Feynman Path Integral Formulation

128 4 Hamiltonian and Wheeler-DeWitt Equation
T 00 = 1 (
2 E
i
a Ea i + B i aB i a)
(4.127)
and
T 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 construct
the quantum Hamiltonian, were it not for the fact that some of the degrees of
freedom, as shown below, are unphysical.
It is convenient to rewrite the gauge field Lagrangian in first order form (see
for example Itzykson and Zuber, 1980), For concreteness we will discuss here the
SU(2) case with f abc = ε abc (in the following bold-face vectors will therefore refer
to iso-vectors with color index a). The Lagrangian is then
L = 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 give
and
with time evolution equations
F 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 having
been defined as Ea i = Fa i0 ).
On the other hand the field canonically conjugate to A 0 vanishes, since there is
no ∂ 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 is
called a primary constraint. From the second equation of motion, Eq. (4.131) one
has
(∇ k + gA k ×)F k0 = 0 , (4.134)
which is the analog of Gauss’s equation ∇ · E = 0 in electrodynamics. Equivalently
the 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 use
of the equations of motion. Eq. (4.134) tells us that not all conjugate momenta F k0
are independent, and one needs therefore to impose a gauge condition, such as
∇ k A k = 0 , (4.136)