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complex numbers in QM

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  • David Strayhorn
    ... There s a discussion that is sorta brewing in the group, qm2, on the topic of complex numbers in QM. Maybe worth looking at if you re interested. I ve been
    Message 1 of 23 , Mar 4, 2004
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      --- In bell_bohm@yahoogroups.com, "brannenworks" <brannenworks@y...>
      wrote:

      > When I was a graduate student, I had two problems with QM. ... the
      > second was that the theory uses complex numbers.
      >
      > Sure, E&M (and many other physics theories) can be (and are) written
      > with complex numbers, but they can also be written without them --
      > the complex numbers are only there to ease calculations. Quantum
      > mechanics, in contrast, has complex numbers at its core, with no
      > explanation. ...

      > So instead of working at this deep philosophical level, I began
      > working at the problem from the other end, from the point of view of
      > trying to remove the complex numbers from QM. I began by spending a
      > few years trying to put the Dirac equation into a real form, rather
      > than a complex form, because I do not believe that complex numbers
      > make an ontologically correct description of reality. It turns out
      > that there are many ways that you can do this, but none of them tell
      > you much, at least as such. I ended up becoming very proficient at
      > manipulating the Dirac equation, but I made no progress at putting
      > it into a form which would match the stresses in a believable space.

      There's a discussion that is sorta brewing in the group, qm2, on the topic of
      complex numbers in QM. Maybe worth looking at if you're interested.

      I've been fiddling around lately with the path integral approach, and one of the
      manipulations that I did with it was to rework the basic approach in a way that
      makes no use of complex numbers. The fundamental problem of the path
      integral approach is to calculate the probability that a particle that starts at x1,
      t1 will end up at x2, t2. There are several steps that involve enumerating all
      paths, calculating the action for each path, calculating the phase for each path
      (which is a complex number), adding all the phases to get the "kernel", and
      then taking the square of the absolute value of the kernel to get the
      (differential) probability. This whole procedure can be summed up by one
      equation for the differential probability of ending up at x2, t2, ie:

      P = | sum (over all paths) e ^ (- i S / h) | ^ 2

      where S is the action. And this technique is general enough that, in principle,
      any QM problem can be solved by this method, iiuc.

      It took me only a few steps to put the above equation into a form so that you
      can calculate the probability without even knowing what complex numbers
      are. The implication (I think) is that, in principle, you should be able to do all of
      QM without ever using complex numbers. (it would be computationally more
      difficult, but possible, in theory.) If you're interested, I uploaded a draft of a
      paper I'm working on in the files of this group, called modified-path-
      integral.pdf -- look at page 5 (which is section 6), equations (41) through
      about (48) or so. I made it with LaTeX, which I recently learned :), so it should
      is easy to read the equations. (btw, much of the rest of the paper is still in draft
      form.)

      DS
    • brannenworks
      Dear David Strayhorn; ... The ether is supposed to be the medium which allows light to propagate. GR doesn t have much to say about light. For example,
      Message 2 of 23 , Mar 5, 2004
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        Dear David Strayhorn;

        > What is the justification for the
        > statement: since there are waves,
        > there must be an ether? GR has waves but no ether.

        The "ether" is supposed to be the medium which allows light to
        propagate. GR doesn't have much to say about light. For example,
        even something as basic as the polaroid filters in sunglasses cannot
        be described in GR alone. The waves that do occur in GR are gravity
        waves, but they've not yet been observed (as far as I know). I'm
        not a GR type, and I don't have any guesses as to whether or not
        those gravity waves will be seen or not.

        > ie, what experiment could tell us
        > that there had to be an ether?

        QM uses a "momentum cutoff" (among other things) to make QED
        calculations work right. If nature has a momentum cutoff, then
        there is a maximum momentum. That says that any object (an
        electron, for example) has a maximum possible momentum. A test for
        this is to accelerate an object to very high momenta. If the
        momentum cutoff is there, then you will eventually reach a limit
        where it is impossible to accelerate any further. Note that this
        would be a violation of Newton's (or Galileo's, I forget which) as
        well as Einstein's relativity.

        To find the ether, repeat the experiment twice, once in the +x
        direction, and once in the -x direction. You are rest with respect
        to the ether when the results from those two experiments match.

        > What makes you say momentum is not "real"
        > in relativity? In general, for X to
        > be a "real" thing (according to the way
        > you define real), does X have to be
        > invariant? In your mind, is GR tainted
        > /tarnished because things that
        > classically seem "real" are viewed in
        > GR as not "real"?

        Momentum in GR is not "real" because it cannot be defined except
        with respect to a particular rest frame. That means that it cannot
        be a fundamental part of a universe made up of "real" things. By
        contrast, if one considers the universe to be a mathematical
        construct, rather than a "real" thing, then there is no problem with
        defining momentum that way.

