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Another attempt to combine QM and relativity

  1. Apr 11, 2003 #1
    Classical Quantum Mechanics
    Abstract

    An analysis of quantum interference and Schroedinger's equation from a classical point of view suggests that an alternative topology to Minkowski space using hidden dimensions will allow a single relatively simple wave equation to have solutions representing multiple particles with all the features of quantum statistics and spin, as well as wave function collapse, but without the complexity of quantum field theory.
    ...
    http://brannenworks.com/Quant.html

    Looking forward to any comments or questions, good or bad.

    Carl
     
  2. jcsd
  3. Apr 11, 2003 #2

    dg

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    I haven't gone through the whole web page but for what I have seen so far I am happy to see someone else which has been wondering about the same issues and that is not happy with the current "understanding" of physical laws.

    Since you are still online let me ask a few questions:
    I agree with your point of view that the use of complex field hides physical meaning; my conclusion is that it is just a useful mathematical tool that can simplify formalism; it can introduce a further dimension in a simple way (e.g. Minkowsky space) or simplify calculus and differential equations in two dimensions or introduce antisymmetric components of operators and still use theorems for symmetric/hermitian operators, etc. Why do you think complex field is representative of certain symmetries?

    I have worked on the real Schroedinger equations and I was wondering if you know what is the exact geometrical meaning of the additional potential energy (the only term containing hbar^2); is it some sort of curvature of the surface of constant probability density? Any reference?

    More will follow

    Thank for now, Dario
     
  4. Apr 11, 2003 #3
    Hi Dario, thanks for pointing out that odd term.

    Re "Why do you think complex field is representative of certain symmetries?"

    The symmetry is usually called a <b>U</b>(1). Here are a lot of links about it:
    http://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=complex+symmetry+U(1)

    Re: "I have worked on the real Schroedinger equations and I was wondering if you know what is the exact geometrical meaning of the additional potential energy (the only term containing hbar^2); is it some sort of curvature of the surface of constant probability density?"

    Wish I knew. My (primitive) view of it is that it is a dispersion term. That is, it is proportional to how much room/reason there is for the probability distribution to "randomize" and spread out to be even. The thing that kind of bugs me is that it seems most simple when written in terms of the square root of the probability density.

    One possible explanation for the term, from the perspective of the topology I'm exploring, is that the quantization of velocities makes it impossible for a particle to follow differential equations of motion exactly. Instead small errors are forced into the movement, and the sum effect of all those small errors is the dispersion term. But I should note that I haven't thought this through very carefully.

    I think a good time to cogitate about interpretations is after I have a derivation that gives that term. Humans are usually pretty good at justifying stuff after the fact, LOL.

    Carl
     
  5. Apr 11, 2003 #4

    dg

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    Hi Carl,
    well it is really about time that I decide to complement my coursework with some decent group theory. Thank you for clarifying...

    As of your second answer the idea of considering it as a dissipative terms sounds reasonable also in (rough) analogy with the viscous term of Navier-Stokes equation but at the same time it does not really fit with my memory of dissipative forces as not having a potential. What am I missing?

    A little additional note for now.
    When you describe the interference of two particles in [5.b]
    You started using S (potential of momentum) in unit of hbar so it might be appropriate, since you are paying some attention to keep explicit physical significance, to revise your paper and say either that your description of quantum intereference is locally invariant under the addition of h (not of 2*pi) to S or to use a different simbol for the phases S1 and S2 in the intereference term...

    It might also be worth it to note that, in the classical limit hbar->0, the phase of the wave function oscillate without any limit. This of course in the hypotheses that S can somehow be considered indepependent of hbar in the course of this limit. The trick of introducing hbar directly in the definition of the wave function is justified a posteriori by the fact that we obtain the right hamilton-jacobi equation otherwise the limiting process for hbar->0 would give quite a different result. Is there any better reason to explain why S is independent of hbar? Or equivalently that the wave phase has to be inversely proportional to it?

    Another interesting question that might be worthwhile to address is why can we decide that an arbitrary velocity/momentum can be described by a gradient? Doesn't this require that the velocity field is conservative? What about a rotational velocity field?

