Modular KMS Relations and Modular Correlators

Every faithful normal state is an equilibrium state for its own modular dynamics: two ordered correlators are the boundary values of one analytic function across a unit strip. The statement is intrinsic to the algebra–state pair. Its sign, strip, and operator order become unambiguous only after the modular-flow convention has been fixed.

Required background. Thermal density operators and the KMS condition supply the Gibbs calculation and physical-time KMS condition used for comparison; Tomita–Takesaki flow supplies the standard pair and modular automorphism group used in the theorem.

Helpful background. Relative modular operators and cocycles explain how the same analytic methods compare two states rather than one state with itself.

The chapter’s route from standard pairs to geometric flow locates the intrinsic KMS theorem. The representative-flow table keeps algebraic, wedge, ball, and regulated formulas separate, while the claim-licensing checklist tests any proposed physical interpretation.

One theorem, two strip conventions

Use the site convention

$$\sigma_s(A)=\Delta^{is}A\Delta^{-is}, \qquad K=-\log\Delta,$$

for the modular group of a standard pair $(\mathcal A,\Omega)$, with $\omega(A)=\langle\Omega,A\Omega\rangle$. For every bounded $A,B\in\mathcal A$, there is a bounded function $F_{A,B}$, continuous on the closed strip and analytic inside it, whose boundary values are

$$\begin{aligned} F_{A,B}(s)&=\omega\!\left(A\sigma_s(B)\right),\\ F_{A,B}(s-i)&=\omega\!\left(\sigma_s(B)A\right), \qquad s\in\mathbb R, \end{aligned}$$

with $-1<\operatorname{Im}z<0$. Entire analytic elements are needed only if one wants to write the interior value literally as $\omega(A\sigma_z(B))$; the theorem for arbitrary bounded elements asserts the existence of the strip function. This is the modular KMS boundary theorem in Takesaki 1970, “The Modular Automorphism Group and the Kubo–Martin–Schwinger Boundary Condition,” pp. 63–72.

The same statement appears in three common forms:

evolution	correlator on the real edge	analytic strip	opposite edge
site flow $\sigma_s$	$F(s)=\omega(A\sigma_s(B))$	$-1<\operatorname{Im}z<0$	$F(s-i)=\omega(\sigma_s(B)A)$
reversed flow $\alpha_s=\sigma_{-s}$	$G(s)=\omega(A\alpha_s(B))$	$0<\operatorname{Im}z<1$	$G(s+i)=\omega(\alpha_s(B)A)$
Gibbs Heisenberg flow $\tau_t$	$H(t)=\omega(A\tau_t(B))$	$0<\operatorname{Im}z<\beta$	$H(t+i\beta)=\omega(\tau_t(B)A)$

For $\rho=Z^{-1}e^{-\beta H}$ and $\tau_t(A)=e^{itH}Ae^{-itH}$,

$$\sigma_s(A)=\rho^{is}A\rho^{-is}=\tau_{-\beta s}(A).$$

Thus the lower modular strip maps to the upper physical-time strip because physical time is reversed. Modular $s$ is dimensionless; a temperature in physical units exists only after a particular physical evolution has been identified. The original infinite-system equilibrium formulation of the KMS boundary condition is developed in Haag, Hugenholtz, and Winnink 1967, §§1–3, pp. 215–229.

A faithful two-level calculation

Let $\rho=\sum_n p_n\lvert n\rangle\langle n\rvert$ with every $p_n>0$. Direct matrix multiplication gives

$$F_{A,B}(z)= \sum_{m,n}p_m^{\,1-iz}p_n^{\,iz}A_{mn}B_{nm}.$$

At $z=s-i$, the coefficient becomes $p_n(p_n/p_m)^{is}$; relabeling $m$ and $n$ yields the required reversed order.

For a numerical convention check, choose

$$\rho=\begin{pmatrix}3/4&0\\0&1/4\end{pmatrix}, \qquad A=\lvert0\rangle\langle1\rvert, \qquad B=\lvert1\rangle\langle0\rvert.$$

Then

$$F(z)=\frac34\,3^{-iz}, \qquad F(s-i)=\frac14\,3^{-is} =\operatorname{Tr}\!\left(\rho\,\sigma_s(B)A\right).$$

The equality is exact, so a reproducible implementation can use

$$R_{\mathrm{KMS}}(s)= \frac{\left\lvert F(s-i)- \operatorname{Tr}(\rho\,\sigma_s(B)A)\right\rvert} {\max(1,\lvert F(s-i)\rvert)}$$

with analytic target $R_{\mathrm{KMS}}=0$. Report the eigenvalue floor $\min p_n$, the sampled $s$ interval, arithmetic precision, and the maximum residual. Faithfulness is not cosmetic: if a $p_n$ vanishes, the complex powers and the modular group must first be defined on the support corner.

