Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

February 27, 2007

Quantization and Cohomology (Week 16)

Posted by John Baez

This week in our course on Quantization and Cohomology we considered some fancier path integrals. Then, fortified by these examples, we returned to the more abstract issues this course is really about:

  • Week 16 (Feb. 27) - More examples of path-integral quantization. The particle in a potential on the real line. The Lie-Trotter Theorem. The particle in a potential on a complete Riemannian manifold. Back to general questions: how do we get a Hilbert space from a category equipped with an action functor? The problem of Cauchy surfaces.

Last week’s notes are here; next week’s notes are here.

An interesting anecdote related to our discussion of Cauchy surfaces and (not mentioned in the notes) closed timelike loops.

I mentioned that Einstein and Gödel were friends at the Institute of Advanced Study in Princeton back in the 1950s. Gödel learned general relativity, and found a solution called the Gödel universe, where there’s a field of observers who each trace out closed timelike loops and each see the rest of the universe revolving around him — perhaps a subtle comment on life at the Institute. The interesting thing is that he did this just to provide evidence for his conviction that time might not be described by a partial ordering on the set of events!

One of the students replied with this anecdote: Rudy Rucker visited the Institute for Advanced Study and asked Gödel why the illusion of the passage of time is so convincing. Gödel replied something like "For me, it’s not".

Posted at February 27, 2007 11:00 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1182

29 Comments & 2 Trackbacks

Re: Quantization and Cohomology (Week 16)

That’s a wonderful anecdote. And since I’m hearing it at least third-hand, on the Internet it must be true!

Posted by: Blake Stacey on February 27, 2007 11:45 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

I am very interested in the issue of Cauchy-surfaces and spaces of states that you briefly mention at the end of this week’s lecture notes.

Lately I have begun thinking about this issue a little.

It is closely related to the time-slice axiom on AQFT (saying that the algebra of observabes of any neighbourhood of a Cauchy surface already coincides with that of the full space), and in fact to most of the AQFT axioms.

But, I don’t know, somehow it feels like these axioms are missing some deeper meaning. Something is strange.

These “nets of observables” are really pre-cosheaves with a funny extra property, which says that all co-restriction maps are injections.

But together with the time-slice axiom this essentially does imply a proper co-sheaf property, I think, even though I have never seen that made explicit in the literature.

But wouldn’t it be more natural to expect a general (co-)sheaf in the first place? Without any extra condition on the injectivity of the corestriction morphisms?

Somehow it feels like the Tao is being intervened with too much in these axioms.

Maybe the deepest formulation of what the AQFT axioms are really about is the formulation given in

The generally covariant locality principle.

At least there the fixation on Minkowski space is done away with. Again, on the very first pages one sees something like a site of Minkowski spaces (without it being explcitly addressed this way) and something that almost looks like a cosheaf, only to find just pre-cosheafs with a couple of conditions put on by hand.

On the other hand, the time-slice axiom seems to be the most crucial aspect of what all this really is about, and maybe it does not want to live in a world of (pre)(co)sheaves at all. Maybe it lives in a world that has not been identified yet.

I don’t know, that’s just my feeling. I cannot yet put my finger more concretely on what I am wondering about, am only just in the process of coming to terms with the nature of algebras of observables #.

I am also worried about putting in all that Minkowski-structure by hand. I’d rather see a general definition of a QFT, say by an nn-functor, and then a notion of “internal causality” derived from its properties. Why exclude Euclidean field theories and topological field theories from the get-go? The best examples of tractable field theories we have are of the very kind that are excluded by the AQFT axioms!

I’d rather wish you’d hand me a QFT, and I could probe it for its notion of locality. Every QFT will associate Hilbert spaces (possibly nn-Hilbert spaces) to subsets of its parameter space, so that’s certainly not something that should a priori be linked to pseudo-Riemannian structure.

Er, why am I going on about AQFT?? Ah, right, because you mentioned Hilbert spaces and their relation to Cauchy surfaces… I need to understand that better.

Posted by: urs on February 28, 2007 3:04 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

The following comment is not directly related to the above lecture, but it is related to the general course.

Ever since I mentioned it (here) I have become more and more convinced that it would be very desireable to have a good general understanding of the deeper meaning of Schrödinger versus Heisenberg picture.

I know that this may sound like a silly tiviality at first, something every student understands completely – but maybe there is more to it.

Something that should make one suspicious is a curious pedagogical inconsistency in quantum mechanics textbooks: when it comes to quantum field theory, they will inevitably behave as if the Schrödinger picture never played any role whatsoever.

The AQFT axioms take this to the extreme: there is the operator algebra and just the operator algebra.

That would be fine with me. If it were not for a curious suspicion I have, that something about the choice of axioms in AQFT is not quite the way nature intended it to be.

So before talking about (n>1)(n \gt 1)-dimensional QFT, let’s maybe concentrate on 1-dimensional QFT, aka quantum mechanics.

In the Schrödinger picture, the theory is a functor U S:Σ()Hilb U_S : \Sigma(\mathbb{R}) \to \mathrm{Hilb} from the worldline to Hilbert spaces: it sends every point on the worldline to a Hilbert space, and every interval to the corresponding propagator.

In the Heisenberg picture the theory is also a functor U H:Σ()C *, U_H : \Sigma(\mathbb{R}) \to C^* \,, now from the wordline to C *C^*-algebras: it sends every point to the C *C^*-algebra of bounded operators on that Hilbert space (the “algebra of observables”, which is generated by the Weyl algebra) and sends every interval to the C *C^*-algebra isomorphism obtained by conjugating with the propagator.

We can think of the passage from the Schrödinger picture to the Heisenberg picture as being induced by the functor B:HilbC *, B : \mathrm{Hilb} \to C^* \,, which sends each Hilbert space HH to the C *C^*-algebra B(H) B(H) of bounded operators on that Hilbert space.

So, the Heisenberg picture is the push-forward of the Schrödinger picture along BB: U H=B *U S. U_H = B_* U_S \,.

It’s a triviality. But still.

By the Gelfand-Naimark theorem, BB is essentially surjective. By way of GNS this is even constructive.

Does B *B_* have a weak inverse? Or at least an adjoint? If so, is that given by the GNS construction?

I expect something like this should be true. But what would be the point of that?

My point would be that this seems to indicate that it is very natural to associate a C *C^* algebra to a point on the worldline.

But every point on the worldline is a Cauchy surface.

So let me say that again: this seems to indicate that it is very natural to associate a C *C^* algebra to a Cauchy surface.

What is, at this point, not really natural is to associate an algebra to an interval of the worldline.

