Configuration space vs physical space

[Demystifier]

The question (or puzzle) that I want to pose essentially belongs to classical (not quantum) physics. Nevertheless, there is a reason why I post it here on the forum for quantum physics, as I will explain at the end of this post.

As a simple example, consider the following Hamiltonian:
H=\frac{p_1^2}{2m}+\frac{p_2^2}{2m}
What does this Hamiltonian describe? Is it one free particle moving in two dimensions, or two free particles moving in one dimension? Clearly, this Hamiltonian describes both. But then, how to distinguish between these two physically different cases? Is there a FORMAL (not purely verbal!) way to distinguish the configuration space from the "physical" space?

This is essentially a classical question, but there are two reasons why I ask this question here:
First, people here are much more clever than people on the forum for Classical Physics.
The second, more important reason is that, although essentially classical, the motivation behind this question is actually quantum. Namely, the idea is that nonlocality of quantum mechanics could be avoided by noting that, ultimately, QM is nonlocal because it is formulated in the configuration space rather than in the "physical" space. For if the configuration space is reinterpreted as a "true physical" space (whatever that means), then QM becomes local in that "true physical" 3n-dimensional space, where n is the number of particles. But then the problem is to explain why the world looks to us as if it was only 3-dimensional (for simplicity, I ignore relativity). To understand that one needs first to understand what exactly makes the standard 3-dimensional physical space more "physical" than the 3n-dimensional configuration space, which is my motivation to ask the question above.

 

[malawi_glenn]

I don't even know what "physical" space is, can you definie it FORMALLY?

 

[Demystifier]

I don't even know what "physical" space is, can you definie it FORMALLY?

If I could, I would not pose the question above in the first place.
Indeed, the problem can be reduced to the problem of finding the appropriate formal definition of the "physical" space, given that we already understand intuitively what the physical space should be. (You know, the 3-dimensional space on the top of which everything else seem to exist ...)

 

[ThomasT]

3D Euclidian space is the space that corresponds to our sensory apprehension of reality. I don't know if that qualifies as a formal definition of physical space, but sensory data are the criteria by which we evaluate claims about reality. Anyway, it seems reasonable to assume that the reality that we can't directly sense is also 3D Euclidian. Maybe there isn't any deep explanation for this, any more than there is an explanation for the universal scale expansion that is deeper than the expansion itself.

 

[nughret]

But then the problem is to explain why the world looks to us as if it was only 3-dimensional (for simplicity, I ignore relativity).

This is a question of biology not physics.

 

[Tac-Tics]

But then the problem is to explain why the world looks to us as if it was only 3-dimensional (for simplicity, I ignore relativity). To understand that one needs first to understand what exactly makes the standard 3-dimensional physical space more "physical" than the 3n-dimensional configuration space, which is my motivation to ask the question above.

When you have a 3n-D configuration space, you're really working with a product of n spaces joined together. It simple describes the number of variables required to completely specify the state of your system.

Similarly, in statistics, if you take the height of a thousand people in a city, your resulting data will be a 1000-D space. You can see the mathematics treating it this way when you look at the standard deviation, which is the shortest distance, given by the Euclidean norm (ie: root of the sum of squares), from the point in 1000-D space describing your sample to the "average" line described by {(t, t, t, ..., t) in R^1000 | t in R}. Does this mean that you can "create" extra dimensions by polling more people. Not really... It's just a mathematical model.

Similarly, a (classical) single particle in space must be specified with three reals. If you have n particles, you need three variables each. That doesn't change the playing field they are in, though, since each particle lives in a 3D world, not a 3n-D world.

 

[strangerep]

[...] Is there a FORMAL (not purely verbal!) way to distinguish the configuration
space from the "physical" space?[...]

I've resisted the temptation to attempt an answer in case this was one of your
rhetorical questions/puzzles for which you already have an answer. But the
thread has become idle without any resolution so I'll risk making a fool of myself...

