Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories

A. Connes; C. Rovelli

REVIEW

Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories

A. Connes, C. Rovelli

General Relativity and Quantum Cosmology · Submitted 14 Jun 1994 · 20 references

Pith's verdict: The origin paper of the thermal time hypothesis: a clearly labelled postulate with correct supporting math, and one real gap the authors half-acknowledge — the flow depends on the subalgebra, not just the state.

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CitationA. Connes, C. Rovelli. "Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories." arXiv:gr-qc/9406019 (1994). https://pith.science/paper/PFU32LHW

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Short version: this is where thermal time comes from, and it reads as a proposal, not a derivation. The authors say so themselves, repeatedly, and that honesty is the main reason to take it seriously rather than dismiss it. If you work on the problem of time or on algebraic QFT, read it. If you want a finished theory, wait for the follow-ups — several of which the paper points to and does not deliver here. What is genuinely new is the identification itself: in a theory with no preferred dynamics, take the modular group of the thermal state as the physical time flow (abstract:s1, §3.1). The supporting observations are not new but they are assembled well. The Gibbs computation in §4.1, getting α_t = γ_{βt} out of the polar decomposition of S, is standard and I checked it line by line; it is a real independent check, not a fit. The remark that Hamilton's equations and the Gibbs relation both fall out of Tomita-Takesaki (§4.2, eqs. 48–50) is the nicest thing in the paper. The outer-automorphism observation (§4.3, abstract:s4) is a genuine pointer, though it is a paragraph, not an argument. Soft spots, in order of weight. First, the flow is fixed by state plus subalgebra, not by state alone. §4.3 shows the Minkowski vacuum giving the boost flow on the Rindler subalgebra and nothing on the full algebra; the paper notes this and speculates about phases in footnote 7 but never confronts it. That choice of subalgebra is exactly the observer/system split general covariance was supposed to remove, and §5 concedes vagueness about state selection without noticing that subalgebra selection is a separate and harder problem. Second, §4.2's commutator-to-Poisson step is asserted, not justified — minor, and probably fixable. Third, the Hawking extension is stated and explicitly not worked out — also minor, since it is flagged. On the reader's worry about faithfulness and Type III: the framing is off. A non-Type-III algebra does not kill the modular flow, it makes it inner, which is still a flow. The load-bearing gap is underdetermination, not existence. One overreach worth noting: the line about throwing away the Standard Model lagrangian if you knew the thermal state is too strong, since the kinematics still has to be given. Recommendation: send it out. The idea is significant, the math that is there is correct, and the honest labelling of the postulate makes it a fair target for referees rather than a desk reject.

Referee report

Significance

If the identification holds, it supplies a general-covariant substitute for the missing parameter of evolution and unifies the Gibbs relation, Hamilton's equations, the Unruh temperature and (by assertion) Hawking radiation under a single structural fact about von Neumann algebras. The individual derivations that are carried out are correct and checkable line by line: the Gibbs computation in §4.1 is a standard and correctly executed Tomita-Takesaki exercise, the classical limit in §4.2 is sound, and the Rindler step correctly invokes Bisognano-Wichmann and reproduces T = a/2πk_B. The observation in §4.4 that the modular flow projects to a canonical subgroup of Out(R) is a genuine structural insight and is the part of the paper most likely to outlive the specific hypothesis. The paper is honest that the central step is a postulate rather than a theorem, and it does not overclaim mathematical novelty for the operator-algebraic ingredients, which are attributed to Connes, Takesaki and Haag. The value of the work lies in the identification and in the consistency checks, not in new mathematics.

