Is the Hamilton-Jacobi Formalism a Classical Analogue to Quantum Wave Functions?

[fanieh]

First of all. I’m aware that the space of all wave functions is a Hilbert space. And states are not waves - but elements of a Hilbert space. When expanded in terms of eigenfunctions of position the coefficients can sometimes look like waves (only sometimes) but they are not waves - they are elements of a Hilbert space.

In the 1930s, John von Neumann consolidated ideas from Bohr, Heisenberg and Schrodinger and placed the new quantum theory in Hilbert space.

In Hilbert space, a vector represents the Schrodinger wave function.
I know they are equivalent..

Furthermore… When describing more than one particle in the Schrodinger wave function.. the wave no longer occur in 3D space, but in higher dimensional mathematical space.

Now the Hamiltonian, I’m aware that The Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system. For different situations or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation.

Now let’s go to classical Hamilton-Jacobi equation. This is used by Bohmian Mechanics to
represent the wave function. But first let’s go to classical Hamilton-Jacobi formalism.

http://www.jh-inst.cas.cz/~kapralova/QUANTCLASS/goldstein.html

“Usually, the classical mechanics is understood in terms of trajectories, that is in terms of delta functions in the phase space. However, the Hamilton- Jacobi formalism of classical mechanics allows us to understand even the purely classical mechanics in terms of moving wavefronts in space.”

How do you understand the classical Hamilton- Jacobi formalism. Is it trying to model all the wave function in higher dimensional mathematical space as waves in 3D space?? How does it differ to the pure Hamiltonian formalism?

 

[vanhees71]

First of all the Hamilton-Jacobi partial differential equation has nothing to do with quantum theory. It is purely classical and a powerful and elegant method to solve classical equations of motion of Hamiltonian systems. The idea is to find a canonical transformation such that for a system with $f$ configuration-space degrees of freedom (i.e., $2f$ phase-space degrees of freedom) you get $f$ conserved quantities, which serve as the new canonical momenta. Then the Hamiltonian is cylcic for all the corresponding new configuration-space variables, and the problem is solved. This set of special phase-space variables are called "action-angle variables". Of course this only works, if the system has $f$ conserved quantities, i.e., if the system is integrable having $f$ first integrals. It turns out that the generating function for the canonical transformation is the action.

The relation to QT can be understood in orthodox minimally interpreted QT. You don't need the de Broglie-Bohm pilot wave approach. There are (at least) to ways to see this relation. The conventional one is to take the Schrödinger equation of the problem at hand and solve it in singular perturbation theory, which is also known as (S)WKB method, standing for (Sommerfeld)-Wentzel-Kramers-Brillouin method. The idea is to make an ansatz
$$\psi(t,q)=\exp \left (\frac{\mathrm{i}}{\hbar} S(t,q) \right)$$
and then do an expansion in powers of $\hbar$. In leading order you get the Hamilton-Jacobi partial differential equation for $S$, and $S$ becomes the action of the corresponding classical mechanical problem.

The other way is much more elegant and makes the physics behind this approximation much more clear: You start from the Feynman path integral for the propagator of the quantum system. Then the classical action appears from the very beginning, and you evaluate at path integral using the saddle-point approximation, valid if the classical action in the exponential of the integral is typically much larger than $\hbar$. Then in the integral over all trajectories in phase space (or in configuration space if you can integrate out the momentum-path integral first as is often the case for non-relativistic QT) almost all cancel out against each other, because the exponential function is rapidly oscillating, only the trajectories close around the one, where the action becomes stationary contributes, and this shows that the WKB method is indeed an approximation "around the classical solution" of the mechanical problem at hand.

 

[fanieh]

First of all the Hamilton-Jacobi partial differential equation has nothing to do with quantum theory. It is purely classical and a powerful and elegant method to solve classical equations of motion of Hamiltonian systems. The idea is to find a canonical transformation such that for a system with $f$ configuration-space degrees of freedom (i.e., $2f$ phase-space degrees of freedom) you get $f$ conserved quantities, which serve as the new canonical momenta. Then the Hamiltonian is cylcic for all the corresponding new configuration-space variables, and the problem is solved. This set of special phase-space variables are called "action-angle variables". Of course this only works, if the system has $f$ conserved quantities, i.e., if the system is integrable having $f$ first integrals. It turns out that the generating function for the canonical transformation is the action.

