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March 19, 2007

QFT of Charged n-Particle: The Canonical 1-Particle

Posted by Urs Schreiber

A category of histories for the 1-particle, whose Leinster measure reproduces (a discretized approximation of the Euclidean version of) the path integral measure for the charged 1-particle on the real line.

I present something which should be at least part of the solution to the exercise that we started to think about in Canonical Measures on Configuration Spaces: the goal is to set up the categories encoding target space and history space of the charged 1-particle in such a way that the canonical Leinster measure on the category of histories provides the right measure for the path integral that yields the quantum dynamics.

General Preliminaries: the Construction put in Perspective.

The quantum theory of the charged nn-particle is a setup which (see also John Baez’ description of this idea)

a) reads in a diagram of nn-categories of the form

(1)(parγconftartraphas) \left( \mathrm{par} \stackrel{\gamma \in \mathrm{conf}}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} \right)

and supposed to encode

- the shape of an nn-particle (par\mathrm{par})

- the background spacetime it propagates in (tar(\mathrm{tar})

- the background gauge field it is charged under (tra)(\mathrm{tra})

and

b) spits out the corresponding quantum theory.

Remarkably, given the diagram (1), everything about this procedure is apparently rather canonical, in that it involves only the natural and obvious pull-push diagrams that can be obtained from (1).

But what may be natural for nature might not always be obvious to us (as nicely pointed out by David Corfield and John Baez here and here).

So let’s be careful. In the present context, a mighty subtlety resides in the way the background gauge field is encoded in its parallel transport nn-functor tra:tarphas. \mathrm{tra} : \mathrm{tar} \to \mathrm{phas} \,.

Concentrate on n=1n=1. Usually, we consider 1-particles coupled to vector bundles with connection and hence consider vector transport tra:tarVect \mathrm{tra} : \mathrm{tar} \to \mathrm{Vect} associated to some principal transport tra prin:tarΣG \mathrm{tra}_{\mathrm{prin}} : \mathrm{tar} \to \Sigma G for gauge group (structure group) GG.

This looks all fine – but may be too simple minded:

by the reasoning reviewed in nn-Transport and Higher Schreier Theory, the right principal transport is not a functor with values in the group, but a pseudofunctor with values in the group’s 2-group of inner automorphisms tra pin:tarInn(ΣG). \mathrm{tra}_{\mathrm{pin}} : \mathrm{tar} \to \mathrm{Inn}(\Sigma G) \,.

For principal 1-transport the difference is essentially invisible, as described in nn-Curvature.

But for the associated 1-vector transport this makes a big difference. The naïve representation ρ:ΣGVect \rho : \Sigma G \to \mathrm{Vect} which we use to associate vector transport to principal transport, will have to to be replaced by some suitable domain phas\mathrm{phas} of “phases” ρ˜:Inn(ΣG)phas, \tilde \rho : \mathrm{Inn}(\Sigma G) \to \mathrm{phas} \,, which now is a 2-category itself.

The important consequence is that this implies that the space of sections of the background field sect=Hom(I,tra) \mathrm{sect} = \mathrm{Hom}(I,\mathrm{tra}) is then a (1-)category, instead of a mere set.

In a context where everything is done using pulling and pushing, this difference is huge: for objects in a category, there are canonical notions of these procedures, while for elements of sets one has to introduce extra structure by hand (measures, in particular), in order to perform analogous constructions.

To see this more clearly, consider the special case where the background gauge field is trivial, such that tra=I\mathrm{tra} = I is the tensor unit in the category of all these background fields.

Then a section, ψ\psi, of this bundle is an assignment

(2)ψ:(xγy)I Id I ψ(x) ψ(γ) ψ(y) I Id I \psi : (x \stackrel{\gamma}{\to} y) \mapsto \array{ I &\stackrel{\mathrm{Id}}{\to}& I \\ \psi(x) \downarrow\;\; &\;\;\Leftarrow^{\psi(\gamma)}& \;\;\downarrow \psi(y) \\ I &\stackrel{\mathrm{Id}}{\to}& I }

of 2-cells in phas\mathrm{phas} to paths xγyx \stackrel{\gamma}{\to} y in target space.

Here the assignment on objects ψ:xψ(x) \psi : x \mapsto \psi(x) is the usual assignment of values to points, that you’d expect from a section of a bundle. But since we are being sophisticated and non-naïve, the section now also assigns morphisms between these values to paths in target space.

So we find that a section is not just a function (locally) on target space, but in fact a functor.

(Notice that it is really a contravariant functor. This is not important for the general point of view currently described, but becomes crucial as soon as we perform concrete computations with these sections.)


A Canonical Measure for Quantization: the Leinster Measure

These functors, in turn, have morphisms between them. This implies that there is, in general, a natural way to push such sections forward to a point, for instance. At the decategorified level, such a procedure amounts essentially to integrating these sections. In the more familiar approach, where we do not realize our background field parallel transport by a pseudofunctor, but just by a 1-functor, sections are mere elements of sets, and integrating them requires specifying (“by hand”) a measure.

Interestingly, once we allow sections to be functors themselves, there is a god-given structure that plays the role of this integration measure, otherwise chosen by hand. This is the Leinster Measure on the corresponding domain category.

The observation that the Leinster measure seems to induce a canonical measure for the quantization procedure was discussed in Canonical Measures on Configuration Spaces.

There, the general concept was described, and some attempts at relating the structure of the target space category tar\mathrm{tar} with the expected measure needed for the quantization of the relativistic particle were listed. Making this example more concrete had remained an unfinished exercise:

Exercise. Find (1) such that the entirely canonical quantization procedure applied to it, in particular using the Leinster measure as described above, reproduces (a discrete approximation to) the ordinary textbook quantum theory of the charged 1-particle.


Solution to the Exercise: the Non-Relativistic Charged 1-Particle

Here I want to spell out what seems to be a solution to this exercise which applies to the non-relativistic particle.

I shall consider the particle propagating on a 1-dimensional target space, for simplicity and clarity. Generalizations should, however, be rather obvious.


Caveat: the Nature of our Quantum Phases

There are two main points where the solution that I present below eventually needs to be filled in with more details. So really it is a solution only modulo these pending issues, which I shall simply trust can be dealt with. Hopefully I am right.

While in the naïve setup it is clear that our charged particle should couple to a parallel transport with values in Vect\mathrm{Vect}, it is not fully clear to me at the moment what exactly best replaces this choice as we do the non-naive version.

One option I considered, inpired by Jeffrey Morton’s work on categorified quantum theory, is passing from the 1-category phas:=Vect =Mod\mathrm{phas} := \mathrm{Vect}_\mathbb{C} = \mathbb{C}-\mathrm{Mod} to the 2-category phas:=SetMod \mathrm{phas} := \mathbb{C}\mathrm{Set}-\mathrm{Mod} by replacing, throughout, complex numbers by sets with maps to the complex numbers. I made some remarks on how to proceed in this case here.

After thinking about this for a while, I began to wonder if something more linear might be closer to the truth. Such as Vect[U(1)]Mod\mathrm{Vect}[U(1)]-\mathrm{Mod}, the 2-category of module categories for U(1)U(1)-graded (assuming we can deal with that) vector spaces.

At least up to the detail of the grading, and always assuming that the Squares(phas)\mathrm{Squares}(\mathrm{phas})-valued functors that we shall be dealing with can be taken to satisfy the assumptions that go into Tom Leinster’s theorem, both of these options seem to make the following construction work: direct sums of sets as well as of vector spaces acts as ordinary sum on the isomorphism classes (cardinality and dimension, respectively).

But I am convinced that nature has a completely natural answer concerning the right choice of the 2-category phas\mathrm{phas} of quantum phases, and I am not convinced that I am at the point of seeing this yet.

Therefore I’ll leave the deatails of this issue for later, and tentatively propose either of the above two choices as a working definition that helps us to proceed with what shall be the main issue here: the determination of a category of histories with the right Leinster measure on it to reproduce the quantization measure for the charged 1-particle.


The Target Space Category

After all these preliminaries, here finally the construction itself.

As a model for 1-dimensional space, take the category freely generated from the directed graph of the form tar=((x3) (x2) (x1) (x) (x+1) (x+2) (x+3)). \mathrm{tar} = \left( \array{ (x-3) &\stackrel{\leftarrow}{\rightarrow}& (x-2) &\stackrel{\leftarrow}{\rightarrow}& (x-1) &\stackrel{\leftarrow}{\rightarrow}& (x) &\stackrel{\leftarrow}{\rightarrow}& (x+1) &\stackrel{\leftarrow}{\rightarrow}& (x+2) &\stackrel{\leftarrow}{\rightarrow}& (x+3) } \right) \,.

