Lorentz invariance and the vacuum expectation value of fields with spin > 0

Question

I had a question about Moduli space, which I was reading about here, but then I read this sentence:

"Lorentz invariance forces the vacuum expectation values of any higher spin fields to vanish."

Can someone explain how exactly this happens? Or at least suggest an exercise to carry out?

Answer 1

There is a Lorentz transformation that maps a spacelike vector $u$ to $-u$. If $A(x)$ is a field of spin 1 with $\langle u \cdot A(0)\rangle = c$ then applying the Lorentz transform we find $-c=c$ and hence $c=0$. Doing this for all spacelike vectors implies $\langle A(0)\rangle = 0$, and translation invariance then gives $\langle A(x)\rangle =0$ for all $x$.

For other spins the argument is similar. You are welcome to try the spinor case as an exercise.