The relation to QT can be understood in orthodox minimally interpreted QT. You don't need the de Broglie-Bohm pilot wave approach. There are (at least) to ways to see this relation. The conventional one is to take the Schrödinger equation of the problem at hand and solve it in singular perturbation theory, which is also known as (S)WKB method, standing for (Sommerfeld)-Wentzel-Kramers-Brillouin method. The idea is to make an ansatz
$$\psi(t,q)=\exp \left (\mathrm{i}{\hbar} S(t,q) \right)$$
and then do an expansion in powers of $\hbar$. In leading order you get the Hamilton-Jacobi partial differential equation for $S$, and $S$ becomes the action of the corresponding classical mechanical problem.

The other way is much more elegant and makes the physics behind this approximation much more clear: You start from the Feynman path integral for the propagator of the quantum system. Then the classical action appears from the very beginning, and you evaluate at path integral using the saddle-point approximation, valid if the classical action in the exponential of the integral is typically much larger than $\hbar$. Then in the integral over all trajectories in phase space (or in configuration space if you can integrate out the momentum-path integral first as is often the case for non-relativistic QT) almost all cancel out against each other, because the exponential function is rapidly oscillating, only the trajectories close around the one, where the action becomes stationary contributes, and this shows that the WKB method is indeed an approximation "around the classical solution" of the mechanical problem at hand.

Can you kindly share any example of a classical system where this Hamilton-Jacobi equation is used versus the plain Hamiltonian? So the Hamilton-Jacobi still retain the phase space of the Hamiltonian? This means the Hamilton-Jacabi can still be formulated into Hilbert Space quantum language? Because it looks Hamilton-Jacobi is attempt to make the vectors in Hilbert space become like 3d wave only? please use actual application and not on the derivations.. thank you for your assistance!

 

[fanieh]

First of all the Hamilton-Jacobi partial differential equation has nothing to do with quantum theory. It is purely classical and a powerful and elegant method to solve classical equations of motion of Hamiltonian systems. The idea is to find a canonical transformation such that for a system with $f$ configuration-space degrees of freedom (i.e., $2f$ phase-space degrees of freedom) you get $f$ conserved quantities, which serve as the new canonical momenta. Then the Hamiltonian is cylcic for all the corresponding new configuration-space variables, and the problem is solved. This set of special phase-space variables are called "action-angle variables". Of course this only works, if the system has $f$ conserved quantities, i.e., if the system is integrable having $f$ first integrals. It turns out that the generating function for the canonical transformation is the action.

To the Bohmians like Demystifier.. please share even in few words what would happen if Bohmian Mechanics doesn't use the Hamilton-Jacobi equations. Can't you model BM anymore? Is there no alternative equation?

What exactly is the function of the Hamilton-Jacobi equation in BM? Is it to replace the Hilbert Space.. or produce quantum potential to control the particle?

Thank you!

The relation to QT can be understood in orthodox minimally interpreted QT. You don't need the de Broglie-Bohm pilot wave approach. There are (at least) to ways to see this relation. The conventional one is to take the Schrödinger equation of the problem at hand and solve it in singular perturbation theory, which is also known as (S)WKB method, standing for (Sommerfeld)-Wentzel-Kramers-Brillouin method. The idea is to make an ansatz
$$\psi(t,q)=\exp \left (\frac{\mathrm{i}}{\hbar} S(t,q) \right)$$
and then do an expansion in powers of $\hbar$. In leading order you get the Hamilton-Jacobi partial differential equation for $S$, and $S$ becomes the action of the corresponding classical mechanical problem.

