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August 24, 2007

That Shift in Dimension

Posted by Urs Schreiber

John Baez and I spent the evening in Café Einstein in Vienna (remember, we are at that conference), drinking beer and talking about Stokes’ theorem, natural nn-transformations and the holographic principle, and how it is all the same thing, really.

Well, okay, I admit we didn’t quite finish proving that theorem, or even making that statement precise. But we had those printouts with us, and after sufficiently many beers, we very much enjoyed figuring out some simple underlying principle of the – at first sight apparently mind bogglingly weird – statement of

Maxim Kontsevich
Deformation quantization of Poisson manifolds, I

which says that any classical kinematics is canonically quantizable by some strange formula which involves lots of sums over lots of diagrams…

…and the maybe even more surprising explanation of this formula (which Kontsevich apparently knew but didn’t bother to talk about) in terms of correlators of a topological two-dimensional theory of quantum fields with values in the original phase space (hence something that is everywhere of one dimension higher than one would think it should be) as given in

Alberto S. Cattaneo, Giovanni Felder
A path integral approach to the Kontsevich quantization formula
math/9902090 .

John went to bed, while I carefully carried our little insight back to the institute, in my bare hands, so as to talk about it here. The following is supposed to, in turn, help explain, in elementary terms understandable by anyone who understands all or either of

- high school quantum mechanics

- Stokes’ theorem

why on earth the quantization of an nn-dimensional theory may be obtained by a topological (n+1)(n+1)-dimensional field theory.

This is meant for those who enjoy things like Kindergarten Quantum Mechanics. If you are not among these, don’t bother continue reading.

The very basics.

While I want to do it the Kindergarten way, we need at least some basic idea about what a phase space in classical mechanics is, and which role the symplectic 2-form on that space plays in physics.

But really all the information we need is recalled in John’s lecture

Quantization and Cohomology (Week 6).

If you feel you need a little more details, just go back to

Quantization and Cohomology (Week 5).

By ignoring lots of more general cases which we could just as well take into account, but which would just distract us from a nice simple main point, we will be content with considering the following basic setup:

The configurations of some physical system form a space X. \mathrm{X} \,. The configuration space. Clearly.

But a physical system weren’t a physical system if it would just sit there, pointlike as it is, and do nothing. Instead, physical systems want to undergo processes and move from here to there in the space of configurations.

A first simple approximation to the notion of a space of configurations XX equipped with a way to move from here to there is TX T X the tangent bundle to XX. This provides us for each configuration xx of the system with an inkling of what it would be like to evolve that system in time in various way: a tangent vector at xx.

But some processes may be more likely than others. The dynamics of the system is encoded in a function which assigns a measure of likeliness of each trajectory in XX that a system traces out over time. Therefore, as soon as the system indeed does start processing in time, we assign a weight to that. This means we are really looking at the cotangent space P:=T *X. P := T^* X \,. That space is called phase space. For no good reason. But that’s what it’s called. (Okay, everybody feel free to teach me about the deep reason to call phase space phase space.)

By abstract magic, the very definition of the cotangent bundle indeed provides us with a canonical way to weight each process. This is given by the canonical 1-form α \alpha on PP, which John describes in week 5, and which is best known under its local name, which reads α=p idq i. \alpha = p_i \; d q^i \,.

So given any path γ:[0,1]P \gamma : [0,1] \to P in phase space, we can immediately assign a weight to it simply by integrating α\alpha over it. γ γα. \gamma \mapsto \int_\gamma \alpha \,.

Around here, in the nn-Café I suspect it doesn’t come as a surprise to anyone that assigning weights to paths in a manner which nicely respects composition of paths makes us want to think of parallel transport of a connection along that path.

(Okay, if you’re really just a Kindergarten graduate, then you should learn about how to color handrails before reading on. Or else, get used to reading stuff which you don’t understand. It’s never too early to develop that skill.)

In that context, we are tempted to turn the above weight into an exponential, γexp(i γα). \gamma \mapsto \mathrm{exp}(i \int_\gamma \alpha) \,.

Well, first I am trying to be so extraordinarily elementary here, and now I introduce an imaginary unit and an exponential for apparently no good reason!

