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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 [itex]B_{a}[/itex] that interacts with fermions such that it affects their energy dispersion. It appears in the energies in the form

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

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

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

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

Thanks in advance!

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# Shifting integration variable when determing population densities

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