To amplify this point a little, let’s just very slightly lift the dimension of parameter space from n=1n=1 to n=2n=2. The standard AQFT text on 2-diemnsional QFT might state the general axioms, but will then quickly derive the fact that everything is in fact already determined by a net of algebras on the circle. Which is the Cauchy surface.

Of course, that’s not really surprising, physically.

But what is maybe surprising, though, is that, from only knowing 1- and 2-dimensional QFT, this fact makes the AQFT axioms look a little ill-motivated. While they talk about assigning algebras to subsets of all of parameter space, in practice we only ever want to assign them to Cauchy-surfaces, here.

(Hopfully some AQFT expert will read this and set me straight.)

We know one way to check our axioms in higher dimensions knowing those in lower dimensions: using categorification.

We know, at least roughly, that we can consider 2-dimensional QFT as a categorified version of quantum mechanics.

Now, interestingly, from that perspective we are lead to find exactly the described outcome: the categorified theory will associate a “space of 2-states” to 1-diemsnsional parts of parameter space. This will have a monoidal category of endomorphisms. From this we can extract an algebra. This is then naturally associated to the 1-dimensional part of parameter space. And only to that.

Given that, I am looking for two things:

1) a precise statment about whether or not the above 1-functor B *B_* has an adjoint or a weak inverse.

2) a categorification of that statement.

Posted by: urs on March 2, 2007 8:52 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

While composing the above comment, I already received a reply over at the NCG blog.

(Which does of course not mean that I am not looking for further replies! :-)

Posted by: urs on March 2, 2007 9:28 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Today I had some very good discussion with Jens about the principles of AQFT, the phenomenon of gerbes of chiral differential operators, Heisenberg and Schrödinger and how it should all fit together. Here is one very simple but potentially helpful observation.

So recall that I allow myself to be puzzled about what might look like a non-issue:

do algebras of quantum fields (in the Heisenberg picture, really) naturally live on pieces of the bulk or on pieces of codimension 1?

In AQFT it almost looks as if people want to consider a sheaf of these fields. Though nobody ever admits that.

In Euclidean field theory, on the other hand, people are quite fond of thinking of sheaves of chiral algebras and even gerbes of them! (See for instance this).

On the other hand, in AQFT we have the time slice axiom, which roughly says something like that the algebras in the bulk are already fully determined by those sitting on certain codimension 1 surfaces.

And this is actually quite natural from the point of view of obtaining the Heisenberg picture from a Schrödinger picture evolution functor, as I tried to say above.

So what’s going on?? Codimension 0 or codimension 1? Sheaf, gerbe, or nothing like this?

After thinking a little, maybe the answer is this:

we do have an (nn-)sheaf of quantum fields on spacetime, true – but this comes to us as a sheaf of flat sections of something. And this flatness is responsible for the fact that the data in the bulk may already be specified by that on the boundary.

I’ll make this precise and illustrate this for (n=1)(n=1)-dimensional QFT, i.e. for quantum mechanics.

So here is a worldline \mathbb{R} and here is the category of paths on the worldline: P 1(). P_1(\mathbb{R}) \,. A quantum mechanical system gives rise to a propagation functor U:P 1()Hilb. U : P_1(\mathbb{R}) \to \mathrm{Hilb} \,. In other words, this is a bundle of Hilbert spaces over \mathbb{R} with connection Hdt, H \,dt \,, where HH is the Hamiltonian. (Compare the discussion with Bruce).

To go from this Schrödinger picture to the Heisenberg picture of QM, we hit UU with the functor B *:[P 1(),Hilb][P 1(),C *], B_* : [P_1(\mathbb{R}),\mathrm{Hilb}] \to [P_1(\mathbb{R}),C^*] \,, which sends every Hilbert space to its algebra of bounded operators, and sends the propagator U(t)U(t) to its adjoint action Ad U(t)\mathrm{Ad}_{U(t)} on these algebras. (This is what I talked about above).

The resulting Heisenberg transport B *U:P 1()C * B_* U : P_1(\mathbb{R}) \to C^* is now an algebra bundle with connection over \mathbb{R}!

(Each fiber should be thought of as the closure of the Weyl algebra characterizing the phase space of the system. )

The connection 1-form now reads ad Hdt. \mathrm{ad}_H \, d t \,.

To be explicit, assume the QM system is that of a particle propagating on n\mathbb{R}^n (I could consider 1\mathbb{R}^1 for simplicity, but I want to keep us from confusing target space with parameter space).

Then all the Hilbert space fibers of the original bundle were canonically identified with L 2( n)L^2(\mathbb{R}^n).

Accordingly, all the algebra fibers are canonically identified with the Weyl algebra, which I think of, for convenience, as canonically generated infinitesimally from elements x ix^i and p ip_i with [x i,p j]=iδ j i[x^i,p_j] = i \delta^i_j.

Now consider the algebra of flat sections of our algebra bundle over the worldline.

Better yet, consider the sheaf of algebras of flat sections of this bundle.

To every open interval (a,b)(a,b) \subset \mathbb{R}, this will associate all flat section of B *U| (a,b):P 1(a,b)C *B_* U|_{(a,b)} : P_1(a,b) \to C^*.

But every such flat section is completely determined already by its value over any one point.

Assume the points 00 and tt are inside our interval.

Over 00, we have the algebra element x ix^i. We might call that x i(0) x^i(0) to make explicit that we think of x ix^i as sitting in the fiber B *U(0)B_* U(0) and not anywhere else.

This element determines an entire flat section over all of (a,b)(a,b). Over tt, the value of this flat section will be (for the free particle) x itmp ix^i - \frac{t}{m}p_i, now regarded as an element of B *U(t)B_* U(t). So we should, following the above convention, write this as x i(t)tmp i(t). x^i(t) - \frac{t}{m}p_i(t) \,.

Let me write e x i(0)Γ(B *U) e_{x^i(0)} \in \Gamma(B_* U) for the flat section uniquely specified by that fact that its value at 00 is x ix^i.

Similarly I then write e x i(t)tmp i(t)e_{x^i(t) - \frac{t}{m}p_i(t)} for the flat section uniquely specified by having the value x i+tmp ix^i + \frac{t}{m}p_i over tt.

Since both x i(0)x^i(0) and x i(t)+tmp i(t)x^i(t) + \frac{t}{m}p_i(t) uniquely specify the same flat section of our algebra bundle, we have e x i(0)=e x i(t)+tmp i(t). e_{x^i(0)} = e_{x^i(t) + \frac{t}{m}p_i(t)} \,. If we were in a more relaxed mood, we might simply write x i(0)=x i(t)tmp i(t). x^i(0) = x^i(t) - \frac{t}{m}p_i(t) \,.