Parameter space for the 1-particle is, as usual, just the discrete category on a single object par={}. \mathrm{par} = \{\bullet\} \,. Configuration space is simply the space of all maps of parameter space into target space, i.e. conf=Hom(par,tar)tar. \mathrm{conf} = \mathrm{Hom}(\mathrm{par},\mathrm{tar}) \simeq \mathrm{tar} \,. For the particle, configuration space and target space coincide. (Notice that I am not, at this point, using the extended configuration space. Compare the discussion here.).

What is the space of histories, i.e. of paths of the 1-particle par\mathrm{par} in target space tar\mathrm{tar}?

The worldvolume of the 1-particle is a Riemannian 1-manifold, which I shall model by the category freely generated as worldvol t:=(123t). \mathrm{worldvol}_t := (1 \to 2 \to 3 \to \cdots \to t) \,. We think of this as a cobordism worldvol t out in par par \array{ & & \mathrm{worldvol}_t \\ & {}^{\mathrm{out}}\nearrow && \nwarrow^{\mathrm{in}} \\ \mathrm{par} &&&& \mathrm{par} } between two copies of our parameter space, which form the two boundaries of the worldline.

The category of histories is a subcategory of that of all maps of this into target space hist tHom(wordlvol t,tar). \mathrm{hist}_t \subset \mathrm{Hom}(\mathrm{wordlvol}_t,\mathrm{tar}) \,. I discuss below how to determine this subcategory.


Pull-Push of States

By precomposing with the injection maps in\mathrm{in} and out\mathrm{out}, we may restrict any history to the configuration at its beginning and at its end, respectively: hist t out * in * conf conf, \array{ & & \mathrm{hist}_t \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} &&&& \mathrm{conf} } \,,

Propagation of the 1-particle over a time interval tt is the operation on states obtained by pull-pushing through this correspondence.

As recalled in (2), a state, here, is a transformation ψ:Itra \psi : I \to \mathrm{tra} between transport functors on target space, which model the background field that the particle couples to. While it is immediate what the pullback of a functor tarsomehwhere\mathrm{tar} \to \mathrm{somehwhere} along in *\mathrm{in}^* would be, the pullback of a transformation of functors through this correspondence needs a little more care:

The fact that a history interpolates between two configurations is very concretly realized in the existence of a transformation

(3) hist t×worldvol out * in * conf×par conf×par ev ev tar \array{ & & \mathrm{hist}_t\times \mathrm{worldvol} \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf}\times\mathrm{par} & &\;\;\Leftarrow& & \mathrm{conf}\times\mathrm{par} \\ & {}^{\mathrm{ev}}\searrow && \swarrow^{\mathrm{ev}} \\ && \mathrm{tar} }

between the projection onto its boundary components.

A moment of reflection shows that the pullback of states that we need is the operation of pasting this 2-morphism to that of a state ψ\psi to obtain

The precomposition with the 1-morphism in *\mathrm{in}^* accomplishes the naive pullback from configurations to histories. The composition with the 2-morphism cyl\mathrm{cyl} then takes care of the parallel transport of all these sections to the other end of our cobordism.

I have described this in more detail in QFT of Charged n-Particle: Dynamics.


Evolution over Small Time Intervals

Now, the standard propagator for the (Euclidean) particle on the line is U(t)=exp(tΔ), U(t) = \exp( t \Delta ) \,, where Δ= \Delta = \nabla^\dagger \nabla is the covariant Laplace operator on target space, coming from the vector bundle with connection \nabla that is encoded in our parallel transport functor tra\mathrm{tra}.

We want to find the category hist t\mathrm{hist}_t such that pulling sections back to hist t\mathrm{hist}_t along the above lines, and then pushing forward from the space of histories to the space of configurations along out *\mathrm{out}^* reproduces this propagator (for more on the relation of this propagator to the corresponding path integral see John Baez’s lecture “Quantization and Cohomology ” week 14 and week 15).

Since we have approximated target space by a lattice, we need the corresponding lattice version of the covariant Laplace operator, which reads (Δf)(x)=tra x1,x(f)(x1)2f(x)+tra x+1,x(f)(x+1). (\Delta f)(x) = \mathrm{tra}_{x-1,x}(f)(x-1) - 2 f(x) + \mathrm{tra}_{x+1,x}(f)(x+1) \,. It is helpful to first consider propagation over “infinitesimal” time intervals, for which we approximate exp(Δ)Id+Δ. \exp( \Delta ) \simeq \mathrm{Id} + \Delta \,. Hence we need to find hist 1\mathrm{hist}_1 such that the pull-push through (3) produces

(4)ψ(U(1)ψ:xtra x1,x(f)(x1)f(x)+tra x+1,x(f)(x+1)). \psi \mapsto (U(1)\psi : x \mapsto \mathrm{tra}_{x-1,x}(f)(x-1) - f(x) + \mathrm{tra}_{x+1,x}(f)(x+1)) \,.

I claim that this is the case if we let hist 1Hom(wordlvol 1,tar), \mathrm{hist}_1 \subset \mathrm{Hom}(\mathrm{wordlvol}_1,\mathrm{tar}) \,, be that full subcategory, which contains all functors that send the unit length worldline at most to an edge of the target space category (instead of to a morphism obtained as an arbitrary concatenation of edges), and whose morphisms only contract, never expand their image.

More precisely, hist 1 out * in * conf conf \array{ && \mathrm{hist}_1 \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} &&&& \mathrm{conf} } is the subcategory hist 1:=Mor(worldvol 1,tar) \mathrm{hist}_1 := \mathrm{Mor}(\mathrm{worldvol}_1,\mathrm{tar}) of all “maps” (functors) from the 1-unit worldvolume worldvo 1:={12} \mathrm{worldvo}_1 := \{1 \to 2\} into target space tar:={(x1)x(x+1)(x+2)}, \mathrm{tar} := \{ \cdots (x-1) \stackrel{\leftarrow}{\to} x \stackrel{\leftarrow}{\to} (x+1) \stackrel{\leftarrow}{\to} (x+2) \} \,, definined by the two restrictions

a) objects are only those functors that are at the same time morphisms of the underlying graphs, i.e. which map the single edge 121 \to 2 either to an identity in tar\mathrm{tar} or to a single edge in tar\mathrm{tar}.

b) morphisms are all natural transformations between these, except for those which would go from an identity image (12)(xx)(1 \to 2)\mapsto (x\to x) to a non-identity image (12)(xx+1)(1 \to 2)\mapsto (x\to x+1).

To see that pull-push propagation through hist 1\mathrm{hist}_1 does indeed reproduce propagation by the lattice Laplace operator, notice that the subcategory hist 1 x \mathrm{hist}_1^x of all paths ending at xx simply looks like hist 1 x={(x1x) (x+1x) (xx)}. \mathrm{hist}_1^x = \left\{ \array{ (x-1 \to x) &&&& (x+1 \to x) \\ & \searrow && \swarrow \\ && (x \to x) } \right\} \,.

Pulling a state ψ\psi back to this by means of (3) produces the transformation given by the assignment ((x1x)(xx)(x+1x))I Id I Id I ψ(x1) ψ(x1,x) ψ(x) ψ(x+1,x) ψ(x+1) tra(x1) tra(x1,x) tra(x) tra(x+1,x) tra(x+1) tra(x1,x) Id tra(x+1,x) tra(x) Id tra(x) Id tra(x). ( (x-1\to x) \to (x\to x) \leftarrow (x+1\to x) ) \;\; \;\; \mapsto \;\; \;\; \array{ I &\stackrel{\mathrm{Id}}{\to}& I &\stackrel{\mathrm{Id}}{\leftarrow}& I \\ {}^{\psi(x-1)} \downarrow \;\; &\Leftarrow^{\psi(x-1,x)}& {}^{\psi(x)} \downarrow \;\; &\Rightarrow^{\psi(x+1,x)}& {}^{\psi(x+1)} \downarrow \;\; \\ \mathrm{tra}(x-1) &\stackrel{\mathrm{tra}(x-1,x)}{\to}& \mathrm{tra}(x) &\stackrel{\mathrm{tra}(x+1,x)}{\leftarrow}& \mathrm{tra}(x+1) \\ {}^\mathrm{tra}(x-1,x)\downarrow\;\; && {}^{\mathrm{Id}} \downarrow \;\; && {}^\mathrm{tra}(x+1,x)\downarrow\;\; \\ \mathrm{tra}(x) &\stackrel{\mathrm{Id}}{\to}& \mathrm{tra}(x) &\stackrel{\mathrm{Id}}{\leftarrow}& \mathrm{tra}(x) } \,.

