When I made my previous two posts about ideas of “state”, one thing I was aiming at was to say something about the relationships between states and dynamics. The point here is that, although the idea of “state” is that it is intrinsically something like a snapshot capturing how things are at one instant in “time” (whatever that is), extrinsically, there’s more to the story. The “kinematics” of a physical theory consists of its collection of possible states. The “dynamics” consists of the regularities in how states change with time. Part of the point here is that these aren’t totally separate.

Just for one thing, in classical mechanics, the “state” includes time-derivatives of the quantities you know, and the dynamical laws tell you something about the *second* derivatives. This is true in both the Hamiltonian and Lagrangian formalism of dynamics. The Hamiltonian function, which represents the concept of “energy” in the context of a system, is based on a function , where is a vector representing the values of some collection of variables describing the system (generalized position variables, in some configuration space ), and the are corresponding “momentum” variables, which are the other coordinates in a phase space which in simple cases is just the cotangent bundle . Here, refers to mass, or some equivalent. The familiar case of a moving point particle has “energy = kinetic + potential”, or for some potential function . The symplectic form on can then be used to define a path through any point, which describes the evolution of the system in time – notably, it conserves the energy . Then there’s the Lagrangian, which defines the “action” associated to a path, which comes from integrating some function living on the tangent bundle , over the path. The physically realized paths (classically) are critical points of the action, with respect to variations of the path.

This is all based on the view of a “state” as an element of a set (which happens to be a symplectic manifold like or just a manifold if it’s ), and both the “energy” and the “action” are some kind of function on this set. A little extra structure (symplectic form, or measure on path space) turns these functions into a notion of dynamics. Now a function on the space of states is what an observable is: energy certainly is easy to envision this way, and action (though harder to define intuitively) counts as well.

But another view of states which I mentioned in that first post is the one that pertains to statistical mechanics, in which a state is actually a statisticial distribution on the set of “pure” states. This is rather like a function – it’s slightly more general, since a distribution can have point-masses, but any function gives a distribution if there’s a fixed measure around to integrate against – then a function like becomes the measure . And this is where the notion of a Gibbs state comes from, though it’s slightly trickier. The idea is that the Gibbs state (in some circumstances called the Boltzmann distribution) is the state a system will end up in if it’s allowed to “thermalize” – it’s the maximum-entropy distribution for a given amount of energy in the specified system, at a given temperature . So, for instance, for a gas in a box, this describes how, at a given temperature, the kinetic energies of the particles are (probably) distributed. Up to a bunch of constants of proportionality, one expects that the weight given to a state (or region in state space) is just , where is the Hamiltonian (energy) for that state. That is, the likelihood of being in a state is inversely proportional to the exponential of its energy – and higher temperature makes higher energy states more likely.

Now part of the point here is that, if you know the Gibbs state at temperature , you can work out the Hamiltonian

just by taking a logarithm – so specifying a Hamiltonian and specifying the corresponding Gibbs state are completely equivalent. But specifying a Hamiltonian (given some other structure) completely determines the dynamics of the system.

This is the classical version of the idea Carlo Rovelli calls “Thermal Time”, which I first encountered in his book “Quantum Gravity”, but also is summarized in Rovelli’s FQXi essay “Forget Time“, and described in more detail in this paper by Rovelli and Alain Connes. Mathematically, this involves the Tomita flow on von Neumann algebras (which Connes used to great effect in his work on the classification of same). It was reading “Forget Time” which originally got me thinking about making the series of posts about different notions of state.

Physically, remember, these are von Neumann algebras of operators on a quantum system, the self-adjoint ones being observables; states are linear functionals on such algebras. The equivalent of a Gibbs state – a thermal equilibrium state – is called a KMS (Kubo-Martin-Schwinger) state (for a particular Hamiltonian). It’s important that the KMS state depends on the Hamiltonian, which is to say the dynamics and the notion of time with respect to which the system will evolve. Given a notion of time flow, there is a notion of KMS state.

One interesting place where KMS states come up is in (general) relativistic thermodynamics. In particular, the effect called the Unruh Effect is an example (here I’m referencing Robert Wald’s book, “Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics”). Physically, the Unruh effect says the following. Suppose you’re in flat spacetime (described by Minkowski space), and an inertial (unaccelerated) observer sees it in a vacuum. Then an accelerated observer will see space as full of a bath of particles at some temperature related to the acceleration. Mathematically, a change of coordinates (acceleration) implies there’s a one-parameter family of automorphisms of the von Neumann algebra which describes the quantum field for particles. There’s also a (trivial) family for the unaccelerated observer, since the coordinate system is not changing. The Unruh effect in this language is the fact that a vacuum state relative to the time-flow for an unaccelerated observer is a KMS state relative to the time-flow for the accelerated observer (at some temperature related to the acceleration).

