I'm a mathematician slowly trying to teach myself quantum field theory. To test my understanding, I'm trying to tell myself the whole story from a Lagrangian to scattering amplitudes for scalar $\phi^4$ theory in a way that feels satisfying. There's one thing that's kind of bugging me.

One of the hypotheses of the LSZ formula is that the fields you're using have vanishing vacuum expectation value, so in the present case that $$\langle \Omega|\phi(x)|\Omega\rangle = 0.$$ For non-scalar fields, I buy that this follows from Lorentz-invariance, but I've also seen it claimed in a few books (e.g. in Section 10.4 of Ticciati's QFT for Mathematicians) that for $\phi^4$ theory in particular it follows from the fact that $\phi\mapsto-\phi$ is a symmetry of the Hamiltonian.

My question is: How exactly does this argument go?

The only idea I've been able to come up with is that maybe there's a unitary operator $U$ which flips the sign of $\phi$ (that is, I guess, it has the property $U^{-1}\phi(x)U=-\phi(x)$) and then you'd argue that $U$ commutes with the Hamiltonian and therefore preserves its eigenstates, but I don't see why (or if) I should expect $U$ to exist, or if this is even the right idea at all.

Thanks, and of course please let me know if I made a mistake somewhere earlier in the process of arriving at this question!

I'm a mathematician slowly trying to teach myself quantum field theory. To test my understanding, I'm trying to tell myself the whole story from a Lagrangian to scattering amplitudes for scalar $\phi^4$ theory in a way that feels satisfying. There's one thing that's kind of bugging me.

One of the hypotheses of the LSZ formula is that the fields you're using have vanishing vacuum expectation value, so in the present case that $$\langle \Omega|\phi(x)|\Omega\rangle = 0.$$ For non-scalar fields, I buy that this follows from Lorentz-invariance, but I've also seen it claimed in a few books (e.g. in Section 10.4 of Ticciati's QFT for Mathematicians) that for $\phi^4$ theory in particular it follows from the fact that $$\phi\mapsto-\phi$$ is a symmetry of the Hamiltonian.

My question is: How exactly does this argument go?

The only idea I've been able to come up with is that maybe there's a unitary operator $U$ which flips the sign of $\phi$, that is, I guess, it has the property $$U^{-1}\phi(x)U=-\phi(x),$$ and then you'd argue that $U$ commutes with the Hamiltonian and therefore preserves its eigenstates, but I don't see why (or if) I should expect $U$ to exist, or if this is even the right idea at all.

Thanks, and of course please let me know if I made a mistake somewhere earlier in the process of arriving at this question!

I'm a mathematician slowly trying to teach myself quantum field theory. To test my understanding, I'm trying to tell myself the whole story from a Lagrangian to scattering amplitudes for scalar $\phi^4$ theory in a way that feels satisfying. There's one thing that's kind of bugging me.

One of the hypotheses of the LSZ formula is that the fields you're using have vanishing vacuum expectation value, so in the present case that $\langle \Omega|\phi(x)|\Omega\rangle = 0$. For non-scalar fields, I buy that this follows from Lorentz-invariance, but I've also seen it claimed in a few books (e.g. in Section 10.4 of Ticciati's QFT for Mathematicians) that for $\phi^4$ theory in particular it follows from the fact that $\phi\mapsto-\phi$ is a symmetry of the Hamiltonian.

My question is: How exactly does this argument go?

The only idea I've been able to come up with is that maybe there's a unitary operator $U$ which flips the sign of $\phi$ (that is, I guess, it has the property $U^{-1}\phi(x)U=-\phi(x)$) and then you'd argue that $U$ commutes with the Hamiltonian and therefore preserves its eigenstates, but I don't see why (or if) I should expect $U$ to exist, or if this is even the right idea at all.

Thanks, and of course please let me know if I made a mistake somewhere earlier in the process of arriving at this question!

I'm a mathematician slowly trying to teach myself quantum field theory. To test my understanding, I'm trying to tell myself the whole story from a Lagrangian to scattering amplitudes for scalar $\phi^4$ theory in a way that feels satisfying. There's one thing that's kind of bugging me.

One of the hypotheses of the LSZ formula is that the fields you're using have vanishing vacuum expectation value, so in the present case that $$\langle \Omega|\phi(x)|\Omega\rangle = 0.$$ For non-scalar fields, I buy that this follows from Lorentz-invariance, but I've also seen it claimed in a few books (e.g. in Section 10.4 of Ticciati's QFT for Mathematicians) that for $\phi^4$ theory in particular it follows from the fact that $\phi\mapsto-\phi$ is a symmetry of the Hamiltonian.

My question is: How exactly does this argument go?

The only idea I've been able to come up with is that maybe there's a unitary operator $U$ which flips the sign of $\phi$ (that is, I guess, it has the property $U^{-1}\phi(x)U=-\phi(x)$) and then you'd argue that $U$ commutes with the Hamiltonian and therefore preserves its eigenstates, but I don't see why (or if) I should expect $U$ to exist, or if this is even the right idea at all.

Thanks, and of course please let me know if I made a mistake somewhere earlier in the process of arriving at this question!