This assignment of 2-morphisms to 1-morphisms is a contravariant functor on hist 1 x\mathrm{hist}_1^x. Therefore the colimit of this is controlled by the Leinster measure on the oppositecategory, (hist 1 x) op(\mathrm{hist}_1^x)^\mathrm{op}, of hist 1 x\mathrm{hist}_1^x: (hist 1 x) op={(x1x) (x+1x) (xx)}. (\mathrm{hist}_1^x)^\mathrm{op} = \left\{ \array{ (x-1 \to x) &&&& (x+1 \to x) \\ & \nwarrow && \nearrow \\ && (x \to x) } \right\} \,. The Leinster measure on this simple category is 1 1 1. \array{ 1 &&&& 1 \\ & \nwarrow && \nearrow \\ && -1 } \,.

This means that, assuming that our state is such that the morphisms ψ(x1,x)\psi(x-1,x) and ψ(x+1,x)\psi(x+1,x) are suitably monic and denoting by |||\cdot| the isomorphism class in our choice of phas\mathrm{phas} (compare the discussion above) then the colimit of the state ψ\psi pulled back to hist 1 x\mathrm{hist}_1^x, according to (3), is |colim hist 1 xin *ψ|=|tra(x1,x)ψ(x1)||ψ(x)|+|tra(x+1,x)ψ(x+1)|. | \mathrm{colim}_{\mathrm{hist}_1^x} \mathrm{in}^*\psi | = |\mathrm{tra}(x-1,x)\psi(x-1)| - |\psi(x)| + |\mathrm{tra}(x+1,x)\psi(x+1)| \,. This is our path integral! Over paths of “very small length”. By (4) we can rewrite this as hist 1 xin *ψ=|exp(Δ)ψ(x)|. \int_{\mathrm{hist}_1^x} in^* \psi = | \exp(\Delta) \psi (x) | \,.


Evolution over two Small Time Intervals

The category of 2-step histories hist 2 out * in * conf conf \array{ && \mathrm{hist}_2 \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} &&&& \mathrm{conf} } should be the composite span of hist 1\mathrm{hist}_1 with itself, i.e. the pullback hist 2 hist 1 hist 1 out * in * out * in * conf conf conf. \array{ &&&& \mathrm{hist}_2 \\ &&& \swarrow && \searrow \\ && \mathrm{hist}_1 &&&& \mathrm{hist}_1 \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} && {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} &&&& \mathrm{conf} &&&& \mathrm{conf} } \,.

I think hist 2 x\mathrm{hist}_2^x looks, explicitly, as follows.

Let now worldvol 2=(123) \mathrm{worldvol}_2 = (1 \to 2 \to 3) be the worldline of length two units of time and consider hist 2 x. \mathrm{hist}_2^x \,. The subcategory of tar\mathrm{tar} that these histories (paths) are allowed to map into is

The category of 2-step histories ending at xx looks like

The Leinster measure on the opposite of the above category (remember that our states ψ\psi are contravariant functors) is

(Notice that all parallel morphisms in hist\mathrm{hist}, hence also in its opposite, are equal, in the present context).

But this means that the path integral now yields |colim hist 2 xin *ψ|= |tra x2,xψ(x2)|2|tra x1,xψ(x1)|+3|ψ(x)| 2|tra x+1,xψ(x+1)|+|tra x+2,xψ(x+2)|. \begin{aligned} | \mathrm{colim}_{\mathrm{hist}_2^x} \mathrm{in}^*\psi | = & |\mathrm{tra}_{x-2,x}\psi(x-2)| -2 | \mathrm{tra}_{x-1,x} \psi(x-1) | + 3 | \psi(x) | \\ & - 2 | \mathrm{tra}_{x+1,x} \psi(x+1) | + |\mathrm{tra}_{x+2,x}\psi(x+2)| \end{aligned} \,. This one checks, by applying (4) twice, is the same as hist 2 xin *ψ=|exp(Δ)exp(Δ)ψ(x)|, \int_{\mathrm{hist}_2^x} in^* \psi = | \exp(\Delta)\exp(\Delta) \psi (x) | \,, if we again approximate the exponential by its leading contribution.

Posted at March 19, 2007 2:30 PM UTC

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37 Comments & 5 Trackbacks

Re: QFT of Charged n-Particle: The Canonical 1-Particle

I’ve already gushed about how awesome I think all this is, so I won’t repeat myself other than to ask why you insist on calling this an “approximation” to the continuum. Perhaps the continuum is the approximation ;)

Posted by: Eric on March 19, 2007 10:40 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: The Canonical 1-Particle

ask why you insist on calling this an “approximation” to the continuum. Perhaps the continuum is the approximation ;)

Yes, right, that’s certainly a tempting speculation.

Right at the moment, though, I think it is important to make contact with standard stuff. Maybe the continuum theory in the textbooks is an approximation, but then I want to see the formalism here approximate this approximation!

By the way, it is noteworthy how very close the construction I discuss above is indeed to your emphasis on how the binary tree already encodes the Euclidean quantum mechanics, hence the statistical mechanics of the Brownian particle.

The above argument is certainly different from the argument about how discrete differential forms on the binary tree give rise to the heat equation. Yet, it extracts the same result from the same graph.

This means that there is something possibly deep going on here, which I do not understand yet.

It is presumeably related to the deeper relation between category algebras (path algebras) and the stuff we have been talking about at length in the context of the Euler Characteristic of a category.

An important point seems to be that a section e:Itrae : I \to \mathrm{tra} for II and tra\mathrm{tra} pseudofunctors (i.e. with values in a 2-category) is two things at once:

a) an ordinary function (or section) over objects

and

b) the covariant derivative of this function over edges.

This is something I noticed at the very beginning of my thinking about arrow-theoretic quantization here. Somehow I am feeling, though, that the real meaning of this phenomenon still escapes me.

Posted by: urs on March 19, 2007 11:00 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: The Canonical 1-Particle

Yes, I think something deep is going on. You might find Section 5.1 interesting

Financial Modeling Using Discrete Stochastic Calculus

Don’t let the word “finance” disctract you. It is basically about discrete stochastic calculus, which is what you partially just reinvented.

Posted by: Eric on March 19, 2007 11:44 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: The Canonical 1-Particle

Just for the record: a related discussion is on p. 5271 of Dimakis&Tzanakis, Dynamical evolution in non-commutative discrete phase space and the derivation of classical kinetic equations.

Posted by: urs on March 20, 2007 11:26 AM | Permalink | Reply to this

Ito formula and time evolution (continuum and discrete versions)

Here and across several posts Urs has discussed the (Euclidean) evolution

(1)U(t)=exp(tH) U(t) = exp(t H)

where

(2)H= . H = \nabla^\dagger \nabla.

Stating the obvious to such a crowd where everyone is a million times smarter than me feels silly, but that has rarely stopped me before. So here it goes…

The above is the solution to

(3)ψt=Hψ. \frac{\partial\psi}{\partial t} = H \psi.

In an old (by now) paper, I fleshed out and expanded a bit on some ideas first proposed by Dimakis and Mueller-Hoissen and applied them to mathematical finance:

Noncommutative Geometry and Stochastic Calculus: Applications in Mathematical Finance

The basic idea there is that by introducing commutative relations among 0-forms and 1-forms in a particular manner, you recover the Ito formula, which otherwise requires probabilistic and statistical considerations (which I was never fond of mostly due to lack of brain cells).

The necessary relations are

(4)[dx,x]=dt [dx,x] = dt

and

(5)[dx,t]=0. [dx,t] = 0.

From the latter expression you can show that

(6)[dt,x]=0 [dt,x] = 0

since

(7)d(tx)=dtx+tdx d(tx) = dt x + t dx

(and the order matters!).

Due to the noncommutativity, if you want to express a 1-form in components, you have a choice of writing the components on the left or the right of the basis 1-forms. I called this left-component and right-component forms.

