Nine formulations of Quantum Mechanics

In summary, the article describes the difference between density matrices and the measurement algebra, and how Schwinger never mentioned the relationship between the two. The author is rewriting his book on density matrix formalism, and he is trying to come up with a more radical version of the measurement algebra that is halfway between the measurement algebra and Feynman's path integral formulation.
  • #1
CarlB
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Nice short article:

http://hubcap.clemson.edu/~daw/D_Realism/AmJPhys.v70p288y02.pdf [Broken]
 
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  • #2
Excellent.Thanks, Carl.
 
  • #3
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 [tex](p^\mu\gamma_\mu+m)/(p^2-m^2+i\epsilon)[/tex] 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.
 

1. What are the nine formulations of Quantum Mechanics?

The nine formulations of Quantum Mechanics are: wave mechanics, matrix mechanics, Dirac's transformation theory, Heisenberg's matrix mechanics, Schrödinger's wave mechanics, Feynman's path integral, Tomonaga's quantum field theory, Schwinger's quantum action principle, and Bell's formulation.

2. What is the difference between wave mechanics and matrix mechanics?

Wave mechanics describes particles as waves, while matrix mechanics describes particles as discrete packets of energy. Wave mechanics is based on the Schrödinger equation, while matrix mechanics is based on the Heisenberg uncertainty principle.

3. How does Feynman's path integral formulation differ from other formulations?

Feynman's path integral formulation is based on the idea that the path of a particle is not determined by one specific path, but rather by all possible paths. This formulation takes into account the probabilistic nature of quantum mechanics and allows for the calculation of complex systems.

4. What is the significance of Bell's formulation in Quantum Mechanics?

Bell's formulation, also known as Bell's theorem, is a mathematical proof that shows that local hidden variables cannot reproduce all of the predictions of quantum mechanics. This has important implications for our understanding of reality and the role of randomness in the universe.

5. How do the different formulations of Quantum Mechanics relate to each other?

While the nine formulations of Quantum Mechanics may seem distinct, they are all mathematically equivalent and describe the same physical phenomena. They just provide different ways of understanding and calculating these phenomena. Each formulation has its own advantages and may be more suitable for certain situations.

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