Schwinger Quantum action

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In summary, the given path integral formula can be used to show that the partial derivative of the path integral with respect to the source term J is equal to the expectation value of the derivative of the Lagrangian integrated over space. This can be applied to a Klein-Gordon scalar field with an interaction of the form \phi^{4}. By using integration by parts, the action can be written in a form where the variation can be found using ordinary derivatives. It is recommended to attempt the solution before seeking help.
  • #1
Given the path integral

[tex] < -\infty | \infty > = N \int D[\phi ]e^{i \int dx (L+J \phi ) } [/tex]

then , it would be true that (Schwinger)

[tex] \frac{\partial < -\infty | \infty >}{\partial J }= < -\infty |\delta \int L d^{4}x | \infty > [/tex]

If so, could someone provide an exmple with the Kelin-gordon scalar field plus an interaction of the form [tex] \phi ^{4} [/tex]
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  • #2
This is a consequence of the fact that
[tex]\int\!\mathcal{D}\phi\, \frac{\delta}{\delta \phi} \left( \dots \right) = 0[/tex]
Essentially, one integrates by parts and hopes that the exponential damps out the boundary in function space, if such a boundary exists. Thus, the equations of motion for [tex]\phi[/tex] are obeyed inside the expectation value.

Anyway, using (space) integration by parts, write your action as
[tex]\int\!d^4x\, \frac{1}{2}\phi ( - \partial^2 - m^2 ) \phi + \frac{\lambda}{4!} \phi^4[/tex].
You should now be able to find the variation in this action just like taking ordinary derivatives. If you get stuck, tell us what you've done and we can see about help.

I hope someone doesn't just blurt out the answer. This forum seems to be full of keeners like that =)
  • #3

I can confirm that the equation presented is known as the Schwinger Quantum action. It is a useful tool for calculating the probability amplitude of a particle moving from one point to another in space-time.

The equation states that the probability amplitude < -\infty | \infty > is equal to the integral over all possible paths of the particle, denoted by D[\phi ], multiplied by the exponential of the action, which is the sum of the Lagrangian (L) and the external source (J) multiplied by the field ( \phi ).

The second equation states that the derivative of the probability amplitude with respect to the external source (J) is equal to the expectation value of the operator \delta \int L d^{4}x between the initial and final states. This means that by varying the external source, we can calculate how the probability amplitude changes.

To provide an example, let's consider the Klein-Gordon scalar field with an interaction term of the form \phi ^{4}. The Lagrangian in this case would be:

L = \frac{1}{2}(\partial_{\mu}\phi)^2 - \frac{1}{2}m^2\phi^2 - \frac{\lambda}{4!}\phi^4

where m is the mass of the particle and \lambda is the coupling constant for the interaction term.

Using the Schwinger Quantum action, we can calculate the probability amplitude for a particle to move from an initial state at x=-\infty to a final state at x=\infty. This would be represented as < -\infty | \infty >.

Taking the derivative with respect to the external source, we get:

\frac{\partial < -\infty | \infty >}{\partial J }= < -\infty |\delta \int L d^{4}x | \infty >

Substituting the Lagrangian for our example, we get:

\frac{\partial < -\infty | \infty >}{\partial J }= < -\infty |\delta \int \left[\frac{1}{2}(\partial_{\mu}\phi)^2 - \frac{1}{2}m^2\phi^2 - \frac{\lambda}{4!}\phi^4 + J\phi\right] d^{4}x | \infty >

Using the properties of the

1. What is the Schwinger Quantum action?

The Schwinger Quantum action is a mathematical framework used in quantum field theory to describe the behavior of systems with both quantum mechanical and classical properties. It is based on the idea of a path integral, which involves summing over all possible paths that a system can take between two points in space and time.

2. Who is Julian Schwinger and what is his contribution to the Schwinger Quantum action?

Julian Schwinger was a Nobel Prize-winning American physicist who made significant contributions to quantum field theory. He developed the Schwinger Quantum action, which is also known as the Schwinger action or the Schwinger variational principle, in the 1950s.

3. How does the Schwinger Quantum action differ from other quantum field theory formulations?

The Schwinger Quantum action differs from other formulations, such as the Feynman path integral, by using an action principle rather than a Hamiltonian or Lagrangian formalism. This makes it particularly useful for studying non-linear systems and is often used in quantum gravity and string theory.

4. What are some applications of the Schwinger Quantum action?

The Schwinger Quantum action has been applied to a wide range of physical systems, including electromagnetism, quantum chromodynamics, and quantum gravity. It is also used in condensed matter physics and has been applied to study phenomena such as superconductivity and the quantum Hall effect.

5. Are there any current developments or advancements in the use of the Schwinger Quantum action?

Yes, there are ongoing developments and advancements in the use of the Schwinger Quantum action. These include its application in high-energy physics, particularly in the study of quantum chromodynamics, and in condensed matter physics to investigate topological phases of matter. Additionally, there are efforts to extend the Schwinger Quantum action to include quantum effects in curved spacetime.

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