Consider a spinor field ψ(x). Its vacuum expectation value is given by v=⟨0|ψ(x)|0⟩.
Using the fact that the vaccum is invariant under Lorentz transformation, we get, v=⟨0|ψ(0)|0⟩.
Why is it that, if v≠0, the Lorentz invariance is broken?
Answer
The v you write is itself a spinor, not a scalar. A non-zero spinor is obviously not invariant under Lorentz transformations, so a non-zero spinorial VEV breaks Lorentz invariance of the 1-point function.
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