1. Introduction

In both classical and quantum theories defined on fixed background space-times, the physical flow of time is represented in much the same way. Time translations correspond to a continuous one-parameter subgroup of space-time symmetries, and the dynamics are implemented either as a parametrized flow on state space (Schrödinger picture) or a parametrized group of automorphisms of the algebra of observables (Heisenberg picture). In generally covariant theories, where diffeomorphisms of the underlying space-time manifold are treated as gauge symmetries, this picture breaks down. There is no longer a canonical time-translation subgroup at the global level, nor is there a gauge-invariant way to represent dynamics locally in terms of the Schrödinger or Heisenberg pictures. Without a preferred flow on the space of states representing time, the standard way to represent physical change via functions on this space taking different values at different times also fails. This is the infamous problem of time.Footnote ¹

Connes and Rovelli (Reference Connes and Rovelli1994) propose a radical solution to the problem: the flow of time (not just its direction) has a thermodynamic origin. Any coarse-grained, statistical state naturally defines a notion of time according to which it is in equilibrium. The thermal time hypothesis (TTH) identifies this state-dependent thermal time with physical time. Drawing on tools from Tomita-Takesaki modular theory, Connes and Rovelli demonstrate how the TTH can be rigorously implemented in generally covariant quantum theories.

The idea is an intriguing one that, to date, has received little attention from philosophers. Not only does the TTH represent a striking conjecture about the origins of time, it also supplies tantalizing clues about the physical significance of Tomita-Takesaki modular theory. One of the most powerful mathematical tools we have to study the structure of operator algebras used in quantum theory, modular theory has found an increasingly diverse array of physical applications.Footnote ² Despite its importance, the basic physical ideas behind modular theory remain murky. If the TTH is right, then modular automorphism groups are employed by generally covariant quantum theories to characterize emergent dynamics.

This article represents a modest initial attempt to sally forth into rich philosophical territory. Its goal is to voice a number of technical and conceptual challenges faced by the TTH and to propose some strategies to respond. In section 2, I provide a self-contained introduction to the TTH, emphasizing the connection between Connes and Rovelli's original proposal and Rovelli's later work on timeless mechanics. (This enables us to clearly separate out various components of the TTH that are easily conflated.) In sections 3 and 4, I explore two technical challenges concerning the relationship between thermal time and proper time conjectured by the TTH and the possibility of implementing the TTH in classical theories. Finally, in section 5, I examine a trio of deeper conceptual problems concerning the flow of time in nonequilibrium states and the extent to which the TTH is background independent and gauge invariant. The outlook is mixed. I argue that while there are potentially viable strategies for addressing the two technical challenges, the three conceptual problems present a tougher hurdle for the defender of the TTH.

2. The Thermal Time Hypothesis

We usually think of theories of mechanics as describing the evolution of states and observables through time. Rovelli (Reference Rovelli2011) advocates replacing this picture with a more general timeless one that conceives of mechanics as describing relative correlations between physical quantities divided into two classes, partial and full observables. Partial observables are quantities that physical measuring devices can be responsive to but whose value cannot be predicted given a state alone. Proper time along a world line is an example of a partial observable. A clock carried by astronauts measures their proper time, but the question “What is the astronaut's proper time?” is ill posed. In contrast, a full observable is a coincidence or correlation of partial observables whose value can be determined given a state. Proper time along a world line between points where it intersects two other world lines is an example of a full observable. The question, “What is the proper time our astronaut experiences between launching from Earth and landing on the moon?” is well posed and can be determinately answered once a state is specified. In general, only measurements of full observables can be directly compared to the predictions made by a mechanical theory.

For Hamiltonian systems, Rovelli's proposed timeless mechanics takes the following form: 𝒞 is a configuration space of partial observables. A motion of the system is given by an unparametrized curve in 𝒞, representing a sequence of correlations between partial observables. The dynamically possible motions are determined by the equation $H=0$, where $H:T^{\ast }𝒞\to ℝ$ is a suitably smooth function, the timeless Hamiltonian, and the cotangent bundle, $T^{\ast }𝒞$, represents the space of partial observables and their conjugate momenta.Footnote ³

If the system has a privileged time variable, the Hamiltonian can be naturally decomposed into

(1)$\begin{array}{}H=p_t+H_0(q^i,p_i,t)\text{,}\end{array}$

where t is the partial observable in 𝒞 that corresponds to time, and p_t is its conjugate momentum (i.e., the energy). The value of t can then be used to naturally parametrize the motions of the system in a manner similar to standard Hamiltonian mechanics.

In generally covariant systems, however, there is no privileged time variable, and the Hamiltonian lacks a canonical decomposition of the form (1). Although these systems are fundamentally timeless, it is possible for a notion of time to emerge thermodynamically. According to the second law of thermodynamics, a closed system will eventually settle into a thermal equilibrium state. Such states possess a range of unique properties. They are invariant with respect to the flow of time, stable under perturbations, and passive (i.e., mechanical work cannot be extracted by cyclic processes). Viewed as part of a definition of equilibrium, this thermalization principle requires an antecedent notion of time. The TTH inverts this definition and uses the notion of an equilibrium state to select a partial observable in 𝒞 and identify it as time.