Modulo some technical details (which could invalidate the whole idea, but who cares :)), when you write the differential of a 0-form ψ\psi in both left and right component forms, you get

(8)dψ=(ψx μ)dx μ+(ψt+Hψ)dt=dx μ(ψx μ)+dt(ψtHψ) d\psi = \left(\frac{\partial \psi}{\partial x^\mu}\right) dx^\mu + \left(\frac{\partial \psi}{\partial t} + H \psi\right) dt = dx^\mu \left(\frac{\partial \psi}{\partial x^\mu}\right) + dt \left(\frac{\partial \psi}{\partial t} - H \psi\right)

A Martingale is (roughly) a process for which the drift, i.e. time component, is zero. Since the right component and left components are not necessarily the same, I called these “left-Martingales” and “right-Martingales”.

Note that a left-Martingale is a time-reversed diffusion and a right-Martingale is a diffusion (I could have that backwards… details).

In other words, if ψ\psi is a right-Martingale, then we have

(9)ψt=Hψ. \frac{\partial\psi}{\partial t} = H \psi.

Look familiar? :)

The really neat thing is that all of this carries through beautifully on a 2-diamond, i.e. binomial tree, including a “discrete Ito formula”, which I discuss in the paper I linked to in the previous post. This, to me anyway, suggests that evolution is built naturally into the very structure of the 2-diamond. It is not surprising that Urs is able to demonstrate things so neatly on a 2-diamond.

It occurs to me that I should have finished reading the rest of the comments before posting because I may have said this very exact thing before. Getting old sucks :)

Anyway, I think all this is very neat and it is all interconnected somehow. When Urs talks about “fundamental” vs “effective” quantum theories, I am stricken by what appears to be a fact that “fundamental” quantum theories have natural formulations on an n-diamond and the evolution is essentially driven by the Ito formula via right-Martingales.

I could be, and probably am, spewing nonsense, but that is the best I could do while Sophia takes her nap. I can hear her rustling now! :D

Best regards, Eric

Posted by: Eric on June 10, 2007 9:22 PM | Permalink | Reply to this

Re: Ito formula and time evolution (continuum and discrete versions)

It is not surprising that Urs is able to demonstrate things so neatly on a 2-diamond.

The fact you mention, that differential forms on the lattice automatically know about the heat equation, is indeed quite cool.

In some sense, this makes it less of a surprise that the above pull-push quantization procedure produces a Leinster measure which also gives the heat propagator.

But on the other hand, I don’t really understand this relation – between the lattice deRham differential and the canonical pull-push quantization on that lattice – at all at the moment!

If this is more than a coincidence (and I tend to expect that it is), then something profound is going on here which still escapes me.

I wish I would be investing more time into thinking about this…

Posted by: urs on June 11, 2007 2:30 PM | Permalink | Reply to this

Re: Ito formula and time evolution (continuum and discrete versions)

In the “discrete stochastic calculus” paper, I show that simply applying the discrete differential to 0-form basis elements gives a 1-form whose values correspond to the Leinster measure you computed above (which you are well aware of). I then show the relation between that and the “heat equation” or “Ito formula” which require an introduction of some “arbitrary” coordinates called “time” and “position” :)

Not sure if that helps…

Posted by: Eric on June 11, 2007 5:23 PM | Permalink | Reply to this

Re: Ito formula and time evolution (continuum and discrete versions)

a 1-form whose values correspond to the Leinster measure you computed above

Wait, what do you mean by that? This measure lives on vertices, meaning: it is a function from vertices to numbers. In which sense are you thinking of the 1-form de x d e_x as “corresponding” to that measure?

Posted by: urs on June 11, 2007 5:39 PM | Permalink | Reply to this

Re: Ito formula and time evolution (continuum and discrete versions)

It is quite likely that I misspoke regarding Leinster measure since I don’t really understand it.

Let’s imagine a single branch of a binary tree with values assigned to the three nodes, which I guess I will denote: ϕ(+)\phi(+), ϕ()\phi(-), and ϕ(0)\phi(0). Hopefully the meanings are clear.

(1)e(0)dϕ=[ϕ(+)ϕ(0)]e(0,+)+[ϕ()ϕ(0)]e(0,), e(0) d\phi = [\phi(+)-\phi(0)] e(0,+) + [\phi(-)-\phi(0)] e(0,-),

where e(0)e(0) is the vertex node of the branch, e(0,+)e(0,+) is the edge pointed toward the node e(+)e(+), and e(0,)e(0,-) is the edge pointed toward the node e()e(-).

I was wanting to think of the assignment of ±1\pm 1 to each node’s value above as a “weight”, but that was probably wrong.

Unfortunately, I have about spurts of 15 seconds at a time to make these comments :)

Eric

Posted by: Eric on June 11, 2007 7:29 PM | Permalink | Reply to this

Re: Ito formula and time evolution (continuum and discrete versions)

[…] but that was probably wrong.

Maybe it’s right! It’s very suggestive, certainly. All I am saying is that the general principle escapes me.

For instance, given any other graph, I don’t see how it’s Leinster measure would be directly read off from the algebra of discrete differential forms on it.

Maybe it’s possible. But currently I don’t see it. But I’ll keep it in mind. One day we’ll have this figured out. :-)

Posted by: urs on June 11, 2007 9:41 PM | Permalink | Reply to this

Alain Connes’ “Intrinsic Time”

By a quoted remark by Alain Connes in the recent entry on the Noncommutative Geometry Blog I was reminded that Alain Connes is fond of the fact that certain noncommutative algebras come equipped with a canonical notion of time evolution.

I know roughly what this refers to. But I should have a closer look at this, because it sounds vaguely like it could be related to our effort here of finding the canonical time evolution induced by the structure of configuration space.

All I know at the moment is this:

for von Neumann algebra factors AB(H)A \subset B(H) (i.e. certain subalgebras of bounded operators on a Hilbert space with trivial center) we have what is called Tomita-Takesaki theory.

This says that, under suitable conditions, the linear map which sends the action of any algebra element a a on the “vacuum vector” Ω\Omega to the action of the adjoint a *a^* of aa, i.e. the linear map S:aΩa *Ω S : a \Omega \mapsto a^* \Omega has a (“polar”) decomposition as S=JΔ 1/2 S = J \Delta^{1/2} where J 2=IdJ^2 = \mathrm{Id} and Δ\Delta is some positive operator (usually unbounded, i.e. in particular not an element of AA).

This induces a 1-parameter action by outer automorphisms on AA: Ad Δ i:Out(A) \mathrm{Ad}_{\Delta^{i \cdot}} : \mathbb{R} \to \mathrm{Out}(A) given by aΔ itaΔ it. a \mapsto \Delta^{i t}a \Delta^{-i t} \,. This action is nontrivial if and only if the vN algebra AA is what is called a type III factor.

In this case, Alain Connes apparently refers to this flow as the intrinsic time evolution on the algebra AA (“of observables”), thinking if Δ it\Delta^{i t} as Δ it:=exp(itH), \Delta^{i t} := \exp(i t H) \,, for HH the “Hamiltonian”.

(I had trouble finding a good quick reference for this. The above is extracted from p. 55-56 of Stolz&Teichner Wiaeo?).

If anyone reading this here knows more about this concept of “intrinsic time”, I’d be grateful for some hints.

For instance: are there any examples known where this “intrinsic time flow” coincides with the one obtained by standard means? (But isn’t that standard evolution always induced by a bounded operator?)

These type III factors of observables appear in particular in 2-dimensional CFT. Usually in these cases the standard time evolution is known. Does it coincide with the “intrinsic time evolution”?

Probably not. But if not, is there any way to understand the relation to the standard time evolution?

If not, what really is the motivation behind addressing Ad Δ it\mathrm{Ad}_{\Delta^{i t}} as “time evolution”?

Posted by: urs on March 20, 2007 12:05 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”


This action is nontrivial if and only if the vN algebra A is what is called a type II factor.

Oh, are you sure? I’d have guessed it was non-trivial only for type III factors.

I had trouble finding a good quick reference for this.

Not sure if this helps, but I remember reading a brief review by Summers on the archive, I’ve no time to do the search for you right now though ;-)

For instance: are there any examples known where this “intrinsic time flow” coincides with the one obtained by standard means? (But isn’t that standard evolution always induced by a bounded operator?)


These type III factors of observables appear in particular in 2-dimensional CFT. Usually in these cases the standard time evolution is known. Does it coincide with the “intrinsic time evolution”?

