Understanding Faddeev-Popov in Schwartz QFT Book

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In summary, the Faddeev-Popov procedure in Schwartz's QFT book involves a functional integral extension of the usual relation for delta functions. This expression equals one due to the n-dimensional vector field being a function of another vector. This is represented by the functional integral extension and involves the determinant of the partial derivatives of the vector function.
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Lapidus
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In Schwartz QFT book, while explaining the Faddeev-Popov procedure, he presents this following observation at (25.99):
upload_2015-1-13_13-16-6.png


Can someone help understanding me why this expression equals one?

THANK YOU!
 
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This is the functional integral extention of the usual relation for a delta functions. Suppose you have an n-dimentional vector field which is a function of another vector, say ##\vec v\equiv \vec v(\vec x)##. Then you have:

$$
1=\int d^nv \delta^n(\vec v)=\int d^nx\det\left(\frac{\partial v^i}{\partial x^j}\right) \delta^n(\vec v(\vec x))
$$

It's quite intuitive that the functional integral extension is the one you wrote.
 
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1. What is Faddeev-Popov gauge fixing in quantum field theory?

Faddeev-Popov gauge fixing is a mathematical technique used in quantum field theory to simplify calculations and remove redundant information. It involves choosing a specific gauge (or mathematical representation) for the fields in a quantum field theory, which allows for the elimination of unphysical degrees of freedom and simplifies the equations of motion.

2. Why is Faddeev-Popov gauge fixing important in quantum field theory?

Faddeev-Popov gauge fixing is important because it allows for a consistent and well-defined mathematical framework for performing calculations and making predictions in quantum field theory. Without gauge fixing, the equations of motion can become ambiguous and calculations can become extremely complicated.

3. How does Faddeev-Popov gauge fixing work in practice?

In practice, Faddeev-Popov gauge fixing involves choosing a specific gauge condition (such as the Lorenz gauge or the Coulomb gauge) and using mathematical operators to transform the quantum fields into that gauge. This results in a simplified set of equations of motion, which can then be used to make calculations and predictions.

4. What are the benefits of using Faddeev-Popov gauge fixing?

The main benefit of using Faddeev-Popov gauge fixing is the simplification of calculations in quantum field theory. By removing unphysical degrees of freedom and choosing a specific gauge, the equations of motion become easier to solve and make predictions from. Additionally, gauge fixing helps to ensure that calculations are consistent and unambiguous.

5. Are there any limitations to using Faddeev-Popov gauge fixing?

One limitation of Faddeev-Popov gauge fixing is that it introduces a dependence on the choice of gauge. Different gauge choices can result in different physical predictions, although in most cases these differences are negligible. Additionally, gauge fixing can become more complicated in certain scenarios, such as when dealing with non-abelian gauge theories.

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