Understanding Weinberg's QFT Proof of Annihilation/Creation Operator Sum

[scottbekerham]

Hi . In weinberg's QFT book 1st volume he states that any operator can be expressed as a sum of products of annihilation and creation operators but I can't understand the proof . can simeone simplify this please? ( page 175)

 

[tiny-tim]

hi scottbekerham!

Hi . In weinberg's QFT book 1st volume he states that any operator can be expressed as a sum of products of annihilation and creation operators but I can't understand the proof . can simeone simplify this please? ( page 175)

he defines an operator O as something with a given value for each (Φ_M,OΦ_N),

and then shows by induction how to get the required coefficients C_NM …

the inductive equation is the one ending "+ terms involving C_NM with N < L etc" …

what do you not follow about that?

 

[dextercioby]

Hi, Scott. Forget about Weinberg's proof for a while and picture this line of thought. Relativistic QFT comes nicely by generalizing single particle relativistic quantum mechanics which in turn generalizing the old quantum mechanics of Schroedinger, Dirac and Heisenberg. The latter is of course a mind-blowing extension of the classical mechanics of Hamilton. What can you say about an observable O in Hamilton mechanics ? It's defined on the phase space (here O becomes a function of p and q) by the property that, when evaluated on the surface of the solutions of Hamilton's equation, it's a mere numerical constant.

Going in reverse, p and q become operators in quantum mechanics, no longer variables of the phase space, so do the quantum mechanical observables become functions of the 'fundamental' operators p, q. But p and q in a single Hilbert space (actually in a RHS, but that's a finesse you won't need) can be linked to a and a^dagger. So the observables become functions of a and a^dagger. And now you go from a single HS to a Fock space and voila', you find Weinberg's statement.