The KMS state for a von Neumann algebra with a given Hamiltonian operator has a density matrix , which is again, up to some constant factors, just the exponential of the Hamiltonian operator. (For pure states, , and in general a matrix becomes a state by which for pure states is just the usual expectation value value for A, ).

Now, things are a bit more complicated in the von Neumann algebra picture than the classical picture, but Tomita-Takesaki theory tells us that as in the classical world, the correspondence between dynamics and KMS states goes both ways: there is a flow – the Tomita flow – associated to any given state, with respect to which the state is a KMS state. By “flow” here, I mean a one-parameter family of automorphisms of the von Neumann algebra. In the Heisenberg formalism for quantum mechanics, this is just what time is (i.e. states remain the same, but the algebra of observables is deformed with time). The way you find it is as follows (and why this is right involves some operator algebra I find a bit mysterious):

First, get the algebra acting on a Hilbert space , with a cyclic vector (i.e. such that is dense in – one way to get this is by the GNS representation, so that the state just acts on an operator by the expectation value at , as above, so that the vector is standing in, in the Hilbert space picture, for the state ). Then one can define an operator by the fact that, for any , one has

That is, acts like the conjugation operation on operators at , which is enough to define since is cyclic. This has a polar decomposition (analogous for operators to the polar form for complex numbers) of , where is antiunitary (this is conjugation, after all) and is self-adjoint. We need the self-adjoint part, because the Tomita flow is a one-parameter family of automorphisms given by:

An important fact for Connes’ classification of von Neumann algebras is that the Tomita flow is basically unique – that is, it’s unique up to an inner automorphism (i.e. a conjugation by some *unitary* operator – so in particular, if we’re talking about a relativistic physical theory, a change of coordinates giving a different parameter would be an example). So while there are different flows, they’re all “essentially” the same. There’s a unique notion of time flow if we reduce the algebra to its cosets modulo inner automorphism. Now, in some cases, the Tomita flow consists entirely of inner automorphisms, and this reduction makes it disappear entirely (this happens in the finite-dimensional case, for instance). But in the general case this doesn’t happen, and the Connes-Rovelli paper summarizes this by saying that von Neumann algebras are “intrinsically dynamic objects”. So this is one interesting thing about the quantum view of states: there is a somewhat canonical notion of dynamics present just by virtue of the way states are described. In the classical world, this isn’t the case.

Now, Rovelli’s “Thermal Time” hypothesis is, basically, that the notion of time is a state-dependent one: instead of an independent variable, with respect to which other variables change, quantum mechanics (per Rovelli) makes predictions about correlations between different observed variables. More precisely, the hypothesis is that, given that we observe the world in some state, the right notion of time should just be the Tomita flow for that state. They claim that checking this for certain cosmological models, like the Friedman model, they get the usual notion of time flow. I have to admit, I have trouble grokking this idea *as fundamental physics*, because it seems like it’s implying that the universe (or any system in it we look at) is always, *a priori*, in thermal equilibrium, which seems wrong to me since it evidently isn’t. The Friedman model does assume an expanding universe in thermal equilibrium, but clearly we’re not in *exactly* that world. On the other hand, the Tomita flow is definitely there in the von Neumann algebra view of quantum mechanics and states, so possibly I’m misinterpreting the nature of the claim. Also, as applied to quantum gravity, a “state” perhaps should be read as a state for the whole spacetime geometry of the universe – which is presumably static – and then the apparent “time change” would then be a result of the Tomita flow on operators describing actual physical observables. But on this view, I’m not sure how to understand “thermal equilibrium”. So in the end, I don’t really know how to take the “Thermal Time Hypothesis” as physics.

In any case, the idea that the right notion of time should be state-dependent does make some intuitive sense. The only physically, empirically accessible referent for time is “what a clock measures”: in other words, there is some chosen system which we refer to whenever we say we’re “measuring time”. Different choices of system (that is, different clocks) will give different readings even if they happen to be moving together in an inertial frame – atomic clocks sitting side by side will still gradually drift out of sync. Even if “the system” means the whole universe, or just the gravitational field, clearly the notion of time even in General Relativity depends on the state of this system. If there is a non-state-dependent “god’s-eye view” of which variable is time, we don’t have empirical access to it. So while I can’t really assess this idea confidently, it does seem to be getting at something important.

