Configuration space vs physical space

[pellman]

I am reading Quantum Gravity by Rovelli. He draws a somewhat similar conclusion--from GR, not QM--that our experience of spacetime is illusory. There are only relations and the relations don't require a "where", but they give the impression that two dynamical entities which are relating are happening in the "same place." He does not use the word "illusory" I think, but that is the gist. That since we could in principle do the physics (macro, GR physics in this case) without introducing spacetime, spacetime itself is not truly physical, it is a construct by the observer.

He doesn't go so far to say that it is construct by the observer, but that seems to be logical result of his argument to me.

Check section 2.3.2 "The disappearance of spacetime", pg 52. http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

 

[Demystifier]

I love the Rovelli's book you mention and I think your remark is a good analogy.

 

[Maaneli]

Quantum physics is supposed to be more fundamental than classical physics. This suggests that configuration space is more fundamental than the "physical" space. But if it is more fundamental, then it should be more physical as well. The problem then is to explain why then the 3-space looks more "physical" to us (despite the fact that actually the configuration space is more physical). What is the origin of this illusion?

Dear Demystifier,

I'd like to suggest an alternative route to a possible solution to the above problem (which I certainly agree is a problem). As you may know, the Einstein-Smoluchowski theory of Brownian motion describes a single massive point particle undergoing a discrete random in one time direction (the +t direction), and has a microscopic description in terms of binary Bernoulli paths of the form (1,0) in 1+1 dimensions. The simplest stochastic differential equation of motion for such a particle is of the form

dX(t) = sqrt(D)*dW(t), (1)

where dW(t) is the Wiener process with mean and autocorrelation function,

< dW(t) > = 0.

< dW(t)^2 > = 2*D*dt.

Equation (1) has an equivalent representation as a classical diffusion equation of the form

d[P(X,t)]/dt = -(D/2)*grad^2[P(X,t)], (2)

where P(x,t) = [1/(4*pi*D*t)]*exp[-x^2/(4*pi*D*t)] is the transition probability density solution. It is a function on "physical space", AKA, 3-space.

However, for N-particles, equations (1) and (2) are in configuration space. In other words,

d(X1...Xn,t) = sqrt(D)*dW(t), (1a)

d[P(X1...Xn,t)]/dt = -(D/2)*grad^2[P(X1...Xn,t)]. (1b)

So the transition probability density for N-particles is instead a function on configuration space.

Now, even though the transition probability density for 1 particle does not correspond to an ontological entity 'out there' in the physical world like the electromagnetic field does, we know that the 3-space it is a function on is still the physically real space that corresponds to our experiences. However, for the N-particle transition probability being a function on configuration space, we know that the configuration space here cannot possibly be physically real, and instead is just an abstract mathematical encoding of the transition probability density distribution for N-particles undergoing a stochastic process defined by equation (1a).

Now, I'm sure you are familiar with the formal similarities between the classical diffusion equation and the non-relativistic Schroedinger equation. In fact, mathematically, the *only* difference between the two equations is the fact that the diffusion constant in the Schroedinger equation is complex-valued, whereas in the classical diffusion equation, it is real-valued; and this difference corresponds to wave solutions for the Schroedinger equation, and diffusive solutions for the classical diffusion equation. Moreover, it is well-known that a Wick rotation, t => i*t, of the Schroedinger equation converts it into a diffusion equation (in imaginary-time), while the same Wick rotation converts the classical diffusion equation into a Schroedinger equation (in imaginary-time). Mathematically, the Wick rotation is breaking the time-symmetry of the Schroedinger equation, while introducing time-symmetry into the classical diffusion equation. In terms of the solutions to the respective equations of motion, this turns the wave solutions of the Schroedinger equation into diffusive solutions of the diffusion equation, and vice versa. These formal mathematical relations suggest that one can perhaps interpret the Schroedinger equation as a "time-symmetric diffusion equation". Indeed, it turns out that if one allows for time-reversal in the discrete random walk (in other words, motion in the -t direction as well as the +t direction) of a single massive point particle in the Einstein-Smoluchowski theory of Brownian motion, then the microscopic description of such a time-symmetric Brownian motion is no longer given by the binary Bernoulli paths, (1,0), but rather the anti-Bernoulli paths given by (-1,0,1). Garnet Ord and Robert Mann have shown how by just forcing time-reversal in the random walk of a single massive point particle, one can obtain, in the continuum limit, the Schroedinger or Pauli or Klein-Gordon or Dirac equation in 1+1 dimensions, instead of the classical diffusion equation or Telegraphs equation:

The Dirac Equation in Classical Statistical Mechanics
Authors: G.N. Ord
Comments: Condensed version of a talk given at the MRST conference, 05/02, Waterloo, Ont.
http://arxiv.org/abs/quant-ph/0206016

The Feynman Propagator from a Single Path
Authors: G. N. Ord, J. A. Gualtieri
Journal reference: Phys. Rev. Lett. 89 (2002) 250403
http://arxiv.org/abs/quant-ph/0109092

Entwined Pairs and Schroedinger 's Equation
Authors: G.N. Ord, R.B. Mann
(unpublished)
http://arxiv.org/abs/quant-ph/0206095

Entwined Paths, Difference Equations and the Dirac Equation
Authors: G.N. Ord, R.B. Mann
(unpublished)
http://arxiv.org/abs/quant-ph/0208004

The Schroedinger and Diffusion Propagators Coexisting on a Lattice
Authors: G.N. Ord
<< The Schroedinger and Diffusion Equations are normally related only through a formal analytic continuation. There are apparently no intermediary partial differential equations with physical interpretations that can form a conceptual bridge between the two. However if one starts off with a symmetric binary random walk on a lattice then it is possible to show that both equations occur as approximate descriptions of different aspects of the same classical probabilistic system. This suggests that lattice calculations may prove to be a useful intermediary between classical and quantum physics. The above figure shows the appearance of the diffusive and Feynman propagators at fixed time as the space-time lattice is refined. Both these functions are observable characteristics of the same physical system. >> (J. Phys. A. Lett. 7 March 1996)

Bohm Trajectories, Feynman Paths ans Subquantum Dynamical Processes
Speaker(s): Garnet Ord - Ryerson University
http://pirsa.org/05100011/

What is a Wavefunction?
Speaker(s): Garnet Ord - Ryerson University
<< Abstract: Conventional quantum mechanics answers this question by specifying the required mathematical properties of wavefunctions and invoking the Born postulate. The ontological question remains unanswered. There is one exception to this. A variation of the Feynman chessboard model allows a classical stochastic process to assemble a wavefunction, based solely on the geometry of spacetime paths. A direct comparison of how a related process assembles a Probability Density Function reveals both how and why PDFs and wavefunctions differ from the perspective of an underlying kinetic theory. If the fine-scale motion of a particle through spacetime is continuous and position is a single valued function of time, then we are able to describe ensembles of paths directly by PDFs. However, should paths have time reversed portions so that position is not a single-valued function of time, a simple Bernoulli counting of paths fails, breaking the link to PDF's! Under certain circumstances, correcting the path-counting to accommodate time-reversed sections results in wavefunctions not PDFs. The result is that a single `switch' simultaneously turns on both special relativity and quantum propagation. Physically, fine-scale random motion in space alone yields a diffusive process with PDFs governed by the Telegraph equations. If the fine-scale motion includes both directions in time, the result is a wavefunction satisfying the Dirac equation that also provides a detailed answer to the title question. >>
http://pirsa.org/08110045


The key result from these papers and talks is that the derived wavefunctions just encode (as a complex-valued vector) the real-valued transitions probabilities for the particle undergoing Brownian motion forward and backward in time.

So far these results are for only 1 particle, and therefore the corresponding wavefunctions derived from the model are on 3-space. Ord and others have yet to work out 2 particles in their binary random walk model. However, since it is already possible in the Einstein-Smoluchowski theory to construct the two-particle transition probability solution to the diffusion equation from the Bernoulli counting of two particles starting from the same initial position and undergoing the standard random walk forward in time, there doesn't seem to be any reason why they shouldn't be able to construct the two-particle wavefunction in configuration space, R^6, by just considering two particles starting with the same initial condition, and undergoing the time-symmetric random walk between two separate spacetime points. If and when this is done, I would propose that this would be a "deeper" explanation for why wavefunctions in configuration space describe quantum particles. It would just be an epistemic means of encoding the forward and backward transition probabilities of two or more particles starting with the same initial condition, and undergoing a time-symmetric "binary" random walk between two separate spacetime points, instead of a time-asymmetric random walk as in the standard Einstein-Smoluchowski theory. From this point of view, nonlocality in the sense of instantaneous action at a distance in deBB theory would not necessarily be fundamental - it would be a property of the configuration space structure of the N-particle wavefunction guiding the two deBB particles, but the underlying ontology would be a sort of retro-causality from these two particles which are actually executing a time-symmetric random walk between their initial and final boundary conditions (the latter of which is assumed to be randomly determined, and not determined by the dynamics of the theory itself). Clearly there are plenty of open questions one can ask about this approach, but I'll leave it here for now.