That’s what it might look like. While I don’t feel awake enough to try to indicate this all the way down to the very roots of it all (that would involve mentioning nn-transformations and their relation to holography, underlying all this), I feel that it will actually prove so very useful for understanding the gist of the Kontsevich-Cattaneo-Felder insight to think of those “weights” which we are assigning to paths as parallel transport of some connection, that I do want to adopt that point of view now. For a gentle introduction to this way of thinking go to

Quantization and Cohomology (Week 22).

So those who know how it works are kindly invited to think of our 1-form α\alpha (locally) as containing the contribution of a connection of a line bundle on configuration space XX. All others simply ignore this.

What we shall not ignore is the Stokes theorem, otherwise known as the Fundamental theorem of calculus.

Suppose our path γ\gamma happens to be a loop. Then we are entitled to compute the parallel transport of α\alpha along γ\gamma tra α(γ)=exp(i γα) \mathrm{tra}_\alpha(\gamma) = \exp(i \int_\gamma \alpha) using – guess what –

a) a curious shift in dimension

b) together with a certain “topological” invariance in that higher dimension.

Yes, I am indeed just overselling the Kindergarten fact that for any disk Σ:DX \Sigma : D \to X in XX which co-bounds our loop γ\gamma, we may compute the transport of α\alpha along γ\gamma equivalently by computing the surface transport of the 2-form ω=dα \omega = d \alpha over Σ\Sigma: exp(i γα)=exp(i Σω). \exp( i \int_\gamma \alpha) = \exp( i \int_\Sigma \omega) \,.

For those readers (anyone?) who learned how to do all this by painting doorknobs, handrails and walls: think of solving the task of painting a handrail that is tightly attached to a wall by painting the entire wall above the handrail, being sufficiently careless while doing so such that the handrail gets painted, too, in the process.

With all that out of the way, let’s finally pass on to the genuine point of what this here is supposed to be about.

A jump in dimension

With the ingredients mentioned above

- the configuration space XX

- the corresponding phase space T *XT^* X

- our canonical “weight” α\alpha on that

we can finally cook up the true dynamics of our system.

Let’s do some reverse engineering of something fancy to something obvious.

Suppose our system evolves for a while. That means we have a path γ:[0,1]P. \gamma : [0,1] \to P \,. Suppose further that after time 1/31/3 we want to check how the system is doing, using a probe we call F^\hat F.

Whatever that means (and in fact I will explain in a moment what it means), it makes us want to write F^(t=1/3). \hat F(t= 1/3) \,. When that is done, we let the system evolve a little more. But, nervous as we are, we check again how it is doing a little later. To keep track of that, we write F^(t=1/3)G^(t=2/3). \hat F(t= 1/3) \hat G(t = 2/3) \,. After that is done, we wait still a little further. Then, let’s say, we check if the system happens to be in confuguration xx. Somebody in the last millenium thought that a good way to note this down is to write x|F^(t=1/3)G^(t=2/3)|x. \langle x | \hat F(t= 1/3) \hat G(t = 2/3) | x \rangle \,. I could go on at this point about how this funny notation is in fact so very close to the right underlying nn-categorical arrow theory, which I think is really at the very heart of the phenomenon to be unravelled here, but I did so elsewhere and don’t have the nerve to do so again here. Instead I just tell you what this funny notation is actually really suppposed to mean: When we write x|F^(t=1/3)G^(t=2/3)|x \langle x | \hat F(t= 1/3) \hat G(t = 2/3) | x \rangle we are imagining we have a measure dμd\mu which in this context people like to write [Dγ] [D\gamma] on the space of maps {[0,1]P} \{[0,1] \to P\} of sorts, and that we have functions F,GC (P) F, G \in C^\infty(P) which we integrate using this measure, weighted by our weight. So: x|F^(t=1/3)G^(t=2/3)|x:= γ(0)=γ(1)=x[Dγ]F(γ(1/3))G(γ(2/3))exp(i γ(αHdt). \langle x | \hat F(t= 1/3) \hat G(t = 2/3) | x \rangle := \int_{\gamma(0) = \gamma(1) = x} [D\gamma] F(\gamma(1/3)) G(\gamma(2/3)) \exp(i \int_\gamma (\alpha - H d t) \,.

