# Shifting integration variable when determing population densities

• "Don't panic!"
In summary, the conversation discusses the interaction between a space-time dependent field and fermions, and how it affects their energy dispersion. The question is raised about why the integration variable can only be shifted if the field is constant, and it is hypothesized that if the field is not constant, the fermions would fluctuate in time. The conversation concludes with the understanding that a solution can only be attempted with an explicit form for the field.
"Don't panic!"
Hi,

I'm hoping someone can enlighten me on this as I'm a little bit fuzzy on the reasoning:

Say I have a space-time dependent field $B_{a}$ that interacts with fermions such that it affects their energy dispersion. It appears in the energies in the form

$$E\sim\sqrt{\left(\vec{p}+\vec{B}\right)-m^{2}}+B_{0}$$

Why is it, that when I then calculate the number density of fermions in such a scenario, i.e.

$$n\sim\int^{+\infty}_{-\infty}\frac{d^{3}p}{\left(2\pi\right)^{3}}\frac{1}{\exp{\left(E/k_{_{B}}T\right)}+1}$$

(where in this case the chemical potential is negligible) that I can only shift the integration variable, such that $\vec{p}\rightarrow \vec{p}+\vec{B}$ (thus "absorbing" the 3-vector components of $B_{a}$), if I consider $B_{a}$ to be constant?

Apologies for the spelling mistake in the title of the thread by the way, should be "determining" , but don't know how to retroactively edit it!

What do you think would happen to d3p if B is not constant?

Would it be that it becomes time dependent and thus coupled to the fluctuations in B over time?

or more explicitly, that you would also introduce an additional integral over $d^{3}B$?

Slow down with the questions and answer my question in post #3

Last edited:
sorry, they were my attempts at a possible answer (shouldn't have included the question marks)!

I assume that you would have $d^{3}p\rightarrow d^{3}\left(p+B\right)=d^{3}p'$ and so, as B is not constant, one could not talk of set momentum states for the fermions as they would fluctuate in time depending on the fluctuations in B.

"Don't panic!" said:
sorry, they were my attempts at a possible answer (shouldn't have included the question marks)!

I assume that you would have $d^{3}p\rightarrow d^{3}\left(p+B\right)=d^{3}p'$ and so, as B is not constant, one could not talk of set momentum states for the fermions as they would fluctuate in time depending on the fluctuations in B.

Correct. If you have a explicit form for B then you might attempt a solution. You can't go much further with the general expression, I don't think

ok, that's cleared things up a bit. Thanks for your time.

## What is shifting integration variable?

Shifting integration variable is a technique used in mathematical calculations to simplify the integration process by changing the variable of integration. This technique is commonly used in determining population densities in scientific research.

## Why is shifting integration variable important in determining population densities?

Shifting integration variable allows for easier integration of complex functions, making it easier to calculate population densities accurately. It also helps to avoid errors and inconsistencies in the calculations.

## How is shifting integration variable used in determining population densities?

In determining population densities, shifting integration variable involves changing the variable of integration to a more convenient one, usually related to the population distribution. This simplifies the integration process and makes it easier to obtain accurate results.

## What are the benefits of using shifting integration variable in determining population densities?

Using shifting integration variable in determining population densities can reduce the complexity of the integration process, improve the accuracy of the calculations, and provide a better understanding of the underlying population distribution.

## Are there any limitations or drawbacks to using shifting integration variable in determining population densities?

One limitation of shifting integration variable is that it may not always be applicable, especially in cases where the population distribution is unknown or cannot be easily expressed in terms of a different variable. It also requires a good understanding of mathematical concepts and techniques.

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