(I know I require a certain tolerance for sophisticated-looking trivialities here. The point is that this exercise is supposed to make life easier as we go up the dimensional ladder…)

Now, sections of a bundle of algebras form an algebra themselves, simply by pointwise multiplication.

For instance, we find [e x i(0),e p j(0)]=e iδ j iId(0). [e_{x^i(0)}, e_{p_j(0)}] = e_{i\delta^i_j\mathrm{Id}(0)} \,. Or [e x i(0),e x j(t)]=[e x i(0),e x j(0)+tmp j(0)]=tme iδ j iId(0). [e_{x^i(0)}, e_{x^j(t)}] = [e_{x^i(0)}, e_{x^j(0) + \frac{t}{m}p_j(0)}] = \frac{t}{m} e_{i\delta^i_j \mathrm{Id}(0)} \,.

In our more relaxed mood, this reproduces the familiar formulas for the operator algebra of fields on the worldline [x i(0),x j(t)]=tmiδ j i. [x^i(0), x^j(t)] = \frac{t}{m} i\delta^i_j \,.

Bottom line: (sheaves of) operator algebras of fields are (sheaves of) algebras of flat sections of the Heisenberg bundle obtained as the push-forward of the Hilbert bundle obtained in the Schrödinger picture.

Posted by: urs on March 4, 2007 9:22 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Urs wrote about AQFT:

While they talk about assigning algebras to subsets of all of parameter space, in practice we only ever want to assign them to Cauchy-surfaces, here.

But what if no Cauchy surfaces exist? Certainly, there are some Lorentzian manifolds that have no Cauchy surfaces; even in an approximately FRW universe like ours seems to be, the Cauchy surfaces appear only before the collapse of the first black hole. (This is my preferred resolution —or rather, explanation away— of the black hole information paradox.) Then we really need the entire net of algebras on more general subspaces.

Posted by: Toby Bartels on March 2, 2007 11:14 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Hi guys,

I’ve been quiet for a while, but on the subject of the Schrodinger picture, and QFT, I couldn’t resist mentioning my favourite textbook on QFT : Brian Hatfield, The quantum mechanics of point particles and strings .

Hey, everyone has their favourite, but Hatfield is the one textbook which gives more than just a nod to the Schrodinger picture in quantum field theory.

When I was a physicist, I remember having the following idea about doing quantum mecahnics (and even field theory) in a geometric way, without co-ordinates, and without choosing Cauchy surfaces. Perhaps you guys can tell me what you think!

We’re going to do relativistic quantum mechanics on a Lorentizan spacetime manifold MM. We should be doing quantum field theory but lets keep things simple… for now.

We’re going to construct a connection on a bundle of Hilbert spaces over the tangent bundle TMTM.

We think of a point (p,v p)TM(p, v_p) \in TM as the status of an observer in spacetime. Let LT pML \subset T_p M be the vector space orthogonal to v pv_p. Each vector wLw \in L gives rise to a geodesic C wC_w on spacetime MM issuing in the direction of ww. Let C (p,v p)MC_{(p, v_p)} \subset M be the union of all these curves. Thus C (p,v p)C_{(p, v_p)} is what the observer considers to be “space”. Note that C (p,v p)C_{(p,v_p)} inherits a Riemannian metric from spacetime MM. [Ed : I think ! If this is wrong, everything goes wrong.]

We make a bundle of Hilbert spaces HH over TMTM by assigning to each (p,v p)TM(p, v_p) \in TM the space of L 2L^2 functions over C (p,v p)C_{(p,v_p)}:

(1)(p,v p)L 2(C (p,v p)). (p, v_p) \mapsto L^2(C_{(p, v_p)}).

Here we are using the Riemannian metric on C (p,v p)C_{(p, v_p)} to make sense of “L 2L^2”.

So we have an (infinite dimensional) vector bundle HH over TMTM.

Claim : The Schrodinger equation is a not-flat connection on this bundle. The Klein-Gordon equation is a flat connection on this bundle.

The point is that we want to do quantum mechanics geometrically, i.e. express the wave equations as connections on vector bundles.

Being a flat connection, in our context, means that if two friends stand next to each other and look around them, and then one runs around the block while the other remains stationary, then when the former returns they will both agree on what the world looks like. Thus in this picture, relativistic invariance means flat connection .

Okay, its not precise at all, and surely wrong as it stands… but you get the idea. Does that vector bundle make sense?

Posted by: Bruce Bartlett on March 2, 2007 11:59 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Brian Hatfield, The quantum mechanics of point particles and strings .

Hey, everyone has their favourite, but Hatfield is the one textbook which gives more than just a nod to the Schrodinger picture in quantum field theory.

Yes, that’s a good point. I thought about mentioning Hatfield, but then didn’t, for some reason.

I used to go around advertising Hatfield a lot in the good old days of sci.physics.research, for instance here.

Posted by: urs on March 3, 2007 7:09 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

…but you get the idea. Does that vector bundle make sense?

Let me try to look at a simple special case before thinking about it in full generality.

So I assume spacetime to be of the form M=Σ×M = \Sigma \times \mathbb{R} with (Σ,g)(\Sigma,g) some Riemannian space and the standard pseudo-Riemannian structure induced on MM (i.e. ds 2=dt 2ds g 2d s^2 = dt^2 - d s_g^2).

Take an observer at (x,t)(x,t) and the vector (0, t)(0,\partial_t) at that point.

Then we probably want to associate to that the Hilbert space L 2(Σ,dμ g)L^2(\Sigma, d\mu_g).

This would be reproduced by your prescription if Σ\Sigma is the union of all geodesics in Σ\Sigma emanating at xx.

So here would be, in general, somewhat of a difference between the prescription you indicated and the standard quantum mechanics on Σ\Sigma. Right?

Now, in a more standard approach, we would assign the very same Hilbert space to all points (y,t)(y,t), (0, t)(0,\partial_t) for all yΣy \in \Sigma. In other words, we would simply assign one Hilbert space to the Cauchy surface Σ\Sigma, which we can think of as an incoming boundary of a Riemannian cobordism of the form Σ×[0,t]. \Sigma \times [0,t] \,. I could reformulate this as saying that we get a (trivial) Hilbert space bundle over \mathbb{R}, whose fiber over tt \in \mathbb{R} is the Hilbert space associated to Σ×{t}\Sigma \times \{t\}, which is L 2(Σ,dμ g)L^2(\Sigma,d\mu_g) everywhere.

