Nine formulations of Quantum Mechanics

[CarlB]

Nice short article:

http://hubcap.clemson.edu/~daw/D_Realism/AmJPhys.v70p288y02.pdf

 

[Mentz114]

Excellent.Thanks, Carl.

 

[CarlB]

I'm rewriting my book on density matrix formalism. The above reference was the best I found, which is a sad statement. I hadn't read about the phase space formulation before reading the above.

One of the things he missed is Schwinger (1959) measurement algebra. The original (but very readable) papers that introduced it are here:
http://brannenworks.com/E8/SchwingerAlgMicMeas.pdf
http://brannenworks.com/E8/SchwingerGeomQS.pdf

Mathematically, the measurement algebra is a variation of the density matrix formulation in a way similar to how matrix mechanics and density matrices share equations. The difference is that the density matrix formulation [that Styer et al gives in the link in the first post] is defined in terms of pure states but the measurement algebra is defined in terms of quantum measurements. So the measurement algebra falls firmly in the "corpuscular" or particle theory camp while density matrices, at least as defined above, fall in the "undulatory" or wave mechanics camp.

So in a way the measurement algebra is halfway between Heisenberg's matrix mechanics and the usual density matrix formulation. But it's interesting that Schwinger never once mentioned the relationship between his measurement algebra and density matrices though they share the same equations.

For my book, I'd like a slightly more radical version of the measurement algebra. What I'd like would be about halfway between the measurement algebra (in that it's defined in terms of quantum measurements) and Feynman's path integral formulation.

What I'd like to take from the Feynman path integrals is the use of Feynman diagrams for building propagators up from elementary interactions. But rather than deriving them from an action principle, I'd rather have them simply assumed, as Schwinger did in the measurement algebra.

The reason for picking these particular parts is that I'd like to be able to model deeply bound states. I want the Feynman diagram, with its propagators and methods of computing convolutions of propagators, but without specifying the particular functional forms that the usual path integral formalism uses. That is, I don't want to restrict myself to $$(p^\mu\gamma_\mu+m)/(p^2-m^2+i\epsilon)$$ or whatever it is, I want to discuss things in more generality. I typed up a description of the bound states of the hydrogen atom according to this sort of thing a few days ago:
http://carlbrannen.wordpress.com/2008/04/04/quantum-bound-states-the-hydrogen-atom/

One finds spin-1/2 when one takes a look at orbital angular momentum for a quantum system and reduces it to a set of commutation relations, and then looks for the simplest representation in the matrices. What I'd like to do is similar to this; to take a look at the bound states for a quantum system and reduce it to set of algebraic relations, and then look for the simplest representation of those algebraic relations in the matrices.

The problem with the path integral formulation as described is that the various propagators are assumed to be solutions of a particular wave equation. I want to not need to specify what wave equation is being solved, but instead to think of the propagators as defining the phase that a particle picks up when it goes through a long sequence of experiences as part of a complicated bound state.

Since the particle is to be part of a bound state, the formulation has to be oriented around virtual states. So I can't write down the action. From the point of view of Feynman's methods, I'm working on the answer before defining the problem. It's the only way I can see of working out bound states in full non perturbative glory.

Anyway, the above reference makes me wonder if I really should orient the book to be another formulation of quantum mechanics rather than a variation of the density matrix formulation. Maybe I should call it the "virtual bound state" formulation.