Feynman Path Integral Formulation

4.13 Lattice Regularized Hamiltonian for Gauge Theories 131
A lattice regularized Hamiltonian can be defined on a purely spatial lattice, by
taking the zero lattice spacing limit in the time direction (Kogut, 1983). The process
can be regarded as the inverse to the one usually follows in deriving the Feynman
path integral from the matrix elements of the quantum time evolution operator e −iHt .
One starts by writing the lattice partition function Z in terms of a product of transfer
matrices T (U t+1 ,U t ) connecting the fields on successive time slices, labeled by t,
and integrated over all intermediate variables U,
∫
Z = trT L = dU t+1 dU t dU t−1 ...T (U t+1 ,U t )T (U t ,U t−1 )T (U t−1 ,U t−2 ) ...
(4.144)
with L is the total time extent of the lattice. In order to take the zero lattice spacing
limit in the time direction, it is convenient to distinguish the lattice spacing in this
direction by the symbol a 0 . Local gauge invariance further allows one to set all the
link variables in the time direction to unity, U n0 = 1, or A a n0
= 0 in this lattice version
of the temporal gauge. Consequently gauge invariance will need to be imposed as a
constraint, which will eventually take the form of a discrete version of Gauss’s law.
With this choice of temporal gauge, the action can be decomposed in a part that
involves only the spatial plaquettes, and a remainder involving pairs of oppositely
oriented link variables separated by a single time step,
I[U] =− 1
4g 2
− 1
4g 2
∑ tr [ UUU † U † + h.c. ]
spatial ✷
∑ tr
spatial
[
]
U nm U n+a † 0 t,m+a 0 t + h.c.
. (4.145)
After labeling the gauge variables on two neighboring time slices by U and U ′ , one
can write the transfer matrix element T (U,U ′ ) as
{
〈U|T |U ′ 1
〉 = exp
4g
∑ 2 tr [ UUU † U † + h.c. ] [
]
+ tr U ′ U ′ U †′ U †′ + h.c.
spatial✷
+ 1
4g 2
∑ tr
links
[
U nm U †′
nm + h.c.] } . (4.146)
In general the matrix elements of the transfer matrix T have a rather complicated
form, but in the limit a 0 → 0 one can write T ≃ 1 − a 0 H + O(a 2 0 ) and extract from
T an expression for the Hamiltonian H. But one notices further that the last two
terms in Eq. (4.146) involve only temporal loops, with two (gauge fixed) links
pointing in the time direction. When written out explicitly, they contain terms of
the type
tr[U † (t + 1)U(t)+h.c.] , (4.147)