Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories

The temperature detected by a thermometer moving in |0> with acceleration a is beta = 2pi/a, the Unruh temperature.

“We obtain B = 2n/a, namely which is the Unruh temperature 151, or the temperature detected by a thermometer moving in 10) with acceleration a.”

• cites [151]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The thermal time defined by the cosmic background radiation is precisely the conventional Friedman-Robertson-Walker time.

“In [ll], it was shown that the thermal time defined by the cosmic background radiation is precisely the conventional Friedman-Robertson-Walker time.”

• cites [ll]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The physical universe and the observed cosmic background radiation can be modeled by a Robertson-Walker cosmological model.

For a Gibbs density matrix, the modular flow is the time flow generated by the Hamiltonian, with time rescaled as t -> beta t.

“The modular flow of w is (29) namely it is the time flow generated by the Hamiltonian, with the time rescaled as t + pt.”

In a generally covariant theory with no preferred dynamics or Hamiltonian, a flow called the thermal time of p is determined by any thermal state p.

“In a generally covariant theory, in which no preferred dynamics and no preferred Hamiltonian are given, a flow a : .which we will call the thermal time of p, is determined by any thermal state p.”

• AssumptionRests on an assumption: In a generally covariant theory there is no preferred time flow, and the dynamics cannot be formulated in terms of evolution in a single external time parameter.

In a model of gravity plus electromagnetic field, the explicitly constructed state p yields a thermal time that is precisely the conventional Friedman-Robertson-Walker time.

“In [ 1 I], the state p was explicitly constructed.Then, the thermal time corresponding to p was computed.The result turns out to be precisely the conventional Friedman-Robertson-Walker time, namely the flow that we identify as the true time flow in OUT universe.”

• cites [11] Ashtekar A, 1994Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The notion of time in general relativity based on timelike directions and proper time along timelike world lines is state-dependent.

“Notice that this notion of time is state-dependent.”

The relation postulated in the classical context between a thermal state and a time flow is precisely the semiclassical analogue of the Tomita-Takesaki construction.

“The relation postulated in the classical context between a thermal state and a time flow is precisely the semiclassical analogue of the Tomita-Takesaki construction.”

The classical thermal time hypothesis can be extended to quantum field theory via the Tomita-Takesaki construction.

“It follows that on the one hand we can extend the classical thermal time hypothesis to the context of quantum field theory via the Tomita-Takesaki construction.”

The Unruh and Hawking effects can be directly traced to the thermal time postulate.

“We show in section 4 that they can be directly traced to the thermal time postulate.”

The cocycle Radon-Nikodym theorem shows that the modular group associated to a faithful normal state on a von Neumann algebra is independent, modulo inner automorphisms, of the choice of state.

“The co-cycle Radon-Nikodym theorem [12] shows that, given a von Neumann algebra M , the one-parameter group of automorphisms of M , which we denote ot E AutM, Vt E R, associated by the Tomita-Takesaki theorem [ 151 to a (faithful, normal) state 4 on M is independent, modulo inner automorphisms, of the choice of 4.”

• cites [12] Gibbs J W, 1902Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [15] Connes A, 1994Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In the general generally covariant case, one is led to postulate that the thermal time defines the physical time.

“In this general case, one is led to postulate that the thermal time af defines the physical time [41.”

• cites [4] Rovelli C, 1991, Phys. Rev. sourceNot checked — no full text of the source

• AssumptionRests on an assumption: In a generally covariant theory there is no preferred time flow, and the dynamics cannot be formulated in terms of evolution in a single external time parameter.

The thermal time postulate reduces to the Hamilton equations of motion in the case in which the system is not generally covariant.

“this postulate reduces to the Hamilton equations of motion, in the case in which the system is not generally covariant.”

The general features of the time evolution may change substantially by focusing on different observables subalgebras.

“the general features of the time evolution may change substantially by focusing on different observables subalgebras, that is, by probing the theory in different regimest”

Applying the thermal time postulate to the Gibbs state gives a definition of physical time proportional to standard non-relativistic time.

“Thus, if we apply the thermal time postulate to the Gibbs state (27), we obtain a definition of physical time which is proportional to the standard non-relativistic time.”

The unitary group of transformations generated by K1 can be interpreted as the physical time characteristic of the observer O.

“This fact suggests that the unitary group of transformations (54) y , ~ = eiroK,Ae-iraK1 can be interpreted as the physical time characteristic of the observer 0.”

In the non-relativistic limit, physics is well described by small excitations of a quantum field theory around a thermal state |omega>.

“In the regime in which we may disregard the effect of the relativistic gravitational field, and thus the general covariance of the fundamental theory, physics is well described by small excitations of a quantum field theory around a thermal state Iw).”

The Hamilton equations and the Gibbs postulate follow immediately from the modular flow relation (8).

“The Hamilton equations, and the Gibbs postulate follow immediately from the modular flow relation (8).”

Since |omega> is a KMS state of the conventional Hamiltonian time evolution, the thermodynamical time defined by the modular flow of |omega> is precisely the physical time of non-relativistic physics.

“Since Io) is a KMS state of the conventional Hamiltonian time evolution, it follows that the thermodynamical time defined by the modular flow of lo) is precisely the physical time of non-relativistic physics.”

Whatever state one starts with, the modular flow projects down to the same one-parameter group of elements of the group of outer automorphisms Out A.

“Therefore, whatever state one starts with, the modular flow projects down to the same one-parameter group of elements of the group of outer automorphisms Out A.”

