Variatonal principles in quantum Field Theory

Click For Summary

Discussion Overview

The discussion revolves around the application of variational principles in quantum field theory, specifically in relation to calculating the S-matrix coefficients for a nuclear reaction involving electrons and positrons. Participants explore the feasibility of using variational methods in this quantum context.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant presents a nuclear reaction involving electrons and positrons and questions whether a variational method can be used to obtain the coefficients of the S-matrix.
  • Another participant argues that the proposed reaction is not possible due to charge conservation issues and states that variational principles primarily apply to classical systems, not directly to the scattering matrix.
  • A third participant reiterates that variational principles are classical in nature and emphasizes the separation between quantization and the action principle, suggesting that variational methods do not provide insights into quantized behavior.
  • A later reply introduces the "Schwinger variational principle" and questions its potential usefulness for obtaining S-matrix elements or for quantizing classical fields, indicating a possible alternative approach.

Areas of Agreement / Disagreement

Participants express disagreement regarding the applicability of variational methods to quantum field theory, with some asserting that these methods are not suitable for calculating the S-matrix while others propose alternative perspectives, such as the Schwinger variational principle. The discussion remains unresolved.

Contextual Notes

Participants note limitations related to the classical nature of variational principles and the need for quantization in calculating S-matrix elements, but do not resolve these issues.

eljose
Messages
484
Reaction score
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.
 
Physics news on Phys.org
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.
 
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.
 
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]
 

Similar threads

Undergrad Why is the coupling factor constant in QFT?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Is Maxwell's electromagnetism an effective (field) theory?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Classical field in quantum field theory?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad QED replacing photon field with current in 3-point function
  • · Replies 1 ·
Replies
1
Views
1K
Graduate What exactly does 'Locality' in Gauge Theory mean?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Measurement in QFT: Mapping Fields to Theory's Math Formalism
  • · Replies 31 ·
2
Replies
31
Views
3K
Graduate Quantum Field theory vs. many-body Quantum Mechanics
  • · Replies 36 ·
2
Replies
36
Views
7K
Graduate Meaning of "symbol" in algebraic field theory?
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Simulating physics: the current status of lattice field theories
  • · Replies 2 ·
Replies
2
Views
2K
Graduate QFT with vanishing vacuum expectation value and perturbation theory
  • · Replies 3 ·
Replies
3
Views
2K