Ground, KMS, and Stationary States

Stationarity permits two distinguished equilibrium notions that do not exist on a generic curved spacetime. A ground state is selected by nonnegative spectrum of the generator of a specified time-translation automorphism group. A Kubo–Martin–Schwinger (KMS) state is selected instead by analyticity and a boundary condition at a specified inverse temperature $\beta$. Neither term is meaningful until the flow, its sign, and its domain of regularity have been declared.

Required background. Partial Differential Equations and Green Functions supplies the invariant Klein–Gordon evolution and its causal propagator. The bounded local bridge used here is that a complete isometric flow commuting with $P$ maps test functions and solutions into test functions and solutions, so it induces automorphisms of the field algebra.

Helpful background. Flat-Spacetime Quantum Field Theory provides Hamiltonian ground and Gibbs states, Causality and Global Structure defines complete stationary flows and their causal character, and Quantum Fields in Curved Spacetime provides the $e^{-i\omega t}$ and Klein–Gordon conventions.

Stationary flow and algebra automorphisms

Let $(M,g)$ admit a complete one-parameter group of time-orientation-preserving isometries

$$\varphi_t:M\longrightarrow M, \qquad \varphi_{t+s}=\varphi_t\circ\varphi_s,$$

generated by a stationary vector field $K^a$. Completeness for all $t\in\mathbb R$ is needed for a global dynamical system. On a smeared scalar field define

$$\alpha_t\big(\Phi(f)\big)=\Phi(f_t), \qquad f_t(x)=f(\varphi_{-t}x).$$

Because the isometry preserves the volume element, the field equation, and the causal propagator, $\alpha_t$ respects every algebraic relation. It therefore extends to a one-parameter group of $*$-automorphisms of $\mathcal A(M)$.

The inverse in $f_t=f\circ\varphi_{-t}$ fixes the generator sign. In an invariant GNS representation we use

$$U(t)=e^{itH}, \qquad U(t)\pi(A)U(t)^{-1}=\pi(\alpha_t(A)).$$

With this convention, a positive-frequency mode evolves as $e^{-i\omega t}$ and has generator eigenvalue $+\omega$. Replacing $U(t)$ by $e^{-itH}$ without changing the spectral inequality would interchange ground and ceiling conditions.

Ground states and the spectral condition

A state $\omega$ is stationary when $\omega\circ\alpha_t=\omega$. If the action is sufficiently continuous, stationarity defines a strongly continuous unitary group on the GNS Hilbert space by

$$U(t)\pi(A)\Omega_\omega =\pi(\alpha_t(A))\Omega_\omega.$$

The state is a ground state for this specified dynamics when

$$\operatorname{spec}(H)\subset[0,\infty), \qquad H\Omega_\omega=0.$$

Equivalently, appropriate correlation functions have no negative energy transfer relative to the declared generator. The definition is relational: the same algebraic state could fail to be a ground state for another automorphism group. It also requires a regularity class in which the unitary generator exists; invariance under a merely discontinuous abstract action would not suffice.

On an ultrastatic spacetime $g=-dt^2+h$, suppose the spatial operator

$$A=-\Delta_h+m^2+\xi R_h$$

has a self-adjoint realization bounded below by a strictly positive constant. Writing $K=A^{1/2}$, the quasifree ground-state two-point kernel is

$$W_0(t,x;t',x') =\left[\frac{e^{-iK(t-t')}}{2K}\right](x,x').$$

The kernel is taken with respect to the spatial measure $d\mathrm{vol}_h$. Its positive spectrum supplies the positive frequencies, and the factor $1/(2K)$ fixes the equal-time commutator. A strict spectral gap is a convenient sufficient assumption here, not a necessary condition for every ground state. If the spectrum reaches zero, the inverse must instead be checked as a distribution on the chosen test functions; thermal factors impose an additional infrared condition. A zero eigenmode or negative spectrum cannot simply be included in the displayed inverse and square root.

KMS states at inverse temperature beta

Let $\beta>0$. A state $\omega_\beta$ is a KMS state at inverse temperature $\beta$ with respect to $\alpha$ when, for suitable algebra elements $A,B$, there is a function $F_{A,B}$ continuous on the closed strip, analytic on its interior,

$$0<\operatorname{Im}z<\beta,$$

and satisfying the boundary relations

$$F_{A,B}(t)=\omega_\beta\!\left(A\alpha_t(B)\right), \qquad F_{A,B}(t+i\beta) =\omega_\beta\!\left(\alpha_t(B)A\right).$$

For a C*-dynamical system it is enough to impose this on a norm-dense algebra of entire analytic elements, together with the usual boundedness condition. Polynomial field algebras involve unbounded generators, so one states the condition on a controlled regular subalgebra or, for quasifree states, on the correlation distributions with well-defined boundary values. The algebraic definition and the needed strip boundary condition are set out in Hollands and Wald 2015, arXiv v2, §2.1, pp. 17–19, PDF.