    Thanks in advance, I will keep on reading today, Dario
     
  6. Apr 11, 2003 #5

    dg

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    Well here I do not really see what you claim: if I use two generic harmonic waves (a generic sine wave can be always put in cosine form adjusting the phase)
    A*cos(a) + B*cos(b)
    and to sum them up I represent them as real parts of a complex numbers, I get an intensity (not amplitude) that goes exactly like the quantum result where only the phase difference appears. This is what happens in classical intereference of light for example... what am I missing here?

    Now I will go into the spin part... Dario
     
  7. Apr 11, 2003 #6
    Dario, much thanks for pointing out the paragraphs where I forgot to include h-bar. By the way, the reason I write "2 pi h-bar" instead of "h" is cause I just can't stand having more than one physical constant running around that does the same thing. Thanks again, I've already made the appropriate changes. Fortunately, I checked units later in the paper while writing it, so the h-bar pops back up where it's needed.

    Complex numbers are so deeply inscribed in our brains in school that by the time we get out of grad school it's hard to remember that they really are just another group, as opposed to the sort of numbers that are used to figure out what we should pay for a beer. Group theory in QM is cool stuff, but I found it difficult to learn, and I was a math major. As a beginning grad student, the book I read on my own as an introduction was "Lie Algebras in Particle Physics" by Georgi, and I could not get past the first 3 or 4 chapters before I had to run to a faculty member for assistance. Since that time I went back to school and studied more math and it's more natural now. But like I say in that paper, I don't think symmetries do us much good when it comes to uncovering how the world is built (as opposed to how stuff in the world interacts, where group theory is irreplaceable). Other than that, the secret I've used for learning difficult things is to work as many practical problems as possible.

    Re: "As of your second answer the idea of considering it as a dissipative terms sounds reasonable also in (rough) analogy with the viscous term of Navier-Stokes equation but at the same time it does not really fit with my memory of dissipative forces as not having a potential. What am I missing?"

    While I'm horrible at misusing terms, a failing which gets worse as I get older, I think that dissipative is the word I want. If you start with a pile of probability in one spot, that term will dissipate it, that is, it will spread it around and flatten the pile. This kind of dissipation is not the same as the frictional dissipation that reduces total kinetic energy. But the equations are the same. That is, if the probability density had instead been a wave, the above term would be a dissipative term (i.e. one that causes the wave to dissipate).

    Re: "It might also be worth it to note that, in the classical limit hbar->0, the phase of the wave function oscillate without any limit."

    This is true. But in my case I'm trying to make classical sense out of the Schroedinger's wave equation with the realistic h-bar value. I see it as an unavoidable part of the universe. I'd just as soon take a limit as h-bar goes to zero as I would take a limit as c goes to infinity or zero. Maybe there's a utility in it, but I don't see it. The usual reason for doing it is to show that quantum mechanics will result in the classical world we live in, a proof that I find rather weak.

    Re: "Another interesting question that might be worthwhile to address is why can we decide that an arbitrary velocity/momentum can be described by a gradient? Doesn't this require that the velocity field is conservative? What about a rotational velocity field?"

    Well, if we consider Schroedinger's equation as two reals, with one being the probability density and the other being the velocity potential, then we're already screwed because there are numerous simple examples of solutions to Schroedinger's equation for which a (single real valued) potential does not exist. See equation (5.8) in my paper for a concrete example, which is, as is typical for angular momentum eigenstates, a rotational velocity field. If, on the other hand, you are willing to have the velocity potential be multiply valued, then you end up with "unphysical" solutions that are not multiples of h-bar. It is only when you push everything back into a complex number version that the velocity potential makes (single valued real) sense.

    Re: "Well here I do not really see what you claim: if I use two generic harmonic waves (a generic sine wave can be always put in cosine form adjusting the phase)
    A*cos(a) + B*cos(b)
    and to sum them up I represent them as real parts of a complex numbers, I get an intensity (not amplitude) that goes exactly like the quantum result where only the phase difference appears. This is what happens in classical intereference of light for example... what am I missing here?"

    Good question, I'll explain more fully in the article. In quantum mechanics, a plane wave is represented by a wave of the form exp(ikx-wt). This wave corresponds to an even distribution of particles. That is, |Psi| is everywhere constant. The wave corresponds to a steady velocity of particles. That is, the momentum P is everywhere constant. In other words, the described physical situation is one that has no sine or cosine dependence.