Suppose we know the full microstate of a generally covariant Hamiltonian system. Since the fundamental dynamics are given by the timeless Hamiltonian, and no special time variable is singled out, the flow of time is absent in our description of the system at the fundamental level. Suppose instead that we have a coarse-grained description of the system. If we somehow knew that this coarse-grained state were an equilibrium state, we could go on to identify the one-parameter group of state space automorphisms with respect to which it is invariant, stable, and passive. The TTH conjectures that this interpretive move is always available. Given an arbitrary coarse-grained, statistical state it is possible to find a privileged one-parameter group of state space automorphisms with respect to which it is in equilibrium. In this sense, the flow of time is a local, emergent phenomenon arising from our ignorance of the system's full state. Rovelli (Reference Rovelli2011) comments: “When we say that a certain variable is ‘the time,’ we are not making a statement concerning the fundamental mechanical structure of reality. Rather, we are making a statement about the statistical distribution we use to describe the macroscopic properties of the system that we describe macroscopically” (1486). This is the theoretical motivation for the TTH. While no partial observable plays a privileged temporal role in the fundamental physics, any coarse-graining will give rise to an emergent equilibrium dynamics that naturally selects a time parameter.

In practice, three hurdles present themselves. The first is providing a coherent mathematical characterization of equilibrium states. The second is finding a method for extracting information about the associated time flow from a specification of the coarse-grained state. Finally, in order for it to count as an emergent explanation of time, one has to show that the partial observable selected behaves as a traditional time variable in relevant limits.

For generally covariant quantum theories, Connes and Rovelli (Reference Connes and Rovelli1994) propose a concrete strategy to overcome these hurdles. Inspired by algebraic formulations of quantum field theory in curved space-time, they propose treating a generally covariant quantum theory as a noncommutative C∗-algebra of diffeomorphically invariant observables, 𝔄, along with a collection of physically possible states, {ρ}.Footnote ⁴ The states are (positive, normalized) linear functionals, $\rho :A\to \mathrm{ℝ,}$, encoding the expectation values of the observables in 𝔄. Choosing a state allows us to naturally expand 𝔄 to form a von Neumann algebra and represent it concretely as an algebra of bounded linear operators acting on a Hilbert space. This extra step is both physically and mathematically important, needed to handle boundary conditions in infinite quantum systems and to develop the tools of Tomita-Takesaki modular theory, but our discussion here will not hinge on the details.

In this setting, the properties of equilibrium states are captured by the Kubo-Martin-Schwinger (KMS) condition. Letting α_t be a one-parameter group of automorphisms of 𝔄 representing the dynamics in the Heisenberg picture and $\beta =1/T$ the inverse temperature, the condition requires that

(2)$\begin{array}{}\rho \left({\alpha }_t\right(A\left)B\right)=\rho \left(B{\alpha }_{t+i\beta }\right(A\left)\right),\end{array}$

for all $A,B\in A$. The left-hand side, ρ(α_t(A)B), represents the correlation between an arbitrary time-evolved observable, A, and another arbitrary observable, B, in the state ρ. In general $\rho \left({\alpha }_t\right(A\left)B\right)\ne \rho \left(B{\alpha }_t\right(A\left)\right)$, but for equilibrium states, the KMS condition says that these reordered correlations can be obtained from one another by substituting $t\mapsto t+i\beta$.Footnote ⁵ The physical significance of this fact is not immediately obvious, but it turns out that (2) guarantees that ρ is stable, passive, and invariant under α_t. It is also a consequence of the more familiar Gibbs postulate,

(3)$\begin{array}{}\rho =\frac{e^{-\beta H}}{\mathrm{Tr}\left[e^{-\beta H}\right]}\text{,}\end{array}$

which can be used to characterize equilibrium states in finite quantum systems governed by a discrete Hamiltonian, H. In order for (3) to be well defined, the trace in the denominator must be finite for all values of β, but in infinite quantum systems where H can have a continuous spectrum (or might not exist at all), this cannot be guaranteed. The KMS condition, however, remains valid and thus provides a mathematically tractable generalization of the Gibbs postulate for infinite quantum systems.Footnote ⁶

The KMS condition allows Connes and Rovelli to clear the first hurdle. In order to clear the second, they appeal to the technical machinery of Tomita-Takesaki modular theory. A state is said to be faithful if $\rho \left(A^{\ast }A\right)=0$ entails that $A=0$. Since every nonzero observable has nonzero expectation value, a faithful state retains information about the full algebra, 𝔄. In this case, the tools of modular theory can be applied.Footnote ⁷ Its main theorem guarantees the existence of two unique modular invariants that depend on ρ, an antiunitary operator, J, and a positive operator, ∆. (Here we will only be concerned with the latter.) The theorem goes on to say that the set of unitary operators, $\left\{{\Delta }^{\mathrm{is}}\right|s\in \mathrm{ℝ?}\}$, forms a (strongly continuous) one-parameter group,

(4)$\begin{array}{}{\sigma }_s\left(A\right)≔{\Delta }^{\mathrm{is}}A{\Delta }^{-\mathrm{is}}\text{,}\end{array}$

called the modular automorphism group. The defining state is invariant under the flow of the modular automorphism group, $\rho \left({\sigma }_s\right(A\left)\right)=\rho \left(A\right)$. Furthermore, it can be shown that $\rho \left({\sigma }_s\right(A\left)B\right)=\rho \left(B{\sigma }_{s+i}\right(A\left)\right)$. Thus, ρ satisfies the KMS condition relative to {σ_s} for inverse temperature $\beta =1$.

