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Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories
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We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the `timelessness' of the hypothetical, fundamental generally covariant quantum field theory with the `evidence' of the flow of time. By using the algebraic formulation of quantum theory, we propose a unifying perspective on these problems, based on the hypothesis that in a generally covariant quantum theory the physical time flow is not a universal property of the mechanical theory, but rather it is determined by the thermodynamical state of the system (`thermal time hypothesis'). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita--Takesaki theorem, which allows us to derive a time flow, namely a one-parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time flow, its classical limit, and we relate it to various characteristic theoretical facts, such as the Unruh temperature and the Hawking radiation. We point out the existence of a state-independent notion of `time', given by the canonical one-parameter subgroup of outer automorphisms provided by the co-cycle Radon--Nikodym theorem.
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In the paper
“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.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > Additional information
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- Not matchedcites [151]
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“In [ll], it was shown that the thermal time defined by the cosmic background radiation is precisely the conventional Friedman-Robertson-Walker time.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 5. conclusions
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- Not matchedcites [ll]
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“The modular flow of w is (29) namely it is the time flow generated by the Hamiltonian, with the time rescaled as t + pt.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 4. I. Non-relativistic limit
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“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.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Connes and C Rovelli
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- Rests 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 the paper
“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.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Connes and C Rovelli
What the sources say
- Not matchedcites [11] Ashtekar A, 1994
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“Notice that this notion of time is state-dependent.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 1. Introduction
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“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.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Connes and C Rovelli
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“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.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Connes and C Rovelli
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“We show in section 4 that they can be directly traced to the thermal time postulate.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Connes and C Rovelli
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In the paper
“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.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 1. Introduction
What the sources say
- Not matchedcites [12] Gibbs J W, 1902
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- Not matchedcites [15] Connes A, 1994
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In the paper
“The stateindependent characterization of time is then given by the co-cycle Radon-Nikodym theorem: thermal time is unique up to inner automorphisms.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Connes and C Rovelli
What the sources say
- Not matchedcites [12] Gibbs J W, 1902
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“The first difficulty is that in general relativity no selection of an internal time is known, which leads to a well defined fully non-parametrized dynamical system.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > X ( t ) = X ( r ( t ) ) .
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“Second, because the corresponding quantum evolution lacks unitarily, and the problem is open whether this fact jeopardizes the consistency of the interpretation of the quantum formalism.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > X ( t ) = X ( r ( t ) ) .
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“if we have a generally covariant system with a very large number of degrees of freedom (say, the universe including the gravitational field), and we describe our best experimental knowledge of its state as a distribution on its state space, then theflow determined by this distribution coincides precisely with what we perceive as the physical flow of time.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Connes and C Rovelli
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“In the canonical formulation of a generally covariant theory, therefore, there is no Hamiltonian defined on the phase space, but the phase space is a highly nontrivial object, and the smooth functions on the phase space, namely the generally covariant observables, contain dynamical information implicitly.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > X ( t ) = X ( r ( t ) ) .
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“We extend the thermal time assumption to the quantum theory, by postulating that: the physical time is the modular flow of the thermal state.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Connes and C Rovelli
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- The authors state this as an assumption: A generally covariant quantum theory can be consistently defined.