This is the infamous path integral.

I have sneaked in here a previously undefined symbol: HH. The cognoscenti immediately recognize this old friend as the Hamiltonian. But all others may immediately forget that I even introduced that symbol – for the following reason:

I do want to consider the case where actually no time passes between our probe F^ \hat F and our probe G^ \hat G of our system. We simply probe it twice in a row, quickly, all at xx.

Those who know what this means may imagine that I write a little limit sign in front of everything, which sends the parameter length of all my paths to zero. All others will hopefully gladly trust me that whatever that limiting procedure means, its effect will simply be to suppress precisely that term which I did not mention in the first place.

So then we are left with

x|F^G^|x:= γ(0)=γ(1)=x[Dγ]F(γ(1/3))G(γ(2/3))exp(i γα). \langle x | \hat F \hat G | x \rangle := \int_{\gamma(0) = \gamma(1) = x} [D\gamma] F(\gamma(1/3)) G(\gamma(2/3)) \exp(i \int_\gamma \alpha ) \,.

If you want to compare this expression with what Cattaneo and Felder have in their footnote on p. 5, you need only note that what I am writing using angular brackets (that’s Dirac’s notation) is what they write using a star (that’s traditional in deformation quantization), so x|F^G^|x:=fg(x). \langle x | \hat F \hat G | x \rangle := f \star g (x) \,.

In any case, we should note three things:

- using our weight on physical processes, we can take the averaged value of the product of two functions on phase space, each evaluated at one point of this process

- while at the same time we assume that this “process” requires no time at all

- but still the order in which we evaluate FF and GG crucially matters, as the formula clearly indicates

(This does make sense and is the way nature works. If you find that part troubling, I cannot help much more right now except for offering the canonical Wikipedia entry: Quantum Mechanics)

Also notice that, since we are integrating over a space of maps whose domain is a 1-dimensional space, we say we are doing the “1-dimensional quantum field theory of the 1-particle”.

(That term makes strict intuitive sense here only if you imagine our configuration space XX to be the space of different ways a particle can sit in some spacetime. If otherwise you feel puzzled by the dimensions here, do make that assumption on XX. If not, all the better: keep in mind that by the same kind of argument we should be able to say what I am going to say next for any dd-dimensional field theory.)

But then – wait a second. Is that definition of dimension well defined? Is there any intrinsic way in which we could determine if any given expression is the integral over a space of paths?

No, there isn’t. Dimension of quantum field theory is not an intrinisic concept. In a way.

We already know this well from the classical differential geometry: there is no really intrinsic sense in which the parallel transport exp(i γα) \exp(i \int_\gamma \alpha) is “1-dimensional”. That’s because the value of this formula is just plain equal to one involving 2-dimensional integration exp(i γα)=exp(i Σdα). \exp(i \int_\gamma \alpha) = \exp(i \int_\Sigma d \alpha) \,. And that very tautological observation we now insert into our path integral – and turn it into a disk integral γ(0)=γ(1)=x[Dγ]F(γ(1/3))G(γ(2/3))exp(i γα) = γ(0)=γ(1)=x,γ=Σ[Dγ]F(γ(1/3))G(γ(2/3))exp(i Σω). \begin{aligned} & \int_{\gamma(0) = \gamma(1) = x} [D\gamma] F(\gamma(1/3)) G(\gamma(2/3)) \exp(i \int_\gamma \alpha ) \\ &= \int_{\gamma(0) = \gamma(1) = x, \gamma = \partial\Sigma} [D\gamma] F(\gamma(1/3)) G(\gamma(2/3)) \exp(i \int_\Sigma \omega ) \end{aligned} \,.

Notice that I am not saying anything non-tautological here. In part, that’s the whole point. In other part it is of course due to the fact that I am evading some more subtle issues.

Like this one: we can easily imagine that instead of integrating over all paths and picking one cobounding disk for each path, as above, we integrate over the space of all disks in the first place – simply by dividing out by the overcounting introduced by the fact that there are many disks (quite a few, actually) with the same boundary.

