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

"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?

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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 $x$.
Okay, I have too many questions now! So you used vector.vector to produce a scalar and showed this will be zero due to Lorentz invariance. What about chiral condensates where you have $<\bar{\psi} \psi> \neq 0$? – Joman Jun 19 '12 at 14:13
@Joman: $\overline\psi\psi$ is a scalar, hence you cannot apply the same argument. – Arnold Neumaier Jun 19 '12 at 15:23