The other way is much more elegant and makes the physics behind this approximation much more clear: You start from the Feynman path integral for the propagator of the quantum system. Then the classical action appears from the very beginning, and you evaluate at path integral using the saddle-point approximation, valid if the classical action in the exponential of the integral is typically much larger than $\hbar$. Then in the integral over all trajectories in phase space (or in configuration space if you can integrate out the momentum-path integral first as is often the case for non-relativistic QT) almost all cancel out against each other, because the exponential function is rapidly oscillating, only the trajectories close around the one, where the action becomes stationary contributes, and this shows that the WKB method is indeed an approximation "around the classical solution" of the mechanical problem at hand.

 

[vanhees71]

Let's take the harmonic oscillator in one dimension. The Hamiltonian is
$$H=\frac{p^2}{2m} + \frac{m \omega^2}{2} q^2.$$
The idea is to find the generator $g(q,P,t)$ of a canonical transformation to new phase-space coordinates $(Q,P)$,
$$p=\frac{\partial g}{\partial q}, Q=\frac{\partial g}{\partial P}, \quad H'(Q,P,t)=H(q,p,t)+\partial_t g(q,P,t) \stackrel{!}{=}0,$$
leading to the Hamilton-Jacobi PDE,
$$H(q,\partial_q g,t)=0.$$
For our case we have
$$\frac{1}{2m} (\partial_q g)^2+\frac{m \omega^2}{2} q^2+\partial_t g=0.$$
Since here $H$ is not explicitly time dependent, we know that $H=E=\text{const}$ along the solutions of the equations of motion. Thus we can choose $P=H$, and the HJPDE becomes
$$\partial_t g=-E \; \Rightarrow \; g=-E t + S(q,E).$$
Putting this again into the HJPDE, you get a simple integral for $S$ (you only need one solution not the complete one!):
$$S(q,E)=\int \mathrm{d} q \sqrt{2mE-m^2 \omega^2 q^2}=\frac{q}{2} \sqrt{2mE-m \omega^2 q^2}+\frac{E}{\omega} \arcsin \left (\sqrt{\frac{m}{2E}} \omega q \right ).$$
Then you now that
$$Q=\text{const}=\frac{\partial S}{\partial E}=-t +\frac{1}{\omega} \arcsin \left (\sqrt{\frac{m}{2E}} \omega q \right).$$
solving for $q$ gives the expected harmonic solution
$$q=\sqrt{2 E}{\sqrt{m} \omega} \sin[\omega (t+Q)].$$
This is of course not very efficient. For the harmonic oscillator the direct method via the equations of motion is much simpler, but it illustrates how the HJPDE works.

 

[Demystifier]

To the Bohmians like Demystifier.. please share even in few words what would happen if Bohmian Mechanics doesn't use the Hamilton-Jacobi equations. Can't you model BM anymore?

Yes you can. You don't necessarily need to use HJ equation in BM, but it is useful to see the analogy with classical physics.

What exactly is the function of the Hamilton-Jacobi equation in BM? Is it to replace the Hilbert Space.. or produce quantum potential to control the particle?

HJ equation in BM gives you the phase of the wave function.

 

[fanieh]

Yes you can. You don't necessarily need to use HJ equation in BM, but it is useful to see the analogy with classical physics.HJ equation in BM gives you the phase of the wave function.

I'm goggling and searching for your papers about "Hamilton-Jacobi".. and I read this http://xxx.lanl.gov/pdf/quant-ph/0609163v2: (p5)

"This interpretation consists of two equations. One is the standard Schrodinger equation that describes the wave-aspect of the theory, while the other is a classical-like equation that describes a particle trajectory. The equation for the trajectory is such that the force on the particle depends on the wave function, so that the motion of the particle differs from that in classical physics, which, in turn, can be used to explain all (otherwise strange) quantum phenomena. In this interpretation, both the wave function and the particle position are fundamental entities. If any known interpretation of QM respects a kind of wave-particle duality, then it is the Bohmian interpretation. More on this interpretation (which also provides a counterexample to some other myths of QM) will be presented in subsequent sections."

Do you have a paper or know of one where Bohmian Mechanics is formulated in Hilbert Space and would this still use the Hamilton-Jacobi equation?