While this is easily imagined, the main intellectual problem with quantum field theory is that of statements like that, involving integrals over huge spaces divided out by huge quotients.

Cattaneo and Felder give a very detailed discussion of how to make this as precise as the current level of understanding QFT allows. That means mainly: they use BV formalism.

And that’s of course where all the technicalities enter and where true glory can be earned.

So let’s pause for 30 seconds and commemorate all the trouble Cattaneo and Felder went through for figuring out how to deal with all this.

But then let’s allow ourselves to extract the basic mechanism, the underlying structure, all that which makes this here work and which is not a technicality. Imagine, if you like, that everything is taking place in the world of plain old finite cell complexes. Then everything technical evaporates. And we are left with a surprisingly simple and at the same time surprisingly powerful statement:

we can equivalently think of the product of two operators in quantum mechanics as being computed by a “path” integral over surfaces, hence as being computed by a 2-dimensional quantum field theory. fg(x)=[DΣ]f(Σ(1/3))g(Σ(2/3))exp(i Σω). f \star g (x) = \int [D \Sigma] f(\partial \Sigma(1/3)) g(\partial \Sigma(2/3)) \exp(i \int_\Sigma \omega) \,.

And the way this works is just

- apply Stokes’ theorem in the path integral.

How dare I bore you with such trivialities?

I did warn those not interested in Kindergarten quantum mechanics not to read this.

Anyone who did read up to this point but is feeling really annoyed but how trivial it all seems, I would be really interested in asking the following question:

Something to ponder. Characterize the relation between Stokes’ theorem in nn-dimensions and transformations of nn-functors. What does this imply for nn-functorial extended QFT? Is there hence a rather high-brow generalization of the above Kindergarten formula?

It does pay to step back and think about what’s really going on here, I believe. The issue discussed at Making AdS/CFT precise is a case in point.

There needs to go more into this last paragraph here. But I am just way too tired.

Posted at August 24, 2007 11:33 PM UTC

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18 Comments & 4 Trackbacks

Re: That Shift in Dimension

The link to Kindergarten Quantum Mechanics should presumably be to here?

That’s how it might look like.


Posted by: Allen Knutson on August 25, 2007 4:28 AM | Permalink | Reply to this

Re: That Shift in Dimension

Allen Knutson remarked:


For our other readers, I should put this into context:

Allen Knutson has taken up the task of teaching a poor German a little bit of English grammar. Each time I write

how [something] looks like

thinking of

wie [etwas] erscheint

he kindly writes in to remind me that in English it has to read

what [something] looks like

While I do perfectly understand that, I keep making that mistake when typing away. I can’t express how stupid I feel for doing that again and again.

Many thanks to Allen for not giving up on me. :-)

(And yes, it’s fixed now.)

Posted by: Urs Schreiber on August 25, 2007 11:24 AM | Permalink | Reply to this

Re: That Shift in Dimension

You could always say

That’s how it might look.

Posted by: David Corfield on August 25, 2007 12:06 PM | Permalink | Reply to this

Re: That Shift in Dimension

You could always say
That’s how it might look.

As with so many things discussed here, Urs’ phrase is locally standard but there is a global twist.

Posted by: Allen Knutson on August 25, 2007 1:00 PM | Permalink | Reply to this

flip-side; Re: That Shift in Dimension

This is, perhaps, a natural transformation of:

“In this world in which we live in.”

[Paul McCarney, “Live and Let Die”]

Posted by: Jonathan Vos Post on August 25, 2007 6:51 PM | Permalink | Reply to this

Re: That Shift in Dimension

You deserve a citation for dedication to the Café beyond the call of duty.

But can it have been so many beers, bearing in mind how few typos there were?

Posted by: David Corfield on August 25, 2007 10:00 AM | Permalink | Reply to this

Re: That Shift in Dimension

bearing in mind how few typos there were?

Ah, thanks for noticing. ;-)

I think the amount of typos I make while typing is constant on average. (For instance, one in the previous sentence, now corrected.) The number remaining there in the end is then inverse proprtional to how much time I take to proof-read, plus a constant offset which contains those typos I make not due to being careless but due to being ignorant.

