Imaginary Time and Matsubara Frequencies

When equilibrium correlators admit the required Euclidean continuation and field representation, the KMS boundary relation becomes a compact Euclidean-time circle of circumference $\beta$ . In the ordinary thermal trace, bosonic fields are periodic and fermionic 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 ordinary 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

Algebraic KMS by itself does not manufacture a Euclidean path integral or a field-mode expansion. The formulas below assume an eligible Euclidean correlator or field representation, a declared operator parity and trace insertion, and the displayed compact-time Fourier convention.

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,

the inverse operator is $-i\gamma_E^0\omega_n+i\gamma_E^i p_i+m$ in the displayed transform convention. With $\{\gamma_E^\mu,\gamma_E^\nu\}=2\delta^{\mu\nu}$ , its Gaussian propagator is

S_E(i\omega_n,\mathbf p) =\frac{i\gamma_E^0\omega_n-i\gamma_E^i p_i+m} {\omega_n^2+\mathbf p^2+m^2}, \qquad \omega_n=(2n+1)\pi T.

Euclidean gamma-matrix and Green-function sign conventions must be translated together before comparing numerators. The occupation factor can be checked without spinor algebra. For one fermionic mode $H=E f^\dagger f$ , define

G_F(\tau)=-\langle \mathrm T_\tau f(\tau)f^\dagger(0)\rangle, \qquad n_F(E)=\frac1{e^{\beta E}+1}.

Then

G_F(\tau)= \begin{cases} -(1-n_F)e^{-E\tau}, & 0<\tau<\beta,\\ +n_Fe^{-E\tau}, & -\beta<\tau<0. \end{cases}

Because $n_Fe^{\beta E}=1-n_F$ , this obeys $G_F(\tau-\beta)=-G_F(\tau)$ . The limits $G_F(0^+)=-(1-n_F)$ and $G_F(0^-)=n_F$ both recover the Fermi occupation and give the canonical discontinuity $G_F(0^+)-G_F(0^-)=-1$ .

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

In the ordinary thermal spin structures used above, only bosons have an $n=0$ mode. For momenta $p\ll T$ , that bosonic mode behaves as a $(d-1)$ -dimensional classical field and can invalidate naive loop counting. Bosonic Zero Modes and Infrared Breakdown diagnoses the failure, while Modes, Matching, and Power Counting constructs the dimensionally reduced description. Periodic fermions in a supertrace instead have integer modes and can possess fermion zero modes; Thermal Boundary Conditions and Graded Traces explains why that sector is not an ordinary thermal ensemble. Thermal Propagators and Spectral Representations continues from these discrete Euclidean modes to the shared thermal spectral function.

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 lower panel makes the causal order explicit: a Euclidean representation, parity, and trace insertion fix the circle monodromy, and the monodromy fixes the Matsubara lattice and zero-mode sector.

Finite Gibbs cyclicity realizes the positive-strip KMS condition; local-limit phase analysis and complete passivity are separate theorem-qualified branches, while Euclidean representation, parity, conserved insertion, and Fourier convention determine ordinary, twisted, or graded Matsubara sectors. — The positive-strip KMS boundary relation is common equilibrium input, but the mode expansion requires the additional Euclidean gate shown in the lower panel. For the ordinary heat trace, bosonic periodicity gives integer modes and permits a zero mode, while fermionic antiperiodicity gives half-integer modes. Conserved or graded insertions alter the monodromy and hence the lattice; algebraic KMS alone does not select a field path integral. The diagram is schematic and not to scale.

The surrounding formulas develop the transform normalization, propagators, and residue checks that follow downstream from the mode sectors summarized in the figure.

Exercise

For fermionic $\omega_n=(2n+1)\pi T$ , evaluate

T\sum_{n\in\mathbb Z}\frac1{\omega_n^2+E^2}

by contour residues. State the summation kernel, contour orientation, physical poles, and zero-temperature limit.

Solution

Use

n_F(z)=\frac1{e^{\beta z}+1}, \qquad f(z)=\frac1{E^2-z^2}.

The poles of $n_F$ at $z=i(2n+1)\pi T$ have residue $-T$ , so a counterclockwise contour $\mathcal C_F$ around them gives

T\sum_n f(i\omega_n) =-\frac1{2\pi i}\oint_{\mathcal C_F}\mathrm dz\,n_F(z)f(z).

Assuming the large arcs vanish, deforming the contour outward reverses the small contours around the physical poles of $f$ at $z=\pm E$ . Their residues are $-1/(2E)$ and $+1/(2E)$ , respectively. Since $n_F(-E)=1-n_F(E)$ ,

T\sum_n\frac1{\omega_n^2+E^2} =\frac{n_F(-E)-n_F(E)}{2E} =\frac{1-2n_F(E)}{2E} =\frac1{2E}\tanh\!\left(\frac{\beta E}{2}\right).

As $\beta\to\infty$ , $n_F(E)\to0$ and the result approaches $1/(2E)$ , the zero-temperature energy integral. The minus residue of the fermionic kernel is the sign that distinguishes this answer from the bosonic $[1+2n_B(E)]/(2E)$ sum.

References

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.