Thermal Time and the Modular Flow in the Biquaternion Framework

Introduction

The modular flow of a state is the one-parameter group of automorphisms that the state defines on its algebra through the Tomita–Takesaki construction. The thermal time hypothesis of Connes and Rovelli is the proposal that this flow is the physical flow of time: that a state, not a Hamiltonian, is what specifies time, and that the modular flow of the state is the time evolution an observer in that state would measure. The proposal is radical where it is nontrivial: on a type III algebra there is no Hamiltonian and no density matrix, so the modular flow is the only canonical flow available, and it is outer.

This article is the dynamical continuation of the modular programme of the three preceding articles. It asks what thermal time and the modular flow are in the biquaternion framework, and the answer is a definite computation with a definite limitation.

The modular flow is explicit. For a faithful biquaternion state $\tilde\rho$ the modular flow is $$ \sigma_t\big(\tilde A\big)=\tilde\rho^{\,it}\tilde A\,\tilde\rho^{-it}=e^{-t\Gamma}\tilde A\,e^{t\Gamma}, \qquad \Gamma=b\,\hat{\mathbf r}\cdot\mathbf e,\quad b=\mathrm{artanh}\big(|\mathbf r|\big), $$ an inner automorphism generated by the modular Hamiltonian $\tilde K=-\log\tilde\rho\in\mathbb{M}_+$. On the algebra it is the identity on the two directions spanned by $e_0$ and $\hat{\mathbf r}\cdot\mathbf e$, and a rotation by angle $2bt$ in the transverse planes. The flow was computed in closed form and verified on generic states.
Thermal time is available but inner. Because the biquaternion algebra is a finite-dimensional factor of type I, every automorphism of it is inner, and the modular flow is generated by an element of the algebra. The state is a KMS (thermal) state with respect to the inverse of its own modular flow, $\sigma_{-t}$, at the modular inverse temperature. The state therefore does define its own time, and the generator is the modular Hamiltonian, which lies in the informational sector $\mathbb{M}_+$ — the dynamical form of the statement that the modular structure is $\mathbb{M}_+$-valued.
The radical part of thermal time is absent, and the framework says where it lives. The nontrivial content of the thermal time hypothesis — an outer modular flow that is canonical because no Hamiltonian exists — requires a type III algebra and hence an infinite-dimensional local algebra. The finite biquaternion flow is inner and periodic; the field flow is a boost. The two are related by the analytic continuation to imaginary time, which is the material sector's $ict$, and it is that continuation that converts the framework's rotation into the field's boost.

The article proceeds as follows. The modular flow and its generator are recalled, and the inner/outer distinction is drawn. The thermal time hypothesis is stated and its nontrivial case identified. The modular flow of a biquaternion state is computed in closed form, its generator's sector is identified, and its periodicity and pure limit are analyzed. The state-defined time and its KMS thermality are then read off. The field case is recalled, and the article closes with the established/interpretation/open split.

Conventions. Those of the companion articles. The biquaternion algebra is $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, with quaternion basis $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$, scalar imaginary $i$, and isomorphism $\Phi(e_k)=-i\sigma_k$, $\Phi(i)=iI_2$, so that $\mathbb{B}\cong M_2(\mathbb{C})$. The material (anti-Hermitian) subspace is $\mathbb{M}_-$ and the informational (Hermitian) subspace is $\mathbb{M}_+$, with $\mathbb{B}=\mathbb{M}_+\oplus\mathbb{M}_-$ and $i\mathbb{M}_\pm=\mathbb{M}_\mp$. The trace is $\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$, with $\mathrm{Tr}(e_0)=2$. States are $\tilde\rho=\tfrac12(e_0+i\mathbf r\cdot\mathbf e)\in\mathbb{M}_+$ with $|\mathbf r|\le1$; the modular operator, modular conjugation and modular flow are those of The Modular Theory of Tomita–Takesaki under the Biquaternion Framework, the states and the Bloch ball are those of The GNS Construction in the Biquaternion Framework, and the logarithm of a state is that of Relative Entropy and the Biquaternion Framework.

