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Brute force quantization

  1. Aug 25, 2006 #1
    "Brute force" quantization..

    Let's suppose we have the "Equation of motions" for a particle:

    [tex] F(y'',y',y,x)=0 [/tex] my question is if exsit a "direct" method to apply quantization rules..for example simply stating that:

    [tex] F(y'',y',y,x)| \psi (x) >=0 [/tex] or something similar.

    - I'm not talking about the usual method (you use the Hamiltonian operator to get the Wave function) but a method to "Quantize" everything without using Hamiltonians or Lagrangians only with the equation of motion and similar...thanks.
  2. jcsd
  3. Aug 30, 2006 #2
    The Hellman-feynman theorem deals with the quantum mechanics of forces, but no, there is no procedure which involves quantised forces.
  4. Sep 5, 2006 #3
    Well, I don't know about that theorem, but I was once thinking about quantum mechanics as a probability fluidum (the Madelung interpretation), something I never published since I cannot believe it has not been done yet. In this case, let [tex] R^2 [/tex] be the ``mass´´ density and [tex] \partial_{\mu} S [/tex] be the integrable fluid velocity field. Then, the traditional navier stokes equation is:
    [tex] R^2 \partial_t \partial_{\alpha}S + R^2 \partial_{\beta} S \partial_{\beta} \left( \partial_{\alpha} S \right) = \frac{R^2}{m} F_{\alpha} - \partial_{\alpha} p + \partial^{\beta} T_{\beta \alpha} [/tex] and the usual continuity equation
    [tex] \partial_t R^2 + \partial^{\alpha} \left( R^2 \partial_{\alpha} S \right) = 0 [/tex]
    Now, let the pressure [tex] p = - \frac{1}{2m^2} \left( R \partial_{\beta} \partial^{\beta }R - \frac{1}{3} \partial_{\beta} R \partial^{\beta} R \right) [/tex] and the stress tensor
    [tex] T_{\alpha \beta} = - \frac{1}{m^2} \left( \partial_{\alpha}R \partial_{\beta} R - \frac{1}{3} \delta_{\alpha \beta} \partial_{\gamma} R \partial^{\gamma} R \right) [/tex] then it is easy to prove that
    with [tex] F_{\alpha} = - \partial_{\alpha} V [/tex], the Navier Stokes equation gives rise to the Hamilton Jacobi equation of Bohmian mechanics. Hence, this provides a general scheme for quantization of particles in general force fields. If you definetly know this has not been done yet, give me a sign and I will post the ``paper'' on the arxiv.

    It seems to me you cannot quantize general force fields (in the case of instantaneous action at a distance, there are no travelling waves, hence no particles), only those which can be derived from a (eventually distributional) field theory seem to be meaningful.

    Last edited: Sep 5, 2006
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