Then, indeed, the Schrödinger equation describes the parallel transport along \mathbb{R} of sections of this Hilbert space bundle over \mathbb{R}. The corresponding parallel transport functor U:P 1()Hilb U : P_1(\mathbb{R}) \to \mathrm{Hilb} is the propagator U(t 1,t 2)=exp(i(t 2t 1)Δ Σ). U(t_1,t_2) = \exp(i (t_2-t_1) \Delta_\Sigma) \,.

So, would this be reproduced, somehow, by the prescription you indicated? I am thinking that you’d need to explicitly include a way to identify all the different Hilbert spaces associated to (y,t)(y,t) (0, t)(0,\partial_t) for different yy. Clearly, all these want to be related by canonical isomorphisms. But I am not quite sure how these would arise from the point of view you described.

Posted by: urs on March 3, 2007 8:18 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Thanks Urs!

Urs wrote:

I used to go around advertising Hatfield a lot in the good old days of sci.physics.research, for instance here.

Cool - if I had had you around when I was doing theoretical physics, I probably would never have left for mathematics!

I went to that paper The Schrodinger Wave Functional and Vacuum States in Curved Spacetime. I see that they start with a splitting of spacetime into R×ΣR \times \Sigma. This is fine, perhaps… but the setup I outlined is precisely an attempt to avoid this splitting - as you observed. But the article seems really interesting!

Urs wrote:

Then we probably want to associate to that the Hilbert space L 2(Σ,dμg)L^2(\Sigma,d\mu g). This would be reproduced by your prescription if Σ\Sigma is the union of all geodesics in Σ\Sigma emanating at xx. So here would be, in general, somewhat of a difference between the prescription you indicated and the standard quantum mechanics on Σ\Sigma. Right?

I don’t think I get it… I like your setup, this is precisely the way I think about it too. But I don’t see the problem… it seems to me that my prescription gives exactly the same as standard quantum mechanics on Σ\Sigma. Suppose the state of our observer is ((x,t),(0, t))T(Σ×)((x,t), (0, \partial_t)) \in T(\Sigma \times \mathbb{R}). The tangent vectors orthogonal to (0, t)T x,t(Σ×)(0, \partial_t) \in T_{x,t}(\Sigma \times \mathbb{R}) are precisely the tangent vectors along Σ\Sigma, i.e. of the form (v,0)(v, 0). Thus the surface swept out by their geodesics is precisely Σ\Sigma. That’s the point : the construction is designed to identify what the observer considers to be “space”.

Hence the Hilbert space our observer sees is exactly L 2(Σ)L^2(\Sigma).

But if he hadn’t been standing still, i.e. if the spacelike component of his tangent vector had been nonzero, it would have been a slightly different Hilbert space (though of course isomorphic to L 2(Σ)L^2(\Sigma)).

Urs wrote:

I am thinking that you’d need to explicitly include a way to identify all the different Hilbert spaces associated to ((y,t),(0, t))((y,t), (0,\partial_t)) for different yy.

Indeed… that’s the whole point of the connection! My motivation is that I’m uncomfortable with the usual setup where all the quantum mechanics takes place in a fixed Hilbert space . I believe its really a bundle of Hilbert spaces, and the quantum equations (Schrodinger, Klein-Gordon, Dirac, field theory version of Schrodinger) are connections on this bundle.

I believe that if you’re sitting at somepoint in spacetime (I really mean some point in the tangent bundle of spacetime), and me at another point, then our Hilbert spaces are not the same . I could compare them with a worldline which connects me to you…and the answer depends on the worldline chosen, unless the connection is flat (i.e. its a relativistically invariant equation).

Philosophy aside.

To sum it up again : we construct a connection on a Hilbert space bundle over the tangent bundle TMTM of spacetime. Dynamics is parallel transport along your worldline. Thus, if γ:ab\gamma : a \rightarrow b is your worldline, the way you “percieve” the world (that is, the Hilbert spaces and the evolution of a wavefunction you observe inside these Hilbert spaces) is precisely parallel transport of ψ\psi.

Posted by: Bruce Bartlett on March 3, 2007 9:11 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Hey Bruce,

interesting discussion! I believe we pretty much agree on lots of the basic ideas. I was just trying to better understand some technical details which you seem to have left as an exercise for the reader (of the former comment). :-)

it seems to me that my prescription gives exactly the same as standard quantum mechanics

Oh, maybe I should have been more explicit about the point I made here: I was indicating that for general Riemannian space Σ\Sigma and any given point xx on Σ\Sigma, you may find a point yy on Σ\Sigma which is not connected by any geodesic with xx. Hence – at least so I thought – your prescription whould give a Hilbert space of functions possibly only on a subset of Σ\Sigma. No? (Well, certainly if Σ\Sigma is not connected. But even if we assume connectedness.)

I’m uncomfortable with the usual setup where all the quantum mechanics takes place in a fixed Hilbert space

Okay. I am pretty sure that quantum mechanics is about bundles of Hilbert spaces – but not on target space but on parameter space!

I wrote my little exegesis of you original comment in order to indicate how I understand that the Schrödinger equation does give (the equation for parallel transport of) a connection on a bundle of Hilbert spaces on the worldline (\mathbb{R}).

I think I could trivially extend this to a bundle of Hilbert spaces on Σ×\Sigma \times \mathbb{R}, simply by pulling back along the “time coordinate map” p 2:Σ×. p_2 : \Sigma\times \mathbb{R} \to \mathbb{R} \,.

But what I was not so sure about is how you would set up your bundle of Hilbert spaces on Σ×\Sigma \times \mathbb{R} directly. Seemed to me you had some magic in mind which I did not see yet.

I am not saying at all that your construction would not work. Really not! I’d just need maybe a little more help! :-)

Let’s see, I should maybe ask a concrete question to get us started:

let’s assume we locally choose coordinates, such that we can express our connection in terms of a connection 1-form.

In the setup that I am talking about, where the connection is that on a bundle of Hilbert spaces on \mathbb{R}, that 1-form would explicitly be Hdt, H\; d t \,, where HH is the Hamiltonian, regarded as an endomorphisms of Hilbert spaces.

You would need to have a connection on Σ×\Sigma \times \mathbb{R}. How would it look like in components, explicitly?

Posted by: urs on March 3, 2007 9:45 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Urs wrote:

Okay. I am pretty sure that quantum mechanics is about bundles of Hilbert spaces – but not on target space but on parameter space!

Ok, good. Let me cast the setup in the language which will make it clear that we both share the same philosophy.

I’m working in the Stolz-Teichner picture. The setup I outlined was inspired by reading page 11 and the top of page 12 of their manuscript. Specifically, my setup is precisely a special case of “Example 2.16”. I’ll use their notation, simplified a bit, and I’ll drop the word “Euclidean” because it might make people think I’m dealing with “space” rather than with both space and time.

