Heisenberg and Schrödinger’s Mechanics Anticipate P Vs NP, Revealing Knowledge Production across Physics and Computation

The fundamental question of how we acquire and validate knowledge lies at the heart of both physics and computer science, and a new analysis by Galina Weinstein from University of Haifa reveals a surprising connection between the early development of quantum mechanics and the enduring P versus NP problem. This work reinterprets the historical divide between the matrix mechanics of Heisenberg, Born and Jordan, and the wave mechanics of Schrödinger, demonstrating that these seemingly equivalent theories embody fundamentally different modes of knowing, one based on procedural construction and the other on recognitional verification. By reconstructing the mathematical derivations of these pioneering physicists, culminating in von Neumann’s unification of their approaches, Weinstein argues that this divergence isn’t merely a mathematical curiosity, but a structural feature of scientific reasoning itself, reflecting the persistent tension between what we can efficiently create and what we can only efficiently verify. This research offers a novel perspective on the foundations of knowledge, bridging the gap between the abstract realms of theoretical physics and computational complexity.

These differing approaches to knowledge anticipate, in philosophical form, the logical asymmetry expressed by the P versus NP problem in computational complexity. The essay reconstructs the mathematical history of quantum mechanics through the original derivations of Werner Heisenberg, Max Born, Pascual Jordan, Paul Dirac, Erwin Schrödinger, Paul Ehrenfest, and Wolfgang Pauli, culminating in John von Neumann’s unification of both approaches within the formalism of Hilbert space. By juxtaposing Heisenberg’s algorithmic formalism with Schrödinger’s representational one, it argues that their divergence reveals fundamental differences in how we acquire knowledge.

Mechanics, Computation, and Modes of Knowing

This work presents a reinterpretation of the foundational divide between matrix mechanics and wave mechanics, revealing them not as distinct theories, but as different modes of knowing, procedural construction and recognitional verification. Researchers demonstrated that this distinction anticipates the core asymmetry expressed by the P versus NP problem in computational complexity, highlighting parallels between efficient computation and knowledge validation in physics. The team reconstructed the historical development of mechanics, tracing the original derivations of key figures including Heisenberg and Pauli, culminating in von Neumann’s unification within Hilbert space formalism. In 1926, Born, Heisenberg, and Jordan introduced an operator calculus foreshadowing the structure of a Hilbert space, identifying Hermitian matrices as the mathematical representation of physical observables.

The team showed that Hermitian matrices, defined by a = a∗, ensure that algebraic structures automatically produce measurement results consistent with physical experience. They mathematically defined the associated quadratic form as A(x, x∗) = X m,n amn xmx∗ n, demonstrating that if amn is Hermitian, A(x, x∗) is always real, guaranteeing real expectation values for measurable quantities. This work established a crucial link between mathematical formalism and empirical observation. Further investigation revealed that vectors represent states, Hermitian matrices represent observables, and inner products represent measurable averages.

While working with finite matrices due to limitations in formalizing infinite-dimensional spaces, the team cited the work of Hilbert and Hellinger on infinite-dimensional quadratic forms as the theoretical foundation for their Hermitian formalism. Researchers recognized that quantum observables could have arbitrarily large eigenvalues, boldly extrapolating the Hilbert-Hellinger results to these unbounded cases, trusting the internal consistency of the new mechanics. In 1927, von Neumann rigorously proved that every self-adjoint operator on a Hilbert space admits a spectral decomposition, extending the finite-dimensional theorem formulated by Born, Heisenberg, and Jordan. This proof completed the formal foundations of quantum mechanics, legitimizing their framework and establishing a complete mathematical edifice.

Simultaneously, Dirac reformulated Heisenberg’s matrix mechanics into a general algebra of quantum quantities, identifying the correspondence between the quantum commutator and the classical Poisson bracket, defined as xy −yx = ih 2π {x, y}. He also expressed time evolution as ̇x = [x, H] ≡xH −Hx, anticipating the Heisenberg equation of motion in operator language. These developments, alongside Heisenberg’s work on the uncertainty principle, demonstrate a profound convergence of theoretical insights, solidifying the mathematical foundations of quantum mechanics.

Heisenberg, Schrödinger, And Computational Asymmetry

This essay argues for a profound connection between the early debates surrounding the interpretation of quantum theory and the core principles of modern computational complexity. The central thesis is that the epistemological and methodological differences between Werner Heisenberg and Erwin Schrödinger in the 1920s foreshadow the fundamental asymmetry at the heart of computational complexity, the distinction between problems that are easy to verify and those that are easy to solve. Heisenberg’s matrix mechanics is presented as an operational and opaque approach, focusing on observable quantities without necessarily revealing the underlying structure, akin to algorithms that provide answers efficiently without exposing how they arrived at those answers. Schrödinger’s wave mechanics, conversely, is characterized as constructive and transparent, emphasizing the wave function, which directly represents the state of the system, making the underlying structure visible, analogous to problems where a potential solution can be quickly verified.

The essay meticulously traces the development of quantum mechanics, highlighting the shift from classical physics, the contributions of key figures like Bohr, and the ongoing struggle to understand the meaning of the wave function. It explains matrix mechanics and wave mechanics, emphasizing their respective strengths and weaknesses, and details von Neumann’s crucial role in providing a mathematically rigorous foundation for quantum mechanics, unifying the seemingly disparate approaches of Heisenberg and Schrödinger. The essay delves into several key philosophical themes, including realism versus instrumentalism, the nature of knowledge, the role of mathematics in physics, and the idea that nature privileges what can be constructed over what can merely be conceived. The essay draws a direct parallel between the historical debate in quantum mechanics and the modern problem of P versus NP.

P problems are those that can be solved efficiently by a deterministic algorithm, like Heisenberg’s matrix mechanics, while NP problems are those for which a potential solution can be verified quickly, but finding a solution is believed to be much harder, like Schrödinger’s wave mechanics. The emphasis is on the importance of verification as a key characteristic of NP problems, and the implication of procedural realism is that the difficulty of solving NP problems may be a fundamental property of the universe, reflecting a deeper limitation on computation. The essay is a compelling and insightful exploration of the deep connections between the history of quantum mechanics and the frontiers of computer science. It offers a provocative argument that the epistemological debates of the 1920s foreshadowed the computational challenges of the 21st century, suggesting that the universe itself may impose fundamental limits on what can be computed.