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October 27, 2009

“States” and Time – Hamiltonians, KMS states, and Tomita Flow

Posted by Jeffrey Morton under algebra, c*-algebras, musing, noncommutative geometry, philosophical, physics, quantum mechanics, reading
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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 H(q,p), where q is a vector representing the values of some collection of variables describing the system (generalized position variables, in some configuration space X), and the p = m \dot{q} are corresponding “momentum” variables, which are the other coordinates in a phase space which in simple cases is just the cotangent bundle T*X. Here, m refers to mass, or some equivalent. The familiar case of a moving point particle has “energy = kinetic + potential”, or H = p^2 / m + V(q) for some potential function V. The symplectic form on T*X 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 H. Then there’s the Lagrangian, which defines the “action” associated to a path, which comes from integrating some function L(q, \dot{q}) living on the tangent bundle TX, 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 T*X or just a manifold if it’s TX), 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 d\mu around to integrate against – then a function like H becomes the measure H d\mu. 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 T. 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 exp(-H/T), where H 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 T, 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 \omega, which is again, up to some constant factors, just the exponential of the Hamiltonian operator. (For pure states, \omega = |\Psi \rangle \langle \Psi |, and in general a matrix becomes a state by \omega(A) = Tr(A \omega) which for pure states is just the usual expectation value value for A, \langle \Psi | A | \Psi \rangle).

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

(SA)\Psi = A^{\star}\Psi

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

\alpha_t(A) = \Delta^{-it} A \Delta^{it}

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 t 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 \mathcal{A} 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.

 

April 29, 2008

Recent Talk – Enxin Wu on Algebra Deformations

Posted by Jeffrey Morton under algebra, cohomology, quantization, talks
1 Comment 

I’m going up to Ottawa for a few days, in part to talk about spans and groupoids (basically, some cross section of the material in these posts here) at a conference put on by the Ottawa U math department, primarily for grad students and postdocs in the general vicinity. This is nice – gives me a chance to visit my parents and friends there (the fraction of my life I lived in Ottawa is now creeping down toward a mere third, but it probably has as strong a claim to “home” as anywhere). May is also one of the most tolerable months to be there. One of the grad students in our department is also going. Enxin Wu recently decided to start working with Dan Christensen too, so probably in future we’ll have various things to talk about. Last week, he gave a seminar talk on algebra deformation that was a long version of the one he’ll be giving in Ottawa.

Enxin is one of those guys who seems to really understand – it’s tempting to say grok- algebra, which I always find impressive. I’m a predominantly visual thinker, and the kind of symbolic computations common in algebra always seem a little mysterious to me at first until I can find a picture, or at least practice them a lot. Lie groups, for instance, make some sense to me – you can picture rotation groups, or at least keep a geometric picture of a manifold in mind. Lie algebras, being infinitesimal versions of Lie groups, are also not so hard to visualize. General associative algebras? Harder.

The talk was about associative algebras, to give some background on deformation, but the things whose deformations Enxin has been thinking about are A_{\infty}-algebras (see this brief intro, for instance), an “invention” of Stasheff. The talk was about deformation of these algebras – the kind of deformation that pertains to deformation quantization. This has been studied by Kontsevich. Deformation quantization has to do with replacing things valued in some algebra A by new things, valued in the bigger algebra A[[t]] of formal power series in t with coefficients in A, so that the original structure you started with is just the constant part that appears when you set t=0. (The term “quantization” applies when you consider algebras of functions on a manifold, with a Poisson bracket – in other words, algebras of observables of a physical system).

Some of the main results have to do with the Hochschild cohomology for some complex associated to the algebra you start with, and the fact that this cohomology classifies obstructions to the deformation. I expected to get lost in a maze of notation – and there certainly is a lot – but as it turns out, I had some mental pictures to attach to these things, because related things came up a few years ago in the quantum gravity seminar at UCR (week 8 on that page especially), which provides a few pictures that helped a lot. Diagrammatic notation makes algebra a lot more comprehensible to me.

So let’s get more specific.

The point is to replace a multiplication operator m : A \otimes A \rightarrow A with a power series whose coefficients are “multiplication” operators. That is, a deformation of an associative algebra (A,m) (where m : A \otimes A \rightarrow A is the multiplication for A) is (A[[t]],m_t), where the new multiplication m_t is defined (by linearity) by its action on elements of A, which works like this:

m_t(a,b) = \sum_{i=0}^{\infty} {\alpha_i}(a,b){t^i}

for some operators \alpha_i : A \otimes A \rightarrow A. Then there are a bunch of conditions on the \alpha that are needed to make m_t associative. There’s one condition for each power of t, since the coefficients in the associator should be zero:

\sum_{i+j=n\\i,j>0} \alpha_i( (\alpha_j \otimes 1) - (1 \otimes \alpha_j)) = 0

The n=0 condition just says that \alpha_0 is associative – so it’s the m from the original algebra, which you get back when t=0.