Hilbert Space is natural if superposition really exists. But in Bohmian Mechanics. Particle is not really in superposition, only one path or branch is chosen always.. does this mean Hilbert Space is not natural for BM?

Because for PBR Theorem. One consequence of Hilbert Space was if wave function or state vector was real there are higher dimensions not far from that of string theory to house an actual Hilbert Space as atyy suggested but if BM doesn't like Hilbert Space then it means the ontology doesn't need higher dimensions but 3D (like suggested in https://arxiv.org/pdf/1707.08508.pdf shared in another thread) and enough to create the source of the Quantum Potential in BM.

 

[fanieh]

I'm goggling and searching for your papers about "Hamilton-Jacobi".. and I read this http://xxx.lanl.gov/pdf/quant-ph/0609163v2: (p5)

"This interpretation consists of two equations. One is the standard Schrodinger equation that describes the wave-aspect of the theory, while the other is a classical-like equation that describes a particle trajectory. The equation for the trajectory is such that the force on the particle depends on the wave function, so that the motion of the particle differs from that in classical physics, which, in turn, can be used to explain all (otherwise strange) quantum phenomena. In this interpretation, both the wave function and the particle position are fundamental entities. If any known interpretation of QM respects a kind of wave-particle duality, then it is the Bohmian interpretation. More on this interpretation (which also provides a counterexample to some other myths of QM) will be presented in subsequent sections."

Do you have a paper or know of one where Bohmian Mechanics is formulated in Hilbert Space and would this still use the Hamilton-Jacobi equation?

Hilbert Space is natural if superposition really exists. But in Bohmian Mechanics. Particle is not really in superposition, only one path or branch is chosen always.. does this mean Hilbert Space is not natural for BM?

Because for PBR Theorem. One consequence of Hilbert Space was if wave function or state vector was real there are higher dimensions not far from that of string theory to house an actual Hilbert Space as atyy suggested but if BM doesn't like Hilbert Space then it means the ontology doesn't need higher dimensions but 3D (like suggested in https://arxiv.org/pdf/1707.08508.pdf shared in another thread) and enough to create the source of the Quantum Potential in BM.

Demystifier. In case you'd say dBB can also be formulated in Hilbert Space because it is equivalent. I want to get direct what I wanted to know. In another Hamilton-Jacobi thread Denis commented:

"DBB theory does not have an answer to your question. In dBB theory, the wave function describes some really existing field, that's all. A physical description of the nature of this field will be the job of some more fundamental theory."

Reference https://www.physicsforums.com/threa...nature-of-the-pilot-wave.911268/#post-5740087

I want to get idea what is this existing field described by the wave function in dBB or explore this fundamental theory. It should pull of a Hilbert Space stunt such that the electron is simultaneously in all location or a ray in Hilbert Space. Short of suggesting Hilbert Space is real in some actual higher dimensions. Is there a 3D field that can also described the Hilbert Space. I don't want to think yet of the fundamental theory to go to higher dimensions just to support Hilbert Space or describe the field used by the wave function in dBB.. just want to start thinking in terms of lower dimensions.

 

[Demystifier]

Do you have a paper or know of one where Bohmian Mechanics is formulated in Hilbert Space and would this still use the Hamilton-Jacobi equation?

I would say that those are two complementary views of the same thing. If you write the equations such that Hilbert space is manifest, then the HJ equation is not manifest. Likewise, if you write the equations such that HJ equation is manifest, then Hilbert space is not manifest. You cannot see both at once. Something like the duck-or-rabbit illusion:
http://www.independent.co.uk/news/s...-tells-you-how-creative-you-are-a6873106.html

 

[Demystifier]

I don't want to think yet of the fundamental theory to go to higher dimensions just to support Hilbert Space or describe the field used by the wave function in dBB.. just want to start thinking in terms of lower dimensions.

Consider, for instance, classical mechanics of two particles moving in 3 dimensions, interacting by the Coulomb potential
$$V({\bf x}_1,{\bf x}_2)=\frac{const}{|{\bf x}_1-{\bf x}_2|}$$
The vectors ${\bf x}_1$ and ${\bf x}_2$ live in 3 dimensions each, but the potential lives in 3+3=6 dimensions. If you find it mysterious, then you should first spend more time on thinking about classical mechanics.

