What are the distinct mathematical formalisms of quantum mechanics?

Question

Consider the physical theory called non-relativistic quantum mechanics. What are the distinct mathematical formalisms for this physical theory? That is, different mathematical frameworks for this physical theory which for each physical phenomena compute exactly equal predictions.

Here is an example of what I mean by a mathematical formalism of quantum mechanics In textbooks like Sakurai, the formalism used is the following:

1. Systems are represented by Hilbert spaces $\mathcal{H}$ with states as equivalence classes $\{\lvert \psi \rangle\} \in P(\mathcal{H})$.
2. Dynamics are goverened by the Schrödinger equation, which is a differential equation defined over $\mathcal{H}$.
3. Observables are linear operators $\hat{O}: \mathcal{H} \to \mathcal{H}$ that live in a particular algebra endowed with two multiplication operations (multiplication and a Lie bracket).

Answer 1

I will give a brief list of the options that come to mind, but this is most likely incomplete. In all cases I'll assume we are dealing with finitely many degrees of freedom, so we don't get to field theory (which is more complicated).

In the cases in which they are applicable, all of the following approaches yield the same results (you need to make a specific choice of deformation product for deformation quantization). However, it might happen that some of them are more general than others.

1. Sakurai's Approach

You just mentioned this one, and I'm just writing it for the count. Notice it works both for the Heisenberg, Schrödinger, and interaction pictures, since they only correspond to different (but unitarily equivalent) Hilbert spaces.

2. Sakurai's Approach with Density Matrices

You can further develop Sakurai's approach by allowing states to be density matrices on the Hilbert space. This allows you to consider mixed states in addition to pure states.

3. Path Integrals

Path integral quantization is an approach that becomes really useful in field theory, but you can do regular quantum mechanics with it. In this case, states are essentially given by means of boundary conditions on your path integral and you use the path integral itself to compute expectation valuables of observables. It is interesting that many things end up being commutative in this approach, so you don't deal with operators, but you might need to have to deal with anticommuting numbers (Grassmann numbers).

4. Algebraic approach

The algebraic approach is really popular in the mathematical physics community when working with quantum fields, especially in curved spacetime. In this case, you understand that the operators form a *-algebra, which is a vector space generalizing the space of operators on a Hilbert space. States are defined as positive normalized linear functionals in this algebra, and they are naturally understood as being the functions that map an observable to its expectation value in that state. This approach is more general than Sakurai's approach with density matrices and can be used to recover it. Usually there are more algebraic states than there are density matrix states.

5. Wigner quasiprobability function

Wigner's quasiprobability distribution allows you to describe a quantum state as a quasiprobability distribution on the classical phase space ("quasi" refers to the fact it can achieve negative values). I don't know much about this formalism, but it essentially allows you to do quantum mechanics in a setting more similar to classical mechanics.