In the paper
“it means that the non-commutative (von Neumann) algebras are dynamical objects and inherit a natural time evolution from their non-commutativity.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 1. Introduction
What the sources say
- Not checked — no full text of the sourcecites [13] Rovelli C, 1993, Class. Quantum Grav. source
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“There is intrinsically no way of constructing a physical clock that is not atfected by the gravitational field.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > (9)
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In the paper
“The Tomita-Takesaki theorem [15] states that the map 01, defined by”
- Also stated in: § 2.1. Modular automorphism · plain
“leaves R globally invariant and hence defines a one-parameter group of automorphisms of the algebra R.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 2.1. Modular automorphism
What the sources say
- Not matchedcites [15] Connes A, 1994
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- Also stated in: § 2.1. Modular automorphism · plain
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In the paper
“The Hamilton equations (49) and the Gibbs postulate (SO), are both contained in the Tomita-Takesaki relation (8).”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > Additional information
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In the paper
“However, in a generally covariant quantum context this observation provides us with the possibility of capturing an abstract fully state-independent notion of time evolution.In fact, it is the algebra itself that determines the allowed time flows.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Comes and C Rovelli
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In the paper
“The fact that the time is determined by the state, adtherefore the system is always in an equilibrium state with respect to the thermal time flow, does not imply that evolution is frozen, and we cannot detect any dynamical change.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 3.1. The modular group as time
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In the paper
“Thus, within the present scheme the temperature is interpreted as the ratio between thermal time and geometrical time, defined only when the second is meaningfult.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > FA.S(t) = w ( % ( B ) A) (26)
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In the paper
“Thus we obtain the result that there is a classical Hamiltonian H that generates the Hamilton evolution, and that the state p is related to this Hamiltonian by the Gibbs relation.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > Additional information
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“In a quantum system with an infinite number of degrees of freedom, what we generally measure is the effect of small perturbations around a thermal state.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 3.1. The modular group as time
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“The hypothesis that we have put forward in this paper is that physical time has a thermodynamical origin.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 5. conclusions
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“Following Unruh's initial suggestion [5], this construction can be immediately generalized to the Schwarzschild solution-an observer at rest in the Schwarzschild coordinates undergoes a constant acceleration pointing away from the black hol e. -and the result allows one to derive the Hawking temperature [6].”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > Additional information
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“In general, the modular flow is not an inner automorphism of the algebra, namely, there is no Hamiltonian in R that generates it.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > Additional information
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In the paper
“The physical time depends on the state.When the system is in a state o, the physical time is given by the modular group a( of o.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 3.1. The modular group as time
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- Rests on an assumption: In a generally covariant theory, notions of time tend to be state dependent.
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In the paper
“As shown in section 2, while different states over a von Neumann algebra R define distinct time flows, still it is possible to define a state-independent flow, in the following sense.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > Additional information
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“These relations hold, in general, for a generally covariant theory; however, we recall that the Hamiltonian H plays a quite different role than in a non-relativistic theory: it does not determine the Gibbs state, but, rather, it is determined by any thermal state.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > Additional information
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“In a generally covariant situation, the thermal time is the only definition of time available.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > FA.S(t) = w ( % ( B ) A) (26)
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“General covariance implies that the Hamiltonian vanishes weakly.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > X ( t ) = X ( r ( t ) ) .
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“In conventional quantum field theory we can extract all the information in terms of vacuum expectation values of products of fields operators, namely by means of a single quantum state IO).This was emphasized by Wight”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 3.1. The modular group as time
What the sources say
- Not matchedcites Wight
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In the paper
“For instance, as the restriction of the observables algebra to the Rindler wedge determines a time flow, similarly the restriction to a different fixed region of spacetime defines a corresponding time flow.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 5. conclusions
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“In a special relativistic system, a thermal state breaks Lorentz invariance.For instance, the average momentum of a gas at finite temperature defines a preferred Lorentz frame.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > A Comes and C Rovelli
What the sources say
- Not matchedcites [Ill]
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“The hypothesis that we explore in this paper is that the notion of a preferred 'flowing' time has no mechanical meaning at the quantum generally covariant level, but rather has a thermodynamical origin.”
§ Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories > 3.1. The modular group as time
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The authors state this as an assumption.
A generally covariant quantum theory can be consistently defined.
“We simply assume that a generally covariant quantum theory can be consistently defined.”
The authors state this as an assumption.
The ensemble of quantum observables form a non-Abelian C*-algebra.
“We assume that the ensemble of the quantum observables form a non-Abelian C-algebra, which we denote as A.”
The authors state this as an assumption.
The quantum algebra A is a deformation of (a subalgebra of) the classical Poisson algebra.
“assume that the quantum algebra A is a deformation of (a subalgebra of) the classical Poisson algebra.”
The paper relies on this without support.
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 generally covariant theory there is no preferred time flow, and the dynamics of the theory cannot be formulated in terms of an evolution in a single external time parameter.”
The paper relies on this without support.
In the presence of thermal or quantum superpositions of geometries, the spacetime causal structure is lost.
“because in the presence of thermal or quantum 'superpositions' of geometries the spacetime causal structure is lost.”
The paper relies on this without support.
The universe around us is in a state of non-zero temperature.
“If we take into account the fact that the universe around us is in a state of non-zero temperature, this means that the information about the dynamics can be fully replaced by the information about the thermal state.”
The paper relies on this without support.
In field theory, 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.”