October 28, 2009 at 12:46 pm

Einstein was right about the shortcomings of Quantum Mechanics and so therefore String Theory is also the incorrect approach. As an alternative to Quantum Theory there is a new theory that describes and explains the mysteries of physical reality. While not disrespecting the value of Quantum Mechanics as a tool to explain the role of quanta in our universe. This theory states that there is also a classical explanation for the paradoxes such as EPR and the Wave-Particle Duality. The Theory is called the Theory of Super Relativity. This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts. It explains the mysterious.

October 28, 2009 at 4:16 pm

While String Theory, and other proposals living between gravitation and QFT (such as Loop Quantum Gravity, to name one apropos to my original post) have various merits and demerits as physics, which I do find interesting. However, I’m a mathematician; and one thing String Theory and LQG (among others) certainly do is produce a significant body of interesting and deep mathematical structures and theorems. These are valid, and probably of enduring mathematical interest, though whether they pertain to physical reality as suggested is unobvious. The anomalies or inaccuracies of GR and Standard Model QFT, where one would look for empirical tests, are almost frustratingly scarce.

October 30, 2009 at 12:08 am

I think every von Neumann algebra has a ‘time-reversed version’, namely the conjugate vector space (where multiplication by i is now defined to be multiplication by -i) turned into a C*-algebra in the hopefully obvious way. And I think the Tomita flow of this time-reversed von Neumann algebra flows the other way!

I know that every symplectic manifold has a ‘time-reversed version’ where the symplectic structure is multiplied by -1. This is equivalent to switching the sign of time in Hamilton’s equations.

I think it’s cool how time reversal is built into these mathematical gadgets.

October 31, 2009 at 4:23 am

One thing that’s struck me several times is the way time-reversal gets represented as complex conjugation – so the distinction between the two time orientations is really reflecting the Galois group for over . Just to start, the fact that you can treat a costate is a time-reversed state, which you can see nicely in the bra/ket formalism, and reflected in the fact that inner products on Hilbert space are sesquilinear… And here’s another example.

It seems like the time flow for vN algebras has some specially nice properties compared to that for symplectic manifolds. Namely, the fact that the Tomita flow is unique up to inner automorphism for different Hamiltonians. Not the case for symplectic manifolds, right?

Seems there’s a lot to say about how time is related to states…

December 3, 2009 at 1:11 pm

For an isolated system + gravitation (or any other long-scale force) there is no thermodynamic limit, so You basically cannot use Gibbs statistics or Canonical ensemble calculations. The proper one is microcanonical one if system is isolated, which for universe is probably good assumptions, however in reality we do not know it. So I cannot believe in time construction for isolated system witch gravity based on canonical ensemble. Please give me where there is thermostate…

December 3, 2009 at 4:41 pm

Well, it’s not entirely clear to me what the right framework is for talking about the whole universe in a quantum theory anyway, where there is no external observer. Since we’re not talking about a system

withgravity, but a system which incorporates the gravitational field (i.e. spacetime geometry), all the foundational issues in quantum gravity probably arise before one even gets to the fact that the universe is (as you say) not in thermal equilibrium with a particular environment.But in any case, the KMS condition comes into Rovelli’s approach just because it’s part of the characterization of the Tomita flow. So as long as we’re using a framework based on von Neumann algebras, the Tomita-Takesaki theory says this structure is there. But again, the vN-algebra framework is all about algebras of

observables, which is probably not relevant to the universe as a whole, since by definition there isn’t any observer governed by quantum physics outside the universe.I don’t really understand your last question/request.

December 3, 2009 at 5:50 pm

last question, well… it was just rhetoric figure – if we are talking about systems with thermal equilibrium we have o point out larger system we are in equilibrium with. Canonical ensemble/Gibbs statistics is only valid when You have short-scale forces. When in system there is long-scale force effect of boundary is not to omit and probably equilibrium is not thermal one. For example velocity of atmospheric particles is far from Maxwell like distribution when You take into account sufficiently large part of atmosphere because of gravitational potential.

Other interesting example: rubidium laser. There is inversion state for some parts of rubidium crystal which is possible because there is poor contact with larger system. And relaxation is therefore so slow, that we may pump energy into this state. If there where good energy flow between modes in such crystal, there will be n laser action at all. So basically when You cannot point on larger system with thermal equilibrium You are with, there is unreasonable to use any statistics other than micro-canonical. That why I do not believe in any “general”,”cosmological” or “field theory” models describing physical states of quantum system based on Canonical ensemble without larger system which gives us thermal bath. If You cannot point out such system, there is probably mistake within this approach.

