= Auxiliary-field transfer operator for an Ising chain
{title2=$T(m',m)=\exp[-D(m'-m)^2-\{U(m')+U(m)\}/2]$}
For a nearest-neighbor auxiliary-field representation of an <Ising model>, sharing each local potential between adjacent bonds gives the displayed symmetric <integral operator>. Its periodic <partition function> is the trace of its $N$th power, up to Gaussian normalization constants. In the slowly varying continuum limit, a slice of length $\epsilon$ has kernel $[M/(2\pi\epsilon)]^{1/2}\exp[-M(m'-m)^2/(2\epsilon)-\epsilon\{U(m')+U(m)\}/2]$. Expanding its action on a smooth function gives $1-\epsilon\widehat H+O(\epsilon^2)$, with $\widehat H=-(2M)^{-1}\partial_m^2+U$. Thus fixed endpoints give an imaginary-time transition kernel and periodic endpoints give a <Euclidean path integral> trace. The finite-spacing integral operator and the continuum differential operator need not have identical high-energy spectra.
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