    Another way of putting this is to note that if you cover up one of the slits in the 2-slit experiment, the resulting pattern has no sine or cosine dependency. And yet when two are present, there is interference. In other words, while there is a sine/cosine dependency in the complex description of the two waves, that dependency has no physical significance, except in the interaction of the wave with itself (or with another like wave).

    I've got to get back to my daily grind, but I'm working hard on using Kaluza-Klein to get E&M running on this new topology. Do you mind if I thank you in the paper for pointing out the error? I hate to intrude, especially as this is considered crack-pot theorizing by the mainstream. I really can't express how indebted I am to you for pointing out these errors and confusions.

    Carl
     
    Last edited by a moderator: Apr 11, 2003
  8. Apr 12, 2003 #7
    I've added a section [7] to that article that extends the real Schroedinger's wave equation to the new topology (in the obvious way), and explicitly shows that the new topology supports stable real spin-1/2 solutions to the extended Schroedinger's wave equation.

    Next is to push these solutions back into real space, and show that they are equivalent to Schroedinger's equation plus the Pauli Spin Matrix representation of SU(2), thereby uniting orbital angular momentum with intrinsic spin. Also, I'm working on E&M through a copy of Kaluza-Klein, but I figured that showing the spin-1/2 solutions was more important.

    Carl
     
    Last edited by a moderator: Apr 12, 2003
  9. Apr 13, 2003 #8

    dg

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    Hi Carl,

    today I will have some more time to write you about your very interesting paper.

    For the time being your idea of the importance of proper time and the very simple -and hence beautiful- approach of changing the topology of SR sounds really close to some of the things I have been ralizing in the last few months studying more differential geometry.

    For now let me point out that in the table of 4-vectors your definition of 4-velocity is not covariant under Lorentz transfo: the 4-velocity has to be P/m and hence gamma*(c,dr/dt)

    Also you might find really useful to extend to relativity the notion of Darboux vector (that shold be a contraction of velocity on Christoffel symbols). It can actually be considered the source of proper time!!!

    More about this later, Dario
     
  10. Apr 13, 2003 #9
    Thanks Dario; I'm going to leave the definition of velocity the way I have it. If I multiply in a factor of gamma, I could end up with velocities in excess of c. Maybe the thing to note is that velocity is not a covariant vector.

    But the concept of dividing out the mass is one that I suspect should be done more in physics. Certainly the real Schroedinger's equation looks nicer without the mass. It all boils down to one thing, and that is the question of whether you want to work with a potential that depends on the "test" mass (i.e. potential energy, perhaps in units of kg m*m/sec*sec), or you instead want to use a potential that does not depend on the mass of the test mass (i.e. PE/kg in units of m*m/sec*sec). The physics communit uses the first standard, but my philosophical inclination is towards the second.

    By the way, I am slowly convincing myself that mass is not a fundamental part of reality, so much as a convenient marker to make various symmetries happen.

    I want to figure out if my calculations for the "self interference" implies the MOND acceleration, but I can't make the calculation without knowing the characteristic mass to use, and due to the fact that particles carry around clouds of virtual particles one has to know the "bare" mass to do this. Maybe all bare masses are around the mass of the lightest neutrino, and that particle is lightest as it has so little interactions that it carries around the minimum amount of vacuum lint.

    I'll read up on Darboux vectors. As a grad student, we called them other things "Christ-awful" tensors, which is a result of being asked to verify Einstein's equation for stuff like the Schwarzchild solution, LOL.

    Also, I should note that my extended Schroedginer's equation admits far too many stable solutions that do not correspond to anything real. The problem appears to be a failure to change, with time towards the standard Schroedinger's equation solution. This is from the velocities not being generated by a potential, so I'll add a term that calms the velocities down, over time. I'm not sure what time scale to use for this, but it will have the effect of making "classical" precise position/velocity solutions wander towards quantum solutions with time.