Just wanted to say that I am also very interested in being enlightened regarding this. The main reason why I still find these approaches to QFT indigestible is that answers to such obvious (for an outsider that is, maybe they are just stupid for those in the field…) questions are nowhere to be found in the litterature.

/Jens

Posted by: Jens on March 20, 2007 1:35 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Hi Jens!

I’d have guessed it was non-trivial only for type III factors.

Right, thanks. This was a typo in my comment. Will correct it.

By the way: can you say anything about listing those kind of theories whose algebras of observables form a type III factor? Is that the default behaviour for rational 2d CFT, for instance?

Are there any circumstances in which we can find type III factors (or any other factors, for that matter) in just quantum mechanical systems (i.e. 0+1-dimensional QFT)?

Or for Lorentzian 4D QFT, simple ones, like treated in the AQFT literature: under what conditions would the algebras appearing there be (type III) factors?

Can you say anything about this? Or do you maybe have some literature to suggest?

Just wanted to say that I am also very interested in being enlightened regarding this.

I have submitted essentially this question to the NCG blog. With a little luck, we can get an answer from one of the experts there.

Posted by: urs on March 20, 2007 1:53 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Hi again Urs!

By the way: can you say anything about listing those kind of theories whose algebras of observables form a type III factor? Is that the default behaviour for rational 2d CFT, for instance?

I can say something, but I can unfortunately not defend a word of it!
As far as I understand, all factors relevant to ‘local quantum physics’ (irrespective of the dimensionality of spacetime) are type-III_1 factors, which is in particular true for 2d RCFT.
But as I said, I cannot defend this statement, just something that stuck in my head from various conversations.

I am also aware that type-II factors can be used to construct TFT:s, but that must be a different construction.

I’ve absolutely no idea about the case of QM though.

I have submitted essentially this question to the NCG blog. With a little luck, we can get an answer from one of the experts there.

Nice, I’ll keep an eye on that blog then.

Posted by: Jens on March 20, 2007 4:40 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

As far as I understand, all factors relevant to ‘local quantum physics’ (irrespective of the dimensionality of spacetime) are type-III_1 factors

I see. I’ll try to check this in the literature.

What I would like to understand, then, is how the outer part of the canonical time evolution that Connes discusses appears in examples of ordinary time evolution of ordinary QFTs.

What I find a little irritating (but maybe my assumptions are wrong) is that I would expect in standard examples (like, say WZW theory using the Sugawara construction), that the exponentiated Hamiltonian e itHe^{i t H} is in fact an element in the algebra of observables. But that would mean the time evolution it induces is completely by inner automorphisms.

That’s not in contradiction with the “intrinsic time”-statement, as far as I understand. Maybe it just says that for the algebra of observables of, in particular, WZW theory, the canonical flow by outer automorphisms is trivial.

I’ve absolutely no idea about the case of QM though.

Let’s see: for the standard free particle the double commutant of the Weyl algebra is all of B(H)B(H), right?

In as far as I understand, B(H)B(H) itself is a factor, as its center consists only of multiples of the identity operator. What type is it, though?

Posted by: urs on March 20, 2007 4:54 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Any local AQFT is made of type III factors. More precisely, the algebra of observables associated to a finite region, is a type III factor. I think this is a consequence of what is called the Borchers property, but I don’t have a book where to check it here. Anyway, it’s in Haag’s for sure. Also, the first example of a type III factor was given by(I don’t remeber) Powers or Araki and Woods, they were studying models of quantum statistical mechanics.

So, locality is tied with type three property, and to talk about locality we need at least one space dimension.

Notice also that locality and type three are connected with the tensor structure of the category of local endomorphism. Type III means that any projection PP of the algebra is equivalent to the identity, i.e. there exists WW s.t. WW *=PWW^* = P and W *W=IW^*W = I.

Given two local morphisms ρ 1\rho_1 and ρ 2\rho_2 you can construct a direct sum W 1ρ 1W 1 *+W 2ρ 2W 2 *W_1 \rho_1 W_1^* + W_2 \rho_2 W_2^*, with W 1W_1 and W 2W_2 partial isometries orthogonal to each other.

The vacuum vector is cyclic and separating for each of the A(O)A(O), the algebra of observables localised in OO. So it looks like that any AQFT gives examples of non trivial flow. But I am not sure, I might be saying nonsense.

On the other hand, QM, as an algebra, is usually isomorphic to B(H)B(H), and the unitary operators giving rise to time translations are elements of B(H)B(H). So I see only inner automorphisms.

In the AQFT case, these belong to the closure of the union of all the local algebras A(O)A(O), for an increasing sequence of spaces OO. So U(t)U(t) is not in A(O)A(O). In other words, the Hamiltonian is not local.

Posted by: anonymous on March 20, 2007 6:13 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

More precisely, the algebra of observables associated to a finite region, is a type III factor.

Thanks!

So if I understand correctly, when we take the inductive limit over all these local algebras (over all causal diamonds, if we are in flat space Lorentzian AQFT context) two things happen, which are closely related:

a) time evolution becomes an inner automorphism, namely the ordinary adjoint action of the exponentiated Hamiltonian

b) the algebra of observables is no longer a type III factor.

Posted by: urs on March 20, 2007 8:27 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

When I wrote that the Hamiltonian is not local, it seemed reasonable to me. But I am not sure whether it is possible to find another set of unitaries, say, V(t)V(t), which are local (i.e. they belong to A(O)A(O)) and realise the same time endomorphism (only on A(O)A(O), though). It might be, I don’t know. One of the fundamental things I am missing, are the differences when dealing with states generated by the vacuum state (I mean with a finite number of particles) and thermal (i.e. KMS) states.

To conclude that the union of all the A(O)A(O) for an increasing net of OO is not a factor of type III, seems to me wrong. It’s just my instinct, I can’t tell you why. BTW, this “envelope” of algebras carries Fredenhagen’s name, if I remember well.

Furthermore, I doubt that this “canonical time flow” has much to do with boundary conditions. The algebras A(O)A(O) are isomorphic to each other. There is not much information in there. I am even afraid that the isomorphism class does not depend of the physical theory you are dealing with. The interesting structure appears when you look at how these algebras are patched together (i.e. the net, and consequently the tensor category). Some people like to make the parallel with locally trivial fibre bundles. So if this story of flow of isomorphisms has anything interesting for you, I would look at Connes’ cocycles. It’s just a guess.

Posted by: anonymous on March 21, 2007 3:27 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

To conclude that the union of all the A(O)A(O) for an increasing net of OO is not a factor of type III, seems to me wrong.

Okay, that was just a guess based on what we said.

In any case, it is true for (0+1)-dimensional QFT, i.e. quantum mechanics. In that case every A(0)A(0) is already isomorphic to the inductive limit over all of them, time evolution is inner (by adjoint action with the exponentiated Hamiltonian, which is itself in any one of the A(O)A(O)).

Now, say (1+1)-dimensional QFT on, say, the circle times the real axis, is, globally (i.e. if we don’t ask for local algebras) the same (at least up to technical issues, so I am still guessing here) as quantum mechanics on the space of field configurations over the circle.

This kind of reasoning would seem to suggest (or at least make plausible) that on the inductive limit of the A(0)A(0) we have a time evolution by inner automorphisms.

In fact, this is the only case that I am able to relate to standard examples. In simple standard textbook field theory (not the local AQFT-version, but just the standard global thing), time evolution is always given by adjoint action of the exponentiated Hamiltonian.

Therefore I would still be quite interested in seeing a concrete example where the nontrivial outer action of time evolution is visible.

Anyway, that’s the sort of reasoning that made me guess that the inductive limit over the A(O)A(O) is no longer a type III factor. But if you say it sounds wrong, I accept that. I’d need to better understand what’s going on, then.

I doubt that this “canonical time flow” has much to do with boundary conditions. The algebras A(O)A(O) are isomorphic to each other.

I was just thinking of a special simple example like open vs. closed string. In the second case the algebras consist of two copies (left and right moving) of those in the first case.

But, again, I may well be wrong. I am not an expert on the AQFT descriton of CFT.

Some people like to make the parallel with locally trivial fibre bundles. So if this story of flow of isomorphisms has anything interesting for you, I would look at Connes’ cocycles.

Now that sounds very interesting, for instance in light of our discussion here.

However, I am not sure what exactly you are pointing me to, here. Can you give me a reference to that relation to locally trivial bundles? And what are Connes’ cocycles?