Major objections (3)

[§4.3 and footnote 7] The hypothesis as stated in abstract:s1 and §3.1 ('the physical time is the modular group α_t of ω') is underdetermined by the state alone, and the manuscript itself supplies the counterexample. In §4.3 the same state — the Minkowski vacuum — yields the boost flow on the Rindler subalgebra A_R by Bisognano-Wichmann, while on the full algebra the state is pure and the modular operator is trivial. The text acknowledges this ('the same state may give rise to different time flows when restricted to different subalgebras') and footnote 7 speculates about phases, but no principle is offered for which subalgebra is the physically relevant one. Since the choice of subalgebra is precisely the observer/system split that general covariance was introduced to remove, this is load-bearing for the central claim and not merely a technicality. The hypothesis should be restated precisely as a map from the pair (state, subalgebra) to a flow, and the paper should either propose a selection principle for the subalgebra or state clearly that the thermal time hypothesis is incomplete without one. Note that this is a different gap from the state-selection vagueness conceded in §5, and the two should not be conflated.
[§3.1] The rhetorical claim that 'if we knew the thermal state of the fields around us ... we could then throw away the full Standard Model lagrangian without loss of information' overstates what the modular group delivers. The modular flow recovers a one-parameter group of automorphisms up to inner automorphisms and up to the constant β; it does not recover the interaction structure, the multi-point functions, or the field content, all of which remain encoded in the algebra and in the choice of state. The sentence is qualified by 'and we knew the full kinematics', but 'without loss of information' is still too strong and should be softened to a statement about the flow alone.
[Abstract vs §4.3] The abstract (abstract:s1, abstract:s2) advertises a relation to Hawking radiation, but §4.3 disposes of it in a single sentence: 'Following Unruh's initial suggestion this construction can be immediately generalised to the Schwarzschild solution ... We will not elaborate on this here.' No derivation, not even a sketch, is given. Either the Schwarzschild case should be worked out at the same level of explicitness as the Rindler case, or the abstract should be softened to claim only the Unruh result, which is actually derived.

Minor objections (7)

[§4.3 and §4.4] Two consecutive sections are both numbered 4.3 (the Rindler/Unruh section and the state-independent Out(R) section). The second should be renumbered 4.4, and the cross-reference in §5 adjusted.
[§4.2, Eq. (47)] The identity [A, Δ] = [A, ω] is written with ω denoting both the state (a positive linear functional) and, implicitly, its density matrix. A commutator with a state functional is not defined; the equation should be phrased in terms of the density matrix representing ω in the given representation, with the notation distinguished.
[§4.3, Eq. (55)] The displayed relation α_t = β 2π a^{-1} t is not dimensionally transparent and does not read as a statement about automorphisms. It would be clearer to state directly that the modular flow of the vacuum on A_R coincides with the boost flow with parameter 2πt, and then to derive t = 2πa τ as a separate line.
[§4.2, Eq. (46)] The sign in dA/dt = i[A, ln Δ] should be checked against the sign convention fixed in Eq. (8) and footnote 3; as written the two conventions appear to differ by an overall sign.
[§2.2] The claim that the reduced ADM phase space is isomorphic to the covariant phase space of solutions modulo diffeomorphisms is supported by reference [17], which is listed as unpublished lecture notes. Since this claim carries some weight in setting up the 'no Hamiltonian on Γ' picture, a published reference should be supplied if one exists.
[References [9] and [12]] Both are listed as 'to appear'. Their status should be updated to published versions where available, particularly [12], on which §4.4 depends.
[§1 and §3.1] Several typographical errors should be corrected: 'the versus of the time flow' in §3.1 should read 'the direction of the time flow'; 'The aim of this paper aims is solely' in §1 is garbled; and 'the time flow α_t it itself generates' in §3.1 has a duplicated word.

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The paper

The paper claims

Physical time is the modular flow of the thermal state, so a generally covariant quantum theory with no built-in time still produces a time flow once a thermal state is specified.

The reading

General relativity has no preferred time. Its equations describe whole histories at once, and any coordinate can serve as 'time'. Quantum theory makes this worse, because superpositions of geometries have no definite causal structure. Yet we experience a clear flow of time. Where does it come from? Connes and Rovelli propose that it comes from thermodynamics. In the mathematics of operator algebras, a theorem of Tomita and Takesaki attaches a one-parameter family of transformations to any state of a quantum system. For an ordinary thermal state, that family is exactly the usual time evolution, up to a rescaling by temperature. The authors take this seriously: the physical time flow, they suggest, is this 'modular flow' of whatever thermal state the system is in. Time is not a property of the theory; it is a property of the state. They check the idea where it can be checked. For a Gibbs state the modular flow reproduces Hamiltonian evolution, so ordinary physics is recovered. In the classical limit the same relation yields Hamilton's equations and the Gibbs distribution together. For an accelerated observer in the vacuum, the relevant algebra is the Rindler wedge, and a theorem of Bisognano and Wichmann says its modular flow is the boost. Reading the ratio of thermal to geometric time as temperature gives the Unruh temperature a/2π. The same logic is claimed to give Hawking radiation, though that step is not worked out. A further observation: different states give different modular flows, but they all agree up to inner automorphisms, so the algebra itself carries a canonical, state-independent time direction.

