Is the K-G Operator of the Kelin-Gordon Equation a Time Ordered Function?

[Karlisbad]

Let be the Kelin-gordon equation (m=0) with a potential so:

$$(-\frac{\partial ^{2}}{\partial t^{2}}+V(x) )\Phi=0$$

my question is if you consider the wave function above as an operator..is the K-G operator of the form:

$$<0|T(\Phi(x)\Phi(x')|0>$$ T=time ordered

I think that in both cases..we use the same wave function but once is an scalar (or an spinor for electrons) and the other is an escalar...:shy: :shy:

 

[dextercioby]

1.I don't know who Kelin was. Maybe you could supply some reference.

2. Your equation, misses a laplacian.

3. You depicted the Feynman Green function, which is a Green function for the operator written with a Laplacian.

All of course, if you mean "Klein-Gordon"

Daniel.

 

[Karlisbad]

I apologize "DSextercioby"... i missed the keyboard.. yes i was referring Klein-Gordon equation with rest mass m=0 so:

$$(-\frac{\partial ^{2}}{\partial t^{2}}+\nabla +V(x))\Phi=0$$


then if you define the Green function by $$G(x,x')=<0|T(\Phi(x)\Phi(x'))0>$$

then my question were if the "Phi" wave function defined in both G and K-G equation is the same ,but in one case is an operator and in the other is an scalar with T=time ordered product.

- By the way i looked at the paper by Scwinger ..taking the Dirac equation with Electromagnetism:

$$(i\gamma_{\mu}\partial _{\mu}-eA_{\mu}+m)\Psi =0$$

he got the Green function (i don't know how he did it.. ), he got the functional equation:

$$\partial _{\mu}-eA_{\mu}+m+\frac{\delta}{\delta J_{\mu}}G(x,x')=\delta(x-x')$$

 

[dextercioby]

I figure you never read (hence never edit) your posts after hitting "submit reply/thread" button.

In the field eqn, the $\varphi (x)$ is not a wavefunction, it is a classical field.

In the VEV of the time-ordered product, it is an operator acting on a Fock space. It still keeps the scalar behavior wrt restricted Poincare' transformations.

As for the second part of your post, please supply the reference to Schwinger's paper.

Daniel.

 

[Karlisbad]

A brief resume..can be found at:

http://www.pnas.org/cgi/content/full/102/22/7783

with the Dirac equation + magnetic field+ scalar potential V(x) and the functional approach to the Green function involving functional derivatives.