The Modular Flow

Tomita's Flow and Its Generator

Let $M$ be a von Neumann algebra with a cyclic and separating vector $\Omega$, and let $S$ be the Tomita operator $S(\tilde A\Omega)=\tilde A^*\Omega$ with polar decomposition $S=J\Delta^{1/2}$. The modular operator $\Delta=S^*S$ is positive and nonsingular, and the modular flow is the one-parameter group of automorphisms $$ \sigma_t\big(\tilde A\big)=\Delta^{it}\tilde A\,\Delta^{-it}, \qquad t\in\mathbb{R}. $$ Tomita's theorem states that $\sigma_t$ maps $M$ to itself for every $t$, that the modular conjugation $J$ implements $\sigma_t$ on the commutant, $J\Delta J^{-1}=\Delta^{-1}$, and that the flow depends only on the state $\omega(\tilde A)=\langle\Omega|\tilde A|\Omega\rangle$, not on the choice of cyclic and separating vector. The modular Hamiltonian is $$ \tilde K=-\log\Delta, $$ so that formally $\sigma_t(\tilde A)=e^{-it\tilde K}\tilde A\,e^{it\tilde K}$; on a finite-dimensional algebra, and for a state given by a density matrix, $\Delta$ acts as $\tilde\rho\,\cdot\,\tilde\rho^{-1}$ and $\tilde K=-\log\tilde\rho$.

Three properties of the flow are used below.

It is determined by the state. Two states with the same modular flow coincide up to inner automorphism; the flow is the state's own time.
The state is KMS for the inverse flow. At inverse temperature one in the modular parameter, $\Omega$ is a KMS vector for $\sigma_{-t}$: the correlation functions have the analyticity and the boundary relation of the KMS companion article.
Its generator is canonical. The pair $(\Delta,J)$ is unique, so the flow and its generator are unique once the state is fixed, whether or not a Hamiltonian is available.

Inner and Outer Flows

A flow is inner if there are unitaries $u_t\in M$ with $\sigma_t=\mathrm{Ad}(u_t)$, and outer otherwise; the outer part is the image of the flow in the group $\mathrm{Out}(M)=\mathrm{Aut}(M)/\mathrm{Inn}(M)$ of outer automorphisms. The distinction determines whether the thermal time hypothesis has content.

On a type I algebra — matrices, and therefore the biquaternion algebra itself — every automorphism is inner. The modular flow of any state is conjugation by a unitary in the algebra, and its generator is an element of the algebra. A Hamiltonian exists and is the modular Hamiltonian.
On a type II algebra there is a trace, and the modular flow of a state is again inner.
On a type III algebra, which is what the local algebra of a quantum field theory is, no trace exists and the modular flow of a normal state is outer. There is no density matrix, no Hamiltonian, and no inner unitary implementing the flow.

This is why thermal time is a proposal about type III. Where the flow is inner, an ordinary Hamiltonian evolution is available and the state merely provides a preferred one; where the flow is outer, the state provides the only flow there is, and calling it time is a substantive hypothesis rather than a reformulation.

The Thermal Time Hypothesis

The Proposal

The thermal time hypothesis (Connes–Rovelli, 1994) states:

In a theory in which no time evolution is given a priori, a state $\omega$ on the algebra of observables determines the physical time evolution up to inner automorphisms, and that evolution is the modular flow $\sigma_t^\omega$ of the state. The modular parameter $t$ is the thermal time; a Hamiltonian is a derived object, the generator of the flow, defined only where the flow is inner.

The proposal is motivated by the algebraic structure of generally covariant theories, in which the observables form a von Neumann algebra, the state is the only invariant datum, and the modular flow is the unique canonical one-parameter group the state supplies. It recovers the ordinary notion of time in the limits in which a Hamiltonian exists, and it predicts that time is state-dependent: two observers in different states have different thermal times, agreeing only up to the inner automorphisms that the states share.

What Makes It Nontrivial

The hypothesis has content precisely where the flow is outer. For a type III algebra, the modular flow of a state is outer, so it is not conjugation by any algebra element, and it is not derivable from a Hamiltonian. It is nonetheless canonical: it is determined by the state alone, without any additional geometric input. The two ends of the correspondence are then:

Thermal time is nontrivial on type III algebras, where the flow is outer and the state's modular flow is the only canonical time.
Thermal time is trivial on type I and type II algebras, where the flow is inner and a Hamiltonian exists. The state picks out a preferred Hamiltonian but does not create the time.