    Carl
     
    Last edited by a moderator: Apr 13, 2003
  11. Apr 13, 2003 #10

    dg

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    It sounds like a good idea! It could also be good to restore c in the relativistic results when discussing interpretation or classical limits.
    In particular I really think that using s to indicate proper time is a little disturbing for units. What about following Schutz and use tau instead so that ds=c*d(tau)
    Thanks for the reference: I hate to say it but I should have taken that high energy physics course at least for that reason: half of that course was based on the Georgi. High energy physicist were/are monopolizing the resources of my home university and I could not stand them. All they wanted were docile blue-collar scientist to send to CERN in Geneve for low-profile jobs. Too much power in the hands of ignorant people can be really bad for scientific education...
    I have in my hands the Hamermesh (Group theory and its applications to physical problems, Dover): do you know anything about it? Is it worth it?
    Well I strongly feel a geometrical point of view would be very useful... I believe that the term has a strong connection with another (unsolved to my knowledge) problem: the geometric interpretation of capacitance of a conductor. Terms of the form lapl(V)/V appears when you search for such a thing...
    Another strong feeling that I have developed in this area is that there a very strong connection between e.m. energy density and probability density that should be explored.
    I have been working along a line re-opened by Gough with his 'recent' work on Hertz potentials and this has given me some insight that justifies this perception.
    Well going to the limit with constants is always a little problem. You have probably noticed how going c->oo in the Lorentz transfo's does not return Galilean transformations if a system is 'large' enough... and hence classical mechanics is not only for small speed but also for 'small' systems.
    Still I believe it is interesting to understand how S which is just a chracteristic function of hamiltonian mechanics ends up to regulate the phase of the wave function through hbar... I should have a fresh look at Goldstein's explanation of S as a phase of the classical wave associated with a particle motion...
    I do not know this results but I guess studying the mechanism by which complex field make sense of real "unphysical" results can be worthwhile. If you have a reference I will appreciate...
    Well I will re-read that part of the article and also read further ahead, since I had probably misunderstood what items you were comparing...
    Well I am really happy to see a seriuos theory in this rather messy section and I am even more happy to see someone on the same line of my conjectures... Feel free to thank you me ;) I am trying to keep my physics alive now that I have decided to stay out of the academia for some time. Feel free to ask for any further help or opinion...

    Dario
     
  12. Apr 13, 2003 #11

    dg

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    Well this is a possibility; I will read more before coming back to this subject...
    I agree with this inclination; the success of electromagnetic theory (using field instead of force) should be copied by analogy straight into classical gravitational theory and mechanics in general using acceleration fields instead of forces.
    Well I am definetly convincing myself of the same thing as I believe that mass, spin and proper time are somehow aspects of a single physical entity and that constant mass is somehow a result of constant spin.
    I will be happy to discuss with you what I have found so far about Darboux vector; this is to me a key to a better understnding of spin...

    Before going back to read further I will send you a little summary of the main points around which my conjectures run and hopefully you can make some use of it...

    Dario
     
  13. Apr 13, 2003 #12

    dg

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    Summary of ideas, insights, conjecture, etc.

    The Darboux vector can be seen as the angular velocity of the intrinsic triad of motion of a particle. It contains all the curvature information (curvature and torsion) up to a euclidean transformation. It is not a common vector at all. It lives in the principal normal plane or better in its upper half! Such a limitation of the degree of freedom make it a strong cansidate to associate intrinsic angular momentum to a classical particle. Furthermore it is a natural clock associated with the particle and this make of it a natural candidate to relate to proper time. There is anatural parameter that can be associated with darboux vector that could be a classical phase or proper time. The motion of charges in uniform e.m. fields suggests that it has strong analogies to magnetic field. In particular by studying spiraling motions there are some very interesting results that realtes to the theroy of physical measurement... more later...

    Dario
     
  14. Apr 13, 2003 #13
    Hi Dario; I wanted to use space dimensions instead of time dimensions for the proper time dimension because it shows up in the metric with the three space dimensions. I like the feeling of a differentiable manifold, being ABD PhD in math in the subject, and that makes the natural metric R^4.\

    The idea is that our global time is special, but it has little physical significance, and is only a marker for integration. And then for similar reasons to why the physicsts set c=1, it's natural to make the proper time dimension have units of distance.

    Re: "All they wanted were docile blue-collar scientist to send to CERN in Geneve for low-profile jobs."