The KMS condition implies stationarity under standard continuity assumptions, but it is not defined by a Planck factor alone. It encodes equilibrium through analyticity and cyclic reordering. The number $\beta$ refers to the parameter $t$ of the chosen automorphism. If $K^a$ is rescaled, both the generator and the numerical inverse temperature rescale; only the pair has invariant meaning.

Where the Killing field is timelike, an observer following its orbit has proper time $d\tau=\sqrt{-K^aK_a}\,dt$. Thus the inverse temperature measured along that orbit is $\beta_{\mathrm{loc}}=\beta\sqrt{-K^aK_a}$, or $T_{\mathrm{loc}}\sqrt{-K^aK_a}=1/\beta$. This is the Tolman redshift relation. It follows by rescaling the same KMS flow, not by assigning independent local thermal states at neighboring points. It ceases to describe such observers where $K$ is null or spacelike.

Gibbs states and detailed balance

In a representation with a self-adjoint Hamiltonian $H$ for which $e^{-\beta H}$ is trace class, set

$$\rho_\beta=Z_\beta^{-1}e^{-\beta H}, \qquad Z_\beta=\operatorname{Tr}(e^{-\beta H}),$$

and use $\alpha_t(A)=e^{itH}Ae^{-itH}$. Analytic continuation of the exponentials and cyclicity of the trace give the two KMS boundary values. This derivation is shown explicitly in Hollands and Wald 2015, arXiv v2, §2.1, pp. 19–20, PDF.

The trace formula is an example, not the definition. In infinite volume, $e^{-\beta H}$ is commonly not trace class, while a KMS state still exists as an algebraic state. This is one reason the strip condition is the appropriate thermodynamic-limit formulation.

For a self-adjoint observable $A$, define the stationary spectral density using

$$C_A(t)=\omega_\beta(\alpha_t(A)A), \qquad S_A(\nu)=\int_{-\infty}^{\infty}e^{i\nu t}C_A(t)\,dt.$$

Whenever these expressions exist as distributions, the KMS boundary relation yields detailed balance,

$$S_A(-\nu)=e^{-\beta\nu}S_A(\nu), \qquad \nu>0.$$

For the ultrastatic scalar, the corresponding thermal covariance is

$$W_\beta(\tau) =\frac{1}{2K} \left[ \frac{e^{-iK\tau}}{1-e^{-\beta K}} +\frac{e^{iK\tau}}{e^{\beta K}-1} \right],$$

understood as an operator kernel. The positive- and negative-frequency weights have ratio $e^{-\beta K}$. When a suitable limit exists, $\beta\to\infty$ removes the negative-frequency occupation and recovers the ground covariance.

Admissibility, existence, and limitations

The ground and KMS conditions select states only after stationary dynamics has been supplied. Under appropriate hypotheses, regular quasifree ground and KMS states for free fields are Hadamard, but KMS analyticity in an arbitrary abstract dynamical system does not by itself imply the spacetime Hadamard wavefront condition. The stationary geometry, field equation, and regularity of the covariance remain part of the theorem. Kay and Wald’s analysis makes these dependencies particularly consequential near bifurcate Killing horizons; see Kay and Wald 1991, pp. 49–136.

If a spacetime is genuinely nonstationary, a coordinate-dependent “instantaneous Hamiltonian” does not generate one complete isometric automorphism group. Calling its instantaneous lowest vector a global ground state, or inserting a local temperature into a Planck factor and calling the result KMS, omits the defining dynamics. Even on a stationary spacetime, an ergoregion or a generator that becomes spacelike can obstruct a positive-energy ground construction.

Finally, neither definition guarantees uniqueness. Boundary conditions, superselection sectors, chemical potentials, or disconnected stationary regions can matter. A KMS state at one $\beta$ is not a ground state at finite temperature, although a zero-temperature limit may converge to one when spectral and compactness conditions permit.

Check your understanding

For one oscillator of frequency $\omega>0$, take $\alpha_t(a)=e^{-i\omega t}a$ and occupation number $n_\beta=(e^{\beta\omega}-1)^{-1}$. Verify detailed balance for the self-adjoint observable $Q=a+a^*$.

Solution

The two nonzero thermal expectations are $\omega_\beta(aa^*)=n_\beta+1$ and $\omega_\beta(a^*a)=n_\beta$. Substitution into $C_Q(t)=\omega_\beta(\alpha_t(Q)Q)$ gives a positive-frequency term with coefficient $n_\beta+1$ and a negative-frequency term with coefficient $n_\beta$. The Fourier transform therefore has weights proportional to $n_\beta+1$ at $+\omega$ and $n_\beta$ at $-\omega$. Their ratio is $n_\beta/(n_\beta+1)=e^{-\beta\omega}$, which is exactly the detailed-balance factor. This check also fixes the sign convention: exchanging $e^{-i\omega t}$ with $e^{i\omega t}$ while keeping the same frequency labels would invert the ratio and contradict the declared generator convention.

References

• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. doi:10.1016/j.physrep.2015.02.001. Open PDF.
• Kay, Bernard S., and Robert M. Wald. “Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Spacetimes with a Bifurcate Killing Horizon.” Physics Reports 207, no. 2 (1991): 49–136. doi:10.1016/0370-1573(91)90015-E.