 

[Demystifier]

Maaneli, thank you for the interesting remarks. My motivation is also (partially) related to the deBB interpretation.

 

[ThomasT]

Quantum physics is supposed to be more fundamental than classical physics. This suggests that configuration space is more fundamental than the "physical" space. But if it is more fundamental, then it should be more physical as well. The problem then is to explain why then the 3-space looks more "physical" to us (despite the fact that actually the configuration space is more physical). What is the origin of this illusion?

Maybe it isn't an illusion.

Maybe 3-space looks more physical to us than configuration space because physical space is 3-space. At least that's a possibility, isn't it?

Assuming that there's a fundamental dynamic(s) governing phenomena on any and all scales, then quantum physics isn't more fundamental than classical physics. It's just dealing with phenomena whose scale is set by the quantum of action. There's no particular reason to think that the reality of an underlying quantum reality isn't a 3D-Euclidian space.

Representations in non-real space(s) are a consequence of the fact that the media in which quantum scale disturbances are propagating are invisible to us, and disturbances in those media are untrackable. Intermittent quantum scale probings yield aggregate statistical results whose probabilities are described via functions in a non-real, configuration space.

The problem then is to explain why configuration space should be considered a physical space at all, much less more real than the 3-space of our sensory reality.

If the real physical space is 3-space, then quantum nonlocality isn't a physical problem. Or is it?

 

[Chrisc]

Demystifier, I think this points to a fundamental problem I've been struggling with
for a long time. What is measured "as" the properties of particles?
I'll paraphrase your OP for my purposes:
Entanglement is non-local in 3-space and local in configuration space.
Therefore non-locality is a problem in 3-space but not in configuration space.

This raises the question: is entanglement a state of particle properties endowed at creation or the state of configuration space on which we define their creation?

If entanglement can be understood NOT as a property of the particles in question but the geometry
of the configuration space on which they are created, non-locality is then NOT a condition of particle property
but a correlation between the configuration space on which the particles (with their corresponding "assumed" properties) are created and the configuration space on which the particles (and assumed properties) are measured.
Thus hidden variables do not exist as unknown qualities of particles nor can such metaphysical attributes account for the predictions of QM, but as correlations of configuration space between creation and detection entanglement is a local dynamic of the evolution of propagating fields.

I'm afraid my math skills are not up to the task, but as I see it, a particle cannot be defined as a finite point in 3-space except as the definition of detecting "in" 3-space the dynamics we set out to measure. The particle then is not defined by the dynamics measured any more than a baseball is defined by the wind blowing through the window it breaks.
In many ways this idea of measurement boils down to relativistic emergence - what you define as properties of a particle at creation evolves and emerges as (via relativistic field equations) what I define as the properties of detection.

 

[Hurkyl]

There's a glaring omission in the original question -- what are the observables?

 

[Chrisc]

Is this a question of distinguishing configuration space from physical space,
or is it a question of distinguishing degrees of freedom of action from the degrees of freedom a particle?
Two free particles in one dimension each have less degrees of freedom than one particle in two dimensions.
If the two particles express the greater probabilities corresponding to less degrees of freedom, the one particle in two
dimensions has less probabilities and greater degrees of freedom.
The former is the wave-function the latter the measurement as a measurement MUST define dimensionality.
By this analogy, uncertainty is simply a matter of the impossibility of a simultaneous confinement of degrees of freedom.

 

[Demystifier]

Maybe 3-space looks more physical to us than configuration space because physical space is 3-space. At least that's a possibility, isn't it?

It certainly is. In fact, this is the standard view.

If the real physical space is 3-space, then quantum nonlocality isn't a physical problem.

Quite the contrary, I think this is exactly why nonlocality is viewed as a problem.

 

[pellman]

Is this topic connected to the question:

How is the x in a quantum field \psi(x,t) related to the N position operators \hat{x}_1,...\hat{x}_N for a system of N particles?