    I loved studying physics, but there appeared to me to be no way to actually make a living at it, so I went into Electronics. While it's true that the world of physics, like the rest of the world, is largely a function of a blend of "sex, money and murder", (as the rap song says) and this bothered me at the time, I've since realized that this is the underlying structure of the world, and there's no reason to expect it to be absent from the world of physics.

    And having left the physics community, it's damn hard to stay abreast of new work. The problem is that there's no one to talk to about it. And most of the internet denizens are unable to differentiate cosine. Hmmm. Now that I think of it, that was true of a certain number of grad students I knew.

    Re: "Another strong feeling that I have developed in this area is that there a very strong connection between e.m. energy density and probability density that should be explored."

    Well they are identical in E&M, if I recall correctly, at least for photons of a particular frequency when converting between a particle and a wave description of an E&M situation, after taking into account a factor of h*nu, (searches in scraps of memory for actual relation, fails to find it).

    The point of view of particle physics is that there are no fields, such as E&M, but only particles that mediate forces, such as photons. I think another way of putting this is to say that the topology of the world is simple. That is, you can describe the situation at a particular point in the world without needing to describe the force fields of every possible force for that point. Where the complexity shows up is in the waves that move around in the world, and in their interactions with each other.

    At least this is the view I subscribe to (currently). I don't think that the world has complex topological objects running around, just waves. By the way, another question on Hadley's concept of general relativity giving spin 1/2 particles is this: According to Feynmann's path integration theory, to calculate a matrix element, you have to sum over all possible paths. That means that you must sum over all possible paths that the electron can take, and at least conceptually, this means that the electron can be anywhere (as is consistent with Schroedinger's wave equation or any other wave equation).

    Now if the electron is everywhere, then doesn't that imply that the complex topological knot that gives it spin 1/2 is also everywhere? And since the amplitude of the electron's wave function can approach zero, doesn't that mean that the topological knots must also approach zero? The problem is that topological knots are generally not so easily untied. And as long as you're going to posit a nearly vanishing topological electron wave function, you might as well instead simply throw the topological complexity into the base space instead of the wave.

    Re: "If you have a reference [to examples of solutions to Schroedinger's equation for which a (single real valued) potential does not exist] I will appreciate..."

    No one cares about the real Schroedinger's wave equation, as far as I know. You'll either have to work out the facts for yourself or take my word for it, LOL. References for the spin-1 wave states of the hydrogen atom are all over. Just take one of these and convert it into real waves according to the Psi = A exp(iS/h-bar) formula. If you don't get a wave that is what I said you'll get, then come back and tell me immediately.

    But the fact that Bohmian mechanics gets in trouble with bound states is well known. Here's an interesting links:
    http://arxiv.org/PS_cache/quant-ph/pdf/9708/9708026.pdf

    The Darboux vector stuff sounds fascinating, I'll do some web searches. But right now, I've got to earn some keep.

    Carl
     
  15. Apr 14, 2003 #14

    dg

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    Sounds reasonable, it is just that I have always found the notation scarcely intuitive. I prefer to keep s to indicate 3-space arc parameter rather than 4-space one.
    Uh another ABD! Here you have an ABD in Physics, Solid State Physics / Surface Science... God knows how I brought myself in such an (!@#) area!
    Is there a global time? What makes t a special parameter in the parameterization of a curve (I see it better that way than as an integration marker)? Well I believe that t is just the proper time of the observer, of the reference frame in which we describe the curve; this makes it special!
    Well I am still working on my ability to accept the world as is, at least at an emotional level; I agree with you rationally but I can't help feeling 'sick' about it.
    Well I do not even contemplate the idea of staying abreast of new works my coursework was so poor that I had to (re)discover many things that were left aside. The Darboux vector is a good example of this. And this thing about e.m. energy and probability is another good example.
    Many people in science end up becoming superspecialized which means that they are ignorant 'almosr everywhere' else.
    People are unable to make connections or cross-reference or work by analogy because they specialize too much.
    This is what makes a difference in your approach, a mathematician who knows his physics is a real rarity to me. I am really happy to see one out there. ;)

    Time to sleep for me, now, but before let me insist on that table of comparison between SR and proper time topology: I understand that you want to keep 4-velocity constant in your formalism but still 4-velocity under SR column should be a 4-vector and in your definition those components do not transform properly...

    Dario
     
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