Feynman Path Integral Formulation

128 4 Hamiltonian and Wheeler-DeWitt EquationT 00 = 1 (2 Eia Ea i + B i aB i a)(4.127)andT 0i = ε ijk E j a B k a . (4.128)T 00 could be interpreted as a Hamiltonian density, and therefore be used to constructthe quantum Hamiltonian, were it not for the fact that some of the degrees offreedom, as shown below, are unphysical.It is convenient to rewrite the gauge field Lagrangian in first order form (seefor example Itzykson and Zuber, 1980), For concreteness we will discuss here theSU(2) case with f abc = ε abc (in the following bold-face vectors will therefore referto iso-vectors with color index a). The Lagrangian is thenL = 1 4 F μν · F μν − 1 2 F μν · (∂ μ A ν − ∂ ν A μ + gA μ × A ν ) . (4.129)The Euler-Lagrange equations giveandwith time evolution equationsF μν = ∂ μ A ν − ∂ ν A μ + gA μ × A ν (4.130)∂ μ F μν + gA μ × F μν = 0 , (4.131)∂ 0 A i = F 0i +(∇ i + gA i ×)A 0∂ 0 F 0i =(∇ j + gA j ×)F ji − gA 0 × F 0i . (4.132)The field canonically conjugate to A i is F 0i (with the chromo-electric field havingbeen defined as Ea i = Fa i0 ).On the other hand the field canonically conjugate to A 0 vanishes, since there isno ∂ 0 A 0 term in the Lagrangian,π 0 = 0 , (4.133)so this field must be treated as a dependent variable. In Dirac’s language this iscalled a primary constraint. From the second equation of motion, Eq. (4.131) onehas(∇ k + gA k ×)F k0 = 0 , (4.134)which is the analog of Gauss’s equation ∇ · E = 0 in electrodynamics. Equivalentlythe last constraint can be written in terms of canonical momenta π i as(∇ k + gA k ×)π k = 0 , (4.135)which is sometimes referred to as a secondary constraint, since it involves the useof the equations of motion. Eq. (4.134) tells us that not all conjugate momenta F k0are independent, and one needs therefore to impose a gauge condition, such as∇ k A k = 0 , (4.136)