Schwartz QFT book, equation 15.59

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In summary, the author explains the process of rewriting the derivative and performing an integral in the context of a mathematical equation. The steps involve using the chain rule and the quotient rule for derivatives.
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Pnin
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Can someone please explain what the author does here in 15.59? I do not understand both steps. Neither the rewriting of the derivative, nor the integral.

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thank you
 
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Attachment seems hard to open.
 
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I pasted an image.
 
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Pnin said:
Can someone please explain what the author does here in 15.59? I do not understand both steps. Neither the rewriting of the derivative, nor the integral.

View attachment 256512

thank you
I'm guessing the bit that's confusing you about the derivative rewrite involves the following:$$s ~=~ p^2 ~=~ p^\alpha p_\alpha ~,~~~~ \Rightarrow~~ p^\alpha = \hat p^\alpha \, s^ {1/2} ~,$$where the overhat denotes a unit vector. (This assumes ##p^\alpha## is not lightlike.) Then,$$\frac{\partial p^\alpha}{\partial s} ~=~ \frac{\partial s^ {1/2}}{\partial s} \; \hat p^\alpha ~=~ \frac{1}{ 2 s^ {1/2}}\; \hat p^\alpha ~=~ \frac{s^{1/2}}{ 2 s}\; \hat p^\alpha ~=~ \frac{p^\alpha}{2s} ~.$$
So, using the chain rule,$$\frac{\partial M}{\partial s} ~=~ \frac{\partial p^\mu}{\partial s} \; \frac{\partial M}{\partial p^\mu} ~=~ \frac{p^\alpha}{2s} \; \frac{\partial M}{\partial p^\mu} ~.$$
Next, the ##\partial/\partial p^\mu## operator passes through the integral sign (since the variables ##k## and ##p## are independent). Then it's just a matter of performing $$\frac{\partial}{\partial p^\mu} \; \frac{1}{(p - k)^2} $$which is merely an application of the usual quotient rule for derivatives. (Can you do that bit?)
 
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Fantastic! Crystal-clear now. Thanks strangerep, much appreciated!
 

Related to Schwartz QFT book, equation 15.59

1. What is the significance of equation 15.59 in Schwartz's QFT book?

Equation 15.59 in Schwartz's QFT book is commonly known as the Feynman-Kac formula. It is a powerful mathematical tool used to calculate the expectation value of a quantum field theory operator. This equation plays a crucial role in understanding the behavior of quantum fields and has numerous applications in theoretical physics.

2. How is equation 15.59 derived in Schwartz's QFT book?

Equation 15.59 is derived using functional integration techniques, specifically the path integral formulation of quantum mechanics. It involves summing over all possible paths of a quantum field and assigning a weight to each path based on its action. The resulting expression is known as the Feynman-Kac formula.

3. Can equation 15.59 be applied to any quantum field theory?

Yes, equation 15.59 can be applied to any quantum field theory, as long as it satisfies certain conditions such as being a local, Lorentz-invariant theory. This equation is a fundamental result in quantum field theory and is applicable to a wide range of physical systems.

4. What are some practical applications of equation 15.59?

Equation 15.59 has numerous practical applications in theoretical physics, particularly in the study of quantum field theories. It is used to calculate the expectation value of operators, which can then be compared to experimental results. This equation also plays a crucial role in the development of quantum field theory models and in understanding the behavior of quantum systems.

5. Are there any limitations to the use of equation 15.59?

While equation 15.59 is a powerful tool, it does have some limitations. It is based on the assumption that quantum fields are continuous and infinitely divisible, which may not always be the case. Additionally, it can be challenging to apply this equation in systems with strong interactions, and in some cases, numerical techniques may be necessary to obtain accurate results.

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