The x in the quantum field refers to the coordinates on the manifold on which the field lives, while the position operators refer to configuration space.

 

[strangerep]

Is this topic connected to the question:

How is the x in a quantum field \psi(x,t) related to the N position operators \hat{x}_1,...\hat{x}_N for a system of N particles?

The x in the quantum field refers to the coordinates on the manifold on which the field lives, while the position operators refer to configuration space.

Is this topic connected to the question? I'd say yes. Taking it a bit further, there is a
"no-interaction" theorem of Currie, Jordan & Sudarshan. They draw a distinction between
"relativistic invariance" (meaning construction of a representation of the Poincare algebra),
and "manifest invariance" (meaning expressing all physical quantities in terms of 4D
spacetime and the particular ways in which things transform under changes of spacetime
reference frame). They show that these two approaches are not really compatible for
interacting multi-particle theories (hence the name "no-interaction theorem").

In orthodox QFT there's also something called the Reeh-Schlieder theorem which
exposes a related paradox.

IMHO, this "physical space" notion arises because an inertial observer's local
symmetry group is ISO(3,1). The set of measurements he/she can perform corresponds
(in the quantum sense) to the set of projection operators constructible in unirreps of
that group. I.e., the only measurements he/she can do are essentially equivalent to
"apply one of those projection operators". Hence the appearance of 3+1 physical space
for each observer - because he/she doesn't have the ability to wield a larger set of projection
operators. (This also leads to some of the puzzles that the physical spaces perceived
by different observers do not always coincide properly).

 

[ThomasT]

Originally Posted by ThomasT
If the real physical space is 3-space, then quantum nonlocality isn't a physical problem.

Quite the contrary, I think this is exactly why nonlocality is viewed as a problem.

If the real physical space is 3-space,
then if quantum nonlocality only 'occurs' in purely formal, nonphysical space,
then quantum nonlocality isn't a physical problem.

 

[Demystifier]

If the real physical space is 3-space,
then if quantum nonlocality only 'occurs' in purely formal, nonphysical space,
then quantum nonlocality isn't a physical problem.

The point is that quantum nonlocality occurs in a physical, not only purely formal, space. It is a measured effect.

 

[pellman]

there is a
"no-interaction" theorem of Currie, Jordan & Sudarshan. They draw a distinction between
"relativistic invariance" (meaning construction of a representation of the Poincare algebra),
and "manifest invariance" (meaning expressing all physical quantities in terms of 4D
spacetime and the particular ways in which things transform under changes of spacetime
reference frame). They show that these two approaches are not really compatible for
interacting multi-particle theories (hence the name "no-interaction theorem").

In orthodox QFT there's also something called the Reeh-Schlieder theorem which
exposes a related paradox.

This sounds very interesting. Thanks for sharing.

 

[ThomasT]

The point is that quantum nonlocality occurs in a physical, not only purely formal, space. It is a measured effect.

You might say, depending on experimental design and observed instrumental behavior, that IF something is propagating FTL in physical 3-space then there are some limits on that. But there's no reason that I know of to assume that something IS propagating FTL in physical 3-space. All that's known is that quantum entanglement, quantum wavefunction collapse, and quantum nonlocality are creatures of the qm formalism, and that the formalism is a probability calculus which ultimately produces probability densities arising from functions which describe 'propagations' in a nonphysical space.

 

[debra]

The relevant property in configuration space is that of state correlation (and not so much else it seems) which when applied to physical space requires an explanation in terms of retained state relationships applicable from creation time onwards (avoiding local variables etc), or is somehow maintained in some synchronous way. e.g. by a shared timing function underlying space time (physical space) but applies in the configuration space.

 

[pellman]

Is this topic connected to the question:

How is the x in a quantum field \psi(x,t) related to the N position operators \hat{x}_1,...\hat{x}_N for a system of N particles?

The x in the quantum field refers to the coordinates on the manifold on which the field lives, while the position operators refer to configuration space.

Just to elaborate a bit further: In regular quantum mechanics, the dynamical equations for a system of N particles can be expressed in terms of any number of different sets of 3N generalized coordinates \hat{q}_1,...\hat{q}_{3N}. Each set, if we write down the correct Hamiltonian, is just as valid as the others and gives the right answers.

This freedom has nothing to do with different representations, e.g. the momentum representation. Each set of 3N generalized coordinates is an equally valid "position" representation.

