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Shifting integration variable when determing population densities

  1. Mar 26, 2014 #1
    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 [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!
     
  2. jcsd
  3. Mar 26, 2014 #2
    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!
     
  4. Mar 26, 2014 #3
    What do you think would happen to d3p if B is not constant?
     
  5. Mar 26, 2014 #4
    Would it be that it becomes time dependent and thus coupled to the fluctuations in B over time?
     
  6. Mar 26, 2014 #5
    or more explicitly, that you would also introduce an additional integral over [itex]d^{3}B[/itex]?
     
  7. Mar 26, 2014 #6
    Slow down with the questions and answer my question in post #3
     
    Last edited: Mar 26, 2014
  8. Mar 26, 2014 #7
    sorry, they were my attempts at a possible answer (shouldn't have included the question marks)!

    I assume that you would have [itex]d^{3}p\rightarrow d^{3}\left(p+B\right)=d^{3}p'[/itex] 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.
     
  9. Mar 26, 2014 #8
    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
     
  10. Mar 26, 2014 #9
    ok, that's cleared things up a bit. Thanks for your time.
     
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