So let XX be spacetime, and let B 1(X)B^1(X) be the category of worldlines in XX. By a “worldline” I just mean a path Γ:IX\Gamma : I \rightarrow X from an interval II [sorry about this “II”, it should come out as “I”] into XX which is parametrized by arclength, and which never travels faster than light. Basically, B 1(X)B^1(X) is just 1Cob internal to XX.

I’m thinking of a field theory as a functor

(1)E:B 1(X)Hilb, E : B^1(X) \rightarrow Hilb,

just like them, in example 2.1.6.

There they point out that one can construct such a functor from a Riemannian submersion π:ZX\pi : Z \rightarrow X. On objects, EE associates to a point xXx \in X the space of L 2L^2 functions on the fiber at xx:

(2)xL 2(π 1(x)). x \mapsto L^2(\pi^{-1}(x)).

If Γ:IX\Gamma : I \rightarrow X is a wordline (path) from Γ(0)=a\Gamma(0) = a to Γ(1)=b\Gamma(1) = b in XX, we associate to it a linear operator E(γ):E(x)E(y)E(\gamma) : E(x) \rightarrow E(y) by integrating over the space of maps Γ˜:IZ\tilde{\Gamma} : I \rightarrow Z which are lifts of Γ:IX\Gamma : I \rightarrow X, weighted by the action of course (unless we’re doing the free theory like them).

Lets summarize that in terms of the familiar formula. So let |aE(x)|a\rangle \in E(x) be a wavefunction sharply peaked around aπ 1(x)a \in \pi^{-1}(x), and similarly let |bE(y)|b\rangle \in E(y) be a wavefunction sharply peaked around bπ 1(y)b \in \pi^{-1}(y). Then the matrix elements of E(Γ):E(x)E(y)E(\Gamma) : E(x) \rightarrow E(y) are given by

(3)b|E(Γ)|b= Γ˜exp(S[Γ˜])DΓ˜ \langle b | E(\Gamma) | b \rangle = \int_{\tilde{\Gamma}} exp(-S[\tilde{\Gamma}]) D \tilde{\Gamma}

Here we are integrating over all Γ˜:IZ\tilde{\Gamma} : I \rightarrow Z which are lifts of Γ:IX\Gamma : I \rightarrow X, and are such that

(4)Γ˜(0)=a,Γ˜(1)=b. \tilde{\Gamma}(0) = a, \quad \tilde{\Gamma}(1) = b.

Okay. That’s Stolz and Teichner’s picture. I think its nice, and I think you like it too.

Here’s the point : my setup is precisely a special case of their example, except for one small twist. We must choose XX not to be spacetime, but to be the tangent bundle TXTX of spacetime. We set ZZ to be the “bundle of spatial slices” I described earlier, giving us a Riemannian submersion π:ZTX\pi : Z \rightarrow TX.

(Recall : The “bundle of spatial slices” ZZ is a fiber bundle over TXTX whose fiber Z (p,v p)Z_{(p, v_p)} at the point (p,v p)TX(p, v_p) \in TX is that submanifold of XX which the observer (x,v p)(x, v_p) considers to be “space”:

(5)XZ (p,v p):=unionofallgeodesicsemanatingfrompwhichareorthogonaltov p. X \supset Z_{(p, v_p)} := union of all geodesics emanating from p which are orthogonal to v_p.

It inherits a Riemannian [I think!] metric from XX, making π:ZTX\pi : Z \rightarrow TX a Riemannian submersion, as we need for Stolz and Teichner’s example.)

And there we go! We turn the crank like they said, and get our quantum theory

(6)E:B 1(X)Hilb. E : B^1(X) \rightarrow Hilb.

The only sublety here is that, given a morphism Γ:IX\Gamma : I \rightarrow X (i.e. a worldline), we must first convert it into a path Γ^\hat{\Gamma} in TXTX. That’s trivial - just add in the information of the tangent vectors on the path:

(7)Γ^(t)=(Γ(t),ddtΓ(t))TX. \hat{\Gamma} (t) = (\Gamma(t), \frac{d}{dt}\Gamma (t)) \in TX.

P.S. You mentioned that the geodesics might not sweep out all of Σ\Sigma. True, but at this stage I regard this as a technicality. Let’s solve the bigger picture first!

P.P.S. You wanted me to express the connection in local co-ordinates. Here’s the point. There are two pictures : the global picture and the picture of an individual observer on his worldline.

(a) global picture: We have a connection on a bundle of Hilbert spaces HTXH \rightarrow TX over the tangent bundle of spacetime.

(b) observer’s picture: Given a wordline Γ:IX\Gamma : I \rightarrow X, first convert it into a map Γ^:ITX\hat{\Gamma} : I \rightarrow TX as explained above, and then use this to pull back the global bundle HH over TXTX to a bundle with connection over the interval. On the interval, this connection is just

(8)Hamiltoniandt. Hamiltonian dt.

But different worldlines will pullback different Hamiltonians!

P.P.P.S. Its clear that your velocity plays a role in how you perceive the wavefunction. For if we are standing in the same place, but you are still while I am moving, then we will disagree on the wavefunction of say, a stationary car. In my picture these two wavefunctions live in different Hilbert spaces, and there’s no problem.

Posted by: Bruce Bartlett on March 4, 2007 1:10 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Ok, let me just summarize the things I was talking about once again. I want to make it clear that I am not proposing something radically new, but simply an invariant version of the way quantum mechanics on curved backgrounds is currently formulated.

(Soon I must try to understand Isham and Doring’s new paper!)

Right. Currently, quantum mechanics of point particles on curved backgrounds runs something like this: [Ed : Yes I know there is really no such thing as “quantum mechanics of point particles on curved spacetime backgrounds” - one needs to do field theory. That’s fine, its easy to do field theory in the picture I propose as well. Lets just keep it simple for now.]

In the ordinary picture, one starts with a splitting of spacetime (a pseudo-Riemannian manifold) into Σ×\Sigma \times \mathbb{R}, where Σ\Sigma represents “space” and \mathbb{R} represents “time”. On Σ\Sigma, the metric is positive definite. One thinks (at least, I do :-)) of this splitting as a trivialized bundle over \mathbb{R},

(1)Σ× \Sigma \times \mathbb{R} \rightarrow \mathbb{R}

whose fiber at time tt \in \mathbb{R} is what is considered to be “space” at that instant.

Quantum mechanics is about calculating the probability that a particle initially observed at time t=0t=0 at position x 0Σx_0 \in \Sigma will be found at time t=1t=1 at position x 1Σx_1 \in \Sigma.

