It follows then that the Hamiltonian commutes with the constraints since it is gauge invari- ant. Finally, using the identity f(y)δ′(x y)= f(x)δ′(x y)+ f ′(x)δ(x y) (14.23) − − − and the Jacobian identity fabdfcpd + fcadfbpd + fbcdfapd =0 (14.24) one shows that the commutator of two different constraints follow the Lie algebra of the gauge group, C (x),C (y) = f C (x)δ(x y), (14.25) { a b } abc c − and thus form a system of first class constraints. The transition from the classical Poisson bracket to the corresponding commutators is as usual achieved by replacing Poisson brackets ., . by commutators i[., .]/h¯ in the above relations. { } − The path integral for the Yang-Mills Hamiltonian (14.10) is given by analogy with the con- strained quantum mechanical system (13.19) by i 1 1 Z = E A δ(C )δ(F ) det F ,C exp (E A˙ E 2 B 2)dtd3x , (14.26) D aD a a a { a b} h¯ a a − 2 a − 2 a Z h Z i a where the Fa are the gauge fixing depending on Aa. We have seen that θ Ca generates in- finitesimal gauge transformations, and hence F ,C is just an infinitesimal gauge transfor- { a b} R mation with parameters θa stripped off δ F (A(y)),C (x) = δ (F [A(y)]) δ F . (14.27) { b a } δθa(x) θ b ≡ a b ———————————— A. Wipf, Path Integrals CHAPTER 14. GAUGE FIELDS 14.1. Classical Yang-Mills Theories 124 For the constraint δ-function we may insert i δ(C )= const A0 exp A0(DE) a · D a h¯ a a Z h Z i so that i 1 1 Z = E Aa δ(F ) det(δ F ) exp (A˙ E (DA0) E E 2 B 2)d4x (14.28), D aD µ a a b h¯ a a − a a − 2 a − 2 a Z h Z i where we have partially integrated in the exponent. Next we calculate the Gaussian Ea-integral which results in i Z = const Aa δ(F ) det(δ F ) exp (A˙ (DA0) )2 B 2 · D µ a a b h¯ a − a − a Z h i Comparing with (14.4) and (14.5) we find the covariant expression for the partition function i a h¯ S[A] Z = const Aµ δ(Fa) det(δaFb) e . (14.29) · Z D In our derivation the gauge conditions Fa depend only on the spatial components of the gauge potential. Recall that det(δaFb) is the determinant of the scalar-products of the gradient vectors AF (A) with the symmetry-generating vector-fields (generating the θ -gauge orbits). We ∇ b a may now assume that Fb also depends on A0 as long as we guarantee that the determinant keeps this geometric meaning in the enlarged space of the gauge potentials (and not only their spatial components). But also in this enlarged space δ δF δ b c (14.30) a δθFb = c ( a δθAµ)=( Fb,Xa), δθ (x) δAµ δθ ∇ where now the gauge transformation may depend on time as well, and hence in δaFb we must a take the gauge variation of all components of Aµ.
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