Extended reading notes

Core claim

The physical time flow in a generally covariant quantum theory is the modular automorphism group of the thermal state (the 'thermal time hypothesis', abstract:s1 and §3.1), so that time is state-dependent and thermodynamics rather than mechanics supplies the parameter of evolution. If correct, a theory with no preferred dynamics still has a physical time, and the Gibbs relation, Hamilton's equations, the Unruh temperature and the Hawking temperature all follow from one identification.

Load-bearing premise

The construction only works if the state of the universe is a faithful normal state on a Type III von Neumann algebra, so that the Tomita-Takesaki modular group exists and is nontrivial. §2.1 and §3.1 assume this silently, and §4.3 cites Haag for the Type III requirement rather than arguing it for the cosmological case. If the physically relevant state is not faithful, or the algebra is Type I or II, the modular flow degenerates and the identification with physical time collapses. No argument is given that the actual state of the fields around us meets the requirement.

Original abstract

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 general 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 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, as the Unruh temperature and the Hawking radiation. We also point out the existence of a state-independent notion of "time", given by the canonical one-parameter subgroup of outer automorphisms provided by the Cocycle Radon-Nikodym theorem.

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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 general 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 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, as the Unruh temperature and the Hawking radiation. We also point out the existence of a state-independent notion of "time", given by the canonical one-parameter subgroup of outer automorphisms provided by the Cocycle Radon-Nikodym theorem.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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 general covariant quantum field theory with the "evidence" of the flow of time.	background 0.98	0.29	0.05	unclear 0.63
s1	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").	contribution 0.95	0.67	0.70	hedged possibility 0.41
s2	We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita-Takesaki theorem, which allows to derive a time-flow, namely a one-parameter group of automorphisms of the observable algebra, from a generic thermal physical state.	method 0.94	0.75	0.17	asserted 1.00
s3	We study this time-flow, its classical limit, and we relate it to various characteristic theoretical facts, as the Unruh temperature and the Hawking radiation.	contribution 0.94	0.85	0.09	asserted 0.98
s4	We also point out the existence of a state-independent notion of "time", given by the canonical one-parameter subgroup of outer automorphisms provided by the Cocycle Radon-Nikodym theorem.	result 0.75	0.77	0.10	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 0.73 · evidence from proof or derivation 0.89.

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References

Works cited (20)

[1]

It is impossible to refer to all the literature on the noti on of time in physics. The following book is a collection of thousands references o n this subject: S. L. Macey, Time: a bibliographic guide , Garland, New York, 1991

View on Pith 1991
[2]

An excellent discussion and general references can be fo und in C.J. Isham, “Prima facie questions in quantum gravity” , Imperial/TP/93-94/1. 23

View on Pith
[3]

C. Rovelli, in Conceptual Problems of Quantum Gravity ed. A. Ashtekar and J. Stachel, (Birkhauser,Boston,1991); Phys. Rev. D 42 ( 1991) 2638; 43 (1991) 442

View on Pith 1991
[4]

C. Rovelli, Class. and Quant. Grav. 10, 1549 (1993)

View on Pith 1993
[5]

W.G. Unruh, Phys. Rev. D14, 870 (1976)

View on Pith 1976
[6]

S.W. Hawking, Comm. Math. Phys. 43, 199 (1975)

View on Pith 1975
[7]

R. Haag, Local Quantum Physics , Springer Verlag, Berlin 1992

View on Pith 1992
[8]

R.C. Tolman, Relativity, Thermodynamics, and Cosmology , Clarendon Press, Oxford, 1934

View on Pith 1934
[9]