This is Connes and Rovelli's central observation: in a generally covariant quantum theory, every faithful state determines a canonical one-parameter group of automorphisms according to which it is a KMS state. This singles out a partial observable, the thermal time, $t_{\rho }≔s$, parametrizing the flow of the (unbounded) thermal Hamiltonian, $H_{\rho }≔-\ln \Delta$. We can go on to decompose the timeless Hamiltonian, $H=p_{t_{\rho }}+H_{\rho }$. Associated with any faithful state, there is a natural “flow of time”? according to which the system is in equilibrium.

It should be emphasized at this stage that the modular machinery employed by Connes and Rovelli requires ρ to be a mixed state. (Nontrivial C∗-algebras have no pure, faithful states.) Giving mixed states an ignorance interpretation serves to connect their procedure to the guiding idea that a coarse-grained state determines the flow of time. According to this interpretation, it is because ρ is missing some information about the universe that a one-parameter group of automorphisms naturally emerges. A state of maximal information would be pure, and the same machinery could not be used to explain the emergence of dynamics.

There are two important caveats here. First, not every mixed state is faithful, so Connes and Rovelli's proposal does not vindicate the idea that any coarse-graining determines a flow of time.Footnote ⁸ Second, as Wallace (Reference Wallace2012) argues, nothing in the quantum formalism forces us to give mixed states an ignorance interpretation. On this alternative reading, the full microstate of the system could very well be faithful, and the flow of time, while still arising from the unique statistical features of such states, would no longer be a product of our ignorance. Thus, there is a gap between much of the motivational rhetoric surrounding the TTH and the current technical results. Given the latter, all we can say is that given a statistically suitable (i.e., faithful) state, an equilibrium dynamics naturally emerges, whether or not this is a product of informational ignorance.

Regardless, the third hurdle remains. It is crucial to ask how this thermal time flow corresponds to various notions of physical time. In particular, how is thermal time related to the proper time measured by a localized observer? Although they do not establish a general theorem linking thermal time to proper time, Connes and Rovelli do make progress on the third hurdle in one important special case. Consider immortal, uniformly accelerating observers in Minkowski space-time. Because of their acceleration, they are only causally connected to a subregion of space-time known as the Rindler wedge. The Unruh effect, a well-known, if rather mysterious physical effect predicts that our Rindler observers will measure a nonzero temperature in the vacuum state,

(5)$\begin{array}{}{T}_U=\frac{\mathrm{ℝ?}a}{2\pi k_{B}c},\end{array}$

where a is the magnitude of the observers— acceleration, k_B is Boltzmann's constant, $\mathrm{ℝ?}$ is the reduced Planck constant, and c is the speed of light. An inertial observer, in contrast, is causally connected to the entirety of Minkowski space-time and measures $T_U=0$ in the vacuum state.

Connes and Rovelli note that another deep theorem connecting modular theory to space-time physics, the Bisognano-Wichmann theorem, provides a direct link between thermal time and the proper time measured by a Rindler observer. Let 𝔄(W) be the algebra of observables localized in the Rindler wedge, W. The vacuum state is faithful for 𝔄(W), so there is a well-defined modular automorphism group, {σ_s}, associated with vacuum wedge algebra. The Bisognano-Wichmann theorem says that {σ_s} coincides with the group of wedge-preserving Lorentz boosts. Since the latter implement proper time translations along the Rindler observers— world line, we find that thermal time is directly proportional to their proper time,

(6)$\begin{array}{}\frac{s}{\tau }=-T_U\text{,}\end{array}$

leading Connes and Rovelli to propose that the Unruh temperature should be physically interpreted as the ratio between thermal and proper time.Footnote ⁹

We can now summarize the main content of the TTH:

The TTH has three broad pillars: (i) the motivating idea that the flow of time is selected at the level of statistical mechanics in a fundamentally timeless, generally covariant theory; (ii) a quantum mechanical model for such a selection mechanism, identifying thermal time with the modular automorphism group naturally associated with any faithful state; and (iii) a conjecture that in the limit where a geometric notion of proper time exists, the Unruh temperature is interpretable as the ratio of thermal time to proper time. This is a bold idea with numerous potential implications for quantum physics and cosmology. Over the next three sections, we will consider a series of technical and conceptual objections to the TTH.

3. Thermal Time and Proper Time

Much of the theoretical support for the TTH comes from the close connection between thermal time and proper time established by the Bisognano-Wichmann theorem. Consequently, most of the attention that the TTH has received in the mathematical physics literature has focused on the third pillar noted above. But the Rindler observers are highly idealized; they are immortal and uniformly accelerating. Moreover, the Bisognano-Wichmann theorem assumes that the background global state is the vacuum state. Loosening each of these assumptions leads to technical complications that collectively appear quite daunting.

Realistic observers are mortal and therefore have causal access to a different region of space-time, the double cone formed by the intersection of their future light cone at birth and past light cone at death. What relationship holds between thermal time and proper time for uniformly accelerating, mortal observers in the vacuum state? An immediate problem arises due to the fact that their proper time is bounded whereas thermal time is unbounded. We therefore cannot expect a simple, linear relationship like (6).

