Finite-Temperature and Finite-Density QFT

All the machinery so far computes vacuum physics — correlators and S-matrix elements in the ground state $|0⟩$. Many questions instead concern QFT in a thermal ensemble: the early universe, the quark–gluon plasma, neutron-star interiors, electroweak and QCD phase transitions. Finite-temperature QFT extends the path integral to a thermal state, and its central result is a beautiful one: temperature is periodic imaginary time.

Conventions: $\hslash =c=k_B=1$; inverse temperature $\beta =1/T$.

Temperature as periodic imaginary time

The thermal partition function $Z=\mathrm{Tr}{e}^{-\beta H}$ has exactly the form of a time-evolution operator $e^{-iHt}$ with imaginary time $t=-i\beta$. Following the Euclidean rotation, the thermal partition function is a Euclidean path integral over a spacetime whose time direction is a circle of circumference $\beta$:

$$\boxed{Z=\mathrm{Tr}{e}^{-\beta H}={\int }_{\text{periodic}}\mathcal{D}\varphi e^{-S_E[\varphi ]},\varphi (\tau +\beta ,\mathbf{x})=\pm \varphi (\tau ,\mathbf{x}).}$$

The boundary conditions in imaginary time encode the statistics:

• bosons are periodic, $\varphi (\tau +\beta )=+\varphi (\tau )$;
• fermions are antiperiodic, $\psi (\tau +\beta )=-\psi (\tau )$ (from the Grassmann trace).

Zero temperature ($\beta \to \infty$) decompactifies the circle and recovers ordinary vacuum QFT.

Matsubara frequencies

Compactifying Euclidean time to a circle quantizes the corresponding energy: the continuous $p^0$ integral becomes a discrete sum over Matsubara frequencies,

$$\int \frac{d{p}^0}{2\pi }⟶\frac{1}{\beta }\sum _n,{\omega }_n=\{\begin{array}{ll}2n\pi T& \text{bosons (periodic)}\\ (2n+1)\pi T& \text{fermions (antiperiodic)}\end{array}$$

All the Feynman-diagram machinery carries over with this single replacement: propagators use discrete ${\omega }_n$, and loop "integrals" become Matsubara sums. The thermal propagator resums into distribution functions — the Bose–Einstein and Fermi–Dirac occupation numbers emerge automatically from the sum,

$$n_B(\omega )=\frac{1}{e^{\beta \omega }-1},n_F(\omega )=\frac{1}{e^{\beta \omega }+1}.$$

The free energy and thermodynamics

The object of interest is the free energy $F=-T\ln Z$, from which all thermodynamics follows (pressure, entropy, energy density). For a free gas of massless bosons the one-loop free energy reproduces the Stefan–Boltzmann law,

$$\frac{F}{V}=-\frac{{\pi }^2}{90}{g}_{\ast }{T}^4,$$

with $g_{\ast }$ the effective number of degrees of freedom — the quantity that governs the expansion of the early universe. Interactions correct this with a nontrivial expansion (notoriously involving odd powers like $g^3$ from the resummation of soft modes), computed for the quark–gluon plasma of QCD.

Symmetry restoration and phase transitions

The most important qualitative effect: thermal fluctuations restore spontaneously broken symmetries. The effective potential acquires temperature-dependent corrections, and its minimum moves with $T$:

$$V_{\text{eff}}(\varphi ,T)=V(\varphi )+\frac{\lambda T^2}{24}{\varphi }^2+\cdots ,$$

so the thermal mass term is positive and grows with $T$. Above a critical temperature $T_c$ the symmetric vacuum $\varphi =0$ is restored — the field-theory analogue of a ferromagnet losing its magnetization above the Curie point. Two cosmologically crucial cases:

• Electroweak phase transition ($T_c\sim 100$ GeV): the Higgs vev turns on as the universe cools, giving particles mass. Whether it is first-order (bubble nucleation, relevant to baryogenesis) or a smooth crossover depends on the Higgs mass — in the Standard Model it is a crossover.
• QCD phase transition ($T_c\sim 150$ MeV): the transition between the confined hadronic phase and the deconfined quark–gluon plasma, probed at RHIC and the LHC and mapped by lattice QCD.

Real-time thermal field theory

The imaginary-time (Matsubara) formalism computes static thermodynamic quantities cleanly, but transport and real-time response (conductivities, damping rates, spectral functions) require analytic continuation back to real time — the closed-time-path (Schwinger–Keldysh) formalism. This doubles the field content onto a time contour and is the thermal analogue of the in-in problem; it is essential for out-of-equilibrium dynamics and the sign-problem-afflicted real-time regime.

Summary

• Finite-$T$ QFT = Euclidean path integral on a time circle of circumference $\beta =1/T$; bosons periodic, fermions antiperiodic.
• Energies quantize into Matsubara frequencies; Bose–Einstein/Fermi–Dirac distributions emerge automatically.
• The free energy $F=-T\ln Z$ gives thermodynamics (Stefan–Boltzmann $\sim g_{\ast }{T}^4$).
• Thermal corrections restore broken symmetries above $T_c$: electroweak and QCD phase transitions.

Where this leads

• Symmetry breaking whose thermal restoration this describes: Higgs, effective potential.
• Nonperturbative thermodynamics: lattice field theory.
• The cosmological setting: QFT in curved spacetime and the GR tree.

References

• Kapusta & Gale, Finite-Temperature Field Theory.
• Le Bellac, Thermal Field Theory.
• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 9.6 (finite-$T$).
• Laine & Vuorinen, Basics of Thermal Field Theory.