To do this, one sums over all paths which interpolate between x 0x_0 and x 1x_1, weighting each by the action.

That’s the standard picture.

Ok, my picture is the same - after you choose a worldline . In my picture each observer determines his own splitting of spacetime. So, we start with spacetime (a pseudo-Riemannian manifold) XX. Each observer traces out a wordline

(2)γ:X \gamma : \mathbb{R} \rightarrow X

in spacetime.

From the point of the observer, what is space (to him/her) and what is time (to him/her)?

Proposed answer : First construct the fibre bundle of spatial slices π:ZTX\pi : Z \rightarrow TX. At each (x,v p)(x, v_p) in TXTX, the fibre π 1(x,v p)X\pi^{-1}(x, v_p) \subset X is that submanifold of XX that an observer having instantaneous state (x,v p)(x, v_p) considers to be space at that instant of his/her time:

(3)π 1(x,v p)=unionofallgeodesicsemantingfromxorthogonaltov p. \pi^{-1}(x, v_p) = union of all geodesics emanting from x orthogonal to v_p.

Now, given a particular observer’s wordline γ:X\gamma : \mathbb{R} \rightarrow X, we first lift it to a path γ^:TX\hat{\gamma} : \mathbb{R} \rightarrow TX in the obvious way,

(4)γ^(t)=(γ(t),ddtγ(t)), \hat{\gamma}(t) = (\gamma(t), \frac{d}{dt}\gamma (t)),

and then pull back the bundle of spatial slices ZZ to \mathbb{R} (we’re really pulling it back to his/her wordline ) using γ^\hat{\gamma}, to obtain a fiber bundle over \mathbb{R}:

(5)π γ:γ^ *(Z). \pi_{\gamma} : \hat{\gamma}^*(Z) \rightarrow \mathbb{R}.

At time tt \in \mathbb{R}, the fibre π γ 1(t)X\pi_{\gamma}^{-1}(t) \subset X is what the observer considers to be “space” at that instant of his/her time.

And then we do quantum mechanics as normal (i.e.as I outlined above, or for a more detailed description, read my earlier comments).

To summarize : Instead of performing a noncanonical splitting of spacetime X=Σ×X = \Sigma \times \mathbb{R}, I propose that each observer determines his own splitting of spacetime , by pulling back the bundle of spatial slices to his worldline. The equations of evolution (eg. Schrodinger equation, Klein-Gordon, Dirac, field theory) are now connections on the L 2L^2-spaces on the fibers.

For (slightly more) precise statements, read my previous post about how this ties in with Stolz and Teichner’s picture.

Posted by: Bruce Bartlett on March 4, 2007 4:25 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Hi Bruce,

thanks, I get your point now. Nice!

Maybe more comments later, after I thought more about it.

Posted by: urs on March 4, 2007 7:09 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Ah, here is already one more comment:

it is interesting to see you interpret the these maps to XX that appear in Stolz-Teichner.

I once interpreted them as being about time-dependent quantum mechanics (somewhere in the middle of this), like you get when one of the background fields is time-dependent, say.

Of course that’s not at all in contradiction to the way you put it, because in fact you do consider a time-depenent evolution, too, in some nice sense.

I guess one reason why I did not see things your way is that for the non-relativistic case the splitting Σ×\Sigma \times \mathbb{R} is canonical. What you describe is actually a curious mix between non-relativistic and relativistic points of view.

Hm, right. So, actually, the way I would conceive the relativistic case would be quite similar to what I described before, only that I now

a) take Σ\Sigma to be space-time

b) take \mathbb{R} to tbe the abstract worldline

c) concentrate only on the kernel of HH (which is now a hyperbolic operator).

Actually, I wonder if I could ask you to go through the trouble of spelling out how you would think of the relativistic case (Klein-Gordon particle) in your picture…

Posted by: urs on March 4, 2007 7:21 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Urs wrote:

Actually, I wonder if I could ask you to go through the trouble of spelling out how you would think of the relativistic case (Klein-Gordon particle) in your picture…

Ah - now you’ve got me.

In principle , of course, the connection is defined by the path integral formula I outlined above. One just takes the limit as t0t \rightarrow 0 to obtain the Hamiltonian, as you well know.

But that’s evading the question - I suppose you want concrete formulas. The answer is that I don’t know : to do this kind of thing one needs to understand how to write things like Laplacians, Dirac operators, etc. in co-ordinate free geometrical terms. Sadly, I only know the vague rudiments of this stuff - not enough to be able to write things down explicitly. I really must study the book Spin Geometry… I’ve been putting it off for too long!

I can only make a guess at the answer. So recall that we have a bundle of spatial slices π:ZTX\pi : Z \rightarrow TX over spacetime, and that π\pi is a Riemannian submersion.

We take the “L 2L^2” functions on each fiber (or the “spinor functions” on the fiber, or whatever gismo we need) to create a bundle of Hilbert spaces

(1)π:VTX, \pi' : V \rightarrow TX,

in the Stolz-Teichner way. (I’m writing VV for this Hilbert-space bundle instead of HH, for obvious reasons).

Now I should describe to you the connection on VV, in nice geometric terms. Thats all the data one needs : any particular observer will pull this stuff back to his worldline and perform his parallel transport that way.

Ok, so let γT (p,v p)TX\gamma \in T_{(p, v_p)} TX be a tangent vector, which we think of as a little curve in XX running through pXp \in X. And let ψπ 1(p,v p)\psi \in \pi'^{-1}(p, v_p) be a state vector in our Hilbert space sitting above (p,v p)(p, v_p). Thus ψ\psi is some L 2L^2 function on the spatial slice Z (p,v p)Z_{(p, v_p)}, the codimension 1 submanifold of XX defined by (p,v p)(p, v_p).

Recall that this spatial slice Z (p,v p)Z_{(p, v_p)} is actually a Riemannian manifold, and thus it comes with all sorts of natural geometric structures (Laplacian, Dirac operator, and all those gismos) attached to it. To make it clear that these differential operators on L 2(Z (p,v p))L^2(Z_{(p, v_p)}) depend on the spatial slice Z (p,v p)Z_{(p, v_p)}, I’ll refer to eg. the Laplacian like this:

(2) Z (p,v p) 2. \nabla^2_{Z_{(p, v_p)}}.

To give a connection on VV, I must tell you how to infinitesimally parallel transport ψ\psi in the direction of γ\gamma. I think the answer will turn out to be the following.

Suppose in the “bad old” Σ×\Sigma \times \mathbb{R} days (where someone had chosen a space/time splitting from the very beginning), that the time evolution of the wavefunction was expressed as follows:

(3)iψt=H( Σ 2,)ψ. i \frac{\partial \psi}{\partial t} = H(\nabla^2_{\Sigma}, \ldots) \psi.