Since the Rindler wedge can be related to a double cone by a conformal transformation, in conformal field theories the modular group for Rindler observers can be used to explicitly compute the modular group for their mortal counterparts. Using this trick, Martinetti and Rovelli (Reference Martinetti and Rovelli2003) prove that the mortal analogue of (6) is given by

(7)$\begin{array}{}\frac{\mathrm{ds}\left(\tau \right)}{d\tau }=\frac{-\mathrm{ℝ?}L{a}^2}{2\pi K_B{c}^3\left(\sqrt{1+\frac{a^2{L}^2}{c^4}-\cosh \frac{a\tau }{c}}\right)}\text{,}\end{array}$

where L is half the lifetime of the observer who is born at $\tau =-L/c$ and dies at $\tau =L/c$. For both long-lived and highly accelerating mortal observers, (7) converges to the Unruh temperature. Moreover, (7) is approximately constant for most of their lifespan, allowing the temperature to be interpreted as the local ratio between thermal and proper time in accordance with the TTH. Toward their birth and death, however, the two quantities rapidly diverge, calling this interpretation into doubt and raising a host of phenomenological questions.Footnote ¹⁰

Unless our world is conformally invariant, Martinetti and Rovelli's result is of limited applicability. Existing results strongly suggest that double cone modular groups in generic quantum field theories cannot usually be given a dynamical interpretation. Any local dynamics must map double-cone-shaped subregions onto double-cone-shaped subregions while preserving space-like separation and time-like ordering. Trebels (Reference Trebels1997) proves that, if a double cone modular group satisfies these minimal geometric requirements, it will be related to the conformal case by a scaling transformation.Footnote ¹¹ In other words, a relationship like (7) is the best that we can hope for. But nothing guarantees that the double cone modular groups in a generic quantum field theory will have these geometric properties, and Saffary (Reference Saffary2005) has argued that in any theory with massive particles, they will not.Footnote ¹²

The situation only becomes thornier when nonuniform acceleration and nonvacuum states are taken into account. In the latter case, it can be shown that the action of the modular automorphism group for the Rindler wedge in a nonvacuum state will differ from the vacuum action by a nontrivial automorphism.Footnote ¹³ This suggests that we should not generally expect the wedge modular group to have a geometric interpretation either. Even when it does, it will not be related to the vacuum action by some simple rescaling. In the case of a nonuniformly accelerating observer, the flow of proper time will fluctuate with acceleration while the flow of thermal time remains constant. If the TTH is correct, the observer will measure a shifting temperature reflecting the changing ratio between thermal and proper time (an idea supported by limited existing results on the Unruh effect for nonuniformly accelerating observers; e.g., Jian-yang, Aidong, and Zheng Reference Jian-yang, Aidong and Zheng1995). The challenge for the TTH is to explain the phenomenological experience of the observers, who will presumably age along with their fluctuating proper time, not the constant thermal time flow.

In the face of these obstacles, the defenders of the TTH have at least four options on the table. They can hold out hope for a suitably general dynamical interpretation of modular groups for wedges and double cones in a wide class of physically significant states. There is some indication that states of compact energy (including states satisfying the physically motivated Döplicher-Haag-Roberts and Buchholz-Fredenhagen selection criteria) give rise to well-behaved modular structure on wedges (Borchers Reference Borchers2000). It is not clear that this is sufficient to ensure that modular automorphisms act geometrically, however, and in light of the limitations imposed by Trebels's and Saffary's no-go results, this first strategy seems like a long shot.

Alternatively, they could reject the idea that the thermal time flow determines the temporal metric directly. Thermal time would only give rise to the order, topological, and group theoretic properties of physical time. Metrical properties would be determined by a completely different set of physical relations. Some support for this idea comes from the justification of the clock hypothesis in general relativity. Rather than stipulating the relationship between proper time, τ, and the length of a time-like curve ||γ||, Fletcher (Reference Fletcher2013) shows that for any $\epsilon >0$, there is an idealized light clock moving along the curve that will measure ||γ|| within ε. This justifies the clock hypothesis by linking the metrical properties of space-time to the readings of tiny idealized light clocks. If the metrical properties of time experienced by localized observers arise via some physical mechanism akin to light clock synchronization, this would explain why the duration of time felt by the observers matches their proper time and not the geometrical flow of thermal time.

In line with this idea, Rovelli makes a number of allusions to the concept of an entropy clock as discussed by Eddington (Reference Eddington1935). Eddington maintains that the order of temporal events is determined by the thermodynamic arrow of time. An entropy clock measures temporal order by correlating events with decreases in entropy. He describes a simple example: “An electric circuit is composed of two different metals with their two junctions embedded respectively in a hot and cold body in contact. The circuit contains a galvanometer which constitutes the dial of the entropy clock. The thermoelectric current in the circuit is proportional to the difference of temperature of the two bodies; so that as the shuffling of energy between them proceeds, the temperature difference decreases and the galvanometer reading continually decreases”? (101). A reliable entropy clock must be in contact with its environment to work properly. In contrast, a reliable metrical clock must be isolated from thermodynamic disturbances. Since the engineering demands pull in separate directions, it might turn out that our phenomenological experience of time is similarly bifurcated.

