Configuration space vs physical space

[Demystifier]

Anyways, this thread reminds me of a question I have about classical mechanics. In the Hamiltonian formulation, areas in phase space are conserved, i.e., the divergence of the phase space velocity is zero. In the Lagrangian formulation, the phase space is coordinates and their velocity, but the divergence of the phase space velocity is not in general zero. Does this mean that determinism is violated in the Lagrangian formulation, since 10 initial phase states will not go into 10 final phase states in general?

If something is not conserved, it does not mean that it does not behave deterministically. So it certainly does not mean that determinism is violated in the Lagrangian formulation.

 

[RedX]

If something is not conserved, it does not mean that it does not behave deterministically. So it certainly does not mean that determinism is violated in the Lagrangian formulation.

Well, if you have a collection of 5 initial states, then after some time, there should only be at most a collection of 5 final states. If you have a collection of 6 final states, that means the Hamiltonian took one state, and outputed two states, violating determinism. So areas in phase space should never expand, so that if your area is 5 states, then it ought to be no more than 5 states after time translation. Hamilton's q-p space is conserved after evolution in time, but Lagrange's q-dq/dt space is not conserved.

 

[jambaugh]

I thought in classical mechanics all phase spaces with the same # of dimensions are the same. Nothing weird happens: it's just a flat space. Specifying a Hamiltonian just specifies one of many different types of canonical transformations you can perform among the coordinates of the space.

Just a comment...

In the more advanced formulation, one works in an extended phase-space which is flat as you say but not all functions of the canonical coordinates and momenta are observables. Some of the dimensions are gauge degrees of freedom. By imposing gauge constraints you pick out a curved sub-manifold of the extended phase-space, define a Dirac bracket instead of the Poisson bracket with which canonical transformations map this sub-manifold onto itself, and thereby work in a curved physical state manifold. What's more the dimension need not be even.

How this relates to the OP is the observation that configuration space is in this general case may be a mixture of physical and gauge degrees of freedom (as is the canonical momentum space). It is not until the gauge constraints are chosen that physical degrees of freedom are well defined.

Note that (e.g. in electrodynamics) it is the canonical momenta [itex]P = p + ieA(x)[/tex] which are the "flat" coordinates and then the physical momenta [itex]p = P - ieA(x)=mv[/tex] which one defines relative to P when one fixes the gauge. (I may have the +/- signs mixed up but that's a matter of convention.) The p's live on a curved sub-manifold of phase space.<br /> <br /> In a more general case one may introduce a U(1) gauge phase as another canonical coordinate in configuration space with dual momentum corresponding to the particle's charge. The phase connection A then defines what mixtures of canonical coordinates correspond to physical coordinates. The hamiltonian is greatly simplified but the physical forces are due to gauge constraints. It is analogous to gravitation via geometry in GR.<br /> <br /> [Edit: A good reference is "Quantization of Gauge Systems" by Henneaux and Teitelbaum ][/itex][/itex]

 

[Fra]

Reflection on the puzzle

I just bumped into this thread and didn't follow it from start, but relating to Demystifiers original question on the differentiation between what "space" is "more physical". I thikn it's an interesting question. This is related to something I'm also pondering. It also connects to Hurky's remark, about what are the observables? Ie. what questions are more "physical"? I think the mathematical view here is not helping.

I am still working on this in a larger context but to me, this question is really deeply entangled with the microstructure of the observer, as well as the concept of inertia of the structure, and I like to think that the answer to why certain microstructures are favoured (say a 3D space) lies at the level of evolving relations.

Someone pondered the idea that, would the laws of physics "allow" an observer who thinks he lives in a 5D space? I think it does. But the question is, what would happen to such an observer, would it be fit and stable? :) Probably not, to me I think an analogous question is what is the evolutionary mechanism that can explain the plausability of the emergent common structures we see.

I am attacking this from the point of view of picturing interacting microstructure-systems (which to me is the abstraction of "an observer", but which well may represent a physical system, say a particle with given properties, properties that are implicit in the makeup of the microstructure), and the trick would be that the interaction between the systems implies a selective pressure that causes evolution of the structures. Thus each propertiy such as dimensionality is seen as a relation between the systems environment, and has no universal sense beyond it's current evolutionary status.

I'm trying to get my head around exactly how this interacting driven evolution produces the basic structures we konw which, would be spacetime and basic properties of the simplest possible observers (elementary particles) that contains the four forces. So I think the propertis of the simplest structurs around, and the emergence of space go hand in hand.

Any attempt to study one, idealised and disconnected from the other doesn't make sense to me.

So I agree it's a puzzle, and I have no answer either, I only have at least for myself a strategy and plan I'm working along the spirit explained above.

/Fredrik

 

[Dmitry67]

Interesting thread.
Assuming Max Tegmarks MUH, if 2 systems are isomorphic then they are the same. So, if we can map our physical space to configurational space then both are physical... or configurational...

