The transfer matrix and physical time

Prerequisites: the lattice definition, semigroups and reflection positivity.

You will learn to convert transfer eigenvalues into energies with correct units, identify the physical gauge-invariant sector, and distinguish time evolution from Monte Carlo updates.

One step through spacetime

A Euclidean lattice can be sliced into spatial configurations. Integrating over the fields between adjacent slices suggests an operator $T$ that advances by a time spacing $a_t$ . Repeating the step gives $T^n$ , and inserting observables on slices produces correlation functions separated by $t=na_t$ .

In the pure $SU(N)$ Wilson-action setting, this construction must account for the gauge transformations on each spatial slice. The physical space consists of gauge-invariant states; temporal-link integration implements the corresponding constraints. A transfer operator on an unconstrained space and a physical transfer operator are not interchangeable.

The Wilson convention used here is

S_W=\beta\sum_p\left(1-\frac1N\operatorname{Re}\operatorname{tr}U_p\right), \qquad \beta=\frac{2N}{g_0^2}

for isotropic spacing, with each plaquette counted once. If temporal and spatial spacings differ, the couplings for their plaquettes must be specified separately. We do not take a Hamiltonian anisotropic limit by silently replacing $a_t$ with the spatial spacing.

Positivity has an operator meaning

For $T$ self-adjoint and positive, normalize its largest vacuum eigenvalue $\lambda_0$ to one. On a suitable subspace where the logarithm is defined,

H=-\frac1{a_t}\log(T/\lambda_0).

An eigenvalue $0<\lambda_n\leq\lambda_0$ then corresponds to

E_n-E_0=-\frac1{a_t}\log\frac{\lambda_n}{\lambda_0}.

The logarithm is defined by spectral calculus, not by taking logarithms of matrix entries. Zero spectral values require attention to its domain; “positive semidefinite” and “strictly positive with a bounded inverse” are different conditions.

A kernel with positive entries need not be a positive operator. The matrix

\begin{pmatrix}1&2\\2&1\end{pmatrix}

has positive entries but eigenvalues three and minus one. An ordinary positive Boltzmann weight is therefore not an adequate substitute for a transfer-positivity argument. Reflection positivity supplies the underlying kind of structure.

Worked conversion: hold the physical mass fixed

Take the exact two-state model

T(a_t)=\begin{pmatrix}1&0\\0&e^{-a_t m}\end{pmatrix},\qquad m>0.

At $a_t=0.1/m$ , the excited transfer eigenvalue is $e^{-0.1}$ ; at $a_t=0.05/m$ , it is $e^{-0.05}$ . Both give energy $m$ . The eigenvalue approaches the vacuum eigenvalue as the lattice is refined, although the physical gap remains fixed.

A claim that a transfer-eigenvalue separation must remain a fixed positive number as $a_t\to0$ would exclude this elementary massive scaling example. The dimensionless energy is $a_t m$ , not $m$ itself.

Monte Carlo time can change without changing the theory

For a two-state uniform probability measure, consider a reversible sampling matrix

P_\epsilon= \begin{pmatrix}1-\epsilon&\epsilon\\ \epsilon&1-\epsilon\end{pmatrix}, \qquad0<\epsilon\leq\tfrac12.

Its stationary distribution is always $(1/2,1/2)$ , while its nonconstant eigenvalue is $1-2\epsilon$ . The relaxation rate depends on $\epsilon$ . Altering the algorithm changes the rate without changing the equilibrium measure.

This exact toy calculation shows why a sampling gap needs a separate argument before it becomes a statement about physical energies. The auxiliary dynamics in the lattice area-law dossier are used to derive a specified static estimate; they are not simply identified with the physical Hamiltonian.

A transfer eigenvalue ratio is $e^{-0.3}$ at time spacing $a_t=0.1$ in chosen length units. What energy does it give? If that same ratio is retained while $a_t$ is halved, has the physical energy stayed fixed?

Solution. The first energy is $0.3/0.1=3$ in inverse-length units. Keeping the ratio fixed at $a_t=0.05$ gives six. To retain energy three, the ratio must instead become $e^{-0.15}$ . “Same dimensionless eigenvalue” is not “same physics.”

Source trail and next step

Creutz, Physical Review D 15, 1128 (1977) treats gauge fixing and the lattice transfer matrix. Kogut–Susskind, Physical Review D 11, 395 (1975) develops the Hamiltonian formulation. For the explicit transfer-positivity assumptions used in spectral extraction, see Blossier et al., arXiv:0902.1265v2, §2.1.