Similarly, in BM particles live in 3 dimensions but wave function and quantum potential live in more dimensions. Those extra dimensions are fully analogous to the extra dimensions in the classical case above.

 

[fanieh]

Consider, for instance, classical mechanics of two particles moving in 3 dimensions, interacting by the Coulomb potential
$$V({\bf x}_1,{\bf x}_2)=\frac{const}{|{\bf x}_1-{\bf x}_2|}$$
The vectors ${\bf x}_1$ and ${\bf x}_2$ live in 3 dimensions each, but the potential lives in 3+3=6 dimensions. If you find it mysterious, then you should first spend more time on thinking about classical mechanics.

Similarly, in BM particles live in 3 dimensions but wave function and quantum potential live in more dimensions. Those extra dimensions are fully analogous to the extra dimensions in the classical case above.

Yes I know what you mean. For example Hamiltonian phase space for position and momentum needs 6 dimensions but the particle in only in 3 dimensions. Thanks for making that clear.

About this unknown field used by dBB quantum potential produced by a more fundamental theory. I think it is possible quantum gravity can only occur by uniting not only spacetime and quantum field but another fundamental field. Do you know of a math where they are not exactly the math used by General Relativity and not exactly the math used by Quantum Field theory but super math where these two are emergent plus another one.. what is this super math be.. any idea? and I think i'd study it as maybe this is the key to all.

 

[Demystifier]

Do you know of a math where they are not exactly the math used by General Relativity and not exactly the math used by Quantum Field theory but super math where these two are emergent plus another one.. what is this super math be.. any idea? and I think i'd study it as maybe this is the key to all.

Causal set theory comes to my mind: https://en.wikipedia.org/wiki/Causal_sets

Or perhaps matrix theory: https://en.wikipedia.org/wiki/Matrix_theory_(physics)

 

[fanieh]

Consider, for instance, classical mechanics of two particles moving in 3 dimensions, interacting by the Coulomb potential
$$V({\bf x}_1,{\bf x}_2)=\frac{const}{|{\bf x}_1-{\bf x}_2|}$$
The vectors ${\bf x}_1$ and ${\bf x}_2$ live in 3 dimensions each, but the potential lives in 3+3=6 dimensions. If you find it mysterious, then you should first spend more time on thinking about classical mechanics.

Similarly, in BM particles live in 3 dimensions but wave function and quantum potential live in more dimensions. Those extra dimensions are fully analogous to the extra dimensions in the classical case above.

Demystifier, I found this old thread of yours written in 2009 about "Configuration space vs real space" https://www.physicsforums.com/threads/configuration-space-vs-physical-space.285019/

"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)."

This is related to my questions lately and in message above whether configuration space (3n space whatever that means) is the true physical where nonlocality is the norm. So what kind of spacetime can support configuration-space like properties where all are connected?

You wrote further in message 30 that:

"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?"

After 8 years of thinking about this. Please give update of what you think now. Especially if your have found a mathematical structure that can function as the configuration space as the real space where nonlocality is the norm and our 3-space just an illusion or better yet a limitation of our senses. Thank you.

 

[Demystifier]

After 8 years of thinking about this. Please give update of what you think now. Especially if your have found a mathematical structure that can function as the configuration space as the real space where nonlocality is the norm and our 3-space just an illusion or better yet a limitation of our senses. Thank you.

Currently I think that ordinary 3-dimensional space is probably more fundamental, but I am not completely satisfied with this opinion and I am open for new insights.

 

[fanieh]

Currently I think that ordinary 3-dimensional space is probably more fundamental, but I am not completely satisfied with this opinion and I am open for new insights.

Ok. I need some confirmation of something in this thread. Earlier I wrote that denis mentioned in page 5 of https://www.physicsforums.com/threa...nature-of-the-pilot-wave.911268/#post-5740087:

"DBB theory does not have an answer to your question. In dBB theory, the wave function describes some really existing field, that's all. A physical description of the nature of this field will be the job of some more fundamental theory."

