Thermal field theory and imaginary time

For Hamiltonian $H$, define $Z=\operatorname{Tr}e^{-\beta H}$ and $\langle O\rangle_\beta=Z^{-1}\operatorname{Tr}(e^{-\beta H}O)$ with $\beta=1/T$. Real-time evolution is $A(t)=e^{iHt}Ae^{-iHt}$. Analytically continuing to $t-i\beta$ inserts $e^{\beta H}$ on the left of $A(t)$ and $e^{-\beta H}$ on the right. Cyclicity of the trace then moves the remaining thermal factor past the second operator.

This produces the Kubo–Martin–Schwinger relation. The raw operator trace identity does not acquire a universal fermionic minus sign; anticommutation signs enter when one defines time-ordered fermionic correlators and moves fields through the ordering operation. The analytic continuation assumes the correlators have the needed strip of analyticity. In field theory it is understood through appropriately defined correlation functions rather than arbitrary unbounded operators manipulated without domains.

The Euclidean path integral for the trace joins initial and final configurations across an interval of length $\beta$. Bosonic fields are periodic on that circle. Fermionic fields are antiperiodic in the thermal path integral, reflecting the Grassmann trace construction. Fourier modes must respect those conditions. A bosonic factor $e^{i\omega_n\tau}$ returns to itself at $\tau+\beta$, whereas a fermionic factor returns with a minus sign.

Solving those elementary phase conditions gives integer and half-integer Matsubara frequencies. A zero bosonic frequency is allowed; there is no fermionic zero frequency in this thermal setup. This distinction matters at high temperature, where soft bosonic modes can dominate long-distance behavior and demand resummation or a lower-dimensional effective description. The mere discreteness of these Euclidean frequencies does not imply that real-time emitted particles have only those discrete energies.

A single bosonic oscillator of energy spacing $\omega>0$ has occupation weights proportional to $e^{-\beta\omega n}$ for nonnegative integer $n$. Summing a geometric series and differentiating its logarithm gives the Bose occupation $1/(e^{\beta\omega}-1)$. The zero-point factor cancels from the normalized occupation. A fermionic mode allows only zero or one occupation, producing $1/(e^{\beta\omega}+1)$ instead. These formulas describe noninteracting quasiparticle modes when that description applies.

Interactions can change masses, widths, screening, and the validity of a quasiparticle picture. Even a small microscopic coupling does not ensure every thermal momentum scale is perturbative: infrared enhancements require a scale-by-scale analysis. Euclidean correlators also do not directly hand over a viscosity or relaxation rate. Those require real-time spectral information, and reconstructing it from finite noisy Euclidean data is an ill-conditioned inverse problem. Equilibrium thermodynamics, screening, and transport should be treated as related but different targets.