Posted by: Urs Schreiber on August 25, 2007 11:38 AM | Permalink | Reply to this

Re: That Shift in Dimension

I’m slightly confused… if PP is phase space, how do we sneak a dtdt in there?

Posted by: Bruce Bartlett on August 25, 2007 10:05 AM | Permalink | Reply to this

Re: That Shift in Dimension

I’m slightly confused… if PP is phase space, how do we sneak a dt in there?

Right, very good point. As you have seen, this entire issue of how there really is a term HdtH \, d t in the first place which then is taken to vanish I didn’t get into at all very much.

But it’s quite simple, just compare with the standard formulation of the Lagrangian and the path integral:

usually, people talk about paths of different parameter length ll, hence about maps γ:[0,l]P \gamma : [0,l] \to P hence γ:σγ(σ). \gamma : \sigma \mapsto \gamma(\sigma) \,.

In such a context we simply identify t=σt = \sigma and read dt=dσ. d t = d\sigma \,. This is the famous statement that “time is not an observable in non-relativistic quantum mechanics” – it’s an external parameter.

Of course you can pull some tricks and rewrite everything equivalently in a way that time does become a configuration coordinate. John talks about that in his lecture.

Now, the way I presented it, in order to make contact with the Cattaneo-Felder formula, all paths were taken to be parameterized by the standard interval [0,1][0,1].

That’s the same as reading dt=ldσ, d t = l d\sigma \,, of course.

This way I can send l0l \to 0, hence send the parameter length of these paths to zero without actually shrinking the paths to the constant paths.

Hope that doesn’t sound weird. It is really supposed to be precisely the standard path integral prescription.

Posted by: Urs Schreiber on August 25, 2007 11:20 AM | Permalink | Reply to this

Re: That Shift in Dimension

Bruce wrote:

I’m slightly confused… if PP is phase space, how do we sneak a dtd t in there?

Urs’ answer may have been too detailed and polite to be maximally clear.

So, here’s something more rude: call it dsd s if you prefer; it doesn’t matter! It’s just a parameter used to parametrize a loop γ\gamma in phase space.

Of course, the integral

γα\int_\gamma \alpha

is parametrization-independent, because α=p idq i\alpha = p_i d q^i is a 1-form on phase space. The integral

γHdt\int_\gamma H \, d t

is not, because it’s just an ordinary high-school integral of a function from t=0t = 0 to t=1t = 1.

If this annoys you, then maybe you want to think of the combined expression

α˜=αHdt\widetilde{\alpha} = \alpha - H d t

as a 1-form on some space. For this, you need to use the ‘extended phase space’

T *(×X)T^* (\mathbb{R} \times X)

as explained on page 3 of these course notes. The 1-form

α˜=p idq iHdt \widetilde{\alpha} = p_i d q^i - H d t

is of course very familiar in the Lagrangian approach to mechanics. Mathematically, it’s just the canonical 1-form on the cotangent bundle T *(×X)T^*(\mathbb{R} \times X). Using this, we can get the action to be parametrization-independent, but only at a certain price — as explained in those notes.

Posted by: John Baez on August 25, 2007 1:09 PM | Permalink | Reply to this

Re: That Shift in Dimension

A great post! There’s nothing better than waking up late after a long conversation about the deep mysteries of the universe, strolling to the Schrödinger Institut, making a coffee, looking at the nn-Café — and finding last night’s conversation all typed up in TeX. I’m still just waking up, so I just have one very minor comment right now.

Urs wrote:

P:=T *X. P := T^* X \,. That space is called phase space. For no good reason. But that’s what it’s called. (Okay, everybody feel free to teach me about the deep reason to call phase space phase space.)

I’ve never known the historical reason why it’s called ‘phase space’. But, Jim Dolan explained to me the reason it should be called phase space — because each path in it gives a phase!

And indeed, this is just what Urs goes ahead and does:

In that context, we are tempted to turn the above weight into an exponential, γexp(i γα). \gamma \mapsto \mathrm{exp}(i \int_\gamma \alpha) \,.

