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I heard that $d$ dimensional relativistic quantum field theory can be viewed as a $d+1$-dimensional statistical mechanics. Can a relativistic/non-relativistic classical field theory be also looked upon as a statistical mechanical system? If yes, how to write down a partition function for such a system?

EDIT: If there is no such general rule, can in certain special cases, a classical field theory be looked upon as a statistical mechanical system and write down a partition function for it?

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  • $\begingroup$ @claudechuber The question is indeed asked with complete generality and I'm seeking for an answer in general case. But the question is not broad. Right? The answer, in general case, could be either Yes or No, for the first question. If yes, then I ask for how to define the partition function, in general. However, keeping your point in mind, I might edit the question a bit. $\endgroup$ – SRS Dec 26 '16 at 14:29
  • $\begingroup$ Relativity has absolutely nothing at all to do with this. A $d$-dimensional quantum field can be viewed as a $d+1$ classical field theory with temperature regardless of whether or not relativity is involved. $\endgroup$ – DanielSank Dec 26 '16 at 14:53
  • $\begingroup$ @DanielSank Ok. Do you mean a $d$-dimensional QFT can be viewed as a $d+1$-dimensional classical field theory? Or you mean statistical mechanics? $\endgroup$ – SRS Dec 26 '16 at 15:39
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In quantum mechanics, the partition function is given by the path integral

$$ Z = \int Dq \exp \left( i \hbar^{-1} S[q] \right). $$

Its classical limit is

$$ Z_{\hbar \rightarrow 0} \sim \exp \left( i \hbar^{-1} S[q_c] \right), $$

where the classical trajectory $q_c(t)$ is the solution of the equations of motion.

However, it might not play a major role in either the formulation of the theory nor in practical calculations.

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The answer to your question depends on which type of partition function we consider. There are essentially two types: the (complex) time and finite-temperature partition functions, defined in the quantum case as, respectively, $$ Z_\textrm{time}(z) \equiv \textrm{Tr} \; e^{-i z \hat H / \hbar} \qquad \textrm{and} \qquad Z_\textrm{temp}(\beta) \equiv \textrm{Tr} \;e^{- \beta \hat H}.$$ The former, $Z_\textrm{time}(z)$, naturally appears when studying the (time-dependent) properties of quantum theories at zero temperature. Note that $z$ has units of time, which necessitates the $\hbar$ to make the units work out. Commonly, one either considers real time $z =t$ or imaginary time $z = - i \tau$, which are related by a Wick rotation. The latter, $Z_\textrm{temp}(\beta)$, is the object of interest when studying finite-temperature statistical-mechanical properties of a quantum systems. Note that $\beta$ has units of inverse energy (setting $k_B = 1$), hence there is no need for $\hbar$.

It is often sloppily said that the imaginary time partition function coincides with the finite-temperature one. This is true if we set $\hbar = 1$, but we see that more generally $Z_\textrm{time}(-i \tau) = Z_\textrm{temp}(\tau / \hbar)$. In particular, this means their classical limits $\hbar \to 0$ are very different.

To obtain their classical limits, it is useful to express both in a path integral formulation. This can be derived as usual, i.e., by inserting a bunch of resolutions of identity. The outcomes are: $$ Z_\textrm{time}(z) = \int e^{\frac{i}{\hbar} \int_0^z \mathrm d z' \int \mathrm d^d x \left( \frac{1}{2}(\partial_{z'} \varphi)^2 - V(\varphi) \right)} \mathrm D[\varphi(x,z')] ,$$ $$ Z_\textrm{temp}(\beta) = \int e^{- \int_0^\beta \mathrm d \beta' \int \mathrm d^d x \left( \frac{1}{2\hbar^2}(\partial_{\beta'} \varphi)^2 + V(\varphi) \right)} \mathrm D[\varphi(x,\beta')]. $$ (If you are curious how $\hbar$ entered into the second expression (despite the aforementioned definition not containing $\hbar$), it arose by virtue of the resolutions of identity and $\langle \Pi|\varphi\rangle \propto e^{\frac{i}{\hbar} \int \Pi(x) \varphi(x) \mathrm d^d x}$.)

It is now straight-forward to obtain the classical limits. For $Z_\textrm{time}(z)$, the stationary phase/method of steepest descent (depending on whether $z$ is real or imaginary) tells us that the partition function is dominated by the trajectories determined by the Euler-Lagrange equations. For $z=t$, these are exactly the classical equations of motion, hence $\lim_{\hbar \to 0 } Z_\textrm{time}(t)$ is the expression given in Prof. Legolasov's answer. Note that this expression still contains $\hbar$, and hence it is an intrinsically non-classical object, despite being evaluated for classical trajectories. As such, it is perhaps not surprising that this ``classical'' partition function does not really appear on its own right in classical physics.

For $Z_\textrm{temp}(\beta)$, however, we see that $\hbar$ only appears in the kinetic term. Hence, for $\hbar \to 0$, the stationary phase approximation tells us that we basically need to kill this term, i.e., we should only consider static solutions. In conclusion, $$ \boxed{\lim_{\hbar \to 0} Z_\textrm{temp}(\beta) \propto \int e^{- \beta \int \mathrm d^d x \; V(\varphi) } \mathrm D[\varphi(x)] } \;. $$ We thus recover the classical statistical-mechanical partition function! Indeed, $$ Z_\textrm{class}(\beta) \equiv \int e^{-\beta H(\Pi,\varphi)} \mathrm D[\varphi(x)] \mathrm D[\Pi(x)] \propto \int e^{-\beta \int \mathrm d^d x V(\varphi)} \mathrm D[\varphi(x)],$$ since one can always perform the trivial gaussian integral over the momentum field (since the kinetic term $\propto \Pi^2$), which just gives an uninteresting overall prefactor.

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