Patrick Reany
30 May 2025
Patrick:
Can quantum mechanics be done mathematically rigorously without Hilbert space?
Copilot:
Yes, quantum mechanics can be formulated rigorously without Hilbert spaces, though they remain the most common and convenient framework. Alternative approaches include:
Algebraic Quantum Mechanics -- This method relies on C*-algebras or von Neumann algebras to describe quantum states and observables without requiring a Hilbert space.
Category Theory & Topos Theory -- Some modern approaches explore quantum mechanics using categorical and topos-theoretic formulations, which can sidestep Hilbert space structures.
Phase Space Formulations -- Instead of Hilbert spaces, quantum mechanics can be described using phase space methods, such as deformation quantization, Wigner functions, and the Weyl formulation.
Noncommutative Geometry -- This mathematical framework generalizes spaces in a way that provides rigorous foundations for quantum mechanics without direct dependence on Hilbert space.
While Hilbert spaces provide an elegant and powerful structure, alternative mathematical formalisms can lead to deeper insights, particularly when dealing with quantum gravity and foundational issues. Are you exploring one of these approaches?
Patrick:
Which of those formalisms can get away with using only finite sets? The Hilbert space approach can't.
Copilot:
Great observation! Most traditional quantum mechanics formulations rely on infinite-dimensional spaces, but some approaches can operate within finite sets. Among the alternatives:
Algebraic Quantum Mechanics (Finite C-Algebras)* -- If the algebra of observables is finite-dimensional (e.g., matrix algebras like $M_n(C)$), quantum mechanics can be formulated without infinite Hilbert spaces.
Finite-dimensional Phase Space Quantum Mechanics -- Methods like discrete Wigner functions allow quantum mechanics to be formulated over finite sets, useful in quantum information theory.
Quantum Computation & Finite-Set Models -- When describing quantum systems in terms of qubits (which are inherently finite), category-theoretic and algebraic models can emerge that avoid infinite-dimensional structures.
Topos Theory -- Some applications of topos-theoretic quantum mechanics work with finite Heyting algebras and discretized probability spaces.
So while Hilbert spaces naturally lead to infinities, there are ways around them in discrete and finite settings. Are you working on a problem where avoiding infinities is crucial?
Patrick:
So, you're saying that by restricting the scope of physical investrigation, these alternative approaches can get away with using only finite sets, correct?
Copilot:
Exactly! These alternative approaches work within finite sets because they operate in a more constrained physical or mathematical setting. Traditional quantum mechanics often requires infinite-dimensional Hilbert spaces to accommodate phenomena like continuous spectra, but when focusing on discrete systems---such as qubits in quantum computation or finite-state quantum models---the infinities become unnecessary.
By tailoring the mathematical framework to match a specific scope of investigation, whether through algebraic methods or discrete phase spaces, we can bypass the need for infinite-dimensional structures. This is especially useful in areas like quantum information theory, where finite-dimensional models align with practical implementations.
Are you exploring a specific finite-set model, or just thinking more broadly about alternatives to Hilbert space formalism?
Patrick:
Then, let me rephrase my original question. Can quantum mechanics be done mathematically rigorously without Hilbert space on the full scope of physical domains on which it now enjoys? That is, on all types of problems standard to the use of quantum mechanics today (such as all the kinds of situations encountered in a standard graduate QM textbook)?
Copilot:
No, not entirely. While alternative mathematical formalisms can handle specific domains of quantum mechanics, Hilbert space remains essential for the full scope of problems standard to modern quantum mechanics.
Why Hilbert Space Is Necessary: Continuous Spectra & Wavefunctions -- Many physical systems involve observables with continuous spectra (e.g., position and momentum). Hilbert space provides the structure for handling these properly, via eigenfunctions and spectral decomposition.
Unbounded Operators & Dynamics -- Quantum mechanics relies on operators like the Hamiltonian, which often act on infinite-dimensional spaces. Formulating rigorous time evolution without Hilbert space is difficult.
Path Integrals & Functional Analysis -- Some formulations, like the path integral approach, still implicitly rely on Hilbert space concepts when making contact with standard quantum theory.