Consider an idealization where an elementary system corresponds to a unirrep
of some dynamical Lie algebra. For brevity, let's say it's some sort of symplectic
algebra with Hamiltonian, etc, etc. Some of the algebra's basis elements
correspond to "position" or "configuration". Let's say there's 3 linearly independent
of these (i.e., considering the non-relativistic case).

Depending on the details of the algebra there'll be some Casimirs, and these together
with one other generator classify the possible representations and hence quantum
numbers. In a general dynamical algebra, the position generators and Hamiltonian
are likely participants in (some of) these Casimirs.

A single elementary system is a bit boring. We can find a canonical transformation
that puts it at rest (or some other canonical state, depending on the details of the
algebra). So let's consider a tensor product of 2 such systems and demand that it
also be a unirrep of the same dynamical algebra. The basic generators for each
system commute, but when we examine the quadratic and higher Casimirs for
the combined system we find various constraints on how two systems can
tensor together to get another valid unirrep. (This is reverse-analogous to the
way we get non-trivial Clebsh-Gordan coefficients when we analyze coupling
between two sets of angular momentum generators. The J^2 Casimir
makes the decomposition quite non-trivial.)

Now consider a tensor product of 3 elementary systems, #1,#2,#3, that we
require to be a unirrep of the same dynamical algebra. Things get very messy.
But in this case, system #1 can have a "physical space" (i.e., a subset of generators)
in which the interaction and behaviour of the #2 \otimes #3 cluster can be described.
Similarly, each of the other two can have their own "physical spaces". But in
general the three physical spaces do not coincide (cf. the Unruh effect and Rindler
wedges, etc, in accelerating situations).

But you wanted a more rigorous way to distinguish physical space from
configuration space. So I suggest the generators of physical space
corresponds to the sum of all the position generators of the subsystems,
and the "configuration" aspect of the rest of the dynamical behaviour corresponds
to differences between generators of all the different clusters one can construct
that decompose the whole system. E.g., for the 3-subsystem case the (canonical)
physical space corresponds to
<br /> X_{PHYS} ~:=~ X_{(1)} + X_{(2)} + X_{(3)}<br />
where the X's represent vector quantities.

The various other combinations, e.g., X_{(1)} - (X_{(2)} + X_{(3)}),
then describe configuration aspect(s) of the relative dynamics.

Such a description is not unique, of course. In general, an ideal observer is one
of the subsystems and defines an observer-centric "physical space" via interactions
with other subsystems (e.g., radar method). But I presume you wanted something
more akin to the spacetime background used in relativity.

So, (now that I've possibly exposed myself to a public spanking), what is
your answer to the puzzle?

 

[Demystifier]

Strangerep, this time I really do not have my answer.

Concerning your attempt, it is certainly the most serious one so far. Yet, I feel that it is not really satisfying to me, though I need more time to understand why.

 

[dx]

Can't you rule out the case of two particles in one dimension by noticing that there is no interaction term in the Hamiltonian?

 

[Demystifier]

Can't you rule out the case of two particles in one dimension by noticing that there is no interaction term in the Hamiltonian?

No. Particles may be able to travel through each other without a recoil or any other interaction.

 

[dx]

Maybe classical relativistic mechanics can give some insight into this. Influences travel through "physical space" at the speed of light, and not necessarily through configuration space. If we have two configuration space coordinates, and the local dynamical evolution of one coordinate depends on the retarded dynamical evolution of the other, then I think we can tell that these are coordinates of two different particles rather than two coordinates of a single particle. So I'm guessing we can see the physical space in the configuration space by analyzing how the coordinates influence each other and the general causal structure generated by the Hamiltonian.

 

[hellfire]

May be it is meaningful to ask first about the relativistic theory. Shouldn't be a constraint between the components of the linear momentum in case that you are talking about one single particle?

