DarMM said:
Yes, but in QFT it is almost impossible to write the wavefunctions down since they are square integrable functions on an infinite dimensional space. Only in a few simple cases can you actually write down what the wavefunction is, which is why I said "explicitly right down wavefunctions". I wouldn't see this as a source of disagreement, since it isn't really related to what we are discussing and surely, for instance, you couldn't write down the wavefunction \psi\left[A_{\mu},t\right) for a proton. Yes there is no "fundamental" difference in some sense, but QFT is a lot harder, especially with regard to getting explicit results. Of course there are other ways in which QFT is quite different from QM, the presence of Lorentz invariance and locality imply strong analytic constraints on correlation functions in QFT that aren't there in QM. For instance one obtains crossing symmetry and various other properties of the S-matrix, where as in QM the S-matrix can be much more general. Also one has the spin and statistics theorem, e.t.c.
I'm sure Weinberg would agree, since he used to say that imposing analyticity, locality, e.t.c. and keeping the rules of quantum theory is literally what QFT is. Of course since QFT is a quantum theory it will have a lot in common with nonrelativistic QM, but it is also very different in some ways.
I am glad that you've mentioned Weinberg, because his book presents the only approach to QFT that makes sense to me.
On page 49 he writes:
"First, some good news: quantum field theory is based on the same quantum mechanics that was invented by Schroedinger,..."
I don't think he ever considers the necessity to introduce a special Hilbert space for interacting theory. He also does not question the validity of Hermitian particle number operators. Although, I should admit that he doesn't bother to describe in any detail the Hilbert space he is working in. (the term "Fock space" is not even in the index). However, my guess is that he is doing "quantum mechanics in the Fock space", as I described earlier.
The crucial point of Weinberg's approach is indicated on page 200, where he expalins how it is different from the traditional QFT:
"Traditionally in quantum field theory one begins with such field equations or with the Lagrangian from which they are derived, and then uses them to derive the expansion of the fields in terms of one-particle annihilation and creation operators. In the approach followed here, we start with the particles, and derive the fields according to the dictates of Lorentz invariance, with the field equations arising almost accidentally as a byproduct of this construction."
On page 191 he explains why he needs to consider fields at all. The physical requirements of Poincare invariance and cluster separability are key here. Somewhat earlier he showed that interacting theory can be made Poincare-invariant if the interaction operator is represented as a space integral of "interaction density", which is a scalar with respect to the non-interacting representation of the Poincare group. The easiest way to construct such a scalar is to make a product of N "quantum fields". In Weinberg's interpretation, quantum fields appear as auxiliary technical constructs, whose only role is to ensure the Poincare invariance of interaction. Another benefit is that building interactions as products of quantum fields ensures cluster separability (or, as Weinberg calls it - the cluster decomposition principle). The obvious downside of such field theories is the inevitable need for renormalization: even single particle experiences unphysical self-interaction and self-scattering.
Note that Weinberg cannot prove that building interactions from fields is the only possible way to provide Poincare invariance, cluster separability, and changing number of particles. As an example of a failed alternative approach he mentions Bakamjian-Thomas theory. However, he doesn't tell that there are recent developments of this theory (see works by W.N. Polyzou), which show progress in both cluster separability and particle-number-changing interactions.
Weinberg doesn't mention yet another alternative - the "dressed particle" approach - which can easily accommodate Poincare invariance, cluster separability, and particle-number-changing interactions. Its main advantage is that renormalization is never needed. I consider this as a major sign of physical/mathematical consistency. The main disadvantage was mentioned by strangerep in one of recent threads: Since the "dressed particle" approach does not use fields, it does not accommodate the idea of "gauge invariance", so, unlike in QFT, there is no (relatively) simple and regular way to construct interaction Hamiltonians that immediately match observations. Currently we must either fit "dressed" Hamiltonians to experiment or use complicated tricks, like "unitary dressing". Though, this may be not a high price for getting rid of renormalization and divergences.
Eugene.
The way I see it, Weinberg is just saying that if you start with QM (as defined by the Dirac-von Neumann axioms), define what a symmetry is, and impose the condition that there's a group of symmetries that's isomorphic to the proper orthochronous Poincaré group (a very natural assumption of you want a special relativistic theory), we're immediately led to the concept of non-interacting particles.