The presence of the imaginary unit $i$ in quantum mechanics is frequently treated as a computational postulate. We demonstrate that the complex structure of Hilbert space is not a decorative choice but a structural necessity derived from the first principles of modern physics: continuous symmetry, the requirement of unitarity to preserve probability, and the correspondence between generators and observables. By examining the Lie algebra of the unitary group, we prove that a real-valued theory fails to map self-adjoint observables to generators of continuous evolution without sacrificing local tomography or the existence of non-trivial phases. We conclude that $i$ is the unique logical bridge between symmetry and observability.
The presence of the imaginary unit i in quantum mechanics is frequently treated as a computational postulate. We demonstrate that the complex structure of Hilbert space is not a decorative choice but a structural necessity derived from the first principles of modern physics: continuous symmetry, the requirement of unitarity to preserve probability, and the correspondence between generators and observables. By examining the Lie algebra of the unitary group, we prove that a real-valued theory fails to map self-adjoint observables to generators of continuous evolution without sacrificing local tomography or the existence of non-trivial phases. We conclude that i is the unique logical bridge between symmetry and observability.
As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda–Maeda and other authors), from the lattice theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another — in our opinion, deeper — viewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the real one, all selfadjoint operators represent observables in accordance with Solèr’s thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras.
Abstract
The homogeneity symmetry is re-examined and shown to be non-unitary, with no requirement for the imaginary unit. This removes symmetry, as reason, for imposing unitarity (or self-adjointness) -- by Postulate. The work here is part of a project researching logical independence in quantum mathematics, for the purpose of advancing a full and complete theory of quantum randomness.
Keywords
foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary, non-unitary, unitarity, mathematical logic, formal system, elementary algebra, information, axioms, mathematical propositions, logical independence, quantum indeterminacy, quantum randomness.
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. \nIn conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while \nthere exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is \ncompletely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators \nwith quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc \nHilbert spaces. Clearly, this property turns out to be crucial.
This work presents an axiomatic and structural reconstruction of core elements of quantum mechanics. Starting from minimal principles — symmetry under vanishing preference, normalization, consistency of probability assignments, and coherence-preserving dynamics — we derive several central features of the quantum formalism rather than postulating them. Key results include: A geometric derivation of the constants 1/21/\sqrt{2}1/2 and 1/21/21/2 from symmetry and normalization. A theorem-level derivation of the Born rule (including the qubit case) from noncontextuality and additivity. A reconstruction of Schrödinger dynamics from reversible norm-preserving structure. A derivation of the tensor-product structure for composite systems from composition axioms and local tomography, with entanglement emerging as a necessary consequence. Formal treatment of Bell–CHSH nonlocality, Tsirelson’s bound, and no-signalling. A conditional reconstruction of complex Hilbert space structure from intrinsic phase dynamics and compositional consistency. The manuscript is self-contained and framed in standard mathematical-physics language. It does not propose modifications to quantum mechanics, but instead clarifies why many of its defining structures arise inevitably from deeper structural constraints. Prior computational and model-based work by the authors (including diagnostic toy models exploring phase locking, coherence stabilization, and recursive-time dynamics) is acknowledged but intentionally not reproduced here, in order to keep the present work focused on axiomatic structure rather than model-specific architecture. This upload is intended as a foundations and reconstruction contribution, suitable for readers interested in the mathematical and conceptual structure of quantum theory.
Starting from two fundamental assumptions, information conservation and logical closure, this paper demonstrates that the complex number structure is theinevitable mathematical foundation for self-referential systems to achieve infinitesemantic expansion. Firstly, it proves that information conservation requires theevolution operator to preserve the inner product, thus belonging to the unitarygroup; logical closure requires the system to be able to generate new propositionsinfinitely, which forces the evolution operator to have a continuous eigenvalue spectrum. The joint constraint of the two uniquely leads to the complex number fieldas the algebraic framework for self-referential systems. It further elucidates thesemantic loading of the complex phase: the squared modulus encodes propositionalcertainty, and the argument encodes potential logical associations; unitary evolutiongenerates new states through phase reorganization without increasing informationentropy, thus avoiding logical freezing. By examining quantum mechanics and theblack hole information paradox, the core role of complex structures in physical theories is demonstrated. Starting from three constructive principles (countability,reversibility, continuity), the complex number field is independently derived as thesimplest algebraic solution, and it is rigorously proven that the real number fieldand quaternions cannot meet the requirements. Self-referential systems, semanticexpansion, and logical freezing are formally defined, and the uniqueness of complex numbers is rigorously proven using Schur’s lemma, the Frobenius theorem,and recent no-go theorems (Życzkowski et al. [9], Masanes [11]). A self-referentialautomaton model is constructed for numerical simulation, visually showing thatin the complex number case, the phase continues to evolve and semantic entropyfluctuates, while the real number case quickly falls into logical freezing. Finally, thetheoretical significance is summarized and future research directions are prospected.
The indispensable nature of complex numbers in quantum mechanics has beenexperimentally confirmed, yet its deep origin — whether complex numbers are anad hoc introduction or a logical consequence of some more fundamental constraint—remains insufficiently clarified. From the perspective of symmetry analysis, thispaper argues that the complex structure is not an accidental choice of quantumtheory, but an inevitable outcome under the joint constraints of spacetime transformation symmetry and algebraic completeness. Through a rigorous analysis ofthe covariance of the Schrödinger equation under Galilean transformations, thispaper reveals: if the transformation rule of the wave function is required to preserve the probability interpretation and satisfy spacetime symmetry, then the wavefunction must take values in the complex number field. This conclusion can beextended to relativistic cases, forming an intrinsic echo with the role of mass asa central charge in Bargmann’s theorem. On this basis, we establish a quantitative criterion for the strength of symmetry constraints, proving that the transitionfrom a real-valued formulation of quantum theory to a complex-valued formulationcorresponds to a central extension of the symmetry algebra, which is the uniqueway for a physical theory to realize symmetry in the sense of projective representations. This framework is compatible with recent experimental advances regardingimaginarity as a quantum resource and multipath interference tests of hypercomplex theories, and proposes a symplectic geometry-based scheme for measuring thestrength of symmetry constraints. This paper advances the foundational issue ofcomplex numbers in quantum theory from an ”ontological debate” to a new levelof ”symmetry constraint analysis.”