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It is usually said that there are different formulations of QM, for example historically there was Schrodinger's (wave mechanics), and Heisenberg's (matrix mechanics), then Dirac's (which showed they are equivalent)

Since they are all physically equivalent I have a few questions:

1-Is Dirac's formulations considered more fundamental in the sense that it can be reduced to one of the 1st two?

2-I also hear about the path integral and density matrices, are they another formulations?

3-Are there more formulations less known to undergraduate students but known by researchers because they are technically advanced?

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possible duplicate of How many different formulations of QM currently exist? –  Qmechanic Feb 11 '12 at 8:01
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3 Answers

To your 3. question:

  • QM in Hilbert Space formulation
  • QM in $C^*$-algebra formulation
  • QM in lattice formulation (compare Blank, Exner Hilbert space operators in quantum physics Ch. 13)
  • Approach to QM using deformation quantization
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I will add some references if I have more time... –  student Feb 11 '12 at 10:03
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http://www-physique.u-strasbg.fr/cours/l3/divers/meca_q_hervieux/Articles/Nine_form.pdf
See this article by Styer et al entitled Nine formulations of quantum mechanics

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2) Path integral formulation sometimes called another (Fynman) formulation of quantum mechanics, as opposed to Dirak and Heisenberg ones.

Density matrices are to my knowledge never called another formulation.

AFAIK, other more advanced approaches are not called another formulations partly due to the fact that they are too advanced, partly because they were formulated later. Maybe also due to the fact that they are never used for QM itself, but for QFT only.

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