A third strategy would be to argue that the metrical properties of time emerge from modular dynamics in the short-distance/high-energy limit. If a quantum field theory has a well-defined ultraviolet completion, the renormalization group flow in this regime should approach a conformal fixed point. Buchholz and Verch (Reference Buchholz and Verch1995) prove that, in this limit, the double cone modular operators act geometrically like wedge operators implementing proper time translations along the observers— world line. It is unlikely that the physics at this scale would directly affect phenomenology, but the asymptotic connection might turn out to be important for explaining the metrical properties of space-time (which bigger, more realistic light clocks measure) as emergent features of some underlying theory of quantum gravity.

A final option would be to go back to the drawing board. Rovelli and Connes briefly note that since the modular automorphism groups associated with each faithful state are connected by inner automorphisms, they all project down onto the same one-parameter group of outer automorphisms of the algebra.Footnote ¹⁴ The TTH could be revised to claim that this canonical state-independent flow represents the nonmetrical flow of physical time.

It is unclear how viable of a strategy this really is. One significant reason for doubt is the fact that the move only works to recover a flow of time in quantum systems described by certain kinds of algebras, specifically type III (and type II_∞) von Neumann algebras. Only in these cases will the relevant one-parameter group of outer automorphisms be nontrivial. In systems described by type I (and type II₁) algebras there is simply no passage of time according to the revised TTH.Footnote ¹⁵

Although various theorems in algebraic and constructive quantum field theory indicate that the local algebras assigned to double cones and wedges are generically type III, it is not obvious that this will save the revised TTH. The global algebras that appear in quantum field theory are almost always type I, and so the revised TTH lacks a coherent story about the emergence of time at the cosmological level. Moreover, it is far from evident that observers like us actually have experimental and phenomenological access to the type III character of the local algebras at all. At the end of the day, when we make numerical calculations and do experiments, we use effective field theories and nonrelativistic quantum mechanics to describe the world around us. In both cases we rely on type I algebras. The moral at the heart of the TTH is supposed to be that time emerges from a coarse-grained description of a fundamentally timeless reality. A moment's reflection on the kinds of coarse-grained descriptions that we actually give renders the idea that the flow of time originates in the type III character of wedge and double cone algebras rather implausible. It would be amazing if our experience of time had such a delicate source.

Because of these difficulties, it appears that the second strategy outlined above offers the best path forward for the defender of the TTH. Temporal topology and ordering is determined by the state-dependent modular automorphism group, while the temporal metric has a different origin yet to be explained. Although this requires either modifying or abandoning the third pillar of the TTH, it preserve the first two pillars and appears to be more plausible than a fully geometric interpretation of the local modular automorphism groups.

4. Thermal Time in Classical Theories

Turning our attention to the first two pillars, it is interesting to note that the motivating idea that coarse-graining determines a thermal dynamics does not obviously require the underlying timeless theory to be quantum mechanical. The proposed modular selection mechanism does, however, appear to crucially rely on the noncommutativity of quantum observables. Classical observables are typically represented by smooth functions over a state space manifold, and these can naturally be equipped with the structure of a commutative C∗-algebra. But in this case, since all observables commute, every state over the algebra is tracial; that is, $\rho \left(\mathrm{fg}\right)=\rho \left(\mathrm{gf}\right)$ for all observables f, g. As a consequence of the main Tomita-Takesaki theorem, it follows that every modular automorphism group acts as the identity, trivializing the flow of thermal time.

Without a coherent mathematical procedure for extracting the thermal dynamics from a chosen state, a classical version of the TTH remains out of grasp. Does one exist, or is the TTH a uniquely quantum mechanical solution to the problem of time?Footnote ¹⁶ Investigating this question will help us to better understand the scope and content of the TTH. In addition, it could play a significant role in explaining the emergence of time in the classical limit, $\mathrm{ℝ?}\to 0$. Of course, a full understanding of this limit requires grappling with a host of entangled philosophical issues, most notably the measurement problem. Our aim here is more modest, to assess a proposed classical selection mechanism briefly sketched by Connes and Rovelli in their original paper. As we will go on to see, the idea stands on firmer foundational footing than one might initially suspect and can even be linked to a classical analogue of Tomita-Takesaki modular theory.

Arguing by analogy with standard quantization procedures, Connes and Rovelli suggest that in classical theories commutators should be replaced by Poisson brackets. With respect to the Poisson bracket, the classical observables form a noncommutative algebra. Given an arbitrary coarse-grained state, ρ, represented by a probability distribution over state space, and reasoning by analogy with the Gibbs postulate, they introduce the “thermal Hamiltonian,”?

(8)$\begin{array}{}{H}_{\rho }≔-\text{ln}\rho \text{.}\end{array}$

If ρ is nowhere vanishing (an assumption analogous to faithfulness in the quantum case), (8) defines a corresponding Hamiltonian vector field parametrized by thermal time, s. With respect to this vector field, the evolution of an arbitrary observable, f, is given by

(9)$\begin{array}{}\frac{d}{\mathrm{ds}}f=\{-\text{ln}\rho ,f\}\text{,}\end{array}$

where { , } denotes the Poisson bracket.

In fact, the parallels between the classical and quantum versions of the TTH run much deeper than this ansatz hints. It is commonly thought that the essential formal difference between classical and quantum mechanics is whether observables commute, but this emphasis on commutativity significantly obscures the rich algebraic structures employed by each theory.