Posted by: John Baez on August 25, 2007 12:47 PM | Permalink | Reply to this

Re: That Shift in Dimension

Here is a guess concerning the historical motivation for people to call phase space phase space, long before “phases” in the sense of “elements of U(1)U(1)” were introduced after the discovery of quantum mechanics:

Given any classical physical system, a useful way to get an impression for its main characteristics is to draw its phase portrait:

draw all of phase space on a piece of paper. By the very nature of phase space, there is one and only one trajectory of the system going through each point.

The phase portrait of the system is the collection of all these trajectories.

The crucial parts of the phase portrait are the separatrices: those trajectories which sit on the boundary between regions all of whose trajectories are closed and those where none of them are closed.

Alternatively, for chaotic systems one likes to highlight those trajectories which separate trajectories converging to atractors from those which don’t.

In any case, the picture one obtains drawing these separatrices in phase space is quite reminiscent of the phase diagrams chemists draw.

I am imagining that this might be the origin of the term “phase space” in physics. But it’s just a guess.

Posted by: Urs Schreiber on August 25, 2007 1:17 PM | Permalink | Reply to this

Re: That Shift in Dimension

But, Jim Dolan explained to me the reason it should be called phase space - because each path in it gives a phase!

That’s cool, I really like that!

Posted by: Bruce Bartlett on August 25, 2007 5:49 PM | Permalink | Reply to this

Complicated, disambiguated; Re: That Shift in Dimension

Wikipedia both disambiguates “phase space” and complicates the origins and usage.
Phase space can refer to:

* Phase space, a concept in physics, frequently applied in thermodynamics, statistical mechanics, dynamical systems, symplectic manifolds and chaos theory.
* Phase space (linear logic), a mathematical model used to interpret linear logic in phase semantics.
* Phase Space (novel), a collection of thematically-linked short stories in the Manifold Trilogy by Stephen Baxter.
* PhaseSpace - Active Marker LED based real time motion tracking hardware and software for VR, AR, Telerobotics, medical and entertainment applications.
In mathematics and physics, a phase space is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. A plot of position and momentum variables as a function of time is sometimes called a phase diagram. This term, however, is more usually reserved in the physical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, as a function of pressure, temperature, and composition.
In quantum mechanics, the coordinates p and q of phase space become hermitian operators in a Hilbert space, but may alternatively retain their classical interpretation, provided functions of them compose in novel algebraic ways (through Groenewold’s 1946 star product). Every quantum mechanical observable corresponds to a unique function or distribution on phase space, and vice versa, as specified by Hermann Weyl (1927) and supplemented by John von Neumann (1931); Eugene Wigner (1932); and, in a grand synthesis, by H J Groenewold (1946). With José Enrique Moyal (1949), these completed the foundations of phase-space quantization, a logically autonomous reformulation of quantum mechanics. Its modern abstractions include deformation quantization and geometric quantization.

Thermodynamics and statistical mechanics

In thermodynamics and statistical mechanics contexts, the term phase space has two meanings:

* It is used in the same sense as in classical mechanics. If a thermodynamical system consists of N particles, then a point in the 6N-dimensional phase space describes the dynamical state of every particle in that system. In this sense, a point in phase space is said to be a microstate of the system. N is typically on the order of Avogadro’s number, thus describing the system at a microscopic level is often impractical. This leads us to the use of phase space in a different sense.
* The phase space can refer to the space that is parametrized by the macroscopic states of the system, such as pressure, temperature, etc. For instance, one can view the pressure-volume diagram or entropy-temperature diagrams as describing part of this phase space. A point in this phase space is correspondingly called a macrostate. There may easily be more than one microstate with the same macrostate. For example, for a fixed temperature, the system could have many dynamic configurations at the microscopic level. When used in this sense, a phase is a region of phase space where the system in question is in, for example, the liquid phase, or solid phase, etc.

Since there are many more microstates than macrostates, the phase space in the first sense is usually a manifold of much larger dimensions than the second sense. Clearly, many more parameters are required to register every detail of the system up to the molecular or atomic scale than to simply specify, say, the temperature or the pressure of the system.