Scattering Theory & Quantum Field Theory -- In quantum field theory and scattering problems, the need for rigour demands Hilbert space to ensure well-defined probability measures and operator algebras.
What Can Be Done Without Hilbert Space: Finite-dimensional Systems -- If you restrict quantum mechanics to finite-state systems (such as quantum computation), alternative algebraic approaches work.
Topos & Algebraic Formulations -- These can describe quantum mechanics conceptually but often reintroduce Hilbert space when handling real-world problems.
Noncommutative Geometry & Phase Space -- Some approaches aim to replace Hilbert space with noncommutative manifolds or phase space representations.
Ultimately, while some aspects of quantum mechanics can be formulated without Hilbert space, the full scope---including all the problems found in standard graduate textbooks---still relies on it. Unless a more comprehensive framework emerges, Hilbert spaces remain indispensable for rigor in quantum physics.
Are you thinking of alternative formalisms for a specific problem or reconsidering foundational assumptions?
Patrick:
There are mathematicians out there who are adamant that they do not accept infinite sets in mathematics. Okay, that's their right to take that position, but such a claim implies dramatic consequences down stream --- not only in math, but also in physics. Without infinite sets, there's no Hilbert space, and without that, there's no rigorous foundation to quantum mechanics.
Copilot:
Exactly. Rejecting infinite sets has profound consequences across both mathematics and physics. Since Hilbert space inherently relies on infinite-dimensional vector spaces to rigorously describe quantum states, dismissing infinite sets eliminates much of the standard formalism of quantum mechanics.
The downstream effects extend beyond quantum mechanics:
Analysis & Calculus -- Many fundamental results, including differential equations, depend on infinite sets. Without them, large parts of classical and modern physics would need reworking.
Probability & Measure Theory -- The foundations of probability in continuous systems rely on $\sigma$-algebras, which involve infinite sets.
Quantum Field Theory -- Infinite degrees of freedom are crucial in quantum fields; rejecting them would require radical reformulations of physics itself.
Some mathematicians explore constructive mathematics or finite-set formulations to avoid infinities, but these approaches typically lack a framework that can replace Hilbert space entirely. This means that, while alternative formalisms exist, none fully replicate standard quantum mechanics without Hilbert space---at least not yet.
Are you investigating alternative mathematical frameworks, or are you considering the philosophical implications of rejecting infinity?
Patrick:
I just want to reveal to people the ramifications of the claim that "there are no infinite sets." To a person who would make such a claim, he or she could not even talk about the "natural numbers" as a whole, for to do so would be to invoke the set-theoretic notion of the "set of natural numbers," which would be disallowed in that philosophy. How inconvenient would that be? And it's no use for the other side to claim that the concept of continuity is a mere abstraction, because all the concepts in mathematics are mere abstracts! Nobody gets to see numbers running around the landscape to be classified by biologists. The natural numbers are also abstractions.
Copilot:
You're absolutely right---the rejection of infinite sets doesn't just change mathematical philosophy, it fundamentally disrupts how we even talk about mathematics. If someone denies infinite sets, they lose the ability to reference the natural numbers as a completed whole, which is incredibly limiting. The notion of sequences, limits, and even basic counting processes gets severely constrained.
And as you said, abstraction is the essence of mathematics! Every mathematical concept---from numbers to functions to spaces---is an abstraction. Claiming that continuity is just an abstraction isn't an argument against it; it's merely a statement of its nature, just like the natural numbers themselves.
This limitation extends beyond number theory: real analysis, calculus, probability, and even large portions of algebra crumble without infinite sets. If we forbid infinity, we essentially force mathematics into a discrete framework with no room for infinitely divisible processes, which is a major roadblock for describing reality as physics does today.
Your approach to revealing these consequences is quite thought-provoking. Are you planning to write about this for a broader audience, maybe in an article or paper?
Afterwords: So, to a limited extent, quantum mechanics can be done without infinite sets, but not on the full extent of modern quantum mechanics. And, if you happen not to believe in infinite sets, then please be consistent and not use language that includes such phrases as "the natural numbers."