Peskin-Schroeder - Eqn 2.45 derivation

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In summary, the conversation discusses issues with the equations leading up to eqn 2.45 on page 25, specifically with the terms in the \phi(x) and \pi(x) commutators. The missing term in the \pi(x) commutator is identified as the n-1 boundary term, which can be eliminated using integration by parts under certain conditions.
  • #1
giant_bog
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I'm having problems with the equations leading up to eqn 2.45 on page 25. The hamiltonian has a [itex](\nabla\phi)^2 + m^2 \phi^2[/itex] term in the [itex]\phi(x)[/itex] commutator and in the [itex]\pi(x)[/itex] commutator it's [itex]\phi(-\nabla^2 + m^2) \phi[/itex].

I'm aware of a vector calculus identity that makes [itex](\nabla\phi)^2 = 1/2 (\nabla^2[\phi^2]) - \phi \nabla^2 \phi[/itex].

That's almost what we have here, but the [itex]\frac{1}{2}(\nabla^2[\phi^2])[/itex] term is missing in the [itex]\pi(x)[/itex] commutator.

Did anybody see where it went?
 
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  • #2
That's the n-1 boundary term which is taken be zero.
 
  • #3
You can do this kind of thing using integration by parts. E.g., in one dimension:

[tex]\int_{-\infty}^{+\infty} \left(\frac{df}{dx}\right)^2 dx = \left[f(x) \frac{df}{dx}\right]_{-\infty}^{+\infty} - \int_{-\infty}^{\infty}f(x)\frac{d^2 f}{dx^2}dx[/tex]

And if f(x) goes to zero at infinity then the first term on the right drops out.
 
  • #4
I see it now. Thanks, folks.
 

Related to Peskin-Schroeder - Eqn 2.45 derivation

1. What is Eqn 2.45 in the Peskin-Schroeder derivation?

Eqn 2.45 in the Peskin-Schroeder derivation is the equation for the propagator of a free scalar field in momentum space. It represents the amplitude for a particle to propagate from one point to another in spacetime.

2. How is Eqn 2.45 derived in the Peskin-Schroeder derivation?

Eqn 2.45 is derived using the quantum field theory techniques of path integrals and Feynman diagrams. It involves summing over all possible paths of the particle and taking into account the interactions with other particles and the physical conditions of the system.

3. What is the significance of Eqn 2.45 in the Peskin-Schroeder derivation?

Eqn 2.45 is significant because it allows us to calculate the probability amplitude for a particle to propagate from one point to another in spacetime. This is a fundamental concept in quantum field theory and is used in many calculations and predictions for physical phenomena.

4. How is Eqn 2.45 used in practical applications?

Eqn 2.45 is used in practical applications to calculate scattering amplitudes and decay rates in particle physics experiments. It is also used in theoretical calculations and predictions for various physical phenomena, such as particle interactions and quantum field theories.

5. Are there any limitations or assumptions in the derivation of Eqn 2.45 in the Peskin-Schroeder derivation?

Yes, there are limitations and assumptions in the derivation of Eqn 2.45. It assumes a free scalar field, neglecting interactions with other particles, and also assumes a flat spacetime. Additionally, the derivation relies on the use of perturbation theory, which may not be applicable in all cases.

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