On the quantum side, Alfsen and Shultz (Reference Alfsen and Shultz1998) emphasize that the usual operator product in a noncommutative C∗-algebra is really two different products in disguise. It has a natural decomposition,

(10)$\begin{array}{}\mathrm{AB}=A·B-i(A\star B)\text{,}\end{array}$

where $A”?B$ is a commutative, nonassociative Jordan product and $A\star B$ is a noncommutative, associative Lie product. The former, defined $A”?B≔1/2(\mathrm{AB}+\mathrm{BA})$, encodes all spectral information about the observables. The latter, defined using the commutator, $A\star B≔i/2(\mathrm{AB}-\mathrm{BA})$, encodes the generating relationship between observables and state space symmetries. This observation reveals that noncommutative C∗-algebras are special cases of a more general class of Lie-Jordan algebras. Commutative C∗-algebras are not like this. Since the commutator vanishes, there is no natural Lie product, and essentially all that is leftover is a Jordan algebra encoding spectra.

On the classical side, Noether's theorem indicates that we should expect a similar generating relationship between observables and symmetries to hold. Indeed, in classical theories where state space is assumed to be a symplectic manifold, or more generally a Poisson manifold, it turns out that the algebra of observables also has a natural Lie-Jordan structure. Pointwise multiplication of smooth functions defines a commutative, associative Jordan product, $f”?g≔\mathrm{fg}$, encoding spectral information. The Poisson bracket determines a noncommutative, associative Lie product, $f\star g≔\{f,g\}$, encoding how classical observables generate Hamiltonian vector fields.

The moral is this. If we naively choose to model classical systems using commutative C∗-algebras, we lose an important kind of information about the link between symmetries and observables. It is precisely this kind of information that is needed to formulate the technical details of the TTH. Reflecting on the physics, a much better choice is an associative Lie-Jordan algebra, which can be more directly compared to the nonassociative Lie-Jordan algebras employed by quantum theory.

This is the perspective adopted by the deformation and geometric quantization programs, two of the most mathematically rigorous approaches to quantization currently on the table.Footnote ¹⁷ In this setting, Gallavotti and Pulvirenti (Reference Gallavotti and Pulvirenti1976) use the Poisson bracket to define a classical analogue of the KMS condition, and Basart et al. (Reference Basart, Flato, Lichnerowicz and Sternheimer1984) link it to the quantum KMS condition in the $\mathrm{ℝ?}\to 0$ limit. This suggests that many of the tools needed for a classical version of the TTH already exist. Perhaps most compellingly, these include initial strides toward a classical analogue of Tomita-Takesaki modular theory made by Weinstein (Reference Weinstein1997).

Relative to a volume form, µ, on classical state space (assumed here to be a Poisson manifold), Weinstein defines the corresponding modular vector field,

(11)$\begin{array}{}{\varphi }_{\mu }:f\to {\mathrm{div}}_{\mu }{X}_f\text{,}\end{array}$

where X_f is the Hamiltonian vector field associated with a classical observable, f. Weinstein proposes ϕ_µ as the classical analogue of the modular automorphism group. Intuitively, it characterizes the extent to which Hamiltonian vector fields, X_f, are divergence free with respect to the volume form µ, vanishing if and only if all Hamiltonian vector fields are divergence free.

Connecting the dots, we can trace a direct link between Weinstein's classical modular theory and the TTH. In the special case that state space is a symplectic manifold, there is a natural volume form, the Liouville form, defined in terms of the symplectic structure. Letting µ be the Liouville form, a quick calculation reveals that any nowhere-vanishing state, ρ, defines a nontrivial modular vector field,

(12)$\begin{array}{}{\varphi }_{\rho \mu }=X_{-\text{ln }\rho }\text{,}\end{array}$

equivalent to the vector field generated by the Hamiltonian −lnρ.Footnote ¹⁸ We immediately recognize this as the thermal Hamiltonian (8) postulated by Connes and Rovelli. The defining state, ρ, is invariant with respect to the corresponding dynamics and satisfies the Gibbs postulate, $\rho =e^{-\beta H_{\rho }}$, for inverse temperature $\beta =1$. Just as in the quantum case, the classical thermal Hamiltonian can therefore be identified with the generator of state-dependent modular symmetries.

This deep structural parallel suggests that the TTH is not essentially quantum mechanical and that the mathematics of modular theory can provide a coherent mechanism for selecting a preferred thermal time variable in classical theories too.Footnote ¹⁹ In addition, beyond its potential application to the TTH, Weinstein's framework offers a valuable laboratory for exploring the physical significance of modular theory in domains where the interpretational complexities bedeviling quantum theories do not arise.

5. Conceptual Challenges

As we have seen in the previous two sections, the TTH faces a number of technical challenges (some of which look easier to overcome than others). Even if the third pillar needs to be modified in light of the challenges discussed in section 3, the idea that a faithful state determines the nonmetrical flow of time has proven resilient, and there is a plausible modular selection mechanism at play in both classical and quantum theories. There are, however, several deeper conceptual problems looming in the background that pose a more serious challenge to the hypothesis. Three of the most pressing raise questions about the coherence of the motivating idea behind the TTH and its adequacy in providing a solution to the problem of time.

The first is the nonequilibrium problem. While the TTH provides a coherent mathematical mechanism for selecting a nonmetrical time flow, it is not clear that we should always view this flow as physical time. According to the thermal dynamics, the defining state is always a KMS state, but if it is a nonequilibrium state with respect to our ordinary conception of time, thermal time and physical time do not align. Relative to thermal time, a cube of ice in a cup of hot coffee is in an invariant equilibrium state. This is the “incredulous stare”? that often confronts the TTH. Only for states that are true equilibrium states will the thermal time be physical time.

