Relativistic KMS Analyticity and Spectrum

Ordinary KMS analyticity follows one preferred time direction. Relativistic locality can enlarge that strip to a tube in complexified spacetime, with an inverse-temperature four-vector specifying both the rest frame and the thermal scale. The resulting condition replaces—not reproduces—the vacuum spectrum condition: thermal correlations contain positive- and negative-energy components tied by detailed balance rather than support only in the forward cone.

Required background. Tube domains, complex Lorentz covariance, and analyticity supplies the several-complex-variable tools; Haag–Kastler nets and locality supplies local translations; C*-dynamical systems and the KMS condition supplies the one-dimensional boundary relation. Helpful background. Exact Euclidean–real-time analytic continuation develops spectral reconstruction, and thermal OPE and KMS crossing specializes related constraints in conformal theory.

The relativistic thermal tube

Let $e\in V_+$ be a future unit timelike vector, let $\beta>0$, and write the inverse-temperature four-vector as $\boldsymbol\beta=\beta e$. For a translation-covariant local net with automorphisms $\alpha_x$, define

$$\mathcal C_{\boldsymbol\beta} =V_+\cap(\boldsymbol\beta+V_-), \qquad \mathcal T_{\boldsymbol\beta} =\mathbb R^4-i\mathcal C_{\boldsymbol\beta}.$$

A relativistic KMS condition requires, for suitable local $A,B$, an analytic function on this tube whose boundary values exchange the operator order between $y=0$ and $y=\boldsymbol\beta$. Using

$$F_{A,B}(x-iy), \qquad y\in\mathcal C_{\boldsymbol\beta},$$

one boundary is $\omega(\alpha_x(A)B)$ and the other is $\omega(B\alpha_x(A))$, with the translation shifted by $-i\boldsymbol\beta$ according to this sign convention. Restricting $y$ to the line $se$, $0<s<\beta$, recovers the ordinary KMS strip. The full cone intersection encodes a finite maximal propagation speed.

This condition is stronger than ordinary KMS. It must either be imposed or derived from locality plus a precise stability approximation. Bros and Buchholz construct local Gibbs approximants and formulate “resistance to boundary effects”; KMS, locality, and that stability input yield extended correlation tubes through edge-of-the-wedge arguments Bros and Buchholz 1994, § 5, Propositions 5.1–5.3 and Corollary 5.4, pp. 306–315. Ordinary time-strip analyticity by itself does not license the four-dimensional tube.

What replaces positive-energy support

For a translation-invariant bosonic two-point distribution

$$W_\beta(x)=\omega_\beta(\phi(x)\phi(0)),$$

KMS gives the momentum-space detailed-balance relation

$$\widetilde W_\beta(-p) =e^{-\boldsymbol\beta\cdot p}\widetilde W_\beta(p)$$

in the convention where $p\cdot e$ is the rest-frame energy. Positivity of the state still makes $\widetilde W_\beta$ a positive-type measure in the scalar case, but it is not supported only in $\overline V_+$. The thermal commutator spectrum

$$\widetilde\rho_\beta(p) =\left(1-e^{-\boldsymbol\beta\cdot p}\right) \widetilde W_\beta(p)$$

contains the state-independent commutator for a free field and a temperature-dependent spectral density in interacting theories. Relativistic KMS controls analytic continuation and regularity; it does not restore a vacuum or a Lorentz-invariant state. The vector $\boldsymbol\beta$ transforms covariantly, but a fixed thermal state selects its rest frame.

Locality adds further continuation because spacelike-separated fields commute. The relevant theorem uses equality of boundary distributions on real open sets and convexification of tube bases. It does not imply a global entire function, and singularities on thermal mass shells or thresholds remain.

For higher-point functions, the ordering matters even more. Each permutation begins with its own primitive tube, and locality identifies boundary values only where adjacent fields are spacelike separated. Successive edge-of-the-wedge steps can join some of these domains, but the result is an explicitly constructed envelope, not an unrestricted complex configuration space. A two-point check therefore cannot establish the full $n$-point relativistic KMS condition.

Massive free-scalar check

For $m>0$, set $p=(\omega_{\mathbf p},\mathbf p)$ with $\omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$ and $n_\beta(\omega)=(e^{\beta\omega}-1)^{-1}$. In the rest frame $e=(1,\mathbf0)$,

$$W_\beta(x)= \int\frac{d^3\mathbf p}{(2\pi)^3,2\omega_{\mathbf p}} \left[ (1+n_\beta)e^{-ip\cdot x} +n_\beta e^{ip\cdot x} \right].$$

Replace $x$ by $z=x-iy$ with $y\in V_+\cap(\beta e+V_-)$. The first term is damped by $e^{-p\cdot y}$; the second combines its Bose factor with $e^{p\cdot y}$ and remains damped because $\beta e-y\in V_+$. At the upper boundary,

$$e^{-\beta\omega}(1+n_\beta)=n_\beta, \qquad e^{\beta\omega}n_\beta=1+n_\beta,$$

so

$$W_\beta(x-i\beta e)=W_\beta(-x).$$

The two terms are exactly the Bose-weighted positive- and negative-energy boundary components. Their analytic continuation and reconstruction are developed at exact Euclidean–real-time analytic continuation. General thermal propagator representations following from relativistic KMS and locality are stated in Bros and Buchholz 1996, §§ 2–4, pp. 499–516.

Adversarial test: a time-periodic function without a tube

Suppose one has a correlation function analytic only for $0<\operatorname{Im}t<\beta$ and periodic at the strip edges. Extend it trivially in spatial momentum with a nonlocal dispersion that permits arbitrarily fast propagation. The time KMS identity can survive, but there is no locality domain on which opposite orderings coincide and no cone base to convexify. The strongest claim is ordinary KMS in one parameter; the missing hypotheses are relativistic locality and the stability control used to reach the spacetime tube.

Common pitfalls

Calling the thermal state Lorentz invariant. The formulation is Lorentz covariant when $\boldsymbol\beta$ transforms. A fixed nonzero inverse-temperature vector breaks boosts to its stabilizer.

Reusing the vacuum spectral condition. Thermal negative-energy weight is required by detailed balance. What survives is a quantitative relation between the two signs and an analytic tube, not forward-cone support.

Exercises

1. Derive detailed balance from the free-scalar coefficients.

Solution

At $p^0=+\omega_{\mathbf p}$ the coefficient is $1+n_\beta$; at $p^0=-\omega_{\mathbf p}$ it is $n_\beta$. Since $n_\beta/(1+n_\beta)=e^{-\beta\omega_{\mathbf p}}$, one obtains $\widetilde W_\beta(-p)=e^{-\beta p^0}\widetilde W_\beta(p)$ on the positive shell, with the corresponding reversed relation on the negative shell.

2. Show that $\mathcal C_{\boldsymbol\beta}$ reduces to the interval $(0,\beta)e$ when restricted to the line spanned by $e$.

Solution

Write $y=se$. The conditions $y\in V_+$ and $\boldsymbol\beta-y=(\beta-s)e\in V_+$ require $s>0$ and $\beta-s>0$. Thus $0<s<\beta$, exactly the ordinary KMS strip.

References

• Bros, Jacques, and Detlev Buchholz. “Towards a Relativistic KMS Condition.” Nuclear Physics B 429 (1994): 291–318. DOI. Open PDF.
• Bros, Jacques, and Detlev Buchholz. “Axiomatic Analyticity Properties and Representations of Particles in Thermal Quantum Field Theory.” Annales de l’Institut Henri Poincaré, Physique théorique 64 (1996): 495–521. Open PDF.