        I am in no way saying that GR is inconsistent with itself, or
        incompatible with observations. What I'm saying is that its
        consistency is limited to that of a mathematical construct. It does
        not possess the consistency that a description of an object in the
        world possesses. It's an "as if" theory.

        Rather than "tainted or tarnished", I would use the
        word "incomplete". It's somewhat ironic that this is the same
        complaint that Einstein had of quantum mechanics.

        > It seems like what you are doing is to
        > describe what sort of things guide your
        > intuition on your search for something new.
        > ie, certain things are not strictly
        > forbidden, but they are "not beautiful" (?)
        > to you, and thus an indication that
        > some sort of new ideas are needed. IOW, the
        > aspects of a theory that cause
        > you "ontological angst" are the aspects
        > that you seek to replace. These are
        > the rocks that you turn over. Would that
        > be fair?

        It's not beauty that distinguishes between a phenomenological and an
        ontological theory. My movement in this direction is not due to an
        appreciation of beauty. There is nothing more beautiful than SR and
        GR. In fact, I think it is this beauty that has bedazzled the eyes
        of physicists for so many years. We'd all like nature to be a
        beautiful thing, and we all have a strong tendency to believe
        theories that are more beautiful than not. For example, for
        centuries astronomers believed that planets moved on circles, rather
        than ellipses, because circles are more beautiful (or symmetric).
        This is human nature. And it is this human nature that has misled
        us. Instead of more beautiful mathematical constructs, I believe
        that what we need in physics now is more realistic descriptions.

        About a century ago, there was an influential physicist named Ernst
        Mach. He believed in "empiriocentrism", which is pretty much the
        opposite of my point of view. Let me quote from the book "Nature
        Loves to Hide":

        <<
        Science, according to Mach, is nothing more than a description of
        facts. And "facts" involve nothing more than sensations and the
        relations among them. Sensations are the only real elements. All
        the other concepts are extra; they are merely imputed on the real,
        i.e., on the sensations, by us. Concepts like "matter" and "atom"
        are merely shorthand for collections of sensations; they do not
        denote anything that exists.
        >>

        What it all boils down to is this: "A good theory is no more than a
        condensation of observations in accordance with the principle of
        thought economy." If you believe this, then there is no reason to
        suppose that relativity is explained by a hidden dimension. But
        here it is 2004 and the strong and weak forces are still not unified.

        Physics has followed Mach's philosophy for 100 years, and now we're
        stuck. What I'm saying is that we may need to ditch the philosophy,
        and go back and rederive physics without it. And that implies that
        we need to have a physics that is more than just logically or
        mathematically consistent.

        For example, QED is obviously a mathematical construction, not a
        real description of what goes on with electrons and photons. This
        is clear from the way that infinities have to be cancelled out of
        the theory. The great physicists like Feynmann recognize this, as
        he notes in his book on QED. Here's what Landau and Lifshitz says
        about QED:

        <<
        There is as yet no logically consistent and complete relativistic
        quantum theory. We shall see that the existing theory introduces
        new physical features into the nature of the description of particle
        states, which acquires some of the features of field theory (see
        chapter 10). The theory is, however, largely constructed on the
        pattern of ordinary quantum mechanics. This structure of the theory
        has yielded good results in quantum electrodynamics. The lack of
        complete logical consistency in this theory is shown by the
        occurrence of divergent expressions when the mathematical formalism
        is directly applied, although there are quite well-defined ways of
        eliminating these divergences. Nevertheless, such methods remain,
        to a considerable extent, semiempirical rules, and our confidence in
        the correctness of the results is ultimately based only on their
        excellent agreement with experiment, not on the internal consistency
        or logical ordering of the fundamental principles of the theory.
        >>

        The original reason I started delving into these matters was to
        repair the above inconsistency. I felt that it had something to do
        with the appearance of complex numbers in the theory. But as I
        continued to work on it, I was unable to make progress until I gave
        up perfect Lorentz symmetry. And by "gave up", I mean exactly
        that. Relativity was torn from me only by years of failing efforts
        to make QM logically consistent under the assumptions of perfect
        relativity. I couldn't do it. Neither could the rest of the
        physics community.

        The most recent response to these consistency problems in QM are
        called "string theories", and these were what got me interested in
        physics once again. But when I picked up a few books, it rapidly
        became obvious that they had more infinities getting cancelled than
        anything dreamed of in QED. So I began working on physics.

        > Given a mathematical
        > formalism of a theory, do you think that
        > it is necessary (or required, or perhaps
        > merely preferred) that every term of
        > every equation correspond to something
        > that we can "point our finger to"? (ie,
        > to be ontologically palatable).

        To be ontologically correct, a theory need only have its most basic
        units be "real", not every term of every equation. Also, I suppose
        I should mention that if someone did have a unified field theory,
        even one that was only a mathematical construct, I wouldn't be
        searching for an ontologically correct unified field theory.

        Carl Brannen
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