nonstandard solutions of the Schwinger-Dyson equations

very_cryptic_at_hotmail.com
Date: 12/24/04

  • Next message: Dushan Mitrovich: "Re: The Hodge dual: some definitions & examples" 
    Date: Fri, 24 Dec 2004 12:50:01 +0000 (UTC)

    While it is well known that Feynman path integrals form a class of
    solutions to the Schwinger-Dyson equations, they most definitely aren't
    the most general solution.

    Let me give an example:

    In the phi to the 4th theory, the SD equation for the generating
    functional is

    [-i lambda/3! \partial^3/\partial J(x)^3+i(box+m^2) \partial/\partial
    J(x)+J(x)]Z[J]=0

    Let phi_i(\vec{x}) be an arbitrary function at t=t_i. Then, for t>t_i,

    Z[J]=\int \mathcal{D}phi_{phi(t_i)=phi_i} exp{i\int d^4x 1/2 (\partial
    phi)^2-m^2/2 phi^2 - lambda/4! phi^4 + J phi}

    forms the class of solutions of interest.

    However, there are infinitely many other classes of solutions. Take for
    example,

    Z[J]=\int \mathcal{D}phi_{phi(t_i)=phi_i} exp{i\int d^4x -1/2 (\partial
    phi)^2 + m^2/2 phi^2 - lambda/4! phi^4 - i J phi}

    This is also a class of solutions, except that it gives negative
    probabilities, which isn't good.

    There are also classes of solutions which are odd in J, etc.

    But isn't it possible that a linear combination of these two solutions
    will yield positive probabilities? I have to admit I wasn't able to
    check the nonnegativity condition for all possible expectation values
    of A*A where A is an operator, but among those combinations I have
    checked, it appears possible there might be linear combinations of
    standard and nonstandard solutions which MIGHT possibly always satisfy
    the nonnegativity conditions.

    At any rate, what is the physical meaning of these nonstandard
    solutions? After all, there are some physicists who work purely with
    Schwinger-Dyson equations instead of with path integrals.

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