Theorem. Periodic traces and field correlations [fdip-007V]
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  • November 1, 2026

Use the kernel and parameters of Definition [fdip-007T] and Theorem [fdip-007U]. For every integer \(N\ge 1\), \[\operatorname {Tr}T^N =\frac {\lambda _0^N}{1-q^N} =\frac {(2\pi /w)^{N/2}}{2\sinh (N\omega /2)}.\] For \(N\ge 3\), this equals the periodic Gaussian normalizer \(Z_N\). If \(f_0,\ldots ,f_{N-1}\) are bounded complex measurable functions and \(M_f\) denotes multiplication by \(f\), then \[\mathbb E_{\mu _N}\prod _{j=0}^{N-1}f_j(x_j) =\frac {\operatorname {Tr} (M_{f_0}T M_{f_1}T\cdots M_{f_{N-1}}T)}{Z_N}.\] In particular, bounded two-point observables have the usual cyclic trace expression with \(T^{N-t}\) and \(T^t\).

For \(N\ge 3\) and the unbounded coordinate observable, let \(Xf(x)=xf(x)\) on its domain \(\{f\in L^2:xf\in L^2\}\). For integers \(0\le t\le N\), initially on finite Hermite sums, put \[A_t=T^{t/2}XT^{(N-t)/2}.\] It has a unique Hilbert–Schmidt extension. The precise coordinate trace formula is \[\begin {aligned} \mathbb E_{\mu _N}[x_0x_t] &=\frac {\operatorname {Tr}(A_t^*A_t)}{Z_N}\\ &=\frac {q^t+q^{N-t}}{2s(1-q^N)}, \qquad x_N=x_0. \end {aligned}\] Here \(\operatorname {Tr}(A_t^*A_t)\) supplies the meaning of the frequently written unbounded expression \(\operatorname {Tr}(T^{N-t}XT^tX)\).

Proof. 

  • November 1, 2026

The eigenvalues in Theorem [fdip-007U] give the first trace by a geometric sum. The remaining expression uses \(\lambda _0=\sqrt {2\pi /w}\,q^{1/2}\) and \(q^{-N/2}-q^{N/2}=2\sinh (N\omega /2)\). On the ordinary cycle, the precision is \(Q_N=m^2I+wL_{C_N}\). The twisted-cycle determinant at trivial holonomy, with the scalar factor \(w\) restored, gives \[\det Q_N=w^N\bigl (2\cosh (N\omega )-2\bigr ).\] The real Gaussian integral therefore gives exactly the same \(Z_N\). The trace identity for \(N=1,2\) remains an operator identity; no ordinary simple-cycle identification is used for those two periods.

For bounded insertions, \(M_fT\) is Hilbert–Schmidt. The product of two such operators is trace class; multiplying by the remaining bounded factors preserves this property. The Hilbert–Schmidt kernel trace identity \[\operatorname {Tr}(AB)=\int _{\mathbb R^2}A(x,y)B(y,x)\,dx\,dy\] follows by expanding the two kernels in an orthonormal basis and taking their \(L^2\) pairing. Applying it to the cyclic product and composing the intervening kernels yields \[\operatorname {Tr}(M_{f_0}T\cdots M_{f_{N-1}}T) =\int _{\mathbb R^N}\prod _j f_j(x_j) \prod _jK(x_j,x_{j+1})\,d^Nx.\] All repeated integrals are absolutely convergent: the absolute integrand is bounded by \(\prod _j\|f_j\|_\infty \) times the integrable massive Gaussian. The kernel factorization in Definition [fdip-007T] proves the asserted expectation.

For the coordinate insertion, the Hermite generating function gives the recurrence \[Xh_n^s=\frac {\sqrt {n+1}\,h_{n+1}^s +\sqrt n\,h_{n-1}^s}{\sqrt {2s}},\] where the second term is zero at \(n=0\). Consequently the matrix of \(A_t\) has only two adjacent diagonals. Summing their squared entries gives \[\begin {aligned} \|A_t\|_{\mathrm {HS}}^2 &=\frac {\lambda _0^N}{2s} \frac {q^t+q^{N-t}}{(1-q^N)^2}. \end {aligned}\] The sum is finite also at \(t=0,N\). Thus the operator on the dense finite Hermite span extends uniquely to a Hilbert–Schmidt operator. Thus \(A_t^*A_t\) is positive trace class and its trace is this squared norm. Division by \(Z_N\) gives the displayed value. Finally, the periodic covariance calculation, with \(s=w\sinh \omega \), gives this same value for \((Q_N^{-1})_{0t}\). Gaussian covariance therefore identifies it with \(\mathbb E_{\mu _N}[x_0x_t]\), including the periodic endpoint. No cyclic manipulation of an undefined unbounded product is used.

This finite periodic calculation makes the trace notation of [wingate2025lattice, Eq. (5.4), p. 4] explicit for the Gaussian chain. That source subsequently takes an infinite-time limit; the identities here hold at finite \(N\). The Ising transfer and Markov-chain comparison in [friedli2017statistical, Section 3.10.4, p. 139] concerns a different state space and is background for the transfer interpretation, not a premise for this Gaussian proof.