Free-field wedge boundary values

The wedge gives a QFT realization in which modular time has a geometric meaning. For the massless scalar field in four dimensions, use point fields only as shorthand for a limit of spacetime-smeared fields supported in the right wedge. Along a uniformly accelerated orbit $x(\tau)$ of acceleration $a$, write $\phi(\tau)=\phi(x(\tau))$ and set

$$\beta_{\mathrm U}=\frac{2\pi}{a}, \qquad \mathcal G(z)=-\frac{a^2}{16\pi^2\sinh^2(az/2)}.$$

The function is analytic for $-\beta_{\mathrm U}<\operatorname{Im}z<0$. With the site convention, the Bisognano–Wichmann trajectory moves the second field by proper time $-\beta_{\mathrm U}s$; boost invariance then gives

$$\begin{aligned} F(s) &=\langle\Omega,\phi(0)\sigma_s(\phi(0))\Omega\rangle\\ &=\langle\Omega,\phi(\beta_{\mathrm U}s)\phi(0)\Omega\rangle =\lim_{\epsilon\downarrow0}\mathcal G(\beta_{\mathrm U}s-i\epsilon). \end{aligned}$$

Approaching the other edge from inside the strip,

$$\lim_{\epsilon\downarrow0} \mathcal G\!\left( \beta_{\mathrm U}s-i\beta_{\mathrm U}+i\epsilon \right) =\langle\Omega,\sigma_s(\phi(0))\phi(0)\Omega\rangle.$$

The equality follows from $\sinh(w-i\pi)=-\sinh w$; the change from $-i0$ to $+i0$ carries the operator-order reversal. It verifies the unit modular strip and the factor $2\pi$ simultaneously. The same periodicity underlies the accelerated-detector relation in Unruh 1976, §III, especially Eq. (3.10), pp. 883–885.

For a reproducible boundary test, choose $a>0$, a dimensionless regulator $a\epsilon$, and a coincidence exclusion $a\lvert u\rvert\ge a u_{\min}>0$. Normalize the dimension-two correlator as $\widehat{\mathcal G}=\mathcal G/a^2$ and compare

$$R_{\epsilon}(u)= \frac{\left\lvert \widehat{\mathcal G}(u-i\beta_{\mathrm U}+i\epsilon)- \widehat{\mathcal G}(u+i\epsilon) \right\rvert} {\max(1,\lvert\widehat{\mathcal G}(u+i\epsilon)\rvert)}.$$

Analytically $R_\epsilon(u)=0$; numerical uncertainty is therefore only roundoff and cancellation near a pole. The smearing widths, support margin from the wedge boundary, $a\epsilon$, and $a u_{\min}$ are part of the result, not optional implementation details.

Spectral detailed balance

Let $E_K(dk)$ be the projection-valued measure of $K$. Since $K\Omega=0$,

$$F_{A,B}(s) =\int_{\mathbb R}e^{-isk}\,d\nu_{A,B}(k), \qquad \nu_{A,B}(E)= \langle A^*\Omega,E_K(E)B\Omega\rangle.$$

This is generally a complex cross measure, not a probability measure. For analytic elements, the lower boundary inserts $e^{-k}$, while modular invariance gives

$$F_{A,B}(s-i)=F_{B,A}(-s).$$

Fourier uniqueness therefore yields the weak measure identity

$$\boxed{ e^{-k}\,d\nu_{A,B}(k)=d\nu_{B,A}(-k) }.$$

The minus sign in the reflected argument and the factor $e^{-k}$ both follow from the chosen transform $e^{-isk}$ and the lower strip. Changing either convention changes the displayed formula. For nonsmooth boundary data, interpret it after smearing in $k$ rather than as a pointwise identity between spectral densities.

Analytic elements and field domains

The modular theorem concerns bounded algebra elements. A useful entire-analytic approximation is

$$A_\epsilon= \frac{1}{\sqrt{\pi\epsilon}} \int_{-\infty}^{\infty}dt\; e^{-t^2/\epsilon}\sigma_t(A), \qquad \epsilon>0.$$

The integral is strong-$*$ convergent, $\lVert A_\epsilon\rVert\le\lVert A\rVert$, $A_\epsilon$ is entire analytic for the modular group, and $A_\epsilon\to A$ strongly-$*$ as $\epsilon\downarrow0$. This mollification controls modular time; it does not replace spacetime smearing of a local field. Field correlators additionally require test functions, a common invariant domain, and distributional boundary values.

Adversarial order and singularity tests

The two-level example detects a missing KMS shift immediately. At $s=0$,

$$F(0)=\omega(AB)=\frac34, \qquad \omega(BA)=\frac14.$$

If one reverses the order without moving to $z=-i$, the absolute residual is $1/2$, even though the correctly shifted residual is zero. What survives is ordinary stationarity; the cyclic boundary identity has been discarded.

The wedge exposes a different failure. The analytic function has poles at $z=2\pi i n/a$ and behaves near coincidence as $-1/(4\pi^2z^2)$. A path that crosses $z=0$ is not an analytic continuation inside the KMS strip. Approaching the pole from the wrong side replaces $(u-i0)^{-2}$ by $(u+i0)^{-2}$ and hence changes the commutator/contact distribution. A pointwise check at $u\ne0$ can miss this failure, which is why the test must retain spacetime smearing and vary $u_{\min}$ and $\epsilon$ together.

