Definition. Temporal gauge and the residual spatial action [fdip-0089]
AGENTDRAFTED
Use the spatial graph and link space
\(X=G^{E_s}\) and \(\mathcal H_{\mathrm {link}}=L^2(X,dU;\mathbb C)\).
On a finite open time interval with slices \(0,\ldots ,N\), the
temporal link \(h_{j,v}\) transports from \(j+1\) to
\(j\) at spatial vertex \(v\). Allow independent gauge changes at
all slices. For slice gauge changes \(g_j\in K\),
where \(K=G^{V_s}\), it transforms as
\[h_{j,v}\longmapsto
g_{j,v}h_{j,v}g_{j+1,v}^{-1}.\]
Temporal gauge means \(h_{j,v}=1\) on these temporal
edges. Choose \(g_{0,v}\) and recursively set
\(g_{j+1,v}=g_{j,v}h_{j,v}\) to reach this gauge.
A transformation preserving temporal gauge satisfies
\(g_{j+1,v}=g_{j,v}\). Its remaining freedom is one spatial gauge
transformation \(g\in K\), independent of time. This is the
residual gauge action:
\[(g\cdot U)_e=g_{t(e)}U_eg_{s(e)}^{-1},\qquad
(R_gf)(U)=f(g^{-1}\cdot U).\]
These are the same unitary operators used in the transfer
construction, now interpreted on a time slice.
On a periodic time circle, the closure \(g_{N,v}=g_{0,v}\)
changes the conclusion. The ordered temporal holonomy is
\[H_v=h_{0,v}h_{1,v}\cdots h_{N-1,v}.\]
The recursion closes exactly when \(H_v=1\). In general one can
set the first \(N-1\) temporal links to one, but the last becomes
\(g_{0,v}H_vg_{0,v}^{-1}\). Thus an arbitrary periodic field does
not admit a gauge with every temporal link equal to one. The
projected periodic trace keeps its temporal averaging
and does not require such a gauge.
Kogut and Susskind's 1975 Hamiltonian formulation of Wilson's
lattice gauge theories [kogut1975hamiltonian] provides
historical context for this subject. The finite residual
action above follows directly from the displayed transformation
law; the state constraints below are established by averaging and
representation theory.