Do you also believe it? Is it true that our Bohmian Mechanics is independent from whatever is this more fundamental existing field?
Second. How does this relate to spacetime of General Relativity. If there is a more fundamental existing field underlying BM. Would it make spacetime as emergent.. or is this more fundamental existing field akin to higgs field which is still inside spacetime?

 

[Demystifier]

Ok. I need some confirmation of something in this thread. Earlier I wrote that denis mentioned in page 5 of https://www.physicsforums.com/threa...nature-of-the-pilot-wave.911268/#post-5740087:

"DBB theory does not have an answer to your question. In dBB theory, the wave function describes some really existing field, that's all. A physical description of the nature of this field will be the job of some more fundamental theory."

Do you also believe it? Is it true that our Bohmian Mechanics is independent from whatever is this more fundamental existing field?

I don't have a simple yes/no answer to that question. Let's just say I am open minded on that.

Second. How does this relate to spacetime of General Relativity. If there is a more fundamental existing field underlying BM. Would it make spacetime as emergent.. or is this more fundamental existing field akin to higgs field which is still inside spacetime?

Classical spacetime is emergent.

 

[fanieh]

I don't have a simple yes/no answer to that question. Let's just say I am open minded on that.

You mean to say just like the 3 fundamental forces of nature is result of gauge symmetry where it is based on pure math without any physical cause.. then it is possible the wave function in Bohmian Mechanics is also due to some pure math where they is no underlying physical cause or field (described by Denis).. is this what you meant? Without clarifying I have no idea what you mean and what is the right way of thinking about it. Thank you.

Classical spacetime is emergent.

 

[Demystifier]

You mean to say just like the 3 fundamental forces of nature is result of gauge symmetry where it is based on pure math without any physical cause.. then it is possible the wave function in Bohmian Mechanics is also due to some pure math where they is no underlying physical cause or field (described by Denis).. is this what you meant?

Sort of.

 

[fanieh]

Yes you can. You don't necessarily need to use HJ equation in BM, but it is useful to see the analogy with classical physics.HJ equation in BM gives you the phase of the wave function.

In General. Is the Hamilton-Jacobi equation in BM giving the phase of the wave function has to do with the trajectories or the quantum potential? I mean is the "phase" has more to do with the trajectories or quantum potential (or others)?

I was reading this paper on the Hamilton-Jacobi equation: https://arxiv.org/abs/quant-ph/0612217

"In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schrodinger point of view and the HamiltonJacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result of this, a wave function in Hamilton-Jacobi theory can be decomposed into traveling waves in any point in space, not only asymptotically."

 

[Demystifier]

In General. Is the Hamilton-Jacobi equation in BM giving the phase of the wave function has to do with the trajectories or the quantum potential? I mean is the "phase" has more to do with the trajectories or quantum potential (or others)?

The phase has more to do with the trajectories than with the quantum potential. The quantum potential is related to the other "part" of the wave function, namely it's absolute value.

 

[fanieh]

I don't have a simple yes/no answer to that question. Let's just say I am open minded on that.Classical spacetime is emergent.

For two entangled particle to be correlated at millions of light years away. Spacetime with c limit can't do this. So at present what you think cause the correlations? Is it because of something more fundamental than Spacetime? What is it? Or some force of nature that is non-local?

Speaking of force of nature. Quantum field theory gauge theory and lie group symmetry explains the forces of nature as from arising from need to fulfill internal symmetry such as the phase symmetry giving rise to electromagnetic wave. Could there be non-local forces that doesn't come from symmetry? Should all the forces of nature be defined from the requirement of gauge symmetry? What you think?

 

[vanhees71]

Classical spacetime is emergent.

In which sense do you mean this? In standard QT (non-relativistic as well as relativistic QFT) spacetime is described classically. It's just the good old "Galilei-Newton fibre bundle" or the "Einstein-Minkowski pseudo-Euclidean affine manifold", respectively. Spacetime itself is not quantized!

