Variatonal principles in quantum Field Theory

[eljose]

Let be the nuclear reaction:

$$ee \rightarrow e+e+$$ (if not possible a similar one)

Of course we have 2 states |A> with 2 electrons and |B> with two "positrons"..if we wished to compute the transition probability we should know:

$$<B|S|A>$$ where "S" is the S-Matrix..my question is..is there any variational method to obtain the coefficients of the Matrix S?..if not could be one or is mathematically impossible?..thanks.

 

[Parlyne]

The reaction you're considering isn't possible, as it fails to conserve electrical charge.

As for variational principles, they really only apply to the classical behavior of the system. They can get you the equations of motion for the various field, but will not give you the scattering matrix. To do that, you need to quantize the fields.

 

[coalquay404]

Let be the nuclear reaction:

$$ee \rightarrow e+e+$$ (if not possible a similar one)

Of course we have 2 states |A> with 2 electrons and |B> with two "positrons"..if we wished to compute the transition probability we should know:

$$<B|S|A>$$ where "S" is the S-Matrix..my question is..is there any variational method to obtain the coefficients of the Matrix S?..if not could be one or is mathematically impossible?..thanks.

As already pointed out, variational principles are inherently classical in nature. For example, canonical quantization is best approached in terms of a Hamiltonian framework, while the path integral formalism relies crucially on having a Lagrangian for a system. Quantization and calculation of elements of an S-matrix are generally regarded (rightly or wrongly) as being quite separate from the action principle. If it helps, you can regard variational methods as laying the groundwork for the classical behaviour of a given field, not as giving any particular insight into the quantized behaviour.

 

[eljose]

But How about "Schwinger variational principle"?..

$$\delta <A|B>=i>A|\delta S|B>$$

wouldn,t it be useful to obtain the elements of "S" matrix or a quantization for the classical fields?...

I made a mistake perhaps i wanted to say $$ee+ \rightarrow ee+$$