Variatonal principles in quantum Field Theory

In summary, the conversation discusses a nuclear reaction that is not possible due to the conservation of electrical charge. The individual also asks about using variational methods to obtain the coefficients of the S-matrix, but it is mentioned that variational principles are more suited for classical behavior and may not provide insight into quantized behavior. There is also a mention of the Schwinger variational principle and its potential usefulness in obtaining elements of the S-matrix or quantizing classical fields.
  • #1
eljose
492
0
Let be the nuclear reaction:

[tex] ee \rightarrow e+e+ [/tex] (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:

[tex] <B|S|A> [/tex] 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.
 
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  • #2
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.
 
  • #3
eljose said:
Let be the nuclear reaction:

[tex] ee \rightarrow e+e+ [/tex] (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:

[tex] <B|S|A> [/tex] 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.
 
  • #4
But How about "Schwinger variational principle"?..

[tex] \delta <A|B>=i>A|\delta S|B> [/tex]

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 [tex] ee+ \rightarrow ee+ [/tex]
 

1. What is the significance of variational principles in quantum field theory?

Variational principles are fundamental principles used in quantum field theory to derive equations of motion and determine the dynamics of quantum fields. They provide a systematic and elegant approach to understanding the behavior of particles and fields in a quantum mechanical framework.

2. How do variational principles differ from other mathematical techniques used in quantum field theory?

Variational principles are based on the principle of least action, which states that the physical system will take a path that minimizes the action. This differs from other mathematical techniques, such as perturbation theory, which use approximations to solve equations of motion.

3. Can variational principles be applied to all types of quantum field theories?

Yes, variational principles can be applied to all types of quantum field theories, including relativistic and non-relativistic theories. They can also be used to study both bosonic and fermionic fields.

4. How are variational principles used to derive Feynman diagrams in quantum field theory?

Variational principles provide a systematic approach to deriving Feynman diagrams, which are graphical representations of the interactions between particles and fields in quantum field theory. The Feynman diagrams are obtained by calculating the perturbation of the action functional.

5. What are the limitations of variational principles in quantum field theory?

While variational principles are powerful tools in understanding the behavior of quantum fields, they have limitations in certain situations. For example, they cannot be used to describe systems that involve strong interactions, such as nuclear forces. In these cases, other mathematical techniques, such as lattice gauge theory, may be more appropriate.

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