Exploring Implicit Assumptions and Foundations of Quantum Mechanics

[Anton_A_Lipovka]

TL;DR

I would like to start a discussion exploring the foundational aspects of quantum mechanics, focusing on implicit assumptions, Planck’s constant, and the structure of Hilbert space.

Question about the role of postulates and implicit assumptions in quantum mechanics

Hi everyone,

I’m trying to better understand the structure of the postulates of quantum mechanics and whether there is a meaningful hierarchy among them.

In the standard presentation, we usually assume:
- states are vectors in a Hilbert space,
- observables are represented by Hermitian operators,
- measurement outcomes follow the Born rule.

At the same time, when looking at the historical development (for example, Schrödinger’s original work), it seems that several additional assumptions are implicitly introduced, even if not always stated explicitly. For instance:

- the existence of a wavefunction ψ describing the system,
- a specific dynamical law (the Schrödinger equation),
- and the appearance of constants like Planck’s constant setting the scale.

This makes me wonder whether it is meaningful to think of these latter ingredients as more “primitive”, more “fundamental”, in the sense that the usual Hilbert space formalism and measurement postulates might emerge from them, or at least be motivated by them.

So my question is:

Is there a well-defined sense in which the standard postulates can be organized hierarchically, or are they generally viewed as independent axioms of the theory?

I would especially appreciate any clarification or references.

Thanks!

 

[Peter Morgan]

In axiomatic approaches to Quantum Field Theory, which I think any attempt to understand quantum theory will eventually have to engage with, it's commonly the algebra of operators and the states that are taken to be fundamental. That's approximately what you have in your first set of three assumptions. More is needed, however, to make contact with experiment.
I think your two sets of three assumptions are too vague as they are, but the way you have stated them is very reminiscent of Von Neumann's axioms for quantum theory, which is still the basis of most axiomatic presentations of QM 94 years later, so have a close look at them.
You've made this a graduate-level question, so I'll suggest a much cited article by Abramsky&Brandenburger, "The sheaf-theoretic structure of non-locality and contextuality", in NewJPhys 2011 (which is Open Access).
An interesting undergraduate-to-graduate-level textbook that compares algebraic and other formalisms, if you can get access to it, is François David's "The Formalisms of Quantum Mechanics" (not OA, preprint on arXiv.)
Klaas Landsman's "Algebraic quantum mechanics" is a nicely succinct handbook account (not OA; almost everything by Klaas Landsman is worthwhile).
A recent preprint that might be worthwhile is Falco&Matthies, "Vistas of Algebraic Probability, Quantum Computation and Information". They include many references: at the graduate level, the literature is enormous even about only algebraic QM.

 

[PeterDonis]

This makes me wonder whether it is meaningful to think of these latter ingredients as more “primitive”, more “fundamental”, in the sense that the usual Hilbert space formalism and measurement postulates might emerge from them

I think it's the other way around.

Assuming a wave function amounts to assuming a particular Hilbert space (the space of square integrable functions over the relevant configuration space).

Assuming the Schrodinger Equation amounts to assuming a particular choice of reference frame and using the non-relativistic approximation.

Planck's constant fixing the scale arises when we try to correlate the theoretical equations with actual measurements.

So those things aren't more fundamental, they emerge from the other stuff once you start trying to work with it for solving particular problems.