 

[jostpuur]

IMO the key is in the interaction terms, and in some kind of resulting "effective dimension". For example, suppose system is described by a following Lagrange's function

<br /> L:\mathbb{R}^{3N}\times\mathbb{R}^{3N}\to\mathbb{R}<br />

<br /> L(x,\dot{x}) = \sum_{k=0}^{N-1} \frac{1}{2}m_{3k}\big(\dot{x}_{3k}^2 + \dot{x}_{3k+1}^2 + \dot{x}_{3k+2}^2\big) -\underset{k&lt;l}{\sum_{k,l=0}^{N-1}} K_{k,l}\big((x_{3k}-x_{3l})^2 + (x_{3k+1}-x_{3l+1})^2 + (x_{3k+2}-x_{3l+2})^2\big)^{\alpha_{k,l}}<br />

In the end, there is no precise way of telling if this should be a one particle in a 3N-dimensional space, N particles in a 3-dimensional space, or 3N particles in one dimension. However, from the form of the Lagrangian one sees that clearly the interpretation of 3 dimensions is somehow favored.

IMO the same effect occurs in the reality. There is no fundamental answer to a question whether our universe contains extremely large number of particles in 3 dimension, or some small number of particles in an extremely large dimensional space. It is form of the interactions which make the universe appear as if 3 dimensional.

 

[dx]

A more precise way of saying what I said above would be: The configuration space coordinates q_1 and q_2 are coordinates of the same particle if they have the same causal past in the sense of special relativity.

 

[jostpuur]

I just realized that entanglement actually seems to favor the interpretation of extremely large amount of dimensions, in a sense. (edit: hmhmh... but was this what Demystifier already explained in the opening post... it mentions locality, not entanglement, but perhaps this is the same stuff...)

Consider a wave function \psi:\mathbb{R}^2\to\mathbb{C} describing single particle in a two dimensions. It is not mysterious at all, that coordinates x_1 and x_2 are correlated in the amplitudes. However, consider a wave function \psi:\mathbb{R}^2\to\mathbb{C} describing two non-interacting particles in one dimension. If \psi does not separate into product of x_1 and x_2 two depending parts, we get mysterious entanglement, where measurement of the position of one particle affects the position of the other one.

The entanglement start appearing less mysterious when dimensions are increased and number of particles decreased.

 

[Demystifier]

I just realized that entanglement actually seems to favor the interpretation of extremely large amount of dimensions, in a sense. (edit: hmhmh... but was this what Demystifier already explained in the opening post... it mentions locality, not entanglement, but perhaps this is the same stuff...)

Consider a wave function \psi:\mathbb{R}^2\to\mathbb{C} describing single particle in a two dimensions. It is not mysterious at all, that coordinates x_1 and x_2 are correlated in the amplitudes. However, consider a wave function \psi:\mathbb{R}^2\to\mathbb{C} describing two non-interacting particles in one dimension. If \psi does not separate into product of x_1 and x_2 two depending parts, we get mysterious entanglement, where measurement of the position of one particle affects the position of the other one.

The entanglement start appearing less mysterious when dimensions are increased and number of particles decreased.

Yes, that was my original motivation too.

 

[atyy]

This is a question of biology not physics.

I agree.

 

[Demystifier]

Maybe classical relativistic mechanics can give some insight into this. Influences travel through "physical space" at the speed of light, and not necessarily through configuration space. If we have two configuration space coordinates, and the local dynamical evolution of one coordinate depends on the retarded dynamical evolution of the other, then I think we can tell that these are coordinates of two different particles rather than two coordinates of a single particle. So I'm guessing we can see the physical space in the configuration space by analyzing how the coordinates influence each other and the general causal structure generated by the Hamiltonian.

I don't think that relativity is essential, because we see in our everyday lives what the "physical" space is, without being aware of relativistic effects. Still, your idea leads me to another idea: That the difference between two spaces cannot be seen on the level of point-particles, but only on the level of fields. For example, the field configuration describing one point-particle in two dimensions is
\delta(x_1-y_1(t))\delta(x_2-y_2(t))
while that describing two point-particles in one dimension is
\delta(x-y_1(t)) + \delta(x-y_2(t))
Both are defined by two functions y_1(t) and y_2(t), and yet they look mathematically different. One is the product of two delta functions, while the other is a sum of two delta functions.

