Integrating by parts in path integral (Zee)

In summary, the conversation discusses the integration by parts method and how it applies to equations 14 and 15 in Zee. The concept is further explained through the example of Green's identity equation in 3D and its extension to 4D. The conversation concludes with the confirmation that the boundary term vanishes when boundary conditions are imposed.
  • #1
GreyBadger
23
0
Hi all,

I have an exceptionally basic question, taken from P21 of Zee. Eq. 14 is

[tex]Z=\int D\psi e^{i\int d^4x(\frac{1}{2}[(\partial\psi )^2-m^2\psi^2] + J\psi)}[/tex]

The statement is then made that 'Integrating by parts under the [tex]\int d^4x[/tex]' leads to Eq. 15:

[tex]Z=\int D\psi e^{i\int d^4x[-\frac{1}{2}\psi(\partial^2+m^2)\psi + J\psi]}[/tex].

Now, I am being supremely thick, but I don't see how this follows. Could somebody please spell it out in small words?
 
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  • #2
Consider the Green's identity equation (4) in the link which is basically a 3D version of integrating by parts:

http://mathworld.wolfram.com/GreensIdentities.html

and taking the surface integral to be zero implies that

[tex] \int dV \;\nabla \phi \nabla \psi = -\int dV \;\phi \nabla^2 \psi [/tex]

and so in this example, one can extend that to 4D and take the volume boundary term to zero
 
  • #3
"Partially integrating" means here that you use partial differentation to rewrite the integrand.

Here the relevant term becomes

[tex]
\int_{\Omega} \partial \phi \partial \phi = \int_{\Omega} \partial(\phi\partial\phi) - \int_{\Omega} \partial^2 \phi
[/tex]

The first term on the RHS becomes

[tex]
\int_{\Omega} \partial(\phi\partial\phi) = \int_{\partial\Omega} \phi\partial\phi
[/tex]

by Stokes theorem. Imposing boundary conditions, this term vanishes.
 
  • #4
Aha yes. Thank you both.
 

1. What is the purpose of integrating by parts in path integral?

Integrating by parts in path integral is used to simplify and solve complex integrals that arise in quantum field theory. It allows us to transform the original integral into a more manageable form, making it easier to solve and analyze.

2. How is integrating by parts different in path integral compared to other integration methods?

In path integral, integrating by parts involves the manipulation of functional integrals instead of ordinary integrals. This means that the functions involved are not just numbers, but also include the path or trajectory of a particle or field. This makes the process more complex, but also more powerful in solving problems in quantum field theory.

3. What are the steps involved in integrating by parts in path integral?

The first step is to identify the integral that needs to be solved. Then, we apply the integration by parts formula, which involves the product of two functions and their respective derivatives. Next, we simplify the integral using algebraic manipulation, and finally, we solve for the original integral.

4. Can integrating by parts be used for all path integrals?

Integrating by parts can be used for most path integrals, but it may not always result in a simpler integral. In some cases, it may even make the integral more complicated. It is important to carefully consider the integral and the functions involved before deciding to use this method.

5. Are there any limitations to integrating by parts in path integral?

Integrating by parts in path integral may not always work for highly non-linear or non-local theories. In these cases, other methods may need to be used to solve the integral. Additionally, it may not always be possible to find an exact solution, and numerical methods may need to be employed.

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