The biquaternion algebra is a finite-dimensional factor and is therefore of the trivial kind, as the next section shows by computing its flow. The framework's role in thermal time is to provide the exact finite model of the flow — the state-defined time, its generator, and its KMS thermality — while the outerness that makes the hypothesis radical lives on the type III local algebra, which the framework reaches only through the infinite-dimensional module algebra of the Reeh–Schlieder companion article.

The Modular Flow of a Biquaternion State

Closed Form of the Flow

Let $\tilde\rho=\tfrac12(e_0+i\mathbf r\cdot\mathbf e)$ be a faithful state, $|\mathbf r|=r<1$, and write $$ \hat{\mathbf r}=\frac{\mathbf r}{r},\qquad \gamma=\hat{\mathbf r}\cdot\mathbf e, \qquad b=\mathrm{artanh}(r), \qquad b_0=\tfrac12\log\frac{1-r^2}{4} . $$ The logarithm of the state was computed in the relative-entropy companion article, $$ \log\tilde\rho=b_0\,e_0+i\,b\,\gamma, $$ so that $$ \tilde\rho^{\,it} =\exp\big(it\,b_0\,e_0+it\,ib\,\gamma\big) =e^{itb_0}\exp\big(-tb\,\gamma\big) =e^{itb_0}\Big(\cos(bt)\,e_0-\sin(bt)\,\gamma\Big). $$ The step in the middle used $\exp(\theta\gamma)=\cos\theta\,e_0+\sin\theta\,\gamma$, which holds because $\gamma^2=(\hat{\mathbf r}\cdot\mathbf e)^2=\hat{\mathbf r}\cdot\hat{\mathbf r}\,(-e_0)=-e_0$, and the final form is the trigonometric expansion appropriate to that sign. The closed form was checked against the explicit matrix power: with $\mathbf r=(0.6,-0.3,0.4)$ and $t=0.3$, the two agree to machine precision, the largest entrywise difference being of order $10^{-16}$.

The scalar $e^{itb_0}$ drops out of conjugation, so the modular flow is $$ \boxed{\ \sigma_t\big(\tilde A\big) =\tilde\rho^{\,it}\,\tilde A\,\tilde\rho^{-it} =e^{-tb\gamma}\ \tilde A\ e^{tb\gamma}\ } $$ with $$ e^{-tb\gamma}=\cos(bt)\,e_0-\sin(bt)\,\gamma . $$ This is the framework's closed form for the modular flow: conjugation by the unit quaternion $e^{-tb\gamma}$, which is a unit element of the commutative subalgebra spanned by $e_0$ and the pure direction $\gamma$.

The Generator Lies in $\mathbb{M}_+$

The generator of the flow is obtained by differentiating at $t=0$, $$ \frac{d}{dt}\sigma_t\big(\tilde A\big)\Big|_{t=0} =-b\big[\gamma,\tilde A\big] =i\big[\log\tilde\rho,\tilde A\big] =-i\big[\tilde K,\tilde A\big], $$ and it is worth recording where each object lives. The direction $\gamma=\hat{\mathbf r}\cdot\mathbf e$ is a sum of the anti-Hermitian basis elements $e_k$ and therefore lies in the material sector $\mathbb{M}_-$. The modular Hamiltonian $$ \tilde K=-\log\tilde\rho =\tfrac12\log\frac{4}{1-r^2}\,e_0-i\,b\,\gamma \ \in\ \mathbb{M}_+, $$ because $i\mathbb{M}_-=\mathbb{M}_+$ and the scalar part is real. The flow is thus generated by an element of the informational sector acting on the algebra by commutation, and the rate $b=\mathrm{artanh}(r)$ is the state's own parameter: it vanishes at the trace state and diverges on the pure boundary.

This is the dynamical form of the section's reading. The state's modular Hamiltonian is a Hermitian algebra element, hence an object of $\mathbb{M}_+$; the modular flow it generates is the state's thermal time; and the flow moves the material directions of the algebra. The informational sector supplies the generator, the material sector supplies the directions that are moved.

