Finite-Temperature QFT and Euclidean Time

Thermal equilibrium is encoded by the Gibbs state $\rho_\beta = e^{-\beta H}/Z$. Wick‑rotate $t\to -i\tau$: $e^{-iHt/\hbar}\mapsto e^{-H\tau/\hbar}$. With a trace, $Z(\beta)=\mathrm{Tr} \, U(-i\hbar\beta)$, the Euclidean time direction gets glued into a circle of circumference $\hbar\beta=\hbar/(k_B T)$. The KMS condition characterizes equilibrium and implies (anti)periodicity of Euclidean correlators; this is exactly the boundary condition for a classical Euclidean statistical field theory on $S^1_{\hbar\beta}\times \mathbb{R}^d$.

1. Thermal states and the partition function

For a QFT with Hamiltonian $H$, thermal equilibrium at temperature $T$ (inverse temperature $\beta=1/(k_B T)$) is described by the Gibbs state

$$\rho_\beta=\frac{e^{-\beta H}}{Z(\beta)},\qquad Z(\beta)=\mathrm{Tr}\,e^{-\beta H}.$$

Thermal expectation of an observable $A$ is $\langle A\rangle_\beta=\mathrm{Tr}(\rho_\beta A)$.

Two remarks:

• Trace = sum over microstates. In quantum theory, “sum over all microstates” is the trace over Hilbert space. It ensures basis‑independence and correct normalization.
• From real to imaginary time. Real‑time evolution $U(t)=e^{-iHt/\hbar}$ becomes Euclidean evolution $U(-i\tau)=e^{-H\tau/\hbar}$ after a Wick rotation $t=-i\tau$.

2. The KMS condition: the fingerprint of equilibrium

Let $A,B$ be Heisenberg operators, $A(t)=e^{\frac{i}{\hbar}Ht} A e^{-\frac{i}{\hbar}Ht}$. The Kubo–Martin–Schwinger (KMS) relation states (bosonic version)

$$\boxed{\;\langle A(t) B(0)\rangle_\beta = \langle B(0) A(t+i\hbar\beta)\rangle_\beta.\;}$$

For fermionic operators, there is a minus sign on the right (graded KMS).

Derivation (one line).

$$\begin{aligned} \langle A(t)B\rangle_\beta &=\frac{1}{Z}\mathrm{Tr}\!\big(e^{-\beta H} e^{\frac{i}{\hbar}Ht} A e^{-\frac{i}{\hbar}Ht} B\big)\\ &=\frac{1}{Z}\mathrm{Tr}\!\big(A(t+i\hbar\beta)\,e^{-\beta H} B\big) =\langle B\,A(t+i\hbar\beta)\rangle_\beta, \end{aligned}$$

where we used $e^{-\beta H} A(t)=A(t+i\hbar\beta)\,e^{-\beta H}$ and cyclicity of the trace.

Analytic strip. More conceptually, there exists a function $F_{A,B}(z)$ holomorphic on $0<\mathrm{Im}\,z<\hbar\beta$ whose boundary values give the two correlators above. “Analytic in a strip + $i\hbar\beta$ shift” is the mathematical fingerprint of thermal equilibrium.

3. From KMS to Euclidean (anti)periodicity

Define Euclidean operators by analytic continuation $A_E(\tau):=A(t=-i\tau)$. The Euclidean two‑point function is the $\tau$-ordered thermal correlator

$$G^{(E)}_{AB}(\tau) := \langle T_\tau \, A_E(\tau) B_E(0)\rangle_\beta.$$

Insert $t=-i\tau$ into KMS and use the definition of $T_\tau$. For $0<\tau<\hbar\beta$,

$$\boxed{\;G^{(E)}_{AB}(\tau+\hbar\beta)=\pm\,G^{(E)}_{AB}(\tau)\;}$$

with $+$ for bosons and $−$ for fermions. Thus all Euclidean $n$-point functions are (anti)periodic with period $\hbar\beta$.

Interpretation: The correlators depend only on $\tau\bmod \hbar\beta$. The most economical realization is to place the theory on a time circle $S^1_{\hbar\beta}$. That circle’s circumference is $\hbar\beta=\hbar/(k_B T)$.

4. Why Euclidean time has period $\hbar\beta$: trace = closed time

Start with the partition function

$$Z(\beta)=\mathrm{Tr}\,e^{-\beta H}.$$

Use $e^{-\beta H}=U(-i\hbar\beta)$ to write

$$Z(\beta)=\mathrm{Tr}\,U(-i\hbar\beta).$$

In the path‑integral representation the trace glues the initial Euclidean time slice to the final one; the Euclidean time coordinate $\tau$ is therefore compact with circumference $\hbar\beta$.