Yet there is an especially unique set of generalized coordinates (up to global transformations) \{\hat{q}_1,...\hat{q}_{3N}\}=\{\hat{x}_1,...\hat{x}_N\} which we say refer to the "positions of the particles". What is it that is special about this set of coordinates? I think we can consider this to be a first stab at a mathematical formulation of the OP question.

On the other hand, there is the x which appears in a quantum field \psi(x). I know enough QFT to know that \psi(x) is associated with the position representation and that we can transform to, say, the momentum representation and find the associated field \phi(p), but what is the analogy in QFT to a set of generalized coordinates as described above for QM?

 

[strangerep]

Yet there is an especially unique set of generalized coordinates (up to global transformations) \{\hat{q}_1,...\hat{q}_{3N}\}=\{\hat{x}_1,...\hat{x}_N\} which we say refer to the "positions of the particles". What is it that is special about this set of coordinates?

I don't think it's "special" in itself. Only the full specification of the dynamics with its
degrees of freedom is physically relevant.

On the other hand, there is the x which appears in a quantum field \psi(x). I know enough QFT to know that \psi(x) is associated with the position representation and that we can transform to, say, the momentum representation and find the associated field \phi(p), but what is the analogy in QFT to a set of generalized coordinates as described above for QM?

In QFT (Fock space), there are operators to create/annihilate particles with given
momenta, spin, etc. One can inverse-Fourier transform to a position-like basis, or indeed one
can start (in axiomatic QFT) from an irreducible set of field operators defined on Minkowski
space. The generalized set of coordinates you mentioned is loosely analogous to the tensor
products of 1-particle Hilbert spaces used in the construction of Fock space. In both cases,
we build up larger and larger dynamical systems via tensor products.

But either way, you run into the embarrassing Reeh-Schlieder paradox.

 

[Demystifier]

But either way, you run into the embarrassing Reeh-Schlieder paradox.

What is Reeh-Schlieder paradox?

 

[atyy]

So, are you saying that the known physical laws would allow the existence of living beings that would think that they live in, e.g., 5 "physical" dimensions? I don't think so.

Not sure what nughret was thinking, I was thinking along the lines of a soundwave hitting the ear - it's 1D as a function of time. But what we hear is eg. 3D in a keyboard fugue, or 25D in a Strauss tone poem. Do people really look out and see 3D physical space? Or point particles for that matter?

 

[QuantumBend]

What is Reeh-Schlieder paradox?

Reeh Scleider paradox for Quantum Feild Theory: not operator IN particle but field observe operator ONLY.
- no good.

 

[Demystifier]

not operator IN particle but field observe operator ONLY.

I do not understand this sentence.

 

[ThomasT]

Demystifier, I looked at several papers dealing with Reeh-Schlieder theorem (or paradox or property), and can confidently say that I don't fully understand its significance ... yet.

Now I'm thinking that maybe the consideration of your thread is a bit over my head for the foreseeable future. But thanks for tolerating my comments.

One parting comment, before retiring to the peanut gallery (where, of course, I'll continue to follow others comments, and look stuff up).

Do people really look out and see 3D physical space? Or point particles for that matter?

Not point particles. But events in the 'space' of our sensory perception are communicable using 3 spatial and 1 time dimension. It remains to be seen, literally, if there's any physical space other than this.

 

[strangerep]

[...] But either way [in orthodox QFT] you run into the
embarrassing Reeh-Schlieder paradox.

What is Reeh-Schlieder paradox?

It's a theorem applicable to axiomatic QFT. Having started with an
irreducible set of causal fields over Minkowski space carrying a +ve energy
unirrep of the Poincare algebra, the R-S thm then essentially is this:

Let A,B be two disjoint spacelike-separated regions in Minkowski
spacetime. Given only knowledge of the field configuration on region A
it is possible to reconstruct the field on region B. This is embarrassing
because stuff happening in region A should physically have nothing to
do with region B. (I used the word "paradox" because we started with
a supposedly causal theory, yet we derived this theorem, but the
phrase "physical contradiction or puzzle" might be more appropriate.)

This can be restated in various ways. E.g., "local operations applied to
the vacuum state can produce any state of the entire field" [1].
Or "The R-S thm asserts the vacuum and certain other states to be
spacelike superentangled relative to local fields". [2]

It's a rather controversial subject. (E.g., see Wiki's entry.)

