Peter Morgan said:
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.
Thank you for your detailed and thoughtful reply. I particularly appreciate the references you provided, as well as the time you have taken to engage with my questions. It is a pleasure to discuss these issues with you.
You are certainly correct in noting that “
more is needed, however, to make contact with experiment”, and in my opinion the main point here is “experiment”. Indeed, quantum theory involves a broader set of axioms. In my earlier post, I singled out a second group of assumptions—primarily for illustrative, historical reasons—namely:
- the existence of a wavefunction ψ describing the system,
- a specific dynamical law (the Schrödinger equation),
- and the appearance of dimensional constants such as Planck’s constant.
These were taken in the spirit of Erwin Schrödinger’s original formulation, rather than as a complete axiomatization.
I also fully agree with your statement that in any attempt to understand quantum theory, we will ultimately face the need to understand the origins of the axioms in the axiomatic approach to quantum field theory. This is precisely the motivation behind my interest in a possible hierarchy of axioms. In particular, I have in mind programs often described as a geometrization of physics, where physical structures—fields and equations—emerge from properties of the metric (which, in general, may be dynamical).
It is quite possible that my original formulation was not sufficiently clear, so let me restate the point more carefully.
There appear to be (at least heuristically) two distinct approaches to QM and QFT, roughly corresponding to the two groups of axioms mentioned above.
The first is what I referred to as a “top” approach: one postulates abstract mathematical structures—Hilbert spaces, or in QFT Fock spaces—and specifies operators and their algebraic properties. In this way, the formalism is constructed axiomatically at a structural level.
The second, “bottom” approach (historically earlier), is based on experiment and is associated with the original formulations of quantum mechanics. Here, the starting point is a set of empirically motivated postulates, such as those listed above.
I would note that most areas of physics — electrostatics, heat fluxes, diffusion, and many others — are constructed in this “bottom-up” manner, starting from the postulates which are ultimately grounded in experiment. By contrast, in QM and QFT, the “top-down” axiomatic structure has proven extremely powerful as a computational and organizational framework, which may explain its prominence.
However, if the goal is to understand the
physical meaning of the axioms themselves, I don't think this algebraic (“top”) approach will be successful.
On the one hand, algebraic formulations of probability and operator structures are extremely useful for organizing the theory, but they do not, by themselves, explain the origin of probability or the role of constants such as ℏ. For this reason, the “bottom” approach—being more directly tied to experimental structure—may be more informative when addressing foundational questions.
On the other hand, if one aims at a unification of different branches of physics, it seems unavoidable that the starting point must again be the experimentally grounded level, rather than purely formal algebraic constructions.
To illustrate this point, consider the following example.
The Sturm–Liouville problem, which leads to discrete spectra, appears throughout classical physics (electrostatics, heat conduction, diffusion, etc.). A particularly relevant case is the Fokker–Planck equation, whose formal similarity to the Schrödinger equation is well known.
Consider the diffusion equation:
(1) g² ∂²N/∂x² − ∂N/∂t = 0, where g² = D (D is the diffusion coefficient),
together with appropriate initial and boundary conditions defining a Sturm–Liouville problem.
The Schrödinger equation,
(ℏ²/2m) ∂²N/∂x² + iℏ ∂N/∂t = 0,
can be written in the same formal form:
(2) g² ∂²N/∂x² − ∂N/∂t = 0, where g² = D′ = iℏ/2m is a modified diffusion coefficient.
(Here I have replaced ψ with N; this does not affect the structure of the solution.)
With, for example, Neumann boundary conditions, the solution takes the familiar form:
N(x,t) = Σ Aₙ cos(nπx/a) exp[−(nπg/a)² t],
where the coefficients Aₙ are determined from the initial condition:
N(x,0) = f(x), Aₙ = ⟨f(x) | Nₙ(x,0)⟩.
The formal difference between the diffusion and quantum cases is entirely contained in the coefficient:
- Classical diffusion: g² = D,
- Quantum case: g² = iℏ/2m.
Substituting this into the solution yields:
(3) N(x,t) = Σ Aₙ cos(nπx/a) exp[−(i/ℏ) Eₙ t],
with Eₙ = (ℏ²/2m)(nπ/a)².
Thus, essentially the same mathematical structure (Hilbert space, operators, Sturm–Liouville theory) appears in both contexts. However, in diffusion theory the model is clearly constructed from experimentally grounded considerations, whereas in QM the axiomatic algebraic structure is often taken as primary.
Given this, I would like to restate my question more precisely:
Could the “bottom” approach—i.e., constructing QM starting from experimentally motivated axioms—be more effective in clarifying the origin and meaning of those axioms (in particular, the second group mentioned earlier), and perhaps even reducing their number?
If so, then understanding the nature of these fundamental assumptions becomes especially important, since they provide the link to experiment. This includes questions such as the origin of Planck’s constant, the interpretation of the wavefunction, and the implicit assumption that the system under consideration is closed.
The latter point raises additional concerns. On the one hand,
the spacetime metric is not static (e.g., cosmological redshift), whereas the standard formulations of QM and QFT are typically constructed in inertial frames without incorporating such effects into the Hamiltonian. On the other hand, the absence of
explicit transverse electromagnetic degrees of freedom in the most basic formulations (e.g. equations of Schrödinger, of Klein-Gordon, of Dirac) may suggest that the system under consideration is, in some sense, incomplete.