Schwinger trick and following change of variables

In summary, the conversation is about the evaluation of an integral associated with the electron self-energy diagram, following the book "Quantum Field Theory of point particles and strings" by Brian Hatfield. The individual is having difficulty with changing the integration limits from z_2 to 1-z_2, and after further examination, realizes that the previous integration over z_1 with a delta function has automatically changed the integration limits for z_2. The correct integration limits for z_2 should be from 0 to 1, resulting in a straightforward change of variables.
  • #1
André H Gomes
3
0

Homework Statement



We have the evaluation of the integral associated with the electron self-energy diagram and I am following Brian Hatfield's book "Quantum Field Theory of point particles and strings", and I am having problems with the integration limits after changing variables from [itex]z_2[/itex] to [itex]1-z_2[/itex], at p.388.

Homework Equations



[itex]\int_0^\infty \frac{d\beta}{\beta} \int_0^\infty dz_2 \left[2m_o - (1-z_2)\gamma^\mu p_\mu)\right] e^{-\beta[(1-z_2)z_2 p^2 - m^2_o z_2 - \lambda^2(1-z_2)]}[/itex]

Since it may be relevant, some additional information: before this integral, we had an integration over [itex]z_1[/itex], which involved a Dirac delta function, [itex]\delta(1-z_1-z_2)[/itex].

The Attempt at a Solution



I expected the change of variables to be as straightfoward as it seemed to be, resulting in:

[itex]\int_0^\infty \frac{d\beta}{\beta} \int_{-\infty}^1 dz \left(2m_o - z\gamma^\mu p_\mu)\right) e^{-\beta[z(1-z) p^2 - m^2_o(1-z) - \lambda^2z]}[/itex]

But, in Hatfield's book, the lower limit of the integral in [itex]z[/itex] becomes [itex]0[/itex] instead of [itex]-\infty[/itex]. I wonder if the integration from [itex]-\infty[/itex] to [itex]0[/itex] may vanish somehow, but I just can't figure it.

In advance, thank you all for the help!

(English is not my native language. I apologize any mistake.)
 
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  • #2
If think I found my mistake: evaluating the previous [itex]z_1[/itex] integration with the delta function will automatically change the integration limits of the [itex]z_2[/itex] integration.

Looking at the step before the above integral, where we have the integral over [itex]z_1[/itex], from [itex]z_1=0[/itex] to [itex]z_1=\infty[/itex], we note that:

- There is the delta, [itex]\delta(1-z_1-z_2)[/itex], demanding [itex]z_1=1-z_2[/itex].

- But [itex]z_2[/itex] is integrated from [itex]z_2=0[/itex] to [itex]z_2=\infty[/itex], saying that [itex]z_1[/itex] may assume values between [itex]-\infty[/itex] and [itex]1[/itex].

- On the other hand the integration over [itex]z_1[/itex] puts its values between [itex]0[/itex] and [itex]\infty[/itex].

- The range of values [itex]z_2[/itex] may assume so that the delta function argument is satisfied goes from [itex]0[/itex] to [itex]1[/itex].

Therefore, this should be the range of integration for the [itex]z_2[/itex] integration after the evaluation of the delta function: from [itex]z_2=0[/itex] to [itex]z_2=1[/itex]. In such a way, the change of variables from [itex]z_2[/itex] to [itex]z=1-z_2[/itex] is indeed as straightfoward as it seems, and also the integration limits are as expected.
 

Related to Schwinger trick and following change of variables

1. What is the Schwinger trick and how is it used?

The Schwinger trick is a technique used in quantum field theory to simplify calculations involving path integrals. It involves rewriting a product of path integrals as a single path integral with an extra integration over a new variable. This allows for the use of techniques such as Gaussian integration to solve the integral.

2. Can the Schwinger trick be applied in other areas of physics?

Yes, the Schwinger trick can also be applied in statistical mechanics and condensed matter physics, where path integrals are commonly used to calculate partition functions and correlation functions.

3. Why is the Schwinger trick useful in quantum field theory?

The Schwinger trick allows for the simplification of complex path integrals, making calculations more manageable. It also helps to reveal underlying symmetries and allows for the use of powerful mathematical techniques to solve the integral.

4. What is the following change of variables and how is it related to the Schwinger trick?

The following change of variables is a mathematical technique used to transform an integral into a different form. It is often used in conjunction with the Schwinger trick to simplify path integrals and reveal underlying symmetries.

5. Are there any limitations to using the Schwinger trick and following change of variables?

While the Schwinger trick and following change of variables are powerful techniques, they are not applicable to all path integrals. In some cases, the integrals may not be solvable using these methods, and other techniques must be used.

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