 

[Ken G]

I actually take a perspective which is more or less opposite to that of Tegmark's. He seems to feel that the purpose of mathematics is to establish the true ontologies of reality, so if two mathematical descriptions are equivalent, then the true ontologies are the same-- even if they sound different before we understand the mathematical equivalence. My approach is that the goal of physics never was to find true ontologies, so the fact that mathematical ontologies are "always true" (within the mathematical theory involved) demonstrates the different goals of math and physics. Physics borrows mathematical ontologies for specific, contextually dependent purposes, and these ontologies are not unique and are not meant to be unique. There is just no such thing as a "true ontology" in physics, and there is no need for one. We don't use true ontologies, we use effective or useful ontologies, and this is quite demonstrably true about physics. So I don't think we should lose any sleep over what is the "true ontology" that should be associated with a particular Hamiltonian, or model of any kind. A model is a model, not a true ontology, and works for whatever it works for.

That doesn't mean I don't think the OP question is interesting-- indeed, one of the most interesting things about it, in my view, is how it can be used as a device to establish this point.

 

[bohm2]

So, if we can map our physical space to configurational space then both are physical... or configurational...

I thought that can't be done because there are so many different ways of doing it and the choice seems arbitrary. As I understand it, Lewis does attempt to do that here:

But suppose instead that we take seriously the idea of a configuration space as a space of configurations-that is, a space which is intrinsically structured as N sets of three-dimensional coordinates. Mathematically, this is not hard to do. Instead of modeling the space as an ordered 3N-tuple of parameters, <x₁, x₂, x_3N> we model it as an ordered N-tuple of ordered triples:

<<x₁,y₁,z₁>, <x₂y₂,z₂>,...<x_N,y_N.z_N>>

And rather than specifying the coordinates by choosing 3N axes, we choose 3-the x, y and z axis, which are the same for each triple. That is, x₁ through x_N pick out points on the same axis, and similarly for y and z. Then the wavefunction can be regarded as a function of these parameters-as a mathematical entity inhabiting a (3 x N)-dimensional configuration space, rather than a 3N-dimensional plain space. And the basic thesis of wavefunction realism is that the world has this structure-the structure of a function on (3 x N)-dimensional configuration space. Given that configuration space has this structure, then an Albert-style appeal to dynamical laws to generate three-dimensional appearance is impossible, but it is also unnecessary. It is impossible because the dynamical laws take exactly the same form under every choice of coordinates (as they should), so no choice makes the dynamical laws simpler than any other. But it is unnecessary because the outcome of that argument-that the coordinates are naturally grouped into threes is built into the structure of reality, and hence doesn’t need to be generated as a mere appearance based on the simplicity of the dynamics.

http://philsci-archive.pitt.edu/8345/1/dimensions.pdf

But others like Monton question this:

http://spot.colorado.edu/~monton/BradleyMonton/Articles_files/qm%203n%20d%20space%20final.pdf
http://spot.colorado.edu/~monton/BradleyMonton/Articles.html (see "Against-3N -dimensional space")

Having said that I'm really confused about this whole topic because is anyone clear on what "dimensionality" means in configuration space?

 

[qsa]

The question (or puzzle) that I want to pose ...

Maybe you can answer you own question. What was the original idea and intend for introducing the configuration space in classical physics ? Was there a price to pay?

 

[Ken G]

And why are we limiting the discussion to configuration vs. Euclidean space? Why not phase space? If I specify the configuration of N particles, and their Hamiltonian, I still don't know enough about them to predict what happens, so that "must not be the reality" either. Specifying a "rate of change of configuration" for each particle is a bizarre way to provide realism to the picture, it would be much more natural to use a 6N dimensional phase space, and call that the reality.

Of course, as I said above, I think calling any of these things reality is a kind of breakdown of sound scientific thinking. The purpose of science is to replace reality with models of reality that achieve various purposes, and it is both fruitless and unnecessary to ask, which one is the "real reality." For example, consider this from the Lewis quote just above:" And the basic thesis of wavefunction realism is that the world has this structure-the structure of a function on (3 x N)-dimensional configuration space." I would have to say that if that is realism, then he can keep it-- it sure doesn't sound like science to assert that the world has a certain structure. What sounds like science is saying "let us provisionally enter into a state of imagination that the world has this structure, because it serves us in the following ways." Stated like that, doesn't the whole issue just dissipate in the way it should?

 

[Demystifier]

What was the original idea and intend for introducing the configuration space in classical physics ?

Mathematical elegance.

Was there a price to pay?

Yes, obscure visualization in the 3-space.

 

[debra]

Wouldn't a configuration space offer a better alternative to renormalisation?
Then below a specified minimum coordinate range space is 'undefined' - that would help us with some of our cowboy infinities I believe...