Here HH is the Hamiltonian, and perhaps it is expressed in terms of the Laplacian on Σ\Sigma, and some other geometrical stuff defined on Σ\Sigma .

Normally the Σ\Sigma-dependence of the Laplacian is left out, since we fix Σ\Sigma once and for all.

But in our case above, the spatial slices Z (p,v p)Z_{(p, v_p)} (which play the role of Σ\Sigma for our observer) were continually changing.

I claim that the infinitesimal parallel transport along γ\gamma above is nothing but the ordinary Hamiltonian, using the Laplacian defined on Z (p,v p)Z_{(p, v_p)}.

In other words, (and finally I’m coming to the end!), a particular observer, on his worldline γ:X\gamma : \mathbb{R} \rightarrow X, will observe time evolution of wave functions given by

(4)iψt=H( Z γ(t),γ˙(t) 2,), i \frac{\partial \psi}{\partial t} = H(\nabla^2_{Z_{\gamma(t), \dot{\gamma}(t)}}, \ldots),

where t\frac{\partial}{\partial t} refers to the direction of his wordline.

Posted by: Bruce Bartlett on March 4, 2007 10:40 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Hi Bruce,

maybe that’s not quite what I tried to ask. Let me be more specific:

instead of the non-relativistic particle propagating on a Riemannian space (times a canonical time axis), we now want to consider the Klein-Gordon-particle propagating on a arbitrary pseudo-Riemannian space.

The action is, in Nambu-Goto form, just the proper worldlength of a given trajectory.

Or, in Polykov form, it is the action of gravity on the worldline, coupled to a nonlinear scalar field (the embedding field).

The Hamiltonian obtained from that is the (pseudo-Riemannian) Laplace operator Δ\Delta on target space. The Schrödinger equation is iddτψ=Δψ i \frac{d}{d \tau}\psi = \Delta \psi and there is an additional constraint, saying that no state actually depends on the arbitrary parameter on the worldline. This gives the wave equation Δψ=0. \Delta \psi = 0 \,.

You are probably not claiming that you can understand that as a connection on a Hilbert space bundle on spacetime. Or are you? It would be nice if you were, it’s just that I don’t see that at the moment (but I’d gladly be educated).

(This kind of headache is called “the problem of time”, by the way. :-)

Posted by: urs on March 4, 2007 11:01 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Somebody following this conversation asked me, by private email, if I could expand a little on what I said above about how the relativistic particle is described either by a Nambu-Goto-like action or a Polyakov-like action, how one finds from this its Hamiltonian and the Hamiltonian constraint and so on.

Luckily, I have done so before, so I can be lazy and just provide links: Nambu-Goto, Polyakov and back

Posted by: urs on March 5, 2007 9:02 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Ok - looked at the notes “Supersymmetric homogenous Quantum cosmology”. I only understand pages 293-295 (the ones about point particles and the Hamiltonian constraint) about 60%. I’m impressed you wrote such a huge tome!. I guess I must just wait until you can explain it to me in person.

Anyhow, I’m quite happy to stop talking about the “global” Hilbert bundle with connection over the tangent bundle of spacetime (which restricts to each observer’s individual wordline), if you’re unhappy with it. Its not an essential ingredient of what I was saying.

The important thing is the following :

Let γ:X\gamma : \mathbb{R} \rightarrow X be a wordline of an observer in spacetime XX. At each instant tt \in \mathbb{R} of his worldline, there is a codimension 1 Riemannian [I think!] submanifold Z tXZ_t \subset X which he considers to be “space” at that instant. This is described above. These fit together into a “spatial slice bundle” over his worldline (which we think of as parametrized by \mathbb{R}):

(1)π:Z \pi : Z \rightarrow \mathbb{R}

To proceed it will depend on what kind of quantum mechanics the observer he’s doing : let’s just say he’s doing “bog-standard” quantum mechanics of point particles (i.e. not the Dirac equation or field theory or something).

Always remember that, from the observer’s perspective, nothing weird has happened at all : to him it is perfectly obvious what is space and what is time . He is blissfully unaware of the trouble we had to go through to set things up for him on his worldline! pic To him he is standing still .

At each instant tt \in \mathbb{R} he makes a Hilbert space L 2(Z t)L^2(Z_t) of L 2L^2 functions on (what he considers to be) space at time tt. He thinks the notion of “space at this instant” is absolute and obvious. He is blissfully unaware that in fact, what he considers to be space at time tt, i.e. Z tXZ_t \subset X, is actually continuously changing.

Thus he can’t imagine why people say he’s actually constructing a different Hilbert space at each time tt : to him, he just makes one Hilbert space (“L 2L^2” functions on space) and thinks that it stays fixed as time tt varies.

He performs ordinary dynamics on his Hilbert space by the Schrodinger equation:

(2)iψt=H( 2)ψ i \frac{\partial \psi}{\partial t} = H(\nabla^2) \psi

where HH is the Hamiltonian, which probably contains the Laplace operator 2\nabla^2. He thinks that the birds are singing, the bees are buzzing, summer’s in the air and that everything is good in the world [Ed : P.G. Wodehouse?].

Aaah… but we know the truth . What he considers to be a fixed Hilbert space HH is really a bundle of Hilbert spaces over his wordline. At each instant tt \in \mathbb{R}, the Hilbert space is the space of L 2L^2 functions on Z tMZ_t \subset M:

(3)tL 2(Z t). t \mapsto L^2(Z_t).

Moreover, Z tMZ_t \subset M has its own Riemannian metric, and the Laplace operator on L 2(Z t)L^2(Z_t) thus depends on it:

(4) 2 Z t 2. \nabla^2 \rightarrow \nabla^2_{Z_t}.

In summary, his Schrodinger equation is correct, but it must now be interpreted as a connection on this Hilbert bundle over his worldline. If we trivialize this Hilbert bundle by choosing the zero section, then dynamics will look as follows:

(5)iψt=H( Z t 2)ψ. i \frac{\partial \psi}{\partial t} = H(\nabla^2_{Z_t}) \psi.

Finally, what does the “problem of time” look like in this picture? The “problem of time” says that somehow Hψ=0H \psi = 0. This seems suspicously like one has chosen a trivialization of the bundle which is not the zero section (as we did), but rather a flat section : along which parallel transport (given by the Hamiltonian) is zero.

I’d just like to know : does all this make sense, or is there a big flaw somewhere?

Posted by: Bruce Bartlett on March 5, 2007 1:00 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

I’d just like to know : does all this make sense, or is there a big flaw somewhere?

Oh, it makes very good sense to me! I like this.

