Imaginary Time and Matsubara Frequencies

KMS turns equilibrium correlation functions into fields on a compact Euclidean-time circle of circumference $\beta$. Bosonic thermal fields are periodic and fermionic thermal fields are antiperiodic, so their Fourier modes have frequencies $\omega_n=2\pi nT$ and $\omega_n=(2n+1)\pi T$, respectively. This discrete spectrum replaces the vacuum energy integral; the bosonic zero mode is often the source of thermal infrared enhancement.

The imaginary-time construction and its connection to real-time thermal functions are derived in Landsman and van Weert 1987, §§ 2.2–2.4, pp. 157–183.

Required background. Thermal Density Operators and the KMS Condition supplies the imaginary shift. Wick Rotation and Analytic Continuation supplies the Euclidean continuation. Helpful background. Laurent Series, Poles, and Residues supports contour evaluation of thermal sums.

The thermal circle and its Fourier modes

For $0\le\tau<\beta$, expand a periodic bosonic field as

$$\phi(\tau,\mathbf x) =T\sum_{n\in\mathbb Z} \int\frac{\mathrm d^{d-1}p}{(2\pi)^{d-1}} e^{-i\omega_n\tau+i\mathbf p\cdot\mathbf x} \phi_n(\mathbf p), \qquad \omega_n=2\pi nT.$$

An antiperiodic fermion has

$$\psi(\tau+\beta,\mathbf x)=-\psi(\tau,\mathbf x), \qquad \omega_n=(2n+1)\pi T.$$

The prefactor $T=1/\beta$ is the inverse-transform normalization used here; a source may distribute factors of $\beta$ differently. Orthogonality reads

$$\int_0^\beta\mathrm d\tau\, e^{i(\omega_n-\omega_m)\tau} =\beta\delta_{nm}.$$

The global Lorentzian Fourier convention is inherited, but the Euclidean thermal transform is declared independently because $\tau$ is compact and the sign of $i\omega_n\tau$ varies across the literature.

Free thermal propagators

For a real scalar with Euclidean action

$$S_E=\frac12\int_0^\beta\mathrm d\tau \int\mathrm d^{d-1}x\, \phi(-\partial_\tau^2-\nabla^2+m^2)\phi,$$

the Matsubara propagator is

$$G_E(i\omega_n,\mathbf p) =\frac1{\omega_n^2+E_{\mathbf p}^2}, \qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

Transforming back gives, for $0\le\tau\le\beta$,

$$G_E(\tau,\mathbf p) =\frac{(1+n_B)e^{-E_{\mathbf p}\tau} +n_Be^{+E_{\mathbf p}\tau}}{2E_{\mathbf p}}, \qquad n_B=\frac1{e^{\beta E_{\mathbf p}}-1}.$$

At equal time,

$$G_E(0,\mathbf p)= \frac{1+2n_B(E_{\mathbf p})}{2E_{\mathbf p}},$$

which separates vacuum and thermal occupation. Periodicity follows because $(1+n_B)e^{-\beta E}=n_B$.

For a free Dirac field,

$$S_E=\int_0^\beta\mathrm d\tau\,\mathrm d^{d-1}x\, \bar\psi(\gamma_E^0\partial_\tau+\gamma_E^i\partial_i+m)\psi,$$

and the inverse propagator is $-i\gamma_E^0\omega_n+i\gamma_E^i p_i+m$ in the displayed transform convention. Euclidean gamma-matrix conventions must be translated before comparing numerator signs.

Thermal sums from contour residues

A bosonic Matsubara sum can be written schematically as

$$T\sum_{n\in\mathbb Z}f(i\omega_n) =\frac{1}{2\pi i}\oint_{\mathcal C_B} \mathrm dz\,n_B(z)f(z),$$

where $\mathcal C_B$ encloses the poles of $n_B(z)$ at $z=i2\pi nT$. Deforming the contour expresses the result as residues or discontinuities of $f$, plus any contribution at infinity. The contour orientation, singularities, and falloff are part of the derivation. Replacing $\int\mathrm dp^0/(2\pi)$ by $T\sum_n$ without changing the energy argument and boundary conditions is only a mnemonic.

Zero modes and chemical shifts

Only bosons have an $n=0$ thermal mode. For momenta $p\ll T$, that mode behaves as a $(d-1)$-dimensional classical field and can invalidate naive loop counting. Screening and dimensional reduction in Chapter 5 are systematic responses to this fact.

A chemical potential can be represented either by evolving with $K=H-\mu Q$ or by a shift of the Euclidean derivative. For a field of charge $q$, a common convention gives

$$\partial_\tau\longrightarrow\partial_\tau-q\mu,$$

so the displayed $e^{-i\omega_n\tau}$ modes obey $D_\tau\mapsto-i\omega_n-q\mu=-i(\omega_n-iq\mu)$. Equivalently, after removing the connection the complex frequencies are $\widetilde\omega_n=2\pi nT-iq\mu$ for a boson. The sign depends on the charge, transform, and covariant-derivative conventions and must be derived from the displayed grand-canonical action, not memorized.

Checks and pitfalls

• Verify periodicity or antiperiodicity after transforming back to $\tau$.
• Recover the zero-temperature energy integral as $\beta\to\infty$.
• Check equal-time occupation factors against canonical quantization.
• Isolate the bosonic zero mode before infrared expansion.
• State whether a chemical potential appears as a twist or a frequency shift.
• Do not continue a finite noisy Matsubara data set as though it were an exact analytic function.

The schematic below organizes the relationships used on this page. Inspect it with this question in mind: How do Gibbs states, infinite-volume KMS states, thermal boundary conditions, Matsubara modes, and graded traces fit together?

Finite Gibbs traces imply KMS analyticity; the KMS condition survives without a trace in infinite volume, while thermal-circle periodicity and grading follow only with the operator statistics and insertion specified. Solid connections show the primary relation; dashed outlines or arrows mark qualifications and failure boundaries. The diagram is schematic and not to scale.

The surrounding discussion supplies the relevant equations and checks in text form; the figure is a navigational summary.

Exercise

Evaluate $T\sum_n(\omega_n^2+E^2)^{-1}$ for bosonic $\omega_n$.

Solution

Closing the contour around the poles at $z=\pm E$ yields

$$T\sum_n\frac1{\omega_n^2+E^2} =\frac{1+2n_B(E)}{2E}.$$

The $1/(2E)$ term is the vacuum contribution and $n_B/E$ is state dependent. The result equals $G_E(0,\mathbf p)$ and therefore checks both the transform normalization and residue signs.

References

• Kapusta, Joseph I., and Charles Gale. Finite-Temperature Field Theory: Principles and Applications. 2nd ed. Cambridge: Cambridge University Press, 2006. doi:10.1017/CBO9780511535130.
• Landsman, N. P., and Ch. G. van Weert. “Real- and Imaginary-Time Field Theory at Finite Temperature and Density.” Physics Reports 145, nos. 3–4 (1987): 141–249. doi:10.1016/0370-1573(87)90121-9.
• Le Bellac, Michel. Thermal Field Theory. Cambridge: Cambridge University Press, 1996. doi:10.1017/CBO9780511721700.