Definition. Temporal gauge and the residual spatial action [fdip-0089]
Definition. Temporal gauge and the residual spatial action [fdip-0089]
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.