How to Derive the Second Expression in @Peskin Eqn 2.54?

  • Thread starter SuperStringboy
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In summary, the conversation discusses how to derive the second expression from the first one, specifically addressing the issue of the second term in the second expression. The conversation also mentions the possibility of changing the dummy variable and the potential changes in the measure d3p. The solution is eventually provided, explaining the reason why the sign of d3p does not change under p -> -p.
  • #1
SuperStringboy
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Homework Statement


I am facing problem to derive the 2nd expression from the first one. My problem is the 2nd term of the 2nd expression.

Homework Equations


[tex]
\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}[\exp(-ip\cdot(x - y)) - \exp(ip\cdot (x - y))]=\int\ \frac{d^3p} {(2\pi)^3}\ \{ \frac {1}{2E_p}\ e^{-ip.(x-y)}\left|_{p^0 = E_p}\ +\ \frac {1}{-2E_p}\ e^{-ip.(x-y)}\left|_{p^0 = -E_p}\ \}
[/tex]

The Attempt at a Solution


[tex]
p\cdot (x - y)= p^0(x^0 - y^0) - \textbf p\cdot(x-y)
[/tex]

For Po = - Ep we can take
[tex]
p\cdot (x - y)= - p^0(x^0 - y^0) - \textbf p\cdot(x-y)
[/tex]
If i am not wrong yet, then what now?
should i change the dummy variable as p = - p? But if do it then i think another change comes d3p becomes -d3p for the 2nd term and i loose the minus sign before the 2nd term.

i don't know how much wrong i am but i am expecting good solution from you guys.
 
Last edited:
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  • #2
SuperStringboy said:
If i am not wrong yet, then what now?
should i change the dummy variable as p = - p? But if do it then i thing another change comes d3p becomes -d3p for the 2nd term and i loose the minus sign before the 2nd term.

The trick is that the measure d3p actually does not change sign under p -> -p.
 
  • #3
Thanks Ben . But will you please explain that why the sign of d3 does not change. Is it because of spherical co-ordinates : d3p = p2sin(theta)d(theta)d(phi)dp , where p = |p| ?
 
Last edited:

Related to How to Derive the Second Expression in @Peskin Eqn 2.54?

What is the Peskin Equation 2.54?

The Peskin Equation 2.54 is a mathematical equation used in quantum mechanics to describe the dynamics of a quantum field. It is named after physicist Michael Peskin, who first derived the equation.

Why is solving @Peskin Eqn 2.54 important?

Solving @Peskin Eqn 2.54 is important because it allows us to understand the behavior of quantum fields, which are fundamental to our understanding of the universe. It also has practical applications in fields such as particle physics and condensed matter physics.

What are the challenges in solving @Peskin Eqn 2.54?

One of the main challenges in solving @Peskin Eqn 2.54 is the complexity of the equation itself. It involves multiple variables and can be difficult to manipulate mathematically. Additionally, it requires a strong understanding of quantum mechanics and mathematical techniques such as functional analysis.

What methods can be used to solve @Peskin Eqn 2.54?

There are several methods that can be used to solve @Peskin Eqn 2.54, including numerical techniques such as Monte Carlo simulations and analytical methods such as perturbation theory. The choice of method depends on the specific problem being solved and the desired level of accuracy.

What are the potential applications of solving @Peskin Eqn 2.54?

Solving @Peskin Eqn 2.54 has potential applications in various fields of physics, including particle physics, condensed matter physics, and cosmology. It can also be used in the development of new technologies, such as quantum computing and quantum sensors.

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