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Wedge modular bridge implies the Unruh clock temperature

Physical scope

An algebraic KMS statement for the vacuum state of the selected massive scalar right-wedge model, conditional on its adopted modular/boost bridge, and for one uniformly accelerated proper-time orbit. It asserts no detector response, particle production, observed temperature, Hawking effect, or density-matrix representation.

Statement

Assume the Axiom of Choice and the selected massive scalar wedge postulate. Use its metric $η = diag (- 1, 1, 1, 1)$ , length coordinates $x^{0} = c t$ , right wedge $W_{R} = {x^{1} > ∣ x^{0} ∣}$ , vacuum $Ω$ , algebra $M = M (W_{R})$ , modular operator $Δ_{W}$ , first-variable-linear pairing $⟨ \cdot, \cdot ⟩_{1}$ from the postulate, and boost $Λ_{W} (s)$ with dimensionless rapidity $s$ . Define the vector state $ω_{W} (X) = ⟨ X Ω, Ω ⟩_{1}$ for $X \in M$ , the modular automorphisms $σ_{t} (X) = Δ_{W}^{i t} X Δ_{W}^{- i t}$ , and $α_{t} = σ_{- t}$ for dimensionless $t$ . The proper-time curve for any $a > 0$ in $m s^{- 2}$ is $γ_{a} (τ) = (\frac{c^{2}}{a} \sinh \frac{a τ}{c}, \frac{c^{2}}{a} \cosh \frac{a τ}{c}, 0, 0), τ \in R,$ where $τ$ is measured in seconds. Its future-boost rapidity is $s = a τ / c$ . For this unit proper-time parametrization, define the proper-acceleration scalar by $a_{γ} = \sqrt{η ({\ddot{γ}}_{a}, {\ddot{γ}}_{a})}$ ; the theorem verifies $a_{γ} = a$ . Define the proper-time action on $M$ by ${\tilde{α}}_{τ} = α_{a τ / (2 π c)}$ ; equivalently, it is conjugation by the future boost $U (Λ_{W} (a τ / c))$ .

Then for every $A, B \in M$ there is a bounded function $G_{A, B}$ , continuous on the closed strip $0 \leq Im z \leq b_{U}$ and holomorphic inside, whose boundaries satisfy $G_{A, B} (τ) = ω_{W} (A {\tilde{α}}_{τ} (B)), G_{A, B} (τ + i b_{U}) = ω_{W} ({\tilde{α}}_{τ} (B) A) (τ \in R),$ with strip width $b_{U} = 2 π c / a$ seconds. Under the physical-time convention that the strip width is $ℏ β_{U}$ , the associated inverse temperature and KMS temperature parameter are $β_{U} = \frac{2 π c}{a ℏ}, T_{U} = \frac{1}{k_{B} β_{U}} = \frac{ℏ a}{2 π c k_{B}},$ where $ℏ > 0$ is in $J s$ and $k_{B} > 0$ is in $J K^{- 1}$ . This is a conditional algebraic model deduction; it does not assert a detector response, particle production, a thermal density matrix, or an observed temperature.

Statement references

The Axiom of Choice

Free scalar wedge data and modular-geometric bridge

Tomita modular automorphisms and the upper-strip KMS condition

Positive rescaling of an upper-strip W*-KMS flow

Thermodynamics-GR fields, currents, reference normalization and units

Facts & Assumptions

Given: The Axiom of Choice; the selected wedge model and modular/boost interface; the stated uniformly accelerated orbit; and the positive constants $a, c, ℏ, k_{B}$ with their SI units.

[A1]

The Axiom of Choice is assumed as defined in The Axiom of Choice.

[A2]

The selected model defines $M (W_{R})$ as the double commutant of the bounded scalar Weyl observables, assumes $Ω$ cyclic and separating for it, and adopts $Δ_{W}^{i t} = U (Λ_{W} (- 2 π t))$ . The boost has future rapidity $s$ under the explicit matrix convention in the postulate. Free scalar wedge data and modular-geometric bridge

[F1]

For standard modular data on a concrete von Neumann algebra, the Tomita–Takesaki theorem gives $σ_{t} (X) = Δ_{W}^{i t} X Δ_{W}^{- i t}$ , a normal point-ultraweakly continuous automorphism group, and an upper-strip dimensionless KMS state for $α_{t} = σ_{- t}$ with width $1$ . It makes no norm-continuity claim for the whole von Neumann algebra. Tomita modular automorphisms and the upper-strip KMS condition

[F2]

The local rescaling lemma rescales this upper-strip W*-KMS boundary condition directly: if ${\tilde{α}}_{τ} = α_{τ / λ}$ with $λ > 0$ , the strip width becomes $λ$ . Positive rescaling of an upper-strip W*-KMS flow

[F3]

The framework units definition uses length coordinates $x^{0} = c t$ , assigns $c$ units $m s^{- 1}$ , $ℏ$ units $J s$ , and $k_{B}$ units $J K^{- 1}$ . In its physical-time KMS convention, a strip width $b$ seconds corresponds to inverse temperature $β = b / ℏ$ . Thermodynamics-GR fields, currents, reference normalization and units

Proof

Proof technique: Identify the future boost from the adopted sign convention, then rescale the modular KMS strip to the proper-time clock.

Given: [A1]–[A2], [F1]–[F3], $a, c, ℏ, k_{B} > 0$ , and $γ_{a}$ .

1.1A2F1algebra

By [A2], $Δ_{W}^{i t} = U (Λ_{W} (- 2 π t))$ . Since $α_{t} = σ_{- t}$ , we have $α_{t} (X) = Δ_{W}^{- i t} X Δ_{W}^{i t} = U (Λ_{W} (2 π t)) X U (Λ_{W} (2 π t))^{*}$ for $X \in M$ . Thus $α_{t}$ is the future-boost action with rapidity $2 π t$ ; the minus sign in the adopted modular identity is essential for this direction.

1.2givenF3algebra

Put $ρ = c^{2} / a$ and differentiate the curve in the Statement. Then ${\dot{γ}}_{a} (τ) = (c \cosh (a τ / c), c \sinh (a τ / c), 0, 0)$ and ${\ddot{γ}}_{a} (τ) = (a \sinh (a τ / c), a \cosh (a τ / c), 0, 0)$ . Direct contraction with $η$ gives $η ({\dot{γ}}_{a}, {\dot{γ}}_{a}) = - c^{2}$ and $η ({\ddot{γ}}_{a}, {\ddot{γ}}_{a}) = a^{2}$ ; also ${\dot{x}}^{0} = c \cosh (a τ / c) > 0$ . Therefore $τ$ is proper time, the curve is future timelike with proper acceleration magnitude $a$ , and its boost rapidity is $a τ / c$ .

2.1A1A2F1F2F3step 1.1step 1.2algebra∎

Let $λ = 2 π c / a > 0$ , which has seconds as its unit by [F3]. Then $τ / λ = a τ / (2 π c)$ and, by step 1.1, ${\tilde{α}}_{τ} = α_{τ / λ}$ . Apply [F1] and [F2] with dimensionless width $θ = 1$ : $ω_{W}$ has the required bounded continuous upper-strip functions on width $λ = 2 π c / a$ seconds for every $A, B \in M$ . By [F3], $β_{U} = λ / ℏ = 2 π c / (a ℏ)$ and $T_{U} = (k_{B} β_{U})^{- 1} = ℏ a / (2 π c k_{B})$ . The displayed units give $[β_{U}] = J^{- 1}$ and $[T_{U}] = K$ . No density matrix or detector statement is used.

Wedge modular bridge implies the Unruh clock temperature

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources