I 'Formulations' of Physical Theories: Overview?

[Logic Cloud]

I'm trying to understand the various different ways in which we can 'formulate' theories in physics and I am finding it somewhat hard to obtain a bird's-eye view. I hope someone here can help clear up some ambiguities.

I apologize in advance for the 'sketchy' ways I put matters in my descriptions below. However, as the primary goal of my question here is to obtain a general overview of how different terms and concepts are related, I feel it's beneficial not to get too bogged down in details that might distract from the bird's-eye view.

The most high-level distinction between different 'formulations' of physical theories that I know of is between:
*algebraic formulations of physical theories
*geometric formulation of physical theories

Roughly, I feel like I understand this distinction. Algebraic formulation start from an algebra of observables and subsequently notions like 'state' are derived from that, whereas geometric formulations start out by postulating some type of relevant manifold (I want to say 'state space' here, I'm not quite sure whether that's truly accurate, as my questions below will illustrate).

The most basic question I have regarding this twofold distinction is the following:
It is well-known that classical mechanics permits at least three different formulations, viz.
*Newtonian
*Lagrangian
*Hamiltonian

Question #1: Is it correct to say that both Hamiltonian and Lagrangian formulations of classical mechanics qualify as 'geometric' formulations (with the difference being that in one case we work on the tangent bundle of our configuration space and the cotangent bundle in the other)?

Question #2: Is it correct to say that the 'Newtonian' formulation of classical mechanics is neither algebraic nor geometric, because it presupposes a particular coordinate set of coordinate systems?

Now, even if my understanding of things (as expressed in the above two questions) turns out to be correct, there's an additional complication if I try to include general relativity in my considerations. If, say, the Hamiltonian formulation of classical mechanics qualifies as a 'geometric' formulation of that theory, than what kind of 'formulation' is the 'standard formulation' of GR (e.g. as found in an introductory textbook)? Is it also geometric? I would, 'yes', because one point that is always heavily emphasized is that the equations we write down in GR must hold true for all coordinate systems. But GR, as it is usually presented, most definitely does not permit a straightforward 'Hamiltonian formulation'. As far as I'm aware, trying to write down a Hamiltonian formulation of GR is quite a non-trivial undertaking.

Question #3: Is it correct that to say that a 'geometric formulation' of a physical theory need not be either a Hamiltonian formulation or a Lagrangian formulation? (With a relevant example being the standard formulation of GR vs Hamiltonian formulation of GR?)

Turning from GR to quantum physics, there's yet one more complication to my 'big picture'. QM is often presented as flowing naturally from the Hamiltonian formulation of classical mechanics. However, the typical Hilbert space formulation of QM is still quite different from, say, the Hamiltonian formulation of CM when you look at the employed state space. The state space for QM is typically taken to be a Hilbert space, whereas the state space in Hamiltonian CM is the (position-momentum) phase space, i.e. the cotangent bundle of the underlying configuration. This distinction is underscored by the fact that there seems to also exist a 'phase space' formulation of QM that is different from the usual Hilbert space formulation.

Question #4: Is it correct to say that both the standard 'Hilbert space formulation' of QM and the phase space formulation of QM (which replaces the Hilbert space by a position-momentum space) are examples of geometric formulations? If yes, then to what type of construction from differential geometry does the Hilbert space correspond? (E.g. the corresponding construction for the phase space formulation would be the taking of the cotangent bundle.) If no, then what is the relevant distinction between Hilbert-space and phase-space QM?

Lastly, there is another ambiguity regarding QM I'd like to get clear on. So far, I've discussed different 'formulations' of physical theories. But in QM we also find different 'pictures', i.e. the Heisenberg, Schrödinger and Dirac pictures of QM.

Question #5: How do the above three 'pictures' of QM correspond the more general geometric/algebraic distinction for formulations of physical theories? Is the distinction between the three 'pictures' independent of the geometric/algebraic distinction? (E.g. could I articulate an algebraically formulated, Schrödinger picture of QM?) Or are the three 'pictures' just specific ways of unpacking the standard Hilbert-space formulation of QM?

Any help with these issues would be appreciated. Of course, everyone should feel free to answer as many or as few of the above questions as they want.

 

[bhobba]

You have a lot of queries. For now just a few comments. We generally when possible work with the Lagrangian formulation of a theory because you can then use the beautiful Noether theorem and it follows directly from Feynman's Path Integral approach to QM. Since all the different formulations are equivalent I am not sure what the classifications as geometric etc signifies. The different formulations of QM are also equivalent.

Thanks
Bill

 

[dextercioby]

Well, there is no pure "algebraic" formulation of a physical theory, because any theory's equations need to be differential (at least in time).

The Newtonian formulation of dynamics is thus algebraico-differential, but it can be geometrized, as soon as one puts gravity in. Here we have the Newton-Cartan theory of classical gravity.

I wouldn't call the standard Hilbert space formulation of QM geometrical, because the mathematical theory involved is functional analysis, not differential geometry.

And what is usually called "algebraic Quantum Mechanics or algebraic Quantum Field Theory (AQFT)" is also a functional analysis-based formulation, with just more algebraic flavor.