In my PhD dissertation, I emphasized “chemical phase space” by analyzing the Laplace Tranform of certain sets of nonlinear differential equations (Michaelis-Menten equations) that define the dynamics of metabolisms, and found lumped coefficients which were eigenvalues of eignefunctions to the Krohn-Rhodes decomposition of the semigroup of differential operators. Back in 1977, there were only a handful of mathematical Biologists (motsly in the USSR) who were happy with that level of abstraction.

As with “Science Fiction” or “P0rn” the term means what the author wants it to me, and we know it when we see it. Sometimes.

Posted by: Jonathan Vos Post on August 25, 2007 7:06 PM | Permalink | Reply to this

Re: That Shift in Dimension

Answering it backwards

Is there hence a rather high-brow generalization of the above Kindergarten formula?

Yes - unless we have a collective failure of the imagination. See next answer.

Characterize the relation between Stokes’ theorem in nn-dimensions and transformations of nn-functors. What does this imply for nn-functorial extended QFT?

The most important thing I ever learnt on how to deal with physics in general and phase spaces - was look at the conserved volumes.

At least in the case where something like Liouville’s Theorem applies: When someone (not me) goes and make these generalizations I personally wouldn’t expect the volumes to be conserved - but that generalization of the elements that create the volume form a closed set of some type.

Am I totally out of the ball game? Or do I need to go back and really learn this time?

Posted by: Ian Burrows on September 3, 2007 7:42 AM | Permalink | Reply to this
Read the post n-Curvature, Part III
Weblog: The n-Category Café
Excerpt: Curvature is the obstruction to flatness. Believe it or not.
Tracked: October 16, 2007 10:51 PM
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 6:42 PM
Read the post A Groupoid Approach to Quantization
Weblog: The n-Category Café
Excerpt: On Eli Hawkins' groupoid version of geometric quantization.
Tracked: June 12, 2008 6:54 PM
Read the post Eli Hawkins on Geometric Quantization, I
Weblog: The n-Category Café
Excerpt: Some basics and some aspects of geometric quantization. With an emphasis on the geometric quantization of duals of Lie algebras and duals of Lie algebroids.
Tracked: June 20, 2008 5:16 PM

Re: That Shift in Dimension

I don’t know if the first usage is due
to Hamilton or Lagrange but, it seems,
phase diagrams have been associated
with electronics and complex numbers.

for instance, we have three-phase 220v
lines, which break-out to residences
at the transformer to 2-phase 110v lines.

there is even a convention to use “j,”
instead of “i” for the imaginary unit,
since the i is often used for current,
as in Ohm’s Law (or Mho’s –
in units of Siemens .-)

there is a certain argument that Minkowski
– if he’d lived somewhat longer – might
have refined his bizarre ejaculations
about “space-time,” which is just a kind
of phase-space. I mean, the guy was quite
a good mathematician, like in numbertheory
… so, what is a “phasor?”

Posted by: it'sawash on December 10, 2008 10:35 PM | Permalink | Reply to this

Phasor defined; modern art and scientific revolution; Re: That Shift in Dimension

Weisstein, Eric W. “Phasor.” From MathWorld–A Wolfram Web Resource.

That’s a good place to start on “… so, what is a ‘phasor?’”

I’ll not respond to “bizarre ejaculations about ‘space-time’” – because of how keywords are searched. Except to say this about the hyphenate “space-time” in the context of when the word was coined –

Staircase Descending a Nude
Jonathan Vos Post

Naked as the stairway to heaven falls down
Marcel Duchamp leaves retinal art behind
stripped of his academic crown
crushed by his superconscious mind

Staircase and nude are intertwined
in 1913 at the Armory Show
zero over zero: undefined
discrete transform of turbulent flow

Like the shirling stars of Vincent Van Gogh
superposed facets of breast and rump
quantum dynamics from head to toe
the heart is a centrifugal pump

Castle in the air, memory trace,
turning point: intergalactic space

1 Sep 2008

copyright (c) 2008 by Emerald City Publishing

Posted by: Jonathan Vos Post on December 14, 2008 8:19 PM | Permalink | Reply to this

typo fixed; Re: Phasor defined; modern art and scientific revolution; Re: That Shift in Dimension


“Like the swirling stars of Vincent Van Gogh”

Posted by: Jonathan Vos Post on December 14, 2008 8:24 PM | Permalink | Reply to this

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