The Explicit Action on the Algebra

The flow acts on the algebra in a way that can be read off from the closed form. Since $\gamma$ commutes with itself and $e^{-tb\gamma}$ is a function of $\gamma$, $$ \sigma_t\big(\hat{\mathbf r}\cdot\mathbf e\big)=\hat{\mathbf r}\cdot\mathbf e, \qquad \sigma_t(e_0)=e_0, $$ so the two directions $\mathrm{span}\{e_0,\gamma\}$ are fixed. For a unit direction $\mathbf u\perp\hat{\mathbf r}$, put $Y=\mathbf u\cdot\mathbf e$. Then $\gamma Y=-Y\gamma$ and $$ e^{-tb\gamma}\,Y\,e^{tb\gamma} =\cos(2bt)\,Y-\sin(2bt)\,\gamma Y, $$ which was checked on the state $\mathbf r=(0.6,-0.3,0.4)$ and a transverse $\mathbf u$ at $t=0.3$ and $t=0.7$, agreeing with the matrix flow to machine precision (entrywise differences of order $10^{-16}$). So the flow is the identity on the two commuting directions and a rotation by angle $2bt$ in each transverse plane, the planes spanned by $Y$ and $\gamma Y$. The generator's eigenvalues on the transverse directions are $\pm2bi$; the flow is a compact one-parameter group.

Periodicity and the Pure Limit

Two features of the finite flow differ from the field flow and are worth naming.

The flow is periodic. Since the rotation angle is $2bt$, the flow returns to the identity after $$ T=\frac{\pi}{b}=\frac{\pi}{\mathrm{artanh}(r)}, $$ which was checked at $\sigma_T(\tilde A)=\tilde A$ to $10^{-15}$. The period is the reciprocal of the state's rapidity: it is long for a nearly maximally mixed state, $T\to\infty$ as $r\to0$ where the flow becomes trivial, and short for a nearly pure state, $T\to0$ as $r\to1$ where the flow becomes arbitrarily fast. The flow is compact because the modular Hamiltonian of a finite-dimensional state has finite spectrum.
The flow is singular on the pure boundary. The rate $b=\mathrm{artanh}(r)$ diverges as $r\to1$, so the modular flow of a nearly pure state is arbitrarily fast; on the pure boundary itself the state is a zero divisor, $\log\tilde\rho$ is not an algebra element, and the flow is not defined by the formula above. The pure state has no finite modular flow in the finite algebra.

Both features are consequences of finite dimensionality. In the field the modular Hamiltonian has continuous spectrum and the flow is a non-compact one-parameter group — a boost — with no period.

Thermal Time in the Framework

The State Defines Its Own Time

Because the modular flow is determined by the state and is generated by the modular Hamiltonian, the biquaternion framework realizes the thermal time proposal in its inner case exactly: the state is the clock. Two faithful states with different Bloch vectors $\mathbf r,\mathbf r'$ define flows with different rates $b=\mathrm{artanh}(r)$ and different fixed directions $\hat{\mathbf r},\hat{\mathbf r}'$, agreeing only when the states are unitarily related in the way that preserves both. The modular parameter $t$ is the state's thermal time, and the generator is the modular Hamiltonian in $\mathbb{M}_+$.

The statement "the state defines the time" is not a reinterpretation of a pre-existing Hamiltonian here; it is the algebra's content, because the flow is constructed from $\tilde\rho$ alone. What the framework cannot supply is the outer case, where there is no element of the algebra to generate the flow and the hypothesis becomes radical. That case requires the type III local algebra.

The KMS Thermality

The state is a KMS state with respect to the inverse of its own modular flow, at the modular inverse temperature. Concretely, with the orientation convention of the KMS and Tomita–Takesaki companion articles, in which the flow carrying the boundary relation is $\sigma_{-t}$, $$ F_{\tilde A\tilde B}(t)=\mathrm{Tr}\big(\tilde\rho\,\tilde A\,\alpha_t(\tilde B)\big), \qquad \alpha_t\big(\tilde B\big)=\tilde\rho^{-it}\tilde B\,\tilde\rho^{\,it}, \qquad F_{\tilde A\tilde B}(t+i)=F_{\tilde B\tilde A}(-t), $$ and the boundary relation was verified on the faithful state of the Reeh–Schlieder companion article, the two sides there agreeing to below $10^{-14}$. The modular flow and the KMS property are therefore two statements of the same fact — the state is thermal for $\sigma_{-t}$ — and thermal time is the name of the modular flow.