But there is also a problem with that. The many-particle wave function is mathematically a field in 3n dimensions, which would imply that physical space of QM is indeed 3n-dimensional, making QM local in the physical space. That is good, but why then we still see the universe as if only 3 dimensions were really physical? This may be related to the fact that quantum effects are not visible on the macroscopic level, which suggests that it is the phenomenon of decoherence that is responsible for the illusion of 3 dimensions in our everyday lives. But then again, how can we see from our fundamental equations that on the macroscopic level the universe will appear as if it had precisely 3 dimensions? And what exactly these fundamental equations are?

 

[Demystifier]

This is a question of biology not physics.

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.

 

[hellfire]

Consider the lagrangian of a free relativistic particle \mathcal L = m \sqrt{- \dot q^{\mu} \dot q_{\mu}}. How does the problem arise in such a case?

 

[pellman]

As a simple example, consider the following Hamiltonian:
H=\frac{p_1^2}{2m}+\frac{p_2^2}{2m}
What does this Hamiltonian describe? Is it one free particle moving in two dimensions, or two free particles moving in one dimension? Clearly, this Hamiltonian describes both. But then, how to distinguish between these two physically different cases? Is there a FORMAL (not purely verbal!) way to distinguish the configuration space from the "physical" space?

I don't quite see the problem. Isn't this simply a case of noting that math is not reality, rather it models reality? and that the same math can model different physical systems? Hence you can't expect to find the physical system identifiable in the math.

How does this question differ from the questions,

Does the Lagrangian L=\frac{1}{2}m\dot{q}^2-\frac{1}{2}kq^2 describe a bouncing spring or an electric circuit?

Does 3+4=7 describe what happens when you put 3 apples with 4 apples or what happens when you put 3 oranges with 4 oranges?

 

[Demystifier]

Consider the lagrangian of a free relativistic particle \mathcal L = m \sqrt{- \dot q^{\mu} \dot q_{\mu}}. How does the problem arise in such a case?

Let me see!
The action is
m \int d\tau \sqrt{- \dot q^{\mu} \dot q_{\mu}}
If this action is all we know about the system, then we do NOT know that the action is also equal to
m \int d\tau
In other words, we do NOT know that \tau is equal to the proper time and that solutions of the equations of motion should satisfy
- \dot q^{\mu} \dot q_{\mu}=1
Instead, \tau is just an auxiliary affine parameter without an explicit physical interpretation. This, by itself, is not yet a problem.
However, now consider a particular solution of the equation of motion
q^0=\tau, \; q^1=\tau, \; q^2=\tau, \; q^3=\tau
This solution may well be interpreted as 4 NON-relativistic particles moving in 1 spatial dimension, and \tau may well be interpreted as an independent scalar parameter, such as the Newton time.

Perhaps you want to add an additional constraint that removes such unphysical solutions. Depending on how exactly you enforce such a constraint, I may discuss what, if any, problem will remain.

 

[Demystifier]

I don't quite see the problem. Isn't this simply a case of noting that math is not reality, rather it models reality? and that the same math can model different physical systems? Hence you can't expect to find the physical system identifiable in the math.

How does this question differ from the questions,

Does the Lagrangian L=\frac{1}{2}m\dot{q}^2-\frac{1}{2}kq^2 describe a bouncing spring or an electric circuit?

Does 3+4=7 describe what happens when you put 3 apples with 4 apples or what happens when you put 3 oranges with 4 oranges?

This is an excellent point!
Still, if you see the quantum motivation that I explained in the first post, you might see that it is not that trivial. The quantum formalism may also be interpreted as a local theory in 3n dimensions. Such an interpretation may be a solution of the nonlocality problem related to the entanglement in QM, which suggests that such an interpretation could be more than just a play with mathematical symbols. There could be something PHYSICAL about such an interpretation.
So basically, I want to solve the problem which, as you say, is not really a problem at all, because this solution may help me to solve another, more serious problem (the problem of non-locality in QM).

 

[pellman]

I can't help answer it yet myself.

But I can't shake the feeling that is related to this question I posted a couple of days ago https://www.physicsforums.com/showthread.php?t=285051

Here I was posing a question about how a relativistic N-body problem need 4N coordinates (or does it? that's part of the question) with clock for each particle, rather than 3N+1. DaleSpam's answer referred to how one slices up the manifold.

