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 | e^{-\beta H / N}| s_N \rangle \cdots \langle s_2 | e^{-\beta H / N} |s_1 \rangle \end{equation}

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.

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