Posted by: urs on March 21, 2007 3:44 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Since it really also is a reaction to the discussion here, which I had partly forwarded to the Noncomm. Geom. blog, I reproduce below another response by Alain Connes.

Alain Connes writes:

Urs: Yes, what happens in fact is that for any quantum system with infinitely many degrees of freedom the hamiltonian HH does not belong to the algebra of observables. Thus the corresponding automorphisms are not inner.

To see what happens it is simplest to take the case of a system of spins on a lattice. The algebra of observables is the inductive limit of the finite tensor products of matrix algebras one for each lattice site. The hamiltonian HH is, even in the simplest non-interacting case, an infinite sum of the hamiltonians associated to each lattice site. Thus it does not belong to the algebra of observables and the corresponding one parameter group is not inner(both in the norm closure ie the C *C^*-algebra, and in the weak closure)…

In QFT the situation is entirely similar and has of course infinitely many degrees of freedom from the start…

Posted by: urs (forwarding a message from A. Connes) on March 26, 2007 7:15 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Any faithful state on a C *C^* (in particular, v.N.) algebra gives you a GNS representation, a vector Ω\Omega, a self adjoint Δ\Delta and an antiunitary JJ. These give you as well a time evolution, σ\sigma. The state turns out to be a KMS state for this time evolution. All of this structure depends on the choice of the state. In particular, different states give you different time evolutions. But these different time evolutions are connected by a unitary cocycle taking values in the original C *C^*-algebra (this is Conne’s cocycle). So, what is canocical is a “time” flow in Aut/Inn\mathrm{Aut}/\mathrm{Inn}, i.e. automorphisms of the algebra modulo inner automophsims.

Traces give trivial flows, so if your algebra happens to have a finite trace (i.e. you can normalise it so to have tr(1)=1\mathrm{tr}(1)=1), this trace state is trivially KMS. But type III factors have no such traces. It is studying these flows that Connes classified hyperfinte type III factors, almost thirty years ago.

Posted by: anonymous on March 20, 2007 2:05 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Ah, I should have looked again at

Masoud Khalkhali, Lectures on Noncommutative Geometry, where this is briefly discussed on p. 12.

Posted by: urs on March 20, 2007 2:19 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Hi Urs,

I came across this interesting fact (that certain von Neumann algebras come with their own canonical, inbuilt, time evolution) recently. It think it was when I was reading Connes’ “A view of mathematics” . Cool! I will be interested to see if you can come up with a nice “n-cafe” understanding of this phenomenon.

Posted by: Bruce Bartlett on March 20, 2007 6:26 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

I came across this interesting fact (that certain von Neumann algebras come with their own canonical, inbuilt, time evolution) recently. It think it was when I was reading Connes’ “A view of mathematics” . Cool!

Yes, it sounds like a very cool statement indeed.

On the other hand, it is just the outer part of ordinary time evolution in QFT which is canonical in this sense.

I need to get a better feeling for what this outer part of the Heisenberg time evolution operator really encodes. Based on the very helpful anonymous comments above, I am guessing that this outer part is that which knows about (asymptotic) boundary conditions, while the “naive part” of the time evolution is given as usual by the inner automorphism obtained by conjugation with the (local part of the) Hamiltonian.

I don’t know if this intuition is right. But it looks plausible to me: any algebra of observables should certainly depend in a deep structural fashion on the chosen boundary condition, so it would seem plausible that for any fixed such algebra the boundary part of the time evolution is fixed, while only the local bulk part is to be chosen.

That’s my intuition, anyway. I need to think more about this. But in any case it seems as if Alain Connes’ “intrinsic time evolution” can not do for us what we tried to do in the above entry.

I will be interested to see if you can come up with a nice “nn-cafe” understanding of this phenomenon.

Right. With lots of help from all nn-Café guests – and with a little luck – we might some day be serving, exclusively, intrinsic noncommutative time evolution in nn-categorical mugs!

Posted by: urs on March 20, 2007 6:51 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Alain Connes has posted a detailed reply over at the Noncommutative Geometry Blog. I think it is worthwhile to reproduce it here, with all formulas properly typeset.

This is what Alain Connes writes:

I will try to describe in loose terms the steps that lead to the emergence of time from noncommutativity in operator algebras. This hopefully will answer the questions of Paul and Sirix (at least in parts) and of Urs.

First I’ll explain the basic formula due to Tomita that associates to a state LL a one parameter group of automorphisms. The basic fact is that one can make sense of the map xs(x)=LxL 1 x \mapsto \mathbf{s}(x)= L x L^{-1} as an (unbounded) map from the algebra to itself and then take its complex powers s it\mathbf{s}^{it}.

To define this map one just compares the two bilinear forms on the algebra given by L(xy)L(x y) and L(yx)L(y x) . Under suitable non-degeneracy conditions on LL both give an isomorphism of the algebra with its dual linear space and thus one can find a linear map s\mathbf{s} from the algebra to itself such that L(yx)=L(xs(y)) L(y x)=L(x \mathbf{s}(y)) for all xx and yy.

One can check at this very formal level that s\mathbf{s} fulfills s(ab)=s(a)s(b)\mathbf{s}(a b)=\mathbf{s}(a)\mathbf{s}(b): L(abx)=L(bxs(a))=L(xs(a)s(b)) L(a b x)=L(b x \mathbf{s}(a))=L(x \mathbf{s}(a)\mathbf{s}(b))

Thus still at this very formal level s\mathbf{s} is an automorphism of the algebra, and the best way to think about it is as xLxL 1x \mapsto L x L^{-1} where one respects the cyclic ordering of terms in writing Lyx=LyL 1Lx=LxLyL 1L y x= L y L^{-1} L x = L x L y L^{-1}. Now all this is formal and to make it “real” one only needs the most basic structure of a noncommutative space, namely the measure theory. This means that the algebra one is dealing with is a von-Neumann algebra, and that one needs very little structure to proceed since the von-Neumann algebra of an NC-space only embodies its measure theory, which is very little structure. Thus the main result of Tomita (which was first met with lots of skepticism by the specialists of the subject, was then succesfully expounded by Takesaki in his lecture notes and is known as the Tomita-Takesaki theory) is that when LL is a faithful normal state on a von-Neumann algebra MM, the complex powers of the associated map s(x)=LxL 1\mathbf{s}(x)= L x L^{-1} make sense and define a one parameter group of automorphism s L\mathbf{s}_L of MM.

There are many faithful normal states on a von-Neumann algebra and thus many corresponding one parameter groups of automorphism s L\mathbf{s}_L . It is here that the two by two matrix trick (Groupe modulaire d’une algèbre de von Neumann, C. R. Acad. Sci. Paris, Sér. A-B, 274, 1972) enters the scene and shows that in fact the groups of automorphism s L\mathbf{s}_L are all the same modulo inner automorphisms!

Thus if one lets mathrmOut(M)mathrm{Out}(M) be the quotient of the group of automorphisms of MM by the normal subgroup of inner automorphisms one gets a completely canonical group homomorphism from the additive group \mathbb{R} of real numbers δ:Out(M) \delta: \mathbb{R} \to \mathrm{Out}(M) and it is this group that I always viewed as a tantalizing candidate for “emerging time” in physics. Of course it immediately gives invariants of von-Neumann algebras such as the group T(M)T(M) of “periods” of MM which is the kernel of the above group morphism. It is at the basis of the classification of factors and reduction from type III to type II + automorphisms which I did in June 1972 and published in my thesis (with the missing III1_1 case later completed by Takesaki).

This “emerging time” is non-trivial when the noncommutative space is far enough from “classical” spaces. This is the case for instance for the leaf space of foliations such as the Anosov foliations for Riemann surfaces and also for the space of QQ-lattices modulo scaling in our joint work with Matilde Marcolli.

The real issue then is to make the connection with time in quantum physics. By the computation of Bisognano-Wichmann one knows that the s L\mathbf{s}_L for the restriction of the vacuum state to the local algebra in free quantum field theory associated to a Rindler wedge region (defined by x 1>±x 0x_1 \gt \pm x_0) is in fact the evolution of that algebra according to the “proper time” of the region. This relates to the thermodynamics of black holes and to the Unruh temperature. There is a whole literature on what happens for conformal field theory in dimension two. I’ll discuss the above real issue of the connection with time in quantum physics in another post.

Posted by: urs (forwarding a reply by A. Connes) on March 21, 2007 5:50 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

When John Baez and Alain Connes are blogging, you know it is the beginning of a new era :)

Fantastic!