All I would maybe add is a remark on how I think this fits into a bigger picture:

I believe that what you describe is a step somewhere mid-way between starting with the full Klein-Gordon equation for a relativistic particle on a pseudo-Riemannian space, and the non-relativistic limit obtained from it, also known as the Schrödinger equation.

I’d think that slightly more commonly one would perform this limit by first choosing a foliation by spatial hyperslices. This singles out “comoving observers”, namely those which see themselves at rest with respect to this chosen foliation (there will be an entire family of such observers, though, an entire time-like vector field on the pseudo-Riemannian space-time).

This step is what you reformulate in terms of a single observer. I think this is something that makes good sense to consider, even though it requires taking care of some slight subtleties, as we have discussed.

Concerning “the problem of time”: we need to carefully distinguish physics on parameter space from physics on target space.

The Klein-Gordon particle can be conceived as 1-dimensional gravity coupled to scalar fields (which are the maps from the worldline into target space that describe the trajectory the particle traces out in spacetime). As such, it has diff invariance on the worldline and all that, which leads to the Hamiltonian being in fact a constraint. Which is good, because the “Hamiltonian” is now a hyperbolic operator anyway, and that entire operator decomposes into the 2-sides of the Schrödinger equation in the suitable limit.

To be more precise, consider flat Minkowski target 4\mathbb{R}^4 and a massive particle. Then the Hamiltonian constraint simply reads 1c 2 t 2ϕΔϕ+(mc) 2ϕ=0, \frac{1}{c^2} \partial_t^2 \phi - \Delta \phi + \left(\frac{m c}{\hbar}\right)^2 \phi = 0 \,, where Δ\Delta is the Laplace operator on 3\mathbb{R}^3.

You look at the Fourier-modes of this equation (ϕ E,p=exp(itEixp)/\phi_{E,p} = \exp(i t E - i x \cdot p)/\hbar) and find that it expresses the “mass shell constraint” E 2=c 2p 2+(mc 2) 2, E^2 = c^2 p^2 + (m c^2)^2 \,, which, in the nonrelativistic limit, may be expressed as E=mc 2+12mp 2. E = m c^2 + \frac{1}{2m}p^2 \,. By Fourier-transforming hither and thither this gives the Schrödinger equations.

But – and that’s where you come in – the above procedure has secretly chosen a time-slicing of 4-dimensional Minkowski space (namely at the point where I first used the symbol t\partial_t). So it is really a little too naive. More properly, one would have to do this in a way along your lines, where we keep track of various non-canonical choices.

And that’s a good thing!! One really should do that. Especially when the background is not just flat Minkowski.

Posted by: urs on March 5, 2007 1:39 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Bruce wrote:

Hatfield is the one textbook which gives more than just a nod to the Schrödinger picture in quantum field theory.

Actually I helped write a textbook on quantum field theory which puts a lot of emphasis on the Schrödinger picture:

You can get it for free online. The really interesting part is how one can use the Schrödinger picture to rigorously construct scalar fields with polynomial interactions on a 2d spacetime of the form ×S 1\mathbb{R} \times S^1, or on ×\mathbb{R} \times \mathbb{R} with an infrared cutoff. This was done by Segal’s student Edward Nelson.

Later, Zhengfang and I figured out how to rigorously define quantum fields on ×S 1\mathbb{R} \times S^1 satisfying the field equation ( 2t 2 2+m 2)ϕ+:P(ϕ):=0 (\frac{\partial^2}{\partial t^2} - \nabla^2 + m^2)\phi + :P(\phi): = 0 for a certain class of polynomials PP:

This is trickier than you might think, because the normal ordering denoted by the colons in the expression :P(ϕ)::P(\phi): must be done relative to the physical vacuum rather than the vacuum for the free scalar field — but the physical vacuum is defined to be the ground state of the quantum field satisfying the given equation! So, there’s a funny kind of circularity here, and we had to use a fixed-point theorem to cut the Gordian knot.

This is another thing that quantum field theory books never seem to mention: the difficulties in getting an interacting quantum field to satisfy a given field equation!

Posted by: John Baez on March 4, 2007 2:20 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Thanks for this link. The book you have co-authored is great. I remember seeing it before, but I didn’t have enough mathematical training to understand it then pic . I’m not sure I have enough math training to understand it now pic , but I’ll give it a try!


Posted by: Bruce Bartlett on March 4, 2007 4:43 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

The link to the pdf given on that website Introduction to Algebraic and Constructive Quantum Field Theory seems to be broken!

Posted by: urs on March 4, 2007 10:49 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

I’ve fixed it — but beware, the pdf file is incredibly bulky and inefficient, since it was scanned in by an inexpert grad student. The djvu file is much better, if you can handle such files.

Posted by: John Baez on March 5, 2007 6:32 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

But what if no Cauchy surfaces exist?

Well, right, then I would start thinking about piecing things together from data on smaller regions.

But standard AQFT is set up in Minkowski space only.

The subsets to which algebras are associated in AQFT are “double cones”, intersections of the future of one point with the past of another. These all have a lots of Cauchy surfaces.

The only generalization of this to curved spacetimes that I am aware of is math-ph/0112041, which demands the manifolds to which algebras are assigned to be globally hyperbolic, i.e. having a foliation by Cauchy surfaces.

Posted by: urs on March 3, 2007 7:31 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

There’s recently been a lot of nice rigorous work on perturbative quantum field theory on globally hyperbolic spacetimes from an AQFT point of view. Here’s a good review:

See especially section 4, “Progress since the mid-1990’s”.

Posted by: John Baez on March 4, 2007 2:37 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Can you briefly say what “essentially self-adjoint” means, or do I have to look at Reed & Simon?

Posted by: Toby Bartels on February 28, 2007 4:05 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

“Essentially self-adjoint” means “has a unique self-adjoint extension”.

Posted by: urs on February 28, 2007 4:21 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 16)

Urs gave the definition, but you should still read the relevant bits of volume 1 of Reed & Simon, because it gives a very clear and crisp introduction to unbounded self-adjoint operators. Volume 3 goes deeper: it gives lots of methods of proving that operators are self-adjoint, or essentially self-adjoint.

Posted by: John Baez on March 1, 2007 2:15 AM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: Sheaves of Observables
Weblog: The n-Category Café
Excerpt: On the concepts of sheaves and nets of algebras of observables in quantum field theory.
Tracked: March 6, 2007 11:40 PM
Read the post Quantization and Cohomology (Week 17)
Weblog: The n-Category Café
Excerpt: Getting Hilbert spaces and operator algebras from categories.
Tracked: March 7, 2007 10:05 PM

Post a New Comment