Then given an algebra A, you can create the deformation category \mathcal{D} of A whose objects are its deformations. The morphisms are continuous algebra homomorphisms that get along with the multiplication operations. It turns out that since formal power series with nonzero n=0 term are invertible (a consequence of the Lagrange theorem) this \mathcal{D} is actually a groupoid. Then the question is to classify the isomorphism classes of deformations – that is, \Pi_0(\mathcal{D}). One can easily imagine that there might be no nontrivial deformations of some algebra – that is, every one is isomorphic to the deformation where all the \alpha_i are trivial except \alpha_0 = m. So when does this happen? More generally, how can one classify the deformations up to isomorphism?

The answer has to do with Hochschild cohomology, which is related to a complex you can make from A. Taking C^n(A) = hom(A^{\otimes n},A), the space of n-ary multilinear operations on A, you build this complex:

0 \stackrel{d_0}{\longrightarrow} C^0(A) \stackrel{d_1}{\longrightarrow} C^1(A) \stackrel{d_2}{\longrightarrow} \dots

where the differential maps are d_n : C^n(A) \rightarrow C^{n+1}(A) defined by an alternating sum:

d(f)(a_1, \dots, a_n) = a_1 f(a_2, \dots, a_{n+1}) + \sum_{i=1}^{n} (-1)^i f(a_1, \dots, a_i a_{i+1}, \dots, a_{n+1}) + (-1)^{n+1} f(a_1, \dots,a_n) a_{n+1}

(Intuitively: there are too many arguments, so you start with the extra one on the left, push it into the middle as a “lump under the rug” where two arguments are combined, and push the lump all the way to the right. To ensure that d^2 = 0, you do this with alternating signs. This kind of algebraic manipulation is the kind of thing I can do, and clearly works, but I don’t exactly grok.)

Then you take the Hochschild cohomology groups in the standard cohomology way: HH^i = \frac{ker(d_{i+1})}{Im(d_i)}. A cohomology class in one of these groups is a class of multilinear maps from n copies of A to A (up to a factor which is d_n of something). As usual with cohomology, they describe obstructions to something – to exactness. Exactness, in this setting, would mean that A has no interesting deformations at the n^{th} level.

What does “level” mean here? Well, for example, at level 2 we’re talking about maps A \otimes A \rightarrow A, such as the multiplication map. In fact, we have d_3(m) = 0 for an associative algebra – you can check that d(m) is twice the associator a_1(a_2a_3) - (a_1a_2)a_3, which is zero. So m is a cochain. Is it a coboundary? Sure – it’s d_2(1). So m is in the trivial class in HH^2(A). The point then is that it turns out that if this is the only class – if HH^2(A) = 0 – then there are no interesting deformations of the multiplication of A in the sense described above. The groupoid $\mathcal{D}$ has just one object. (One thing that occurs to me is that this makes it a group – which group is something Enxin didn’t discuss. My algebra instincts aren’t quite up to answering that off the top of my head.) For example, if A = \mathbb{C} (as an algebra over \mathbb{R}), there are no nontrivial deformations: HH^2(\mathbb{C}) = 0.

What do the other levels mean? Really, this is where you’d want to look at the generalization from associative algebras to A_{\infty}-algebras. Whereas for an associative algebra A, the associator $a(x,y,z) = x(yz) – (xy)z$ is zero, in general an A_{\infty}-algebra will have an associator map a : A^{\otimes 3} \rightarrow A (that is, a \in C^3 in the complex above), which might not be zero, but which is d_3(m).

This is the beginning of a story relating A_{\infty}-algebras to weak \infty-categories: a bicategory, for example, has an associator for composition of morphisms. In a bicategory, you expect the associator to satisfy a certain identity – the Pentagon identity – but in general you’d just ask for a “pentagonator” (something in C^4), and so on (this is where those seminar notes above help me think in pictures, by the way). An A_{\infty}-algebra is a vector space equipped with maps at all these levels – described by Stasheff’s associahedra – satisfying some relations. The general story of deformation relates the Hochschild cohomology groups at different levels to deformations of A_{\infty}-algebras. Enxin didn’t go into this in his talk, but he did say a little something about the next level:

An infinitesimal deformation of A is a deformation not in A[[t]], but in the quotient A[[t]]/(t^2=0). This only needs two maps, \alpha_0 , \alpha_1. The third Hochschild cohomology measures obstructions to extending an infinitesimal deformation to a full deformation in A[[t]] – if HH^3(A) = 0, then any infinitesimal deformation can be extended to a full deformation.

All in all, I thought the talk was interesting – it tied in much more closely to things I already knew about TQFTs and higher categories than I’d expected. I’ll be really impressed if he can condense it into a 25-minute version…