• Bosons: periodic around the circle.
• Fermions: anti‑periodic (due to the graded nature of the trace/KMS).

Expanding along $S^1$ yields Matsubara frequencies

$$\omega_n^{\rm bos}=\frac{2\pi n}{\hbar\beta},\qquad \omega_n^{\rm ferm}=\frac{(2n+1)\pi}{\hbar\beta}.$$

5. From finite‑$T$ QFT to a classical statistical field theory

The Euclidean functional integral of the finite‑$T$ QFT is

$$\boxed{\quad Z(\beta) = \int_{\text{(anti)periodic}}\!\!\mathcal{D}\Phi\; \exp\!\left[-\frac{1}{\hbar}\,S_E[\Phi]\right], \qquad \tau\in[0,\hbar\beta),\;}$$

which is precisely the partition function of a classical Euclidean statistical field theory on $S^1_{\hbar\beta}\times\mathbb{R}^d$ with weight $e^{-S_E/\hbar}$.

Transfer‑matrix viewpoint

Discretize Euclidean time with spacing $a_\tau$ and define the transfer matrix

$$T := e^{-a_\tau H/\hbar}.$$

Then

$$Z(\beta)=\mathrm{Tr}\,T^{N_\tau},\qquad N_\tau=\frac{\hbar\beta}{a_\tau},$$

which is literally a classical partition sum over configurations on a $(d+1)$-dimensional lattice with local Boltzmann weights. This is the foundation of finite‑temperature lattice QFT.

Dimensional reduction for static physics

Because non‑zero Matsubara modes have gaps $\sim 2\pi n/\hbar\beta$, at high $T$ one can integrate them out. For static observables the effective theory becomes a $d$-dimensional classical field theory for the Matsubara zero modes.

6. Vacuum wave functional from Euclidean projection

The ground‑state wave functional $\Psi_0[\phi]\equiv \langle \phi|0\rangle$ admits a Euclidean representation:

$$\Psi_0[\phi(\mathbf x)] \;\propto\; \int_{\substack{\tau<0\\ \phi(\tau=0,\mathbf x)=\phi(\mathbf x)}}\!\!\mathcal D\phi\;e^{-S_E[\phi]/\hbar}.$$

Reason: the Euclidean kernel $K_E=\langle \phi_2|e^{-HT}|\phi_1\rangle$ admits a spectral sum $\sum_n e^{-E_nT}\Psi_n[\phi_2]\Psi_n^*[\phi_1]$. As $T\to\infty$, the factor $e^{-E_nT}$ projects onto the ground state, leaving the path integral on a half‑space with boundary value $\phi(\tau=0)=\phi$.

7. Subtleties and extensions

• Fermions: anti‑periodic b.c. on $S^1_{\hbar\beta}$ (graded KMS / Grassmann trace).
• Chemical potential: in the grand canonical ensemble, $H\to H-\mu Q$. KMS twists the boundary condition by a phase: $\Phi(\tau+\hbar\beta)=\mp\,e^{\beta\mu q}\Phi(\tau)$ (minus for fermions). Geometrically, a flat $U(1)$ Wilson line along the time circle.
• Gauge theories: include gauge fixing and ghosts; reflection positivity holds at the level of gauge‑invariant observables.
• Sign problems: if $S_E$ is not real and bounded below (e.g., finite density or $\theta$-terms), the probabilistic interpretation as a classical ensemble may fail (Monte Carlo obstruction).
• Geometry link (black holes): near a static horizon, the Euclidean metric looks like a cone; removing the conical singularity fixes the time circle’s period $2\pi/\kappa$, giving Hawking’s $T=\hbar\kappa/(2\pi k_B)$.

8. Summary

• KMS condition $\Rightarrow$ equilibrium correlators are analytic in a strip and related by an $i\hbar\beta$ shift.
• Wick rotation + trace $\Rightarrow$ Euclidean time is a circle of circumference $\hbar\beta$; bosons periodic, fermions anti‑periodic; Matsubara modes follow.
• Functional integral $\Rightarrow$ a finite‑$T$ QFT is equivalent to a classical Euclidean statistical field theory on $S^1_{\hbar\beta}\times\mathbb{R}^d$.
• Transfer matrix / OS reconstruction $\Rightarrow$ this is not a coincidence but a structural, (semi‑)rigorous equivalence under standard assumptions.

Notation & units

We keep $\hbar$ and $k_B$ explicit. In “natural units”, set $\hbar=k_B=1$, so the thermal circle has circumference $\beta=1/T$.