The paper relies on this without support.
In a generally covariant theory, notions of time tend to be state dependent.
“The set of considerations above, and, in particular, the observation that in a generally covariant theory notions of time tend to be state dependent, lead us to make the following hypothesis.”
The authors state this as an assumption.
The physical universe and the observed cosmic background radiation can be modeled by a Robertson-Walker cosmological model.
“let us model the physical universe and the observed cosmic background radiation by means of a Robertson-Walker cosmological model filled”
Citations
Cites
Particle creation by black holes
Hawking, S. W. (1975) · Communications In Mathematical Physics · doi:10.1007/bf02345020 · Cited by 13,106
Notes on black-hole evaporation
Unruh, W. G. (1976) · Physical Review D · doi:10.1103/physrevd.14.870 · Cited by 4,850
On the equilibrium states in quantum statistical mechanics
Haag, R., Hugenholtz, N. M., & Winnink, M. (1967) · Communications in Mathematical Physics · doi:10.1007/bf01646342 · Cited by 763
On the duality condition for a Hermitian scalar field
Bisognano, J. J., & Wichmann, E. H. (1975) · Journal of Mathematical Physics · doi:10.1063/1.522605 · Cited by 648
On the duality condition for quantum fields
Bisognano, J. J., & Wichmann, E. H. (1976) · Journal of Mathematical Physics · doi:10.1063/1.522898 · Cited by 642
Une classification des facteurs de type ${\rm III}$
Connes, A. (1973) · Annales scientifiques de l'École normale supérieure · doi:10.24033/asens.1247 · Cited by 553
What is observable in classical and quantum gravity?
Rovelli, C. (1991) · Classical and Quantum Gravity · doi:10.1088/0264-9381/8/2/011 · Cited by 449
Self-citationTime in quantum gravity: An hypothesis
Rovelli, C. (1991) · Physical Review D · doi:10.1103/physrevd.43.442 · Cited by 445
Self-citationQuantum mechanics without time: A model
Rovelli, C. (1990) · Physical Review D · doi:10.1103/physrevd.42.2638 · Cited by 316
Self-citationRovelli, C. (1991) · Classical and Quantum Gravity · doi:10.1088/0264-9381/8/2/012 · Cited by 250
Self-citation
Research works printed but not linked: 2
- Bergmann P G, 1956, Helv. Phys. Acta Suppl., 4, 79
- Rovelli C, 1994, Nuovo Cimento
Cited by
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Discreteness of area and volume in quantum gravity
Rovelli, C., & Smolin, L. (1995) · Nuclear Physics B · doi:10.1016/0550-3213(95)00150-q · Cited by 1,365
“the thermodynamics of the gravitational field[ 39 ].”
BackgroundSelf-citationQuantum theory of geometry: I. Area operators
Ashtekar, A., & Lewandowski, J. (1997) · Classical and Quantum Gravity · doi:10.1088/0264-9381/14/1a/006 · Cited by 852
(1998) · doi:10.12942/lrr-1998-1 · Cited by 917
Self-citationBlack Hole Entropy from Loop Quantum Gravity
Rovelli, C. (1996) · Physical Review Letters · doi:10.1103/physrevlett.77.3288 · Cited by 660
Self-citationQuantization of diffeomorphism invariant theories of connections with local degrees of freedom
Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J., & Thiemann, T. (1995) · Journal of Mathematical Physics · doi:10.1063/1.531252 · Cited by 528
Metadata
- Authors
- Connes, A., & Rovelli, C.
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1994-12-01
- Volume
- 11
- Issue
- 12
- Pages
- 2899-2917
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W1994766604
- Semantic Scholar
- 86df4cc3e26ad368152d50b8123af98ead2a5030
- Cited by
- 319
- References
- 12
- Citations by year
- 2026: 31 · 2025: 15 · 2024: 11 · 2023: 17 · 2022: 20 · 2021: 17 · 2020: 13 · 2019: 11 · 2018: 14 · 2017: 11 · 2016: 7 · 2015: 10 · 2014: 9 · 2013: 12 · 2012: 7
- FWCI
- 7.73
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9311465)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Keywords
- von Neumann algebras, thermal time hypothesis, passage of time, timelessness, Unruh temperature, Hawking radiation, one-parameter groups of automorphisms, outer automorphisms, algebraic quantum field theory
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