Schwinger variational principle

1. Apr 7, 2006

eljose

What is this used for?..i don,t see any utility on using it.. for Commuting and Anti-commuting operators we would have:

$$\delta{<A|B>}=i<A|\delta{S_{AB}}|B>$$

but i don,t see that it provides a way to obtain Schroedinguer equation or the propagator for the theorie....what is SVP used for?..thanks.

2. Apr 8, 2006

marlon

The Schwinger Variational principle [1] (SVP) is used to calculate the effective Lagrangian in quantum field theory. Effective Lagrangian means the Lagrangian in which you do not have any elementary particles.

For example, in QCD, we use this principle to describe the interaction of quarks with gluons and quark-pair generation. More specifically, to describe the formation of quarkpairs in the colour electric field of two valence quarks due to the fact that the QCD vacuum is not empty : VIRTUAL QUARK-PAIR FORMATION.

You wanna integrate out the quarkfields because you want a theory that does not depend on such parameters. The quarksfields, here, are just the quarks that make up the QCD vacuum.

1) One starts from the Dirac field equations.

2)The probability of interaction between virtual quarks and the colour electric field is expressed in terms of a Green function.

3) The SVP gives you a way the find this Green function and get the right action and Lagrangian. From that you can calculate how "fast" virtual quarkpairs can become real once you introduce a colour electric field into the QCD vacuum.

regards
marlon

[1] Julian Schwinger, “On Gauge Invariance and Vacuum Polarization”, Phys. Rev. Volume 82, Number 5, 1951

3. Apr 8, 2006

eljose

Thank you for your response "Marlon" so as far as i understood you could obtain the Green function (propagator) or at least an approximation to it by minimizing a functional?..(sorry if my reasoning is not true) by the way could you point me a book or a practical toy-example in wich SVP is involved?...

4. Apr 8, 2006

marlon

You are minimizing the action functional with respect to the quark fields, to get rid of them.

Read the paper, it gives a nice explanation and a practical example.
I also recommend the Books of Weinberg

marlon

5. Apr 10, 2006

dextercioby

Read the QM book by Schwinger himself. I'm sure you'll get answers to your problems. Actually Roger Newton wrote an "Advanced Quantum Physics" book in which he also presented Schwinger's approach to quantum mechanics and its equivalence to "ordinary one" by deriving S's Eq from his postulates.

Daniel.