It would be incorrect to infer that the TTH rules out any thermodynamical change. A system in a KMS state can still exhibit fluctuations away from equilibrium. The defender of the TTH could try to argue that local nonequilibrium behavior can be viewed as fluctuations in some thermal background state. On this approach, the local flow of time in my office according to which the ice melts and the coffee cools is defined not by the thermal state of the ice/coffee system but by the thermal state of some larger enveloping system.

Hints in this direction can be found in Rovelli (Reference Rovelli1993). In this paper, Rovelli explores the notion of thermal time in a spatially homogenous, isotropic Robertson-Walker universe filled with blackbody radiation. Such a model is a plausible approximation of a universe much like our own. The radiation represents the cosmic microwave background (CMB), highly redshifted light left over from a phase early in the universe's history during which photons first decoupled from the cosmic plasma. In Robertson-Walker models, there is a natural notion of cosmological time given by the proper time experienced by a privileged class of observers comoving with the expansion of the universe, for whom the universe always appears isotropic. (This is the time parameter usually employed in discussions of standard big bang cosmology.) Rovelli shows that the thermal time induced by equilibrium states of the CMB will be related to this cosmological time by a constant rescaling.

Although this nontrivial result is exciting, there remains a large explanatory gap between the physics it suggests and the temporal phenomenology of human observers. The CMB is a nearly uniform field of microwaves with temperature 2.7 K. Without specialized equipment—radio telescopes, radiometers, spectrophotometers—human observers would not even know it was there. It is highly implausible that our faculties of perception are sensitive to the thermal features of the CMB. The proposed chain of explanation must be longer and more complex. Thermal time explains the emergence of cosmological time, and then cosmological time is shown to be a natural measure of time for a particular class of observers. Even if the first half of this story can be convincingly filled out (see the related background-dependence problem introduced below), the vastly different scales involved in the second half of the story should give us pause. It is not exactly true that the universe appears isotropic to us. In only appears isotropic at the very largest scales, when we look beyond the earth, the solar system, the Milky Way, the local group, and so on. This process requires significant inductive extrapolation from what we directly experience.Footnote ²⁰ At cosmological scales humans, stars, and galaxies might plausibly be viewed as small fluctuations in a largely homogeneous, isotropic background. Consequently, equilibrium thermal dynamics might well describe the universe at this scale. But at the scales humans occupy, the local universe is highly nonhomogeneous and nonisotropic. If our temporal phenomenology is grounded in what we directly experience, and this experience is decidedly nonequilibrium, it is hard to see how the order of explanation could plausibly run from cosmological time to local time.

Even if this challenge can be overcome, there is an additional wrinkle. Probably the most popular explanation for the arrow of time among physicists and philosophers alike, the past hypothesis requires that in one temporal direction the universe is in an incredibly low-entropy state. But if thermal time is identified with physical time, this kind of asymmetric boundary condition is ruled out. The universe is in a KMS state with respect to thermal dynamics. It has high entropy in both of the temporal directions determined by the flow of thermal time. The TTH is sometimes linked to the past hypothesis and motivated by parity of reasoning—if the direction of time has a thermodynamic origin, maybe the underlying flow of time does too—but the past hypothesis and the TTH are in fact deeply at odds with one another. The TTH forces us to adopt a rather unappealing “Boltzmann brain”? view of cosmology as large-scale fluctuations from equilibrium.Footnote ²¹

If defenders of the TTH balk at this conclusion, they have limited options on the table. One is to temper the view by only allowing certain reference states to determine the flow of thermal time, but the challenge of specifying a class of equilibrium states without an antecedent time flow was what prompted the permissiveness of the TTH in the first place. Furthermore, if a system is not actually in one of these reference states, it is hard to envision how a counterfactual state of affairs could determine the actual flow of time. This dilemma might motivate the defender of the TTH to explore the state-independent, outer modular flow as a last-ditch option. Identifying physical time with this flow would render it possible in principle to reconcile the TTH and the past hypothesis; however, the criticisms discussed at the end of section 3 must be overcome. In particular, if the global algebra is type I, then the triviality of the outer modular flow presents a new puzzle for quantum cosmology to grapple with.

A second, closely related worry to the nonequilibrium problem has been voiced by Earman (Reference Earman2011) and Ruetsche (Reference Ruetsche2014). In the physical situations in which we can justify viewing the modular automorphism group as a kind of dynamics, it seems this is only possible because we already have a rich spatiotemporal structure in the background. This casts doubt on whether the TTH can provide a coherent definition of time in situations in which such structure is absent (as required to solve the full problem of time).

In the scenario described by the original Bisognano-Wichmann theorem, we are focused on space-like wedges in Minkowski space-time. We immediately recognize the geometric significance of the modular automorphism group because its flow is everywhere time-like. The orbits of σ_s correspond to a clear class of observer world lines, and $\mathrm{ds}/d\tau$ is constant along those world lines, yielding a simple scaling relation between s and τ. Similarly, in the CMB model discussed above, the geometric interpretation of thermal time is secured by relating it to cosmological time in a highly symmetric Robertson-Walker universe. In other cases, even when the modular operators act geometrically, it can be hard to recognize the modular automorphisms as dynamical. The scaling relation for double cone modular groups in conformal field theories (eq. [7]) shows that the relation between thermal and proper time can be highly nontrivial (and this is the best case scenario for double cones).

