- #1

diracsgrandgrandson

- 6

- 1

- Homework Statement
- Prove that the integral $$\int d^3p(\xi^r)^T\mathbf{p}\mathbf{\sigma}\xi^s(a^{r\dagger}_pa^s_p+a^{s\dagger}_pa^r_p)$$ vanishes.

- Relevant Equations
- Here, ##\xi^{1,2} = (1,0)^T,(0,1)^T## are spinors. We are working with Dirac particles, so the anticommutation relations are given by ##\{a^{r\dagger}_p,a^s_q\} = \delta^{rs}\delta(p-q)##. Note that all in my notation here ##\mathbf{p} = p## (I am lazy but there are no four-momenta in this question, so no risk of confusion).

We use the invariance of the measure under ##p\rightarrow -p## to get $$-\int d^3p\xi^{rT}\mathbf{p}\mathbf{\sigma}\xi^s(a^{r\dagger}_{-p}a^s_{-p}+a^{s\dagger}_{-p}a^r_{-p}) = -\int d^3p\xi^{rT}\mathbf{p}\mathbf{\sigma}\xi^sA(-p).$$ If this pesky ##A(-p)## can be shown to be equal to ##A(p)## or somehow ##p##-independent, then it all works. But I don't know how. I know the anticommutation relations are somehow involved but the complication is that ##p=q## which gives a delta function evaluated at zero.