Now, it would be neat to try to relate Connes’ concept of time with the concept of time that is naturally encoded in a diamond graph.

Posted by: Eric on March 21, 2007 7:59 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Now, it would be neat to try to relate Connes’ concept of time with the concept of time that is naturally encoded in a diamond graph.

Yup, this is exactly what I was trying to get at in the above discussion.

The term “canonical/intrinsic time evolution” induced by an algebra (of observables of a quantum theory) suggests that it could be the Heisenberg picture analog of the canonical time evolution in Schrödinger picture that the above entry is concerned with.

But it is not, apparently. As was pointed out, the canonical “time evolution” that Alain Connes is referring to is a canonical 1-parameter action of outer automorphisms on our algebra, while the “standard” time evolution is, usually at least, a 1-parameter action by inner automorphisms, namely by the conjugation xe itHxe itH x \mapsto e^{i t H} x e^{- i t H} that gives the familiar time evolution in the Heisenberg picture.

This evolution is “global”, in that it propagates the entire system forward in time.

The very helpful anonymous commenter pointed out above that this situation changes as we are looking at local algebras of observables only: they do not contain the global Hamiltonian, hence conjugating with that is now an outer automorphism.

So it seems that what Connes’ “canonical time evolution” could correspond to is, in field theory, that aspect of time evolution which is not local, but depends on global information.

I would like to find out what exactly that purely global aspect is, in concrete examples.

The only plausible thing I can think of is that this non-local part is that which knows about the boundary conditions. I still feel that this is plausible, but the anonymous commenter had different feelings.

Anyway, I would like to see concrete examples of standard time evolution in quantum field theory which has an outer component, in the sense of Connes.

I have submitted the following question to Alain Connes’ latest entry on his blog:

Question:

Can anyone point me to any literature where the action by outer automorphisms that Alain Connes describes is explicitly identified in a concrete field theoretical context?

I gather that all local nets of algebras of observables appearing in AQFT are type III factors and should therefore come equipped with the canonical 1-parameter action by outer automorphisms, as discussed.

At the same time, usually for these field theories we have a notion of time evolution that is derived from the physical specification of the system.

How is this “standard” time evolution related to the canonical one by outer automorphisms?

Has this been discussed anywhere, possibly with concrete examples?

Posted by: urs on March 21, 2007 8:18 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Should the algebra be graded?

In the continuum, I think it makes sense to think of xx and pp and as grade 0 operators on some Hilbert space, but for some reason I taught myself that it was more natural to think of momentum as a 1-form, i.e. it is more natural to work with dpdp than just pp. Then the more natural commutative relation is

[x,dp] = i hbar dt.

I’m not exactly sure where I’m going with this, but I wonder if it make sense to view what Connes said from the viewpoint of a graded algebra.

Is it possible to write something like

L(xdp)=L(dps(x))L(x dp) = L(dp s(x))

??

Posted by: Eric on March 22, 2007 2:47 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

but for some reason I taught myself that it was more natural to think of momentum as a 1-form

Well, while I am not 100 percent sure about the formulas you gave, I think you certainly have the right intuition here.

The Weyl algebra for the particle on the line has two kinds of generators which play a different role: one comes from objects in configuration space (“position operators”), the other from morphisms (momentum operators).

Recently I tried to say this in as general terms as I could come up with, and seemed to find a cool general structure: QFT of Charged n-Particle: Algebra of Observables.

Posted by: urs on March 22, 2007 8:17 PM | Permalink | Reply to this

Positions, Translations and Weyl Algebras

I should maybe add another comment to this (with the reference courtesy of Christian Fleischhack):

The Stone-von Neumann theorem, which says that the only well-behaved irrep of the Weyl algebra corresponding to the particle in n\mathbb{R}^n, i.e. that whose infinitesimal generators are elements x ix^i and p ip_i such that [x i,p j]=iδi j[x^i,p_j] = i \delta i_j, is the space L 2( n)L^2(\mathbb{R}^n) of square integrable functions on n\mathbb{R}^n, has a generalization to the case where

1) the configuration space n\mathbb{R}^n (regarded as an abelian group) is replaced by any locally compact group GG

2) translations, e λ ip ie^{\lambda^i p_i}, in n\mathbb{R}^n, regarded as the left action of n\mathbb{R}^n on itself, are replaced by the left action of that group GG on itself.

The Mackey-Stone-von Neumann theorem says that the unique irrep of the algebra generated by functions on GG and the action by (left) translations of the group on these is, again, the space of square integrable functions on this group.

One can find this nicely reviewed on p. 8 of

J. Rosenberg, A Selective History of the Stone-von Neumann Theorem.

I am just saying this in order to support Eric’s intuition that the Weyl algebra of observables in a quantum theory usually comes to us as an algebra generated by two types of generators that play a distinctively different role: one behaves as functions (0-forms), the other as translations (\simeq 1-forms, very roughly).

Whether or not this has a relevance for Connes’ “canonical time evolution” I can’t tell yet.

What is true, on the other hand, is that this splitting re-appears for instance in the GNS construction: if on a Weyl algebra of the above kind (functions on a group and translations on that group) we choose the standard vacuum state given by the constant function on that group (assuming that group is compact), then as we walk through the GNS prescription an ideal I inside the Weyl algebra appears, which has vanishing exectation value with respect to that state.

This ideal is precisely that of translation operators. GNS factors them out to retain just the position operators (the functions on config space) and turn them again in the original space L 2(G)L^2(G).

Posted by: urs on March 23, 2007 2:52 PM | Permalink | Reply to this

Re: Positions, Translations and Weyl Algebras

I’m not sure if this is relevant, but it smells relevant. In stochastic calculus, you will often see things like

dW 2=dtdW^2 = dt

where dWdW is a Brownian motion. This kind of notational trick allows you to derive the Ito formula in a few lines and is a handy computational trick to keep in mind. However, you can also derive the Ito formula by assuming that 0-forms and 1-forms do not commute in such a way that

[W,dW]=dt[W,dW] = dt.

If you squint your eyes, it looks like you are replacing products in some non-graded algebra with commutators in some graded algebra, i.e.

dW*dW[W,dW]dW*dW \to [W,dW].

So when trying to relate Connes’ stuff, it might help to keep this transformation of algebras in mind.

For example, in the first case we could define a commutative algebra where

dW*dt=dt*dW=dt*dt=0dW*dt = dt*dW = dt*dt = 0 and dW*dW=dtdW*dW = dt.

In the second case, we have a non-commutative algebra,

[W,dt]=[t,dW]=[t,dt]=0[W,dt] = [t,dW] = [t,dt] = 0 and [W,dW]=dt[W,dW] = dt.

Just a thought…

Posted by: Eric on March 23, 2007 5:51 PM | Permalink | Reply to this

Re: Alain Connes’ “Intrinsic Time”

Well, while I am not 100 percent sure about the formulas you gave, I think you certainly have the right intuition here.

Argh!!

All this time I’ve been going on about the importance of the commutative relation

[x,dx]=dt[x,dx] = dt

and then I make such a silly typographical (and mental) error.

Of course

[x,dp]=idt[x,dp] = i \hbar dt

is incorrect. The units don’t even work out right! There should be some relation between

[x,p]=i[x,p] = i \hbar

and

[x,dx]=dt[x,dx] = dt.

This reminds me of the conversation we had over here.

If we multiply both sides by ii \hbar we have

[x,idx]=idt[x,i \hbar dx] = i \hbar dt.

Ok. I was going somewhere with this, but I’m lost again. Help! :)

Posted by: Eric on March 28, 2007 6:24 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: The Canonical 1-Particle

Towards the end of the above entry, right after the picture of the category hist 2 x\mathrm {hist}_2^x of paths of length 2 ending at xx, I wrote:

except that I have removed two morphism which would connect entries on the middle row. These morphisms would come from separating two points on the path which have already merged to a single point. By the principle that these paths come from separate unit steps, this is not to be allowed.

Maybe a better way to say this is: remove all morphisms between paths which amount to mere reparameterizations.

Posted by: urs on March 20, 2007 8:37 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: The Canonical 1-Particle

I have now thought more about the right way to state the definition of hist n\mathrm{hist}_n, and, as so often, the answer is so very obvious once one has it.

