Free scalar wedge modular geometry

Assume the Axiom of Choice. Let $H$ be a complex Hilbert space and let $M\subseteq \mathcal{B}(H)$ be a nonzero unital $\ast$-subalgebra with $M=M^{\mathrm{\prime }\mathrm{\prime }}$, where $\mathcal{B}(H)$ is the bounded-operator algebra and $M^{\mathrm{\prime }}$ is its commutant. Let $\omega$ be a normal state, meaning a bounded positive linear functional with $\omega (I)=1$ that preserves suprema of bounded increasing nets of positive elements. Let $t\mapsto {\alpha }_t$ be a one-parameter group of bijective unital $\ast$-automorphisms of $M$, with dimensionless parameter $t$, such that $t\mapsto \phi ({\alpha }_t(X))$ is continuous for every $X\in M$ and every normal linear functional $\phi$ on $M$ (an element of the predual). Suppose that for some dimensionless $\theta >0$, for every $A,B\in M$ there is a bounded continuous function $F_{A,B}$ on $S_{\theta }=\{z\in \mathbb{C}:0\le \mathrm{Im}z\le \theta \}$, holomorphic inside, with $F_{A,B}(t)=\omega (A{\alpha }_t(B)),F_{A,B}(t+i\theta )=\omega ({\alpha }_t(B)A)\mathrm{.}$ This is the upper-strip convention of the preceding von Neumann KMS definition. For any scalar $\lambda >0$, set ${\tilde{\alpha }}_{\tau }={\alpha }_{\tau \mathrm{/}\lambda }$ for $\tau \in \mathbb{R}$. Then $\tilde{\alpha }$ is again a point-ultraweakly continuous one-parameter group, and $\omega$ satisfies the same upper-strip boundary condition for $\tilde{\alpha }$ on the strip of width $\lambda \theta$.

The Axiom of Choice

Upper-strip KMS condition for von Neumann algebras

Assume both the Axiom of Choice and the Axiom of Countable Choice. Fix a massive real scalar species with rest mass $m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}>0$ in kilograms. Use four-dimensional affine Minkowski spacetime with metric $\eta =\mathrm{diag}(-1,1,1,1)$ and length coordinates $x=(x^0,x^1,x^2,x^3)$, where $x^0=ct$ and $c>0$ has units $\mathrm{m}{\mathrm{s}}^{-1}$. Set $\mu =m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}c\mathrm{/}\mathrm{\hslash }>0$ in ${\mathrm{m}}^{-1}$, where $\mathrm{\hslash }>0$ is in $\mathrm{J}\mathrm{s}$. The future shell, one-particle space, and measure are $H_{\mu }^{+}=\{(p^0,\mathbf{p}):(p^0{)}^2-\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2={\mu }^2,p^0>0\},{\mathcal{H}}_{\mu }=L^2(H_{\mu }^{+},d{\nu }_{\mu }),d{\nu }_{\mu }=\frac{d^3\mathbf{p}}{(2\pi {)}^{3}2\sqrt{\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^2+{\mu }^2}}\mathrm{.}$ Take the scalar Hilbert space to be the bosonic symmetric Fock space $\mathcal{H}={\mathcal{F}}_s({\mathcal{H}}_{\mu })$ and its unit zero-particle vector to be $\mathrm{\Omega }$.

For an open set $O\subseteq {\mathbb{R}}^4$ and real $f\in C_c^{\mathrm{\infty }}(O;\mathbb{R})$, define $K_{\mu }f(p)={\int }_{{\mathbb{R}}^4}f(x)e^{i(p^0{x}^0-\mathbf{p}\cdot \mathbf{x})}{d}^{4}x,W_{\mu }(f)=W_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}(\sqrt{2}{K}_{\mu }f)\mathrm{.}$ Let ${\mathfrak{A}}_s(O)$ be the operator-norm closure of the unital $\ast$-algebra generated by all these bounded Weyl unitaries, and set $W_R=\{x\in {\mathbb{R}}^4:x^1>\mathrm{\mid }{x}^0\mathrm{\mid }\}$ and $M(W_R)={\mathfrak{A}}_s(W_R{)}^{\mathrm{\prime }\mathrm{\prime }}\subseteq \mathcal{B}(\mathcal{H})$. Here $\mathcal{B}(\mathcal{H})$ denotes the bounded linear operators; ${\mathcal{S}}^{\mathrm{\prime }}$ is the set of bounded operators commuting with every member of $\mathcal{S}$, and ${\mathcal{S}}^{\mathrm{\prime }\mathrm{\prime }}=({\mathcal{S}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$. Thus $M(W_R)$ is a concrete von Neumann algebra in the chosen double-commutant convention.