Imaginary Time and the Boost

The framework's material sector has time coordinate $ict$, imaginary by construction, and the analytic continuation in which the KMS condition lives is the continuation along that imaginary direction. The modular flow of this article is a rotation at real modular parameter and becomes a boost when continued to imaginary time: writing $t=is$, $$ \cos(2bt)=\cos\big(2b\,is\big)=\cosh(2bs), \qquad \sin(2bt)\to i\sinh(2bs), $$ so the transverse rotation of angle $2bt$ becomes a hyperbolic rotation of rapidity $2bs$. This is the framework's structural link to the field: the boost that the Bisognano–Wichmann theorem identifies as the wedge modular flow, and whose rapidity the Unruh effect reads as a temperature, is the imaginary-time continuation of the finite algebra's rotation. The continuation is not an extra assumption; it is the direction the KMS condition already requires, and it is the direction in which the material sector's time is imaginary.

The Field Case

In quantum field theory the local algebra $\mathcal A(O)$ of a region is a type III factor, the modular flow of the vacuum state is outer, and the thermal time hypothesis has its content. For a wedge region the flow is the boost, by the Bisognano–Wichmann theorem, and the state is KMS at $\beta=2\pi$ for it; the thermal reading is the Unruh effect; the modular Hamiltonian is the boost generator smeared with the appropriate weight. None of this is re-derived here; it is the subject of the Bisognano–Wichmann and Unruh companion articles.

The biquaternion framework reaches this case through the one-particle module and the CAR algebra built on it: the module supplies the field's mode structure, the infinite dimension supplies the type III character, and the modular flow of the module algebra's vacuum is the field's thermal time. The finite flow computed in this article is the truncation of that flow to a single mode, and its periodicity is the truncation's artifact: a single mode's modular Hamiltonian has finite spectrum, so its flow closes into a circle, while the field's boost does not.

What Is Established and What Is Interpretation

Established (theorem, imported). The Tomita–Takesaki modular flow and its uniqueness for a given state; the inner/outer decomposition and the fact that the modular flow of a normal state on a type III factor is outer; the KMS property of a cyclic and separating state for the inverse of its modular flow; the Connes–Rovelli thermal time hypothesis; the Bisognano–Wichmann wedge result and the Unruh thermal reading. All standard.

Established (recomputed here). The closed forms $\tilde\rho^{it}=e^{itb_0}(\cos(bt)e_0-\sin(bt)\gamma)$ and $\sigma_t(\tilde A)=e^{-tb\gamma}\tilde A e^{tb\gamma}$; the generator identity $\frac{d}{dt}\sigma_t|_0=i[\log\tilde\rho,\tilde A]$; the explicit action (identity on $\mathrm{span}\{e_0,\gamma\}$, rotation by $2bt$ in the transverse planes); the period $T=\pi/b$; the sector location $\tilde K\in\mathbb{M}_+$; and the imaginary-time boost $\cos(2bt)\to\cosh(2bs)$.

Interpretation. That the modular flow of a biquaternion state is read as the state's thermal time, that the state-dependence of the flow is the state-dependence of time, and that the imaginary-time continuation of the finite rotation is the field's boost. The algebra realizes the reading; it does not force it.

Gaps, left visible. The finite algebra is type I, its modular flow is inner and periodic, and the outer case that gives thermal time its content is not reached. The framework supplies no dynamics, no state selection, and no geometric time. No empirical consequence is derived.

Open Questions

1. The outer flow from the module algebra. The Reeh–Schlieder companion article locates the field in the CAR algebra over the biquaternion module. Is the modular flow of the module-algebra vacuum outer, and does its restriction to a single mode reproduce the finite flow computed here as a limit?

2. The period and the continuum limit. The finite flow has period $\pi/\mathrm{artanh}(r)$, which diverges as the state approaches purity. Does the period's divergence reproduce, in a regulated many-mode model, the emergence of the non-compact field flow from the compact one-mode flow?

3. The thermal-time covariance. The hypothesis is often stated as the statement that the flow is covariant under the algebra's symmetries. Which symmetries of $\mathbb{B}$ — the inner automorphisms, the sector exchange $i$, the conjugations — act on the set of thermal times, and what is the orbit structure?

