aiqbench analysis
This paper presents a foundational theoretical argument asserting that the complex structure in quantum mechanics is not merely nominal but a strict requirement for time evolution under energy conservation. The authors clarify that while real-valued formulations exist, they are essentially representations of complex linear algebra rather than true alternatives to complex numbers. This work addresses core mathematical and philosophical questions regarding the necessary algebraic structures underlying quantum theory.
The Requirement of (at least) Complex Structure for Quantum Mechanics
arXiv:2609.23041v1 Announce Type: new Abstract: It is argued that many real-valued constructions of quantum mechanics are only \emph{nominally} real in that the operators and states are restricted to impose a \emph{complex structure} on the Hilbert space. That is, complex algebra between pairs of elements representing complex numbers is preserved in these formulations. It is therefore mistaken to think of these constructions as being `real' as they are actually a representation of complex linear algebra. It is then shown that under the assumptions of state normalisation and strict energy conservation, this complex structure (at the very least) is required to allow for time evolution in quantum mechanics. This is only a \emph{minimum} requirement, as our arguments do not preclude the formulation of quantum mechanics in terms of hyper-complex entities, such as quaternions.
Abstract:It is argued that many real-valued constructions of quantum mechanics are only \emph{nominally} real in that the operators and states are restricted to impose a \emph{complex structure} on the Hilbert space. That is, complex algebra between pairs of elements representing complex numbers is preserved in these formulations. It is therefore mistaken to think of these constructions as being `real' as they are actually a representation of complex linear algebra. It is then shown that under the assumptions of state normalisation and strict energy conservation, this complex structure (at the very least) is required to allow for time evolution in quantum mechanics. This is only a \emph{minimum} requirement, as our arguments do not preclude the formulation of quantum mechanics in terms of hyper-complex entities, such as quaternions.
Quantum Physics (quant-ph); History and Philosophy of Physics (physics.hist-ph)
arXiv:2609.23041 [quant-ph]
(or arXiv:2609.23041v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.23041 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 226 words
