Theorem. Gauge projection and periodic-time traces [fdip-0083]
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  • November 1, 2026

For the spatial transfer data, normalized gauge averaging is the orthogonal projection \[\begin {aligned} (Pf)(U)&=\int _K f(h^{-1}\cdot U)\,dh,\\ \mathcal H_{\mathrm {phys}} &=\{f:R_hf=f\text { for every }h\in K\}=P\mathcal H_{\mathrm {link}}. \end {aligned}\] This defines the physical Hilbert space of neutral states. The restriction \(T_{\mathrm {phys}}=T|_{\mathcal H_{\mathrm {phys}}}\) is positive, injective and trace class. It is nonzero since constants belong to \(\mathcal H_{\mathrm {phys}}\). For every integer \(N\ge 1\), \[Z_N=\operatorname {Tr}_{\mathrm {phys}}T_{\mathrm {phys}}^N =\operatorname {Tr}_{\mathrm {link}}(PT^N)>0.\] The gauge projector is part of the periodic trace.

There is an explicit lattice integral for \(Z_N\). Put \(U_N=U_0\), take independent variables \(U_j\in X\) and \(h_j\in K\) for \(0\le j<N\), and define \[\begin {aligned} B_{j,e}&=U_{j,e}h_{j,s(e)}U_{j+1,e}^{-1}h_{j,t(e)}^{-1},\\ \mathcal W_N(U,h)&= \prod _{j=0}^{N-1}e^{-V(U_j)} \prod _{j=0}^{N-1}\prod _{e\in E_s}k(B_{j,e}). \end {aligned}\] Then, with normalized product Haar measures, \[Z_N=\int _{X^N\times K^N}\mathcal W_N(U,h)\,dU\,dh.\] Thus \(Z_N^{-1}\mathcal W_N\,dU\,dh\) is a probability law. The temporal link \(h_{j,v}\) transports from time \(j+1\) back to time \(j\). For \(N=1,2\), the formula retains the indexed temporal steps; it does not identify them with edges of an ordinary simple cycle.

For independent gauge changes \(g_j\in K\), with \(g_N=g_0\), the transformations are \[\begin {aligned} U_j&\longmapsto g_j\cdot U_j,\\ h_{j,v}&\longmapsto g_{j,v}h_{j,v}g_{j+1,v}^{-1}. \end {aligned}\] Each \(B_{j,e}\) is conjugated by \(g_{j,t(e)}\). Therefore the weight and measure are invariant. If \(f_j:X\to \mathbb C\) are bounded measurable gauge-invariant functions, multiplication by \(f_j\) preserves \(\mathcal H_{\mathrm {phys}}\), and \[\mathbb E_N\prod _{j=0}^{N-1}f_j(U_j) =\frac {\operatorname {Tr}_{\mathrm {phys}} (M_{f_0}T_{\mathrm {phys}}\cdots M_{f_{N-1}}T_{\mathrm {phys}})}{Z_N}.\] This is a finite periodic-time identity for the specified spatial observables, with the temporal links integrated as above.

Proof. 

  • November 1, 2026

Fubini and Jensen define \(P\) on \(L^2\) and give \(\|P\|\le 1\). Haar invariance gives \(P^2=P\); inversion of \(h\) gives \(P^*=P\). Translating the averaging variable shows that its image is fixed by each \(R_h\), while averaging a fixed vector leaves it unchanged. These are Hilbert-space equalities; no common pointwise representative for all group elements is required. By Theorem [fdip-0080], \(T\) commutes with \(P\). Its restriction is therefore positive and trace class, and \(\langle f,Tf\rangle >0\) for nonzero physical \(f\) proves injectivity there. Set \(S=PT=PTP\) on the full link space. It acts as \(T_{\mathrm {phys}}\) on the physical space and as zero on its orthogonal complement, so \(S^N=PT^N\) has the asserted physical trace.

Averaging the first variable of the kernel of \(T\) gives the continuous kernel of \(S\): \[L(U,U')=b(U)b(U')\int _K \prod _e k(U_eh_{s(e)}U_e'^{-1}h_{t(e)}^{-1})\,dh.\] To obtain this expression, use invariance of \(b\) and move \(h_{t(e)}^{-1}\) from the front of the kernel argument to its end by centrality. Every factor is strictly positive.

The diagonal trace formula, including the case \(N=1\), can be justified directly. The rank-one expansion in Theorem [fdip-0080] gives \[S=\sum _I\alpha _I |P(b\phi _I)\rangle \langle P(b\phi _I)|\] in trace norm. Its kernel expansion converges uniformly when summed over finite sets of representation blocks. For one edge, unitarity gives \[\sum _{i,j}|\phi _{\rho ij}(U)|^2=d_\rho ^2.\] For the finite edge product, the diagonal sum over a tail of blocks is therefore the corresponding tail of \(\sum _I\alpha _I=\kappa ^{|E_s|}\), uniformly in \(U\). Cauchy–Schwarz gives the same bound off the diagonal. Multiplying by \(b\) and averaging in both variables preserve uniform convergence. Integrating the diagonal term by term consequently gives \[\int _XL(U,U)\,dU =\sum _I\alpha _I\|P(b\phi _I)\|_2^2 =\operatorname {Tr}S.\] Multiplication by a bounded function permits the same argument for \(\operatorname {Tr}(M_fS)\).

For \(N\ge 2\), the Hilbert–Schmidt kernel trace identity of Theorem [fdip-007V] gives \[\operatorname {Tr}(M_{f_0}S\cdots M_{f_{N-1}}S) =\int _{X^N}\prod _j f_j(U_j)L(U_j,U_{j+1})\,dU.\] All kernels are bounded on compact spaces, so repeated integrals are absolutely convergent. The same formula for \(N=1\) was proved by the diagonal expansion. Gauge-invariant multiplication commutes with \(P\); hence this full-space trace equals the physical-space trace in the statement. Expanding each \(L\) with an independent \(h_j\) gives \(\mathcal W_N\), because the two adjacent factors \(b(U_j)\) combine to \(e^{-V(U_j)}\). Taking every \(f_j=1\) proves the partition function; general \(f_j\) prove the expectation identity. Strict positivity and compactness of the weight give \(0<Z_N<\infty \). The displayed gauge transformation conjugates \(B_{j,e}\) by direct cancellation, which proves the final invariance assertion.

The temporal plaquette based at \((j,t)\) is traversed down the right edge, backwards along the bottom, up the left edge and then along the top. With the transport order, the first transport acts on the right and the product is exactly \(B_{j,e}\). Temporal averaging is retained at every step, so no periodic gauge-fixing assumption removes \(P\).