 

[Demystifier]

For two entangled particle to be correlated at millions of light years away. Spacetime with c limit can't do this. So at present what you think cause the correlations? Is it because of something more fundamental than Spacetime? What is it? Or some force of nature that is non-local?

According to BM, it is entangled wave function (related to nonlocal quantum potential) that causes the correlations. The entangled wave function is caused by Schrodinger equation. The Schrodinger equation is caused by ... well, the chain must stop somewhere, doesn't it?

Speaking of force of nature. Quantum field theory gauge theory and lie group symmetry explains the forces of nature as from arising from need to fulfill internal symmetry such as the phase symmetry giving rise to electromagnetic wave. Could there be non-local forces that doesn't come from symmetry? Should all the forces of nature be defined from the requirement of gauge symmetry? What you think?

I think that some forces may not be related to gauge symmetry or any symmetry at all.

 

[Demystifier]

In which sense do you mean this? In standard QT (non-relativistic as well as relativistic QFT) spacetime is described classically. It's just the good old "Galilei-Newton fibre bundle" or the "Einstein-Minkowski pseudo-Euclidean affine manifold", respectively. Spacetime itself is not quantized!

I mean the curved metric tensor of classical general relativity should ultimately emerge from some quantum theory.

 

[vanhees71]

Ah, ok. Of course, this makes sense. Unfortunately you have to write "should". We have no complete QT of gravitation yet, and given the close connection between gravitation and spacetime structure in the classical domain one can expect to get a quantum theory of spacetime whenever one has found a satisfactory quantum theory of the gravitational interaction.

 

[fanieh]

The phase has more to do with the trajectories than with the quantum potential. The quantum potential is related to the other "part" of the wave function, namely it's absolute value.

In this paper http://xxx.lanl.gov/pdf/quant-ph/0609163v2 page 5, you mentioned "This interpretation consists of two equations.." so the two equations are the Schrodinger Equation and Hamilton-Jacobi Equation, right? You didn't say it directly so trying to confirm. Also if the phase is related to the Hamilton-Jacobi Equation. Is the quantum potential also belong to the HJ equation or Schrodinger Equation? You wrote in page 9:

"This assumption represents the core of the Bohmian deterministic interpretation of QM. To see the most obvious consequence of such a classical-like interpretation of the Schroedinger equation, note that the Schroedinger equation (20) corresponds to a Hamilton Jacobi equation in which V in (16) is replaced by V + Q. This is why Q is often referred to as the quantum potential. The quantum potential induces a quantum force. Thus, a quantum particle trajectory satisfies a modified Newton equation"

About this Schroedinger Equation and Hamilton-Jacobi Equation. When you are working in Bohmian Mechanics. Do you use them separately and combine the two equations.. or do you use just one equation only either (since they are equivalent?) Thanks for your help!

 

[Demystifier]

In this paper http://xxx.lanl.gov/pdf/quant-ph/0609163v2 page 5, you mentioned "This interpretation consists of two equations.." so the two equations are the Schrodinger Equation and Hamilton-Jacobi Equation, right?

Not exactly. One is the Schrodinger equation, the other is Eq. (15).

Is the quantum potential also belong to the HJ equation or Schrodinger Equation?

Quantum potential belongs to quantum Hamilton-Jacobi equation, which is derived from Schrodinger equation.

About this Schroedinger Equation and Hamilton-Jacobi Equation. When you are working in Bohmian Mechanics. Do you use them separately and combine the two equations.. or do you use just one equation only either (since they are equivalent?) Thanks for your help!

It depends on the context, but I usually use a single Schrodinger equation.

 

[fanieh]

Not exactly. One is the Schrodinger equation, the other is Eq. (15).Quantum potential belongs to quantum Hamilton-Jacobi equation, which is derived from Schrodinger equation.

In the Equations.. can the Quantum potential feedback to the wave function or can the quantum potential influence the wave function? Why not as this seems to be the case. Is it hard to change the equations to make the quantum potential affect the wave function? The hidden variable in BM is the non-local quantum potential. So if the hidden variable can influence the wave function.. then PBR theorem can apply to BM..

