====== Transfer matrix ======

The transfer matrix is the object that connects the Hamiltonian formulation of a theory to a path integral. Consider as an example a two-level quantum mechanical system with Hamiltonian:
\begin{equation}
H = -\mu \sigma_x
\text.
\end{equation}

We can derive a (Euclidean) path integral for this Hamiltonian by starting from the thermal partition function and inserting many copies of the identity operator $\left|0\right>\left<0\right| + \left|1\right>\left<1\right|$.
\begin{equation}
Z(\beta) = \operatorname{Tr} e^{-\beta H}
= \operatorname{Tr} \left(e^{-\beta H / N}\right)^N
= \sum_{s_1,\ldots,s_N \in \{0,1\}} \langle s_1 | T | s_N \rangle \cdots \langle s_2 | T |s_1 \rangle
\end{equation}
Here we have introduced $T \equiv e^{-\beta H / N}$, the transfer matrix. The final expression has the form of a path integral, as $\langle s' | T | s \rangle$ can be written $e^{-J s' s}$ for some $J$ (a function of $\mu$). As a result, we have derived a Euclidean action of the form
\begin{equation}
S = -J \sum_{\langle i j \rangle} s_i s_j\text,
\end{equation}
and we see that the two-level system lets us solve the one-dimensional Ising model!

This story can be told in reverse, where we start from an apparently difficult high-dimensional integral (like the Ising model in one dimension), and reduce it to a low-dimension matrix by expressing it in terms of a transfer matrix.

The transfer matrix need not be taken between space-like slices; see for instance [[radial quantization]].

===== External links =====

  * [[https://en.wikipedia.org/wiki/Transfer-matrix_method|Wikipedia]]