In field theory, the fundamental description of the state of a field system is always incomplete and therefore intrinsically thermodynamical, because we cannot perform infinite measurements with infinite precision.

“Notice that in field theory we are always in such a thermodynamical context, because we cannot perform infinite measurements with infinite precision.”

• Also stated in: § 3.1. The modular group as time · plain

  “The observation that a fundamental description of the state of a field system is always incomplete, and therefore intrinsically thermodynamical, in the sense above, is an important ingredient of the following discussion.”

• AssumptionRests on an assumption: In field theory, we cannot perform infinite measurements with infinite precision.

In conventional field theories, the only consequence of the canonical outer flow is that one must choose the correct type of von Neumann algebra to start with, type III.

“In conventional field theories, where we are independently provided with a notion of time evolution, the only consequence of this observation is that we must be sure to choose the correct type of von Neumann algebra to start with (type I I I , as argued in [7], and elsewhere).”

• cites [7] sourceNot checked — no full text of the source

Gibbs states are the states for which the thermal time flow and the geometrical time flow are proportional.

“As we shall see in the examples of the following section, the Gibbs states are the states for which the two time flows are proportional.”

An equilibrium quantum thermal state contains all the information on the dynamics contained in the Hamiltonian, except for the constant factor beta.

“Thus, as we noticed in the introduction in the context of classical mechanics, an equilibrium quantum thermal state contains all the information on the dynamics which is contained in the Hamiltonian (except for the constant factor j3, which depends on the unit of time employed).”

In a quantum generally covariant context, the physical time is determined by the thermal state of the system, as its modular flow.

“In a quantum generally covariant context, the physical time is determined by the thermal state of the system, as its modular flow (8).”

In a non-generally covariant theory, an equilibrium state is a state whose modular automorphism group is the time translation group, with time measured in units of beta/kbT.

“Therefore, in a non-generally covariant theory, an equilibrium state is a state whose modular automorphism group is the time translation group (where the time is measured in units of f i / k b T ) .”

A von Neumann algebra is intrinsically a 'dynamical' object.

“In this sense, a von Neumann algebra is intrinsically a 'dynamical' object.”

The classical limit of the thermal time hypothesis allows one to define a generally covariant statistical theory, and thus a theoretical framework for the statistical mechanics of the gravitational field.

“In 141 it is shown that the classical limit of the thermal time hypothesis allows one to define a generally covariant statistical theory, and thus a theoretical framework for the statistical mechanics of the gravitational field.”

• cites [141]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The true statistical state of the universe that surrounds us and the factual flow of time are related precisely as requested by the thermal time postulate.

“Therefore, the true statistical state of the universe that surrounds us, and the factual flow of time are related precisely as requested by the thermal time postulate.”

• AssumptionThe authors state this as an assumption: The physical universe and the observed cosmic background radiation can be modeled by a Robertson-Walker cosmological model.

Any faithful state is a KMS state with inverse temperature beta = 1 with respect to the modular automorphism it itself generates.

“Any faithful state is a K M s state (with inverse temperature p = 1) with respect to the modular automorphism at it itself generates 1141.”

• cites [1141]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The derivation of the time flow via the modular group of A_R depends only on the trajectory of the observer, which determines A_R, and does not require any a priori choice of coordinates on the Rindler wedge.

“An important aspect of this derivation of the time flow via the modular group a, of the algebra AR is given by the fact that the result depends only on the trajectory of the observer (which determines AR). and does not require any a priori choice of coordinates on the Rindler wedge.”

Applying the thermal time hypothesis to a non-generally covariant system in thermal equilibrium recovers known physics.

“If we apply the thermal time hypothesis to a non-generally covariant system in thermal equilibrium, we recover immediately known physics.”

The thermal time of the system that the observer can reach is proportional to the time flow determined geometrically by its proper time.

“Therefore, the thermal time of the system that the observer can reach is proportional to the time flow determined geometrically by its proper time.”

Due to the co-cycle Radon-Nikodym theorem, the difference between two modular flows is always an inner automorphism.

“Due to the co-cycle Radon-Nikodym theorem, however, the difference between two modular flows is always an inner automorphism.”

• cites [12] Gibbs J W, 1902Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The same state may give rise to different time flows when restricted to different subalgebras of the full observables algebra.

“Furthermore, as the Rindler example points out, the same state may give rise to different time flows when restricted to different subalgebras of the full observables algebra of the theory.”

The canonical flow of outer automorphisms depends only on the algebra itself.

“This flow is canonical: it depends only on the algebra itself.”

Bisognano and Wichmann proved that the modular group of |0> over A_R is precisely alpha_t = gamma_{-2pi t}.

“Bisognano and Wicbmann [21] have proven that the modular group of 10) over d R is precisely at = Yzno-lt.”

• cites [21] Bergmann P G, 1956, Helv. Phys. Acta Suppl.Not checked yet

If we knew the thermal state of the fields around us and the full kinematics, we could throw away the full standard model Lagrangian without loss of information.

“To put it dramatically, we could say that if we knew the thermal state of the fields around us (and we h e w the full kinematics, namely the specification of all the dynamical fields), we could then throw away the full standard model Lagrangian without loss of information.”

• AssumptionRests on an assumption: The universe around us is in a state of non-zero temperature.

Applying the thermal time postulate to a thermal state in a special relativistic system singles out a preferred time, which is the Lorentz time of the Lorentz frame in which the thermal bath is at rest.

“If we apply the postulate to such a state, we single out a preferred time, and it is easy to see that this time is the Lorentz time of the Lorentz frame in which the thermal bath is at rest.”