Sorry for my English – I am not very fluent English writer…

If You are interested in statistical states + gravitation please find related papers withn this: http://arxiv.org/find/cond-mat/1/au:+Gross_D/0/1/0/all/0/1

You have very interesting blog I will definitely subscribe;-) Best regards

January 5, 2010 at 5:38 pm

[…] In any case, this doesn’t exhaust what we know about factors. In his presentation, Ivan Dynov described some examples constructed from crossed products of algebras, which is important later, but for the moment, I’ll finish describing another invariant which helps pick apart the type factors. This is related to Tomita-Takesaki theory, which I’ve mentioned in here before. […]

November 14, 2012 at 5:40 am

I’ve been reading Rovelli’s September 2012 paper “General relativistic statistical mechanics” (1209.0065) and wonder if you’ve looked at and would comment. You mentioned the FQXi essay “Forget time!” and this September paper could be seen as extending from that and the original paper with Connes.

As a math grad student many years back, I had a brief exposure to C* algebras and was just now reviewing the basics (very beautiful) with the help of Landsman math-ph/9807030. There is the appealing idea of states being on a continuum between pure and mixed, with an pureness index tr (rho^2).

Claus Kiefer recently posted a paper about LQC where he used this.

I’m feeling unusually excited and confused. Is it possible that Rovelli and friends are going to reformulate quantum gravity using C* algebras and thermodynamics/statistical mechanics ideas such as thermal time?

Are there some unresolved thermodynamics problems connected with the LQC bounce model of cosmology? Intuitively, in the bounce “gravity becomes repellent” briefly due to quantum effects and horizons dissolve. I don’t satisfactorily understand this. One intuitively expects a pure state and zero entropy. How did this come about? Is entropy even well defined through the bounce? Might these questions be resolved by reformulating LQC in C*algebraic quantum theory language?

I’m reaching out to you because I feel the need of another interested person’s perspective. Feeling excited, bewildered, and even slightly scared.

Hope this is not an imposition.

November 15, 2012 at 3:12 pm

Hi Marcus:

I haven’t had a chance to look at the more recent paper you mentioned except for a quick look at the introduction and a skim through the rest just a moment ago, so keep that in mind. Having said that, my non-expert impression is that this is fairly far from a complete formulation of a quantum theory of gravity – this paper is just taking quantum statistical mechanics and formulating it in a generally-covariant way. But yes, I think the idea that Carlo is pursuing here is indeed that one can come at quantum gravity using the tools of the C*-formulation of quantum mechanics, and that the Tomita flow is supposed to be the origin of physical time.

The obvious problem with this on its face is that any state is in thermal equilibrium, if you consider the time-evolution that comes from the Tomita flow associated to that state. This is a direct result of how you find the flow: in physical terms, it gets a time flow which comes from a Hamiltonian which is picked exactly to make the given state a Gibbs state. The new paper appears to address this by trying to characterize the states which are in physical equilibrium as ones whose Tomita flow is a “flow in spacetime”. For this to make sense, one needs an already-existing notion of spacetime, and the states are associated to spacelike surfaces, like formulating classical GR in terms of the evolution of Cauchy data. In particular, he assumes space is a 3-sphere and spacetime is . The states are thermal distributions on the space of fields on , and the point is that one looks at the flow for such a state, and sees if these give “physical” solutions at all times, i.e. ones satisfying the constraints of the theory. That’s the classical version, and then there’s a quantum version which is in a similar spirit, where one is checking if a flow comes from a local Hamiltonian. I don’t immediately follow all this, but it seems like a nice step in the direction of what these folks are trying to accomplish.

I’m not sure about how these thermodynamic questions interact with the “bounce” model… That’s definitely puzzling. What little I understand from having seen Abhay Ashtekar talk about this model is that the “gravity becomes repellent” principle is a result of something like like the Pauli exclusion principle, which acts like a repellent force in, e.g. neutron stars. The difference being that it is applied to gravitational field excitations, instead of neutrons or other excitation of some field governed by the Dirac equation. My limited physics intuition is telling me that this is a way of saying that when one tries to pack gravitons at a very high density, they repel each other. (Actually, since gravitons are bosons, the “exclusion principle” analogy seems wrong, but I don’t really follow the details much better than that.) I don’t intuitively expect zero entropy near a state like that, since it’s not actually a singularity – it seems like entropy should make as much sense when evolving through a bounce as at other times. Which is, as I say, puzzling – it seems like in a Big Bounce event, entropy is somehow being cleaned away and made low again. So I’m confused as well by how these two ideas might fit together.

November 17, 2012 at 6:07 pm

Jeff, thanks for your response, I was getting desperate for an orthogonal PoV to steady me. I wrote my take on the LQG situation here:

http://physicsforums.com/showthread.php?p=4163232#post4163232

It is a superficial and wobbly overview of work in a halfdozen different directions towards a reformulation of the theory and its cosmology adjunct. I tried to condense my view into a couple of short paragraphs, with links to literature.