It depends on the context, but I usually use a single Schrodinger equation.

 

[Demystifier]

In the Equations.. can the Quantum potential feedback to the wave function or can the quantum potential influence the wave function? Why not as this seems to be the case. Is it hard to change the equations to make the quantum potential affect the wave function? The hidden variable in BM is the non-local quantum potential. So if the hidden variable can influence the wave function.. then PBR theorem can apply to BM..

Do you actually read equations, or do you just read the english? Quantum potential is uniquely determined by the wave function. Quantum potential is not a hidden variable. Your questions do not make much sense.

 

[fanieh]

Do you actually read equations, or do you just read the english? Quantum potential is uniquely determined by the wave function. Quantum potential is not a hidden variable. Your questions do not make much sense.

Oh.. I read this wiki entry on "Hidden variable theory"... https://en.wikipedia.org/wiki/Hidden_variable_theory

"In Bohm's interpretation, the (nonlocal) quantum potential constitutes an implicate (hidden) order which organizes a particle, and which may itself be the result of yet a further implicate order: a superimplicate order which organizes a field.[20]."

I thought the above described the nonlocal quantum potential as a hidden variable.. so there are two meanings of "hidden" in BM.. "hidden" as in hidden variable... and the second "hidden" as in Implicate (hidden) order. I'll keep that in mind next time I read stuff. Thanks a lot for your help.

 

[vanhees71]

Wikipedia is amazingly good but not always reliable. For a good reason it's not accepted as a valid reference in scientific papers/discussions!

 

[fanieh]

Not exactly. One is the Schrodinger equation, the other is Eq. (15).Quantum potential belongs to quantum Hamilton-Jacobi equation, which is derived from Schrodinger equation.It depends on the context, but I usually use a single Schrodinger equation.

The S-function determine the trajectories of the BM particle.. and the trajectories can't affect the S-function.. is this the mathematical statement why you told name123 that the guiding wave of one particle couldn't affect the guiding wave of other particles?

About quantum potential. The S-function determine the quantum potential.. can't the quantum potential affect the S-function? and most importantly.. can't the quantum potential of one particle affect the quantum potential of other particles? Thanks.

 

[Demystifier]

The S-function determine the trajectories of the BM particle.. and the trajectories can't affect the S-function.. is this the mathematical statement why you told name123 that the guiding wave of one particle couldn't affect the guiding wave of other particles?

No.

About quantum potential. The S-function determine the quantum potential.. can't the quantum potential affect the S-function?

The S-function does not determine the quantum potential. But S-function affects quantum potential and is affected by it.

and most importantly.. can't the quantum potential of one particle affect the quantum potential of other particles?

When the particles are not entangled, no. When the particles are entangled, there are no separate quantum potentials for each particle.

 

[fanieh]

No.

In the context of the Hamilton-Jacobi equation.. what is the mathematical reason the guiding wave of one particle couldn't affect the guiding wave of other particles? I thought it was because the trajectories couldn't affect the S-function. Today I happen to be analyzing the work of a physicist with Ph.D. who is suggesting the pilot wave is connected to magnetic monopoles traveling faster than c in reciprocal space or Fourier transform of spacetime. Is there other official pilot wave researchers doing something akin to this. Anyway, your discussions with name123 is very relevant. Thanks.

The S-function does not determine the quantum potential. But S-function affects quantum potential and is affected by it.When the particles are not entangled, no. When the particles are entangled, there are no separate quantum potentials for each particle.

 

[vanhees71]

The entire thing is just splitting the Schrödinger equation in real and imaginary parts and making the ansatz
$$\psi=\rho \exp(\mathrm{i} S/\hbar),$$
leading naturally to a set of coupled partial differential equations for $\rho$ and $S$. This only makes the entire business of wavemechanics more complicated than necessary. As I said several times before, I don't see any merit in the dBB approach for the understanding of the philosophical meaning of quantum mechanics, and you should rather spend your time learning QT (which is impossible without the adequate mathematics) without such unnecessary philosophical detours.