[...] not operator IN particle but field observe
operator ONLY

I do not understand this sentence.

I'm not sure I do either. Possibly QuantumBend was pointing out
that R-S is applicable to QFT, not relativistic particle theory.

---------------------------------
Refs:

[1] Halvorson, "Reeh-Schlieder defeats Newton-Wigner ..."
available as quant-ph/0007060
(see also refs therein)

[2] Fleming, "Reeh-Schlieder meets Newton-Wigner"
Phil Sci 67 (proceedings), S495-515
http://philsci-archive.pitt.edu/archive/00000649/00/RS_meets_NW,_PDF.pdf

 

[Demystifier]

Thanks, strangerep.
But I still cannot say that I understand it.
For example, what if we replace a continuous space by a lattice, is the RH theorem still valid then?

 

[jambaugh]

I have contemplated this issue for many years myself. Here are some of my thoughts:

Recall that in non-relativistic theory time is not an observable but rather a parameter. In the unification of space-time we can choose to re-interpret time as an observable or lose the "observable" status of spatial coordinates and treat them too as parameter, i.e. no different than abstract configuration coordinates in phase space.

I think this second is the correct view and think breaking Born reciprocity is a good thing. Once we move to field theory this transition is complete. The observables are field values (particle type and number charge etc) at a given coordinate position. Thus coordinates are numbers we put on our measuring devices.

This to some is a problem given GR which in the current geometric formulation treats gravitation as curved space-time geometry. However if we read the Equivalence Principle correctly (as I see it) then it is not that "gravity is just geometry" but rather that we only see dynamic evolution of test particles and so the boundary between gravity and geometry is indistinguishable. We can vary our choice of space-time geometry and introduce a "physical" force of gravity and not see any difference in predictions. I think this means rather that it is the geometry which is "not physical" rather than the gravitational force.

I also think failure to see this view has hampered quantum grav. research.

 

[QuantumBend]

I'm not sure I do either. Possibly QuantumBend was pointing out that R-S is applicable to QFT, not relativistic particle theory.

I saying: In QFT we say NOT 'particle here', we saying 'OPERATOR who observe particle here'. You know this. When no observe - no particle, no history, big one area. 3 spaces no good.

This saying, I know in complex mathematic. Why?
https://www.physicsforums.com/showthread.php?t=270084&highlight=photon+exist
Big one, no good - Reeh-Schlieder do this.
https://www.physicsforums.com/showthread.php?t=282289&highlight=single+photon

 

[strangerep]

what if we replace a continuous space by a lattice, is the RH theorem still valid then?

Hmmm, I'm not sure.

I've just had a look at the proof of the RS theorem given in Appendix 4
of Lopuszanski [1]. It relies on some functional-analytic results, together
with an extension to complex spacetime variables to ensure a certain
integral is meaningful.

A strictly discrete spacetime is very different from the foundations
used in axiomatic QFT.

---------------
[1]: Lopuszanski, "An Introduction to Symmetry [...] in QFT",
World Scientific, ISBN 9971-50-161-9.

 

[turin]

I got tired of reading halfway through the thread, so I appologize if I am repeating anyone.

I am considering the very original post, with the Hamiltonian H=p1^2+p2^2. I see two possibile distinguishing features between particle degrees of freedom vs. spacel degrees of freedom, but both of them require extending to other considerations besides the Hamiltonian.

1) The statistics of combining two space degrees of freedom for a single particle is trivial. The statistics of combining two particle degrees of freedom in a 1-D space is nontrivial. E.g., a single fermion in 2-D space doesn't care that it is a fermion, and, in particular, p1=p2 is allowed. Two fermions in a 1-D space care that they are fermions, and, in particular, p1=p2 is not allowed.

2) The topology of 2 space degrees of freedom for a single particle is different than the topology of 2 particle degrees of freedom in a 1-D space. E.g., for a single particle in 2-D space, I could say that one unit of momentum in the p1 direction is \sqrt{2} units of momentum away from one unit of momentum in the p2 direction in momentum space. However, I think it might be less meaningful/physical to talk about the amount of separation between one unit of momentum for particle 1 and one unit of momentum for particle 2.

 

[RedX]

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.

If the phase space is all the same I don't see how you can ask if the configuration space is different because a single rule for converting a phase space into a configuration space should convert the same phase space into the same configuration space?

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?