[Paper Review] How to (Un-) Quantum Mechanics

C. Baumgarten|arXiv (Cornell University)|Oct 12, 2018

Quantum Mechanics and Applications39 references3 citations

TL;DR

This paper argues that quantum mechanics can be derived from classical Hamiltonian mechanics without postulating quantum principles, by abandoning spatio-temporal metaphysics and retaining only time translation invariance. It shows that the Dirac equation and relativistic quantum mechanics emerge naturally from a purely classical phase space formulation using symplectic structures and Clifford algebras, demonstrating that the mathematical distinction between classical and quantum mechanics is illusory.

ABSTRACT

When compared to quantum mechanics, classical mechanics is often depicted in a specific metaphysical flavour: spatio-temporal realism or a Newtonian "background" is presented as an intrinsic fundamental classical presumption. However, the Hamiltonian formulation of classical analytical mechanics is based on abstract generalized coordinates and momenta: It is a mathematical rather than a philosophical framework. If the metaphysical assumptions ascribed to classical mechanics are dropped, then there exists a presentation in which little of the purported difference between quantum and classical mechanics remains. This presentation allows to derive the mathematics of relativistic quantum mechanics on the basis of a purely classical Hamiltonian phase space picture. It is shown that a spatio-temporal description is not a condition for but a consequence of objectivity. It requires no postulates. This is achieved by evading spatial notions and assuming nothing but time translation invariance.

Motivation & Objective

• To challenge the standard view that quantum mechanics is fundamentally different from classical mechanics in its mathematical structure.
• To demonstrate that the mathematics of relativistic quantum mechanics, including the Dirac equation, can be derived from classical Hamiltonian mechanics without additional axioms.
• To show that spatio-temporal realism is not a foundational requirement but a consequence of time translation invariance and symplectic dynamics.
• To argue that the perceived 'weirdness' of quantum mechanics stems from an incorrect presentation, not from intrinsic differences in mathematical structure.

Proposed method

• Formulates classical mechanics using generalized coordinates and momenta in a Hamiltonian phase space, abstracting away spatio-temporal metaphysics.
• Applies time translation invariance as the sole physical postulate, avoiding assumptions about space or objects.
• Uses symplectic dynamics and Lax pairs to describe spinors and their evolution, showing consistency with Dirac-type equations.
• Employs Kronecker products of fundamental spinors (size 4) to construct higher-order spinors and their corresponding Hamiltonians.
• Represents higher-order Hamiltonians via Taylor series expansions in spinor fields, using real Clifford algebras (Cl(N−1,1)) with specific matrix structures.
• Derives the Dirac equation and relativistic quantum mechanics from the symplectic motion of spinors in a classical phase space framework.

Experimental results

Research questions

• RQ1 Can the mathematical structure of relativistic quantum mechanics be derived from classical Hamiltonian mechanics without postulating quantum principles?
• RQ2 Is the distinction between classical and quantum mechanics fundamentally mathematical, or is it a consequence of interpretive metaphysics?
• RQ3 Does a spatio-temporal description of reality emerge from time translation invariance, or is it a foundational assumption?
• RQ4 Can the Dirac equation be derived from a purely classical phase space formulation using symplectic dynamics and Clifford algebras?
• RQ5 What is the role of spinor composition via Kronecker products in generating higher-order quantum-like dynamics from classical systems?

Key findings

• The Dirac equation and relativistic quantum mechanics can be derived from a classical Hamiltonian phase space formulation using only time translation invariance and symplectic dynamics.
• Spinors composed from an odd number of fundamental Dirac spinors (size 4) exhibit symplectic motion, while even compositions yield constants of motion with vanishing linearized Hamiltonians.
• Higher-order Hamiltonians are expressed as Taylor series in spinor fields, with non-vanishing terms constrained by symmetry of γ₀Γ matrices, limiting the number of independent terms to [n(2n+1)]^k for order 2k.
• The matrix size of the Hamiltonian for k-th order moments is 2^{8m+4} for k=2m+1, corresponding to real Clifford algebras Cl(N−1,1) with N=8m+4.
• Real Dirac spinors built from even Kronecker products (k=2m) have size 2^{4m}, but no corresponding real Clifford algebra exists for N=8m, indicating a structural mismatch in higher-order correlations.
• The paper concludes that the perceived 'quantumness' of quantum mechanics is not mathematical but interpretive, and that quantum mechanics emerges naturally from classical mechanics when metaphysical assumptions are removed.