First recall that the category of 1-step histories hist 1 out * in * conf conf \array{ && \mathrm{hist}_1 \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} &&&& \mathrm{conf} } was the subcategory hist 1Mor(worldvol 1,tar) \mathrm{hist}_1 \subset \mathrm{Mor}(\mathrm{worldvol}_1,\mathrm{tar}) of all “maps” (functors) from the 1-unit worldvolume worldvol 1:={12} \mathrm{worldvol}_1 := \{1 \to 2\} into target space tar:={(x1)x(x+1)(x+2)}, \mathrm{tar} := \{ \cdots (x-1) \stackrel{\leftarrow}{\to} x \stackrel{\leftarrow}{\to} (x+1) \stackrel{\leftarrow}{\to} (x+2) \cdots \} \,, definined by the two restrictions

a) objects are only those functors that are at the same time morphisms of the underlying graphs, i.e. which map the single edge 121 \to 2 either to an identity in tar\mathrm{tar} or to a single edge in tar\mathrm{tar}.

b) morphisms are all natural transformations between these, except for those which would go from an identity image (12)(xx)(1 \to 2)\mapsto (x\to x) to a non-identity image (12)(xx+1)(1 \to 2)\mapsto (x\to x+1).

Now, I think the category of 2-step histories that I talked about hist 2 out * in * conf conf \array{ && \mathrm{hist}_2 \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} &&&& \mathrm{conf} } is, as certainly it should be, nothing but the composite span, i.e. the pullback of hist 1\mathrm{hist}_1 along itself hist 2 hist 1 hist 1 out * in * out * in * conf conf conf. \array{ &&&& \mathrm{hist}_2 \\ &&& \swarrow && \searrow \\ && \mathrm{hist}_1 &&&& \mathrm{hist}_1 \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} && {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} &&&& \mathrm{conf} &&&& \mathrm{conf} } \,.

So I’d expect that as we form hist n x\mathrm{hist}_n^x in this fashion, its Leinster measure will reproduce the formula for the nnth power of identity plus lattice Laplace operator. But I still have not checked this beyond n=2n=2.

But is there maybe some general nonsense that ensures this:

Question: Given a collection of spans and some objects that may be pull-pushed through them. Under which conditions is the map from spans to pull-push operations functorial? I.e. is there some general theorem which ensures me that the pull-push through hist n\mathrm{hist}_n is the same as the nn-fold pull-push through hist 1\mathrm{hist}_1?

I know that when the pull-push is done on ordinary functions with the push-forward being induced by a measure, then this functoriality is a condition on that measure.

But here I am talking about pull-push where the push-forward is done using adjoint functors to the pullback functors. So I imagine that functoiality of pull-pushing might here be automatically guaranteed. But I don’t know.

Posted by: urs on March 21, 2007 12:23 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: The Canonical 1-Particle

Mmm… interesting. Something which is slightly relevant to the situation here is something I found yesterday while trying to understand what a “Green functor” was. There’s a talk by E. Panchadcharam, joint work with Ross Street, at CT06 available online, on “Mackey functors and Green functors”.

I’m not sure what the most elegant way to state it is (I believe this talk was exactly trying to find the right “categorical language” for them) but it seems a Mackey functor is, basically, a pair of functors F *F^* and F *F_*, one contravariant and one covariant, from some category CC (for instance, the category of subgroups of a group) to Vect, behaving nicely with respect to induction and restriction.

Thats the old terminology, at least.

A nice fact is that, basically, the category of Mackey functors is equivalent to the category [Span(C), Vect] of functors from Span(CC) into Vect:

(1)Mackey[C,Vect][Span(C),Vect]. Mackey[C, Vect] \simeq [Span(C), Vect].

This works as follows. If (F *,F *)(F^*, F_*) is a Mackey functor (i.e. F *F^* and F *F_* are contravariant (resp. covariant) functors from CC to Vect), then we get a functor F^:Span(C)Vect\hat{F} : Span(C) \rightarrow Vect by sending

(2)F^(AaBbC)=F(A)F *(a)F(B)F *(b)F(C). \hat{F} \left(A \stackrel{a}{\leftarrow} B \stackrel{b}{\rightarrow} C \right) = F(A) \stackrel{F^*(a)}{\rightarrow} F(B) \stackrel{F_*(b)}{\rightarrow} F(C).

In other words, we send a span to a composite of morphisms by the push-pull formula! See page 10 of the talk.

Anyhow, I’m just mentioning this because it gives some additional tobacco into the pipe that people like to smoke around here - it relates push-pull mechanics to Mackey functors.

Posted by: Bruce Bartlett on March 21, 2007 1:34 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: The Canonical 1-Particle

Hi Bruce!

Thanks a lot for this information!

While it does not directly answer my question how to decide when pull-push is functorial, it does reveal a standard name for this question:

in terms of Mackey functors what I am asking is if pullback and pushforward make the Mackey square commute (slide 8).

In more detail (using the variable names from the present example just for convenience), given a composite of spans hist 2 p 1 p 2 hist 1 hist 1 out * in * out * in * conf conf conf \array{ &&&& \mathrm{hist}_2 \\ &&& {}^{p_1}\swarrow && \searrow^{p_2} \\ && \mathrm{hist}_1 &&&& \mathrm{hist}_1 \\ & {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} && {}^{\mathrm{out}^*}\swarrow && \searrow^{\mathrm{in}^*} \\ \mathrm{conf} &&&& \mathrm{conf} &&&& \mathrm{conf} } and the corresponding pull-push of states (sections, living in sect\mathrm{sect}) through the two hist 1\mathrm{hist}_1-factors sect hist 2 pf(p 1) pb(p 2) sect hist 1 sect hist 1 pf(out *) pb(in *) pf(out *) pb(in *) sect conf sect conf sect conf \array{ &&&& \mathrm{sect}_{\mathrm{hist}_2} \\ &&& {}^{\mathrm{pf}(p_1)}\swarrow && \nwarrow^{\mathrm{pb}(p_2)} \\ && \mathrm{sect}_{ \mathrm{hist}_1 } &&&& \mathrm{sect}_{ \mathrm{hist}_1 } \\ & {}^{ \mathrm{pf}(\mathrm{out}^*)}\swarrow && \nwarrow^{\mathrm{pb}(\mathrm{in}^*)} && {}^{\mathrm{pf}( \mathrm{out}^*)}\swarrow && \nwarrow^{\mathrm{pb}(\mathrm{in}^*)} \\ \mathrm{sect}_{\mathrm{conf}} &&&& \mathrm{sect}_{\mathrm{conf}} &&&& \mathrm{sect}_{\mathrm{conf}} } I would like to know if and when the square in the middle of this diagram commutes.

Because if it does, then the map from spans to the corresponding pull-push operations is functorial.

This square is, apparently, called the Mackey square of pb\mathrm{pb} and pf\mathrm{pf}.

Thanks!

P.S. One issue we need to keep in mind is that, for our example, all of the above lives in Cat\mathrm{Cat}, hence all pullback squares ought to be regarded as suitably weak pullbacks, etc.

Due to the simplicity of the setup (i.e. the simplicity of tar\mathrm{tar}, worldvol\mathrm{worldvol}, etc) this seems to have little effect. But it needs to be kept in mind.

Posted by: urs on March 21, 2007 2:45 PM | Permalink | Reply to this

Re: QFT of Charged n-Particle: The Canonical 1-Particle

I will be offline until Sunday evening, when I arrive in Oberwolfach.

Just searched the web for some literature to read on various train trips.

The scary thing is that when one currently types “”AQFT time evolution” into Google the resulting hits start with my question on that matter, this comment section here, and lots of links where the terms happen to coappear just by coincidence…

Posted by: urs on March 28, 2007 5:10 PM | Permalink | Reply to this
Read the post Some Notes on Local QFT
Weblog: The n-Category Café
Excerpt: Some aspects of the AQFT description of 2d CFT.
Tracked: April 1, 2007 5:31 AM
Read the post The n-Café Quantum Conjecture
Weblog: The n-Category Café
Excerpt: Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
Tracked: June 8, 2007 6:24 PM
Read the post QFT of Charged n-Particle: Extended Worldvolumes
Weblog: The n-Category Café
Excerpt: Passing from locally to globally refined extended QFTs by means of the adjointness property of the Gray tensor product of the n-particle with the timeline.
Tracked: August 2, 2007 7:32 PM
Read the post The Canonical 1-Particle, Part II
Weblog: The n-Category Café
Excerpt: More on the canonical quantization of the charged n-particle for the case of a 1-particle propagating on a lattice.
Tracked: August 15, 2007 11:45 AM
Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Tracked: March 27, 2008 4:49 PM

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