Peskin and Schroeder derivation of Klein-Gordon propagator

In summary, the derivation of the Klein-Gordon propagator in Peskin and Schroeder's "An Introduction to Quantum Field Theory" involves changing the dummy variable in an integral, resulting in the conversion of one of the exponentials to its negative form. This explains why p^0=-E_p in the second step of equation (2.54) and why "ip(x-y)" is changed to "-ip(x-y)".
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chern
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In page 30 of book "An introduction to quantum field theory" by Peskin and Schroeder in the derivation of Klein-Gordon propagator, why p^0=-E_p in the second step in equation (2.54). and why change "ip(x-y)" to "-ip(x-y)"? I thought a lot time, but get no idea. Thank you for your giving me an explanation.
 
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chern said:
In page 30 of book "An introduction to quantum field theory" by Peskin and Schroeder in the derivation of Klein-Gordon propagator, why p^0=-E_p in the second step in equation (2.54). and why change "ip(x-y)" to "-ip(x-y)"?
It took me a long time to figure that out too, when I first studied P+S.

First, look at this 1D integral:
$$
\int_{-\infty}^{+\infty} dp \; e^{-ipx} ~.
$$ If you perform a change of dummy variable ##p \to p' = -p##, what do you get?

So in the 2nd step of (2.54), they're just converting the ##e^{ip\cdot(x-y)}## of the 2nd term in the last line on the previous page 29, so that both exponentials are the same, i.e., ##e^{-ip\cdot(x-y)}##. (The latter explains "why p^0=-E_p").
 
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Thank you!
 

Related to Peskin and Schroeder derivation of Klein-Gordon propagator

What is the Klein-Gordon propagator?

The Klein-Gordon propagator is a mathematical expression that describes the propagation of a scalar particle, such as a Higgs boson or a pion, in quantum field theory. It is derived from the Klein-Gordon equation, which is a relativistic wave equation that describes the behavior of spinless particles.

How is the Klein-Gordon propagator derived?

The Klein-Gordon propagator is derived using the path integral formalism in quantum field theory. Specifically, it is derived in the textbook "An Introduction to Quantum Field Theory" by Peskin and Schroeder, which is a commonly used reference for graduate-level courses in particle physics.

Why is the Klein-Gordon propagator important?

The Klein-Gordon propagator is important because it allows us to calculate the probability amplitude for a scalar particle to propagate from one point to another in spacetime. This is a fundamental quantity in quantum field theory and is used in many calculations, such as calculating scattering amplitudes or particle decay rates.

What are some applications of the Klein-Gordon propagator?

The Klein-Gordon propagator has many applications in theoretical and experimental physics. It is used to calculate scattering amplitudes in particle physics experiments, as well as to study the behavior of scalar fields in cosmology and astrophysics. It is also used in condensed matter physics to study the behavior of quasiparticles in materials.

What is the relationship between the Klein-Gordon propagator and the Feynman propagator?

The Klein-Gordon propagator and the Feynman propagator are closely related, as the Feynman propagator can be obtained from the Klein-Gordon propagator by performing a Wick rotation. The Feynman propagator is used in perturbative calculations in quantum field theory, while the Klein-Gordon propagator is used in non-perturbative calculations.

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