4. The generator's observable status. The modular Hamiltonian is an algebra element in the framework and a formal object in the field. Is there a framework measurement whose statistic is the modular energy $\mathrm{Tr}(\tilde\rho\tilde K)$ beyond the relative entropy computed in the companion article?

5. Empirical contact. As everywhere in the subcategory, no prediction distinguishing the reading from standard quantum theory is derived.

Summary

The modular flow of a state is the one-parameter automorphism group $\sigma_t(\tilde A)=\Delta^{it}\tilde A\Delta^{-it}$ that the state defines through the Tomita–Takesaki construction, generated by the modular Hamiltonian $\tilde K=-\log\Delta$. The thermal time hypothesis of Connes and Rovelli proposes that this flow is the physical flow of time: the state, not a Hamiltonian, specifies time. The hypothesis is trivial where the flow is inner — type I and type II algebras, where a Hamiltonian exists — and radical where the flow is outer, which is the case of the type III local algebras of quantum field theory.

For a faithful biquaternion state $\tilde\rho=\tfrac12(e_0+i\mathbf r\cdot\mathbf e)$ the modular flow is the closed-form inner automorphism $$ \sigma_t\big(\tilde A\big)=\tilde\rho^{\,it}\tilde A\,\tilde\rho^{-it} =e^{-tb\gamma}\tilde A\,e^{tb\gamma}, \qquad \gamma=\hat{\mathbf r}\cdot\mathbf e,\quad b=\mathrm{artanh}\,|\mathbf r| . $$ It is the identity on the two directions $\mathrm{span}\{e_0,\gamma\}$ and a rotation by angle $2bt$ in the transverse planes, with generator $\frac{d}{dt}\sigma_t|_0=i[\log\tilde\rho,\cdot]=-i[\tilde K,\cdot]$ and modular Hamiltonian $\tilde K=\tfrac12\log\frac{4}{1-r^2}e_0-ib\gamma\in\mathbb{M}_+$. The flow is periodic with period $T=\pi/b$, is singular on the pure boundary where the state is a zero divisor, and reduces to the trivial flow at the trace state. The state is KMS with respect to $\sigma_{-t}$, the inverse flow, so the modular flow is its thermal time.

The framework thus realizes thermal time in its inner, trivial case exactly, with the generator in the informational sector and the moved directions in the material sector, and it locates the radical case in the type III module algebra. The finite algebra's rotation becomes the field's boost under the analytic continuation to the material sector's imaginary time, which is the direction the KMS condition already carries.

Summary of Notation

Symbol	Meaning
$\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra, $\cong M_2(\mathbb{C})$
$e_0=1,e_1,e_2,e_3$	Quaternion basis, $e_k^2=-e_0$
$i$	Scalar imaginary, $i^2=-1$
$\mathbb{M}_+,\mathbb{M}_-$	Informational (Hermitian) and material (anti-Hermitian) subspaces
$\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$	Trace pairing; $\mathrm{Tr}(e_0)=2$
$\tilde\rho=\tfrac12(e_0+i\mathbf r\cdot\mathbf e)$	State; Bloch vector $\mathbf r$
$r=\|\mathbf r\|$, $\hat{\mathbf r}=\mathbf r/r$, $\gamma=\hat{\mathbf r}\cdot\mathbf e$	State axis and its algebra direction
$b=\mathrm{artanh}(r)$	Flow rate (rapidity)
$b_0=\tfrac12\log\frac{1-r^2}{4}$	Scalar part of $\log\tilde\rho$
$\Delta=S^*S$	Modular operator
$\sigma_t(\tilde A)=\Delta^{it}\tilde A\Delta^{-it}=\tilde\rho^{it}\tilde A\tilde\rho^{-it}$	Modular flow
$\tilde K=-\log\tilde\rho\in\mathbb{M}_+$	Modular Hamiltonian
$e^{-tb\gamma}=\cos(bt)e_0-\sin(bt)\gamma$	Unit-quaternion generator of the flow
$T=\pi/b$	Period of the finite flow
$\alpha_t(\tilde B)=\tilde\rho^{-it}\tilde B\tilde\rho^{it}$	KMS orientation (companion convention)
$t=is$	Imaginary (material-sector) time; rotation becomes boost