Physics

In Srednicki's proof of the spin-statistics theorem for scalar fields ("Quantum Field Theory", section 4), he considers interaction terms, added to the free Hamiltonian, that are Hermitian functions of $$\varphi^+(x)=\int\frac{d^3k}{(2\pi)^32\omega}e^{ikx}a(\mathbf{k})$$ and $$\varphi^-(x)=\int\frac{d^3k}{(2\pi)^32\omega}e^{-ikx}a^\dagger(\mathbf{k}).$$ He then goes on to show that we will only get Lorentz-invariant transition amplitudes if we are dealing with the theory for a real scalar field obeying Bose-Einstein statistics.

Question: Why do we only need to consider interaction terms that are functions of $\varphi^+$ and $\varphi^-$? Couldn't there be other possible interaction terms? For example, $H_1=a^\dagger(\mathbf{k})+a(\mathbf{k})$?

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Interaction terms in Srednicki's proof of spin-statistics theorem

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