= Classical Ising chain as a quantum particle
{title2=$\widehat H_Q=-\frac{J\sinh\kappa}{2}\frac{d^2}{dm^2}+U(m)$}
Hubbard-Stratonovich decoupling of an exponentially coupled Ising chain gives a scalar-field action with a nearest-neighbor <gradient> term and $U(m)=\tanh(\kappa/2)m^2/J-\log[2\cosh(2m+h)]$. Its continuum <partition function> is the <Euclidean path integral> of the displayed quantum particle. At zero field the potential can have two wells, but a finite barrier still permits <quantum tunnelling> and gives a unique symmetric <ground state>. The excitation gap produces a finite classical <correlation length>, so there is no finite-temperature spontaneous magnetization for fixed positive $\kappa$. One classical spatial coordinate becomes imaginary time; a $d$-dimensional classical model corresponds to a quantum model with $d-1$ spatial dimensions.
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