Common pitfalls

Giving a strip width without a flow sign. A unit width does not determine whether the strip is above or below the real axis. Display the adjoint action, the correlator order, and the shifted boundary together.

Calling modular temperature a thermometer reading. Modular time is dimensionless. The Unruh and Gibbs temperatures arise only after a boost or Hamiltonian evolution has been identified with the modular flow.

Treating point fields as bounded analytic elements. A Wightman function is a distributional boundary value. State the spacetime smearing, operator domain, and pole prescription before using the bounded-element theorem.

Exercises

1. A two-level strip and a failed shortcut

For the two-level data above, derive $F(z)$ from matrix multiplication. Verify the lower boundary for every real $s$, and evaluate the residual produced by replacing $F(s-i)$ with $F(s)$ at $s=0$.

Solution

Only $A_{01}B_{10}$ is nonzero. The general finite-dimensional formula therefore gives

$$F(z)=p_0^{1-iz}p_1^{iz} =\frac34\left(\frac13\right)^{iz} =\frac34\,3^{-iz}.$$

At $z=s-i$,

$$F(s-i)=\frac34\,3^{-is-1} =\frac14\,3^{-is}.$$

Moreover $\sigma_s(B)=3^{-is}B$ and $BA=\lvert1\rangle\langle1\rvert$, so

$$\operatorname{Tr}(\rho\,\sigma_s(B)A) =3^{-is}\operatorname{Tr}(\rho BA) =\frac14\,3^{-is}.$$

At $s=0$, the unshifted same-order value is $F(0)=3/4$, whereas the reversed-order value is $1/4$. The failed shortcut has absolute residual $1/2$.

2. Derive the modular detailed-balance sign

Starting from the lower-strip boundary relation and the transform $F_{A,B}(s)=\int e^{-isk}d\nu_{A,B}(k)$, derive the relation between $\nu_{A,B}$ and $\nu_{B,A}$. Then state how the result changes if the Fourier kernel is $e^{+isk}$.

Solution

At the lower edge,

$$F_{A,B}(s-i) =\int e^{-isk}e^{-k}\,d\nu_{A,B}(k).$$

Modular invariance gives

$$\omega(\sigma_s(B)A) =\omega(B\sigma_{-s}(A)) =F_{B,A}(-s) =\int e^{isk}\,d\nu_{B,A}(k).$$

Changing $k\mapsto-k$ in the last integral and using uniqueness of Fourier transforms of finite complex measures gives

$$e^{-k}d\nu_{A,B}(k)=d\nu_{B,A}(-k).$$

With Fourier kernel $e^{+isk}$, the variable called $k$ is replaced by $-k$; the exponential is correspondingly written $e^{+k}$. This is a convention change, not a different physical law.

3. The accelerated strip and its forbidden crossing

List the poles of $\mathcal G(z)=-a^2/[16\pi^2\sinh^2(az/2)]$. Show that the strip $-2\pi/a<\operatorname{Im}z<0$ contains no pole, derive its two boundary prescriptions, and explain why a contour crossing $z=0$ cannot prove the KMS identity.

Solution

The zeros of $\sinh(az/2)$ occur at $az/2=i\pi n$, so the double poles are

$$z=\frac{2\pi i n}{a}, \qquad n\in\mathbb Z.$$

Only the boundary lines at imaginary parts $0$ and $-2\pi/a$ contain poles; the open strip is analytic. Its upper boundary is

$$\lim_{\epsilon\downarrow0}\mathcal G(u-i\epsilon)=G^+(u).$$

Approaching the lower edge from inside and using $\sinh(w-i\pi)=-\sinh w$ gives

$$\lim_{\epsilon\downarrow0} \mathcal G(u-2\pi i/a+i\epsilon) =\lim_{\epsilon\downarrow0}\mathcal G(u+i\epsilon) =G^-(u).$$

The two $i0$ prescriptions encode opposite operator orders. A contour through $z=0$ crosses a double pole, so Cauchy continuation is unavailable and an additional commutator/contact distribution is picked up. Excluding $\lvert u\rvert<u_{\min}$ in a pointwise plot does not remove this distributional issue; a valid test keeps smeared observables and approaches each edge from inside.

References

• Haag, Rudolf, Nico M. Hugenholtz, and Marinus Winnink. “On the Equilibrium States in Quantum Statistical Mechanics.” Communications in Mathematical Physics 5 (1967): 215–236. DOI.
• Takesaki, Masamichi. Tomita’s Theory of Modular Hilbert Algebras and Its Applications. Lecture Notes in Mathematics 128. Berlin: Springer, 1970. DOI.
• Unruh, William G. “Notes on Black-Hole Evaporation.” Physical Review D 14 (1976): 870–892. DOI.