A. Ashtekar, Lectures at Les Houches 1992 Summer School, to appear

View on Pith 1992
[11]

C. Rovelli, Class. and Quant. Grav. 10, 1567 (1993)

View on Pith 1993
[12]

A. Connes, ”Une classiﬁcation des facteurs de type III” , Ann. Sci. Ecole Normale Superieure 6, 133-252 (1973). A. Connes Non commutative geom- etry, Academic Press, to appear

View on Pith 1973
[13]

J. Diximier, Von Neumann algebras , North-Holland, Amsterdam 1981. J. Diximier, C⋆-algebras , North- Holland, Amsterdam 1982. O. Brat- teli, D.W. Robinson, Operator algebras and quantum statistical mechanics , Springer-Verlag, New York 1979

View on Pith 1981
[14]

M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applica - tions , Springer-Verlag Berlin 1970

View on Pith 1970
[15]

A. Einstein, ¨Uber die spezielle und die allgemeine Relativit¨ atstheori e , (Vieweg, Braunshweig, 1920); P. G. Bergmann, Helv. Phys. Ac ta Suppl. 4, 79 (1956); B.S. DeWitt in Gravitation: an introduction to recent re- search, ed. L. Witten (Wiley, New York 1962); J. Stachel, Ein stein search for general covariance 1912-1915 in Einstein and the Histor y o...

View on Pith 1956
[16]

C. Rovelli, Class. Quant. Grav. 8, 297 (1991), 317 (1991)

View on Pith 1991
[17]

A. Ashtekar, L. Bombelli and O. Reula, In: Analysis, Geometry and Me- chanics: 200 Years after Lagrange , eds. M. Francaviglia and D. Holm (North Holland, Amsterdam, 1993). C. Rovelli, An introduction to canon- ical Quantum Gravity. Lecture notes , 1992, Unpublished. 24

View on Pith 1993
[18]

Gibbs, Elementary Principles in Statistical Mechanics , Yale University Press, 1902

View on Pith 1902
[19]

R. Haag, N.M. Hugenholtz, M. Winnik, Comm. Math. Phys. 5 , 215 (1967)

View on Pith 1967
[20]

A.S. Eddington, The nature of the physical world , MacMillan, New York, 1930, pg. 99-102

View on Pith 1930
[21]

J.J. Bisognano, E.H. Wichmann, J. Math. Phys. 16, 985 (1 975); 17, 303 (1976). 25

View on Pith 1976

Forward citations

Cited by 9 Pith papers

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Subregions in semiclassical quantum gravity carry type II von Neumann algebras whose renormalized entropy equals the generalized entropy, up to a state-independent constant.
Reduced density matrix and internal dynamics for multicomponent regions
hep-th 2009-03 accept novelty 7.0 of 10

The vacuum density matrix of a massless 2D Dirac field on several disjoint intervals is diagonalized exactly, and its modular flow is shown to be non-local yet confined to one trajectory per interval.
Large time effective kinetics $\beta$-functions for quantum (2 + p)-spin glass II: Effective vertex expansion, local potential approximation and symmetries
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Bulk operators in the entanglement wedge, including on the Ryu-Takayanagi surface, can be reconstructed from boundary operators evolved by modular flow, with an explicit formula that needs only bulk-to-boundary correlators.
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hep-th 2026-03 unverdicted novelty 5.0 of 10

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Stochastic dynamics for group field theories II: Methods for nonequilibrium renormalization group
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Space from Hilbert Space: Recovering Geometry from Bulk Entanglement
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Entanglement patterns in a quantum state can be used to reconstruct an approximate spatial geometry whose curvature responds to entanglement perturbations in a way that resembles a spatial Einstein equation.

How the literature uses it

42 arXiv papers cite this in 63 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 45 sentences without a confident label

Cites it as background 18 papers

The quantity $H$ defined in this manner is called the “entanglement hamiltonian”, “modular hamiltonian” [ref] or “thermal time” [ref] hamiltonian in various contexts.

Spacetime thermodynamics without hidden degrees of freedom · Entanglement thermodynamics · 0.94
The importance of entanglement between the clock and its environment, and of the modular automorphism in particular, in the emergence of time has been discussed in the past, [ref] .

