I have been confused by two details of Weinberg's delivery of spontaneous symmetry breaking(SSB) in Volume II of his QFT textbook. I'm reading a paperback, so the edition must be after 2005, and both of the confusions come from page 164-166.

(1)On page 164, when Weinberg tries to justify that there are preferred basis among the degenerate vacua for the system to fall into, he starts with a system with a simple symmetry $\phi \to -\phi$, and he writes

For instance, in a theory with a symmetry $\phi \to -\phi$, even if $\Gamma(\phi)$(quantum effective action) has a minimum for some nonzero value $\bar{\phi}$ of $\phi$, how do we know that the true vacuum is one of the states $|\text{VAC},\pm\rangle$ for which $\Phi$ has expectation values $\bar{\phi}$ and $-\bar{\phi}$, and not some linear combination like $|\text{VAC}, +\rangle+|\text{VAC}, -\rangle$ that *would *respect the symmetry under $\phi \to -\phi$? The assumed symmetry under the transformation $\phi \to -\phi$ tells us that the vacuum matrix elements of the Hamiltonian are

\[\langle \text{VAC}, +|H|\text{VAC}, +\rangle=\langle \text{VAC}, -|H|\text{VAC}, -\rangle\equiv a\]

(with $a$ real) and

\[\langle \text{VAC}, +|H|\text{VAC}, -\rangle=\langle \text{VAC}, -|H|\text{VAC}, +\rangle\equiv b\]

(with $b$ real), so the eigenstates of the Hamiltonian are $|\text{VAC}, +\rangle \pm |\text{VAC}, -\rangle$, with energies $a\pm|b|$.

My confusion only concerns the last sentence, shouldn't $a$ be a common eigenvalue of $|\text{VAC}, \pm\rangle$, so $a$ will also be the eigenvalue of $|\text{VAC}, +\rangle \pm |\text{VAC}, -\rangle$? Where does $a\pm|b|$ come from? (In fact, the later discussions seem to remain intact if the eigenvalue is $a$ instead of $a\pm|b|$.)

(2)Starting from the end of page 165 continuing to the beginning of 166, he writes:

For infinite volume, a general vacuum state $|v\rangle$ may be defined as a state with zero momentum

\[\mathbf{P}|v\rangle=0\]

for which this is a *discrete* momentum eigenvalue.(This excludes single-particle or multiparticle states, for which the momentum value zero is always part of a continuum of momentum values in a space of infinite volume.) In general there may be a number of such states. They can usually be expanded in a discrete set, ...

This brings up three questions:

A. Why would he bother to require $\mathbf{P}|v\rangle=0$, shouldn't it be an automatic consequence of the more general requirement that vacuum must be Poincare-invariant?

B. And how do we implement the other requirement that "for which this is a discrete momentum eigenvalue"? I mean, if we only look at 3-momentum spectrum, there's no way to tell on the spectrum if a point represents a vacuum or a zero-momentum multipaticle state, and surely the 3-momentum spectrum is always a continuum.

C. Why would one expect "They can usually be expanded in a discrete set"?