KMS states for a Killing flow in curved spacetime #
This file connects the abstract KMS condition (Physicslib4.AQFT.IsKMSState) to
the curved-spacetime stabilizer action. A Killing flow fixing a region B is a
one-parameter subgroup t ↦ φ_t of the stabilizer Stab(B) = {φ : φ·B = B}. It
induces, via the stabilizer automorphism stabAutHom, a one-parameter group of
*-automorphisms of the single local algebra 𝔘(B), and one can then ask
whether a state on 𝔘(B) is KMS for that flow.
The motivating examples are the Hartle-Hawking state on a Schwarzschild exterior (KMS for the stationary Killing flow at the Hawking temperature) and the Bunch-Davies state restricted to a de Sitter static patch (KMS for the boost Killing flow at the Gibbons-Hawking temperature). Those regions are not diamonds (basis sets), and no such flow fixes a nonempty diamond, so the axioms as stated place no constraint on their algebras: the examples motivate the definitions but are not instances of them.
Main definitions / results #
HaagKastlerNet.flowAut: the one-parameter automorphism family of𝔘(B)induced by a flow intoStab(B).HaagKastlerNet.isOneParameterAut_flowAut: if the flow is a one-parameter subgroup, the induced family is a one-parameter automorphism group.HaagKastlerNet.IsKMSStateForFlow: a state on𝔘(B)is a KMS state for the Killing flow.
The automorphisms t ↦ τ_t = α̂_(φ_t) of the local algebra 𝔘(B) along a curve t ↦ φ_t in the stabilizer Stab(B).
Here α̂_φ is the stabilizer automorphism of 𝔘(B) (stabAutHom). In the motivating examples t ↦ φ_t
is a Killing flow, the one-parameter group of isometries generated by a Killing vector field whose
flow maps B to itself: the stationary flow of a black-hole exterior, or the boost of a de Sitter
static patch. Those regions are not diamonds (basis sets), and no such flow fixes a nonempty
diamond, so the axioms as stated place no constraint on their algebras: the examples motivate the
definition but are not instances of it. When t ↦ φ_t is a one-parameter subgroup, τ is a
one-parameter group of automorphisms, the time evolution of 𝔘(B) along the flow
(isOneParameterAut_flowAut).
Implementation note: flow is any function ℝ → Stab(B); the group laws are hypotheses of the
results that need them.
Blueprint: def:flow-aut-in-curved-spacetime
Equations
- N.flowAut B flow t = N.stabAutHom B (flow t)
Instances For
A Killing flow induces a one-parameter automorphism group. If flow is a
one-parameter subgroup of Stab(B) (flow 0 = 1, flow (s+t) = flow s * flow t),
then the induced automorphisms of 𝔘(B) form a one-parameter group.
A state ω on a local algebra 𝔘(B) is a KMS state at inverse temperature β for a Killing flow t ↦ φ_t fixing B.
The Kubo–Martin–Schwinger condition characterises thermal equilibrium: for all a, b the correlation
function t ↦ ω(a τ_t(b)) extends to a bounded continuous function on the strip 0 ≤ Im z ≤ β,
holomorphic inside, whose values on Im z = β are ω(τ_t(b) a). This definition is the general KMS
condition AQFT.IsKMSState for the automorphisms τ_t = α̂_(φ_t) of 𝔘(B) (flowAut). Curved
spacetime has no quasilocal algebra, so the dynamics must act on the algebra of a single region,
which is why the flow takes values in the stabilizer of B. The Hartle–Hawking state of a
Schwarzschild exterior and the Bunch–Davies state on a de Sitter static patch, KMS at the Hawking
and Gibbons–Hawking temperatures, are the motivating examples. Those regions are not diamonds (basis
sets), and no such flow fixes a nonempty diamond, so the axioms as stated place no constraint on
their algebras: the examples motivate the definition but are not instances of it, and the library
constructs neither state. When t ↦ φ_t is a one-parameter subgroup and β > 0, a KMS state is
invariant under the flow; if moreover the functions t ↦ ω(a* τ_t(b)) are continuous, its GNS space
carries a strongly continuous unitary group implementing the flow
(IsKMSStateForFlow.exists_gns_unitary_strongContinuous).
Blueprint: def:kms-state-for-flow-in-curved-spacetime
Equations
- N.IsKMSStateForFlow B flow β ω = Physicslib4.AQFT.IsKMSState (N.flowAut B flow) β ω
Instances For
The Killing-flow KMS thermal representation. A KMS state ω on 𝔘(B) for
a one-parameter Killing flow t ↦ φ_t into Stab(B), at positive inverse
temperature β, yields a GNS triple (H, π, Ω) reproducing ω together with a
strongly continuous one-parameter unitary group U : ℝ → (H ≃ₗᵢ[ℂ] H)
implementing the flow:
U t (π a Ω) = π (α_t a) Ω, U t Ω = Ω, U 0 = id, U (s+t) = U s ∘ U t,
and t ↦ U t ψ is continuous for every ψ.
This is the curved-spacetime equilibrium (thermal) representation - the role played
by the vacuum representation in Minkowski spacetime. The Hartle-Hawking and
Gibbons-Hawking states motivate it but are not instances, since their regions are
not diamonds (see IsKMSStateForFlow). The hypotheses are: flow is a
one-parameter subgroup of Stab(B), 0 < β, the KMS condition, and weak
continuity of the matrix coefficients t ↦ ω(a⋆ · α_t b). KMS at β > 0 gives
α-invariance (IsKMSState.invariant_of_pos), which feeds the abstract
strongly-continuous GNS unitary construction.
The Killing-flow KMS state set is convex. A convex combination
s·ω₁ + (1-s)·ω₂ (0 ≤ s ≤ 1) of two KMS states on 𝔘(B) for the same Killing
flow at the same inverse temperature β is again a KMS state for that flow. This
specializes the abstract KMS convexity (AQFT.IsKMSState.convexCombo) to the
induced one-parameter group flowAut. Physically the curved-spacetime thermal
equilibrium states for a stationary Killing flow form a convex set.
A ground state for a Killing flow t ↦ φ_t fixing B: a flow-invariant state on 𝔘(B) in which the flow has a positive generator.
A state ω on 𝔘(B) is a ground state for t ↦ φ_t when
- ω(τ_t a) = ω(a) for all t and a, where τ_t = α̂_(φ_t) (
flowAut); and - there is a representation (K, π, Ω) reproducing ω, with unitaries U(t) that fix Ω and implement
the flow, U(t)π(a)Ω = π(τ_t a)Ω, such that t ↦ U(t) has positive energy
(
AQFT.IsPositiveEnergy): it is strongly continuous with positive generator.
The generator is the local Hamiltonian of the Killing flow. The ground state is the
zero-temperature counterpart of the KMS states of IsKMSStateForFlow. Two consequences do not use
the positivity of the generator: a ground state is flow-invariant (IsGroundStateForFlow.invariant)
and its unitary group is strongly continuous
(IsGroundStateForFlow.exists_strongContinuous_unitary).
Implementation note: the representation space K ranges over Type v, the universe of the local
algebras.
Blueprint: def:ground-state-for-flow-in-curved-spacetime
Equations
- One or more equations did not get rendered due to their size.
Instances For
A Killing-flow ground state is invariant under the flow (the first conjunct); the spectrum condition is not used.
The implementing unitary group of a Killing-flow ground state is strongly
continuous: strong continuity is a field of the positive-energy hypothesis
(AQFT.IsPositiveEnergy.strongContinuous).