Inside the Hologram: Reconstructing the bulk observer's experience · Emergent time · 0.83
The idea that nature is built by information serves as a basis for so-called emergent theories in which classical space-time is not a fundamental concept but can be of information-statistical origin [ref] .

Probing the Information-Probabilistic Description · Introduction · 0.88

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

It is impossible to refer to all the literature on the noti on of time in physics. The following book is a collection of thousands references o n this subject: S. L. Macey, Time: a bibliographic guide , Garland, New York, 1991

View on Pith 1991

[2] [2]

An excellent discussion and general references can be fo und in C.J. Isham, “Prima facie questions in quantum gravity” , Imperial/TP/93-94/1. 23

View on Pith

[3] [3]

C. Rovelli, in Conceptual Problems of Quantum Gravity ed. A. Ashtekar and J. Stachel, (Birkhauser,Boston,1991); Phys. Rev. D 42 ( 1991) 2638; 43 (1991) 442

View on Pith 1991

[4] [4]

C. Rovelli, Class. and Quant. Grav. 10, 1549 (1993)

View on Pith 1993

[5] [5]

W.G. Unruh, Phys. Rev. D14, 870 (1976)

View on Pith 1976

[6] [6]

S.W. Hawking, Comm. Math. Phys. 43, 199 (1975)

View on Pith 1975

[7] [7]

R. Haag, Local Quantum Physics , Springer Verlag, Berlin 1992

View on Pith 1992

[8] [8]

R.C. Tolman, Relativity, Thermodynamics, and Cosmology , Clarendon Press, Oxford, 1934

View on Pith 1934

[9] [9]

A. Ashtekar, Lectures at Les Houches 1992 Summer School, to appear

View on Pith 1992

[10] [11]

C. Rovelli, Class. and Quant. Grav. 10, 1567 (1993)

View on Pith 1993

[11] [12]

A. Connes, ”Une classiﬁcation des facteurs de type III” , Ann. Sci. Ecole Normale Superieure 6, 133-252 (1973). A. Connes Non commutative geom- etry, Academic Press, to appear

View on Pith 1973

[12] [13]

J. Diximier, Von Neumann algebras , North-Holland, Amsterdam 1981. J. Diximier, C⋆-algebras , North- Holland, Amsterdam 1982. O. Brat- teli, D.W. Robinson, Operator algebras and quantum statistical mechanics , Springer-Verlag, New York 1979

View on Pith 1981

[13] [14]

M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applica - tions , Springer-Verlag Berlin 1970

View on Pith 1970

[14] [15]

A. Einstein, ¨Uber die spezielle und die allgemeine Relativit¨ atstheori e , (Vieweg, Braunshweig, 1920); P. G. Bergmann, Helv. Phys. Ac ta Suppl. 4, 79 (1956); B.S. DeWitt in Gravitation: an introduction to recent re- search, ed. L. Witten (Wiley, New York 1962); J. Stachel, Ein stein search for general covariance 1912-1915 in Einstein and the Histor y o...

View on Pith 1956

[15] [16]

C. Rovelli, Class. Quant. Grav. 8, 297 (1991), 317 (1991)

View on Pith 1991

[16] [17]

A. Ashtekar, L. Bombelli and O. Reula, In: Analysis, Geometry and Me- chanics: 200 Years after Lagrange , eds. M. Francaviglia and D. Holm (North Holland, Amsterdam, 1993). C. Rovelli, An introduction to canon- ical Quantum Gravity. Lecture notes , 1992, Unpublished. 24

View on Pith 1993

[17] [18]

Gibbs, Elementary Principles in Statistical Mechanics , Yale University Press, 1902

View on Pith 1902

[18] [19]

R. Haag, N.M. Hugenholtz, M. Winnik, Comm. Math. Phys. 5 , 215 (1967)

View on Pith 1967

[19] [20]

A.S. Eddington, The nature of the physical world , MacMillan, New York, 1930, pg. 99-102

View on Pith 1930

[20] [21]

J.J. Bisognano, E.H. Wichmann, J. Math. Phys. 16, 985 (1 975); 17, 303 (1976). 25

View on Pith 1976