The problem, in my mind, was that the dynamical equations --in terms of coordinates-- doesn't refer to the manifold. There is no information about the manifold at all in the equations.

I'll be watching this thread intently.

 

[Demystifier]

I can't help answer it yet myself.

But I can't shake the feeling that is related to this question I posted a couple of days ago https://www.physicsforums.com/showthread.php?t=285051

Here I was posing a question about how a relativistic N-body problem need 4N coordinates (or does it? that's part of the question) with clock for each particle, rather than 3N+1. DaleSpam's answer referred to how one slices up the manifold.

The problem, in my mind, was that the dynamical equations --in terms of coordinates-- doesn't refer to the manifold. There is no information about the manifold at all in the equations.

I'll be watching this thread intently.

That's interesting too. I think you need 4n coordinates in order to maintain the manifest Lorentz covariance. Indeed, it is just a special case of the general rule that if you want to maintain a manifest symmetry, then you need to introduce some redundant degrees of freedom. Gauge fixing removes the redundancy, but also the manifest symmetry.

By the way, you may find interesting to see that recently I have used a 4n-formalism to show that nonlocality of QM is compatible with Lorentz invariance, even if explicitly nonlocal Bohmian hidden variables are involved:
http://xxx.lanl.gov/abs/0811.1905 [accepted for publication in Int. J. Quantum Inf.]

 

[hellfire]

Thanks for your answer to my question, it clarified more to me what you have in mind. I actually was thinking of that lagrangian with the constraint - \dot q^{\mu} \dot q_{\mu}=1 and I still do not see how the problem arises (that condition removes the 4 particle interpretation as you said). I feel this will not answer your question but I would like to understand it.

 

[Demystifier]

Thanks for your answer to my question, it clarified more to me what you have in mind. I actually was thinking of that lagrangian with the constraint - \dot q^{\mu} \dot q_{\mu}=1 and I still do not see how the problem arises (that condition removes the 4 particle interpretation as you said). I feel this will not answer your question but I would like to understand it.

Fine! With this additional constraint you can remove one unphysical degree of freedom, q^0(\tau). After some straightforward manipulations, what remains is an action
for the physical degrees of freedom
q^1(t), \; q^2(t), \; q^3(t)
where t\equiv q^0. The explicit form of this action is not relevant here, it is sufficient to know that it is symmetric under the exchange of
q^1, \; q^2, \; q^3
and that the action is no longer manifestly Lorentz invariant.
But this is completely analogous to the nonrelativistic case. Nothing forbids you to interpret this action as describing 3 particles in 1 spatial dimension.

 

[Haelfix]

I too fail to see the problem or what you are getting at. What differentiates the configuration space in the original OP between the two scenarios? Nothing! But it maps to a completely different physical space (in your case, even a different dimension of the target manifold).

Back in the definitions of mathematical Quantum mechanics, you always have to specify that map. The configuration space itself (eg the cotangent bundle of the classical phase space) can be seen as the space of maps between a parameter space to a target space. Without specifying all three things, you don't have a physical system yet by definition.

Seen in this light, just because one object is the same between two different problems, doesn't really mean anything and generically happens all the time.

 

[pellman]

I think what makes the distinction (possibly) different in the quantum case is that the quantum wave function lives on the configuration space itself. In classical problems, relativistic or not, physical solutions are the particle paths x(t) or fields E(x), which are necessarily mapped back to some physical space. The ambiguity that the same math may apply to different physical systems is still there, but x(t) and E(x) necessarily refer to back to some such physical space. Configuration space is the more artifical construct.

The quantum state doesn't get mapped back to a physical space at all. It lives on configuration space, period. Yet we are supposed to look on it as the ultimate physical "thing" underlying the classical world?

 

[Demystifier]

The quantum state doesn't get mapped back to a physical space at all. It lives on configuration space, period. Yet we are supposed to look on it as the ultimate physical "thing" underlying the classical world?

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?

 

[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?