In general, calculating the explicit action of local modular automorphism groups is a very hard problem. In all of the cases outlined above in which we can perform the calculations, extracting a dynamical interpretation requires antecedent knowledge of the background space-time structure. In generic models of general relativity with no global time-like killing fields and no global isometries, such an interpretation may no longer be possible. Moreover, if the ultimate goal is to use the TTH in conjunction with an eventual theory of quantum gravity to explain the emergence of space-time itself, we cannot even appeal to the local Lorentzian geometry of space-time to aid us. The problem is exacerbated if the TTH is modified in response to the nonequilibrium problem by restricting the set of states in which modular automorphisms define the flow of time. Unless the modular group can always be viewed dynamically, the defender of the TTH will be hard-pressed to find constraints capable of separating the dynamical cases from the nondynamical cases that are suitably independent of background spatiotemporal structure. We will call this second problem, the background-dependence problem.Footnote ²²

The third and final problem is the gauge problem. In spite of all the challenges discussed above, the TTH does succeed in providing a means to select a privileged one-parameter flow on the space of full, gauge-invariant observables of a generally covariant theory. What makes this flow interpretable as a dynamical flow, however, is its description as a sequence of correlations between partial observables. The difficulty is that these partial observables are not diffeomorphism invariant. When an object changes position, we measure two gauge-invariant quantities, the position-of-the-object-at-s ₁ and the position-of-the-object-at-s ₂. We can describe these as measurements of correlations between position and time partial observables, but to do so requires a gauge-dependent deparametrization of the timeless Hamiltonian.

Assuming that we treat diffeomorphisms in generally covariant theories as standard gauge symmetries (which is how we got into the problem of time in the first place), then only diffeomorphism-invariant quantities will represent objective features of our world. The partial observables are just superfluous descriptive fluff. The problem is not the resultant timelessness of fundamental physics. The TTH adopts this dramatic conclusion willingly. The problem is that the TTH is supposed to explain how the appearance of time and change emerge from timeless foundations. But the explanation given is couched in gauge-dependent language, and it is not apparent how we can extract a gauge-invariant story from it.

An analogy with classical space-time physics will serve to illuminate the central issue. It is widely thought that the invariance of Newton's second law with respect to Galilean boosts indicates that there are no objective facts about absolute velocity in classical space-time. There are, however, objective facts about relative velocities. By selecting a preferred reference frame (i.e., fixing a gauge), we can introduce absolute velocities into our theoretical description of the world and use them to compute gauge-invariant relative velocities. But we cannot use correlations between absolute velocities to explain facts about relative velocity. Objectively speaking, there are no such correlational facts to appeal to. Instead, we must restrict ourselves to the gauge-invariant structure of classical space-time.

This structure can be captured by modeling classical space-time as Galilean space-time. Intuitively, Galilean space-time consists of a time-ordered stack of three-dimensional spatial slices. On each spatial slice, there is a metric determining facts about relative spatial distance between objects at that time. Across slices there is a temporal metric determining temporal distances and an affine structure characterizing deviations from inertial trajectories. Crucially, there is no spatial metric across slices, and as a result, there are no objective facts about absolute velocity. Despite this, there are still objective facts about relative velocity. Since the relative spatial distance between two objects at a given time is gauge-invariant, their relative velocity can be defined as the rate of change of this relative distance quantity. The world lines of objects in relative motion correspond to nonparallel four-dimensional curves in Galilean space-time.

Analogously, in a generally covariant setting we can freely introduce partial observables and use correlations between them to calculate and predict emergent dynamical behavior, but we cannot use these correlations to explain that behavior. At this stage we lack a gauge-invariant picture of generally covariant theories akin to the one provided by Galilean space-time in the example above. The TTH, at least in its present form, does not provide one.

A radical option is to reject the standard story about gauge symmetries. Rovelli (Reference Rovelli2014) suggests that gauge-dependent quantities are more than just mathematical redundancies, arguing that they are critical for understanding interactions between physical systems: “they describe handles through which systems couple: they represent real relational structures to which the experimentalist has access in measurement by supplying one of the relata in the measurement procedure itself”? (91). On this picture, gauge-invariant quantities are intrinsically relational. Certain gauge-dependent quantities supply the relata, carrying modal information about the possible ways that free systems can interact with each other. For example, in classical electromagnetism, the interaction term in the Lagrangian, −j ^µA _µ, depends explicitly on the gauge-dependent vector potential. The Lagrangian itself is gauge invariant, but to interpret it as describing the coupling between a charge density characterized by j ^µ and the electromagnetic field, we must recognize A _µ as a genuine feature of the field, the “handle”? to which charge couples (albeit in a manner that ultimately does not depend on the choice of a gauge-dependent coordinate system). The details of Rovelli's new proposal still need to be hammered out.Footnote ²³ It should be emphasized that it marks a significant break from the received view on gauge.

Can a revised form of the TTH provide us with the explanatory tools to understand the flow of thermal time without reference to gauge-dependent partial observables, or does the framework of timeless mechanics require us to revise our conception of how explanation, ontology, and gauge symmetries are related? Whether or not the TTH can save time may ultimately rest on the solutions to these new reincarnations of vexingly familiar philosophical problems.