For dimensionless rapidity $s\in \mathbb{R}$, define the future-directed boost by ${\mathrm{\Lambda }}_W(s)(x^0,x^1,x^2,x^3)=(x^0\cosh s+x^1\sinh s,x^0\sinh s+x^1\cosh s,x^2,x^3)\mathrm{.}$ Use its scalar Fock implementation: on the one-particle space $[u_{\mu }(\mathrm{\Lambda })\psi ](p)=\psi ({\mathrm{\Lambda }}^{-1}p)$, and on the $n$-particle symmetric sector $U(\mathrm{\Lambda })=u_{\mu }(\mathrm{\Lambda }{)}^{{\otimes }_{s}n}$, with the vacuum sector fixed. The QFT Fock pairing is linear in its second argument; for the modular definitions use the equivalent first-variable-linear pairing $⟨u,v{⟩}_1:=⟨v,u{⟩}_2$. Thus the vacuum vector state is ${\omega }_W(X)=⟨X\mathrm{\Omega },\mathrm{\Omega }{⟩}_1=⟨\mathrm{\Omega },X\mathrm{\Omega }{⟩}_2$. Assume explicitly that $\mathrm{\Omega }$ is cyclic and separating for $M(W_R)$, meaning $\overline{M(W_R)\mathrm{\Omega }}=\mathcal{H}$ and $X\mathrm{\Omega }=0$ for $X\in M(W_R)$ implies $X=0$. The Tomita operator is the closure of $S_{W,0}(X\mathrm{\Omega })=X^{\ast }\mathrm{\Omega }$ on $M(W_R)\mathrm{\Omega }$; write its polar decomposition as $S_W=J_W{\mathrm{\Delta }}_W^{1\mathrm{/}2}$, in the standard modular-data convention.

As an additional model interface, adopt ${\mathrm{\Delta }}_W^{it}=U({\mathrm{\Lambda }}_W(-2\pi t))(t\in \mathbb{R})\mathrm{.}$ This geometric identity and cyclic-separating property are explicit adopted assumptions for this selected model. The abstract Tomita–Takesaki theorem supplies modular automorphisms and their KMS property, but this carrier does not use it as a proof that the modular unitaries equal spacetime boosts. No tensor-product factorization of the continuum Hilbert space across the wedge and its complement is assumed.

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Quantum field quantities and compatible SI normalizations

Scalar future mass shell, invariant measure and test restriction

Schwartz/distribution spaces and Hilbert Fock conventions

Specified bosonic scalar and CAR Dirac model adoption

Specified free state and observable interpretation

Strongly continuous scalar one-particle Poincare action

Smeared free scalar field and its Poincare representation

Concrete von Neumann algebra and faithful vector

Tomita graph and standard modular data

Tomita modular automorphisms and the upper-strip KMS condition

The free scalar construction and local Weyl algebra use only the scalar branch of the selected QFT model. The pairing conversion above makes the QFT vector state identical to the state used by the first-variable-linear modular convention. No charge-number domain, Dirac field, detector instrument, observation, or thermal density matrix is imported.

Assume the Axiom of Choice and the selected massive scalar wedge postulate. Use its metric $\eta =\mathrm{diag}(-1,1,1,1)$, length coordinates $x^0=ct$, right wedge $W_R=\{x^1>\mathrm{\mid }{x}^0\mathrm{\mid }\}$, vacuum $\mathrm{\Omega }$, algebra $M=M(W_R)$, modular operator ${\mathrm{\Delta }}_W$, first-variable-linear pairing $⟨\cdot ,\cdot {⟩}_1$ from the postulate, and boost ${\mathrm{\Lambda }}_W(s)$ with dimensionless rapidity $s$. Define the vector state ${\omega }_W(X)=⟨X\mathrm{\Omega },\mathrm{\Omega }{⟩}_1$ for $X\in M$, the modular automorphisms ${\sigma }_t(X)={\mathrm{\Delta }}_W^{it}X{\mathrm{\Delta }}_W^{-it}$, and ${\alpha }_t={\sigma }_{-t}$ for dimensionless $t$. The proper-time curve for any $a>0$ in $\mathrm{m}{\mathrm{s}}^{-2}$ is ${\gamma }_a(\tau )=(\frac{c^2}{a}\sinh \frac{a\tau }{c},\frac{c^2}{a}\cosh \frac{a\tau }{c},0,0),\tau \in \mathbb{R},$ where $\tau$ is measured in seconds. Its future-boost rapidity is $s=a\tau \mathrm{/}c$. For this unit proper-time parametrization, define the proper-acceleration scalar by $a_{\gamma }=\sqrt{\eta ({\ddot{\gamma }}_a,{\ddot{\gamma }}_a)}$; the theorem verifies $a_{\gamma }=a$. Define the proper-time action on $M$ by ${\tilde{\alpha }}_{\tau }={\alpha }_{a\tau \mathrm{/}(2\pi c)}$; equivalently, it is conjugation by the future boost $U({\mathrm{\Lambda }}_W(a\tau \mathrm{/}c))$.

Then for every $A,B\in M$ there is a bounded function $G_{A,B}$, continuous on the closed strip $0\le \mathrm{Im}z\le b_U$ and holomorphic inside, whose boundaries satisfy $G_{A,B}(\tau )={\omega }_W(A{\tilde{\alpha }}_{\tau }(B)),G_{A,B}(\tau +i{b}_U)={\omega }_W({\tilde{\alpha }}_{\tau }(B)A)(\tau \in \mathbb{R}),$ with strip width $b_U=2\pi c\mathrm{/}a$ seconds. Under the physical-time convention that the strip width is $\mathrm{\hslash }{\beta }_U$, the associated inverse temperature and KMS temperature parameter are ${\beta }_U=\frac{2\pi c}{a\mathrm{\hslash }},T_U=\frac{1}{k_B{\beta }_U}=\frac{\mathrm{\hslash }a}{2\pi c{k}_B},$ where $\mathrm{\hslash }>0$ is in $\mathrm{J}\mathrm{s}$ and $k_B>0$ is in $\mathrm{J}{\mathrm{K}}^{-1}$. This is a conditional algebraic model deduction; it does not assert a detector response, particle production, a thermal density matrix, or an observed temperature.

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