Abstract We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary operator which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution only for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex quantum theory, we reproduce that the correspondence dictated by the Schrödinger equation is the only possible choice, up to a shift of the global phase.
Abstract We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary operator which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution only for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex quantum theory, we reproduce that the correspondence dictated by the Schrödinger equation is the only possible choice, up to a shift of the global phase.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second 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 quaternionic 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 quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, 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. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.
We present Unified Quantum Mechanics (UQM) by unifying real and standard complex quantum theories via real energy and momentum operators derived from the Hilbert Transform. This paper especially presents a fully real-valued formulation of quantum mechanics, showing that while complex numbers are commonly used in the standard theory, they are not fundamentally necessary for describing quantum phenomena. We develop Real Quantum Mechanics (RQM) by defining the real energy and momentum operators using the temporal Hilbert transform which shifts the phases of wave functions while preserving all essential quantum behavior. These real operators unify the proposed RQM and standard Complex Quantum Mechanics (CQM), as these are valid for both real and complex wavefunctions. We introduce a real-valued algebraic structure constructed from the identity operator and the Hilbert transform operator. This framework is mathematically equivalent to the system of complex numbers, preserving all fundamental algebraic properties while operating entirely within the realm of real operators. The isomorphism provides a novel representation of complex quantum mechanics using only real-valued components, offering potential advantages for both theoretical analysis and physical implementations. Importantly, we establish that the Hilbert transform operator is a generalization of the imaginary unit. The key insight is that the Hilbert transform generalizes the role of imaginary numbers in standard quantum mechanics, allowing us to construct real operators for both real and complex wave functions that produce the same physical predictions. We derive real versions of energy and momentum operators, leading to a modified wave equation that replaces the Schrödinger equation. Despite its non-local time evolution, this framework retains local spatial behavior and reproduces the usual energy-momentum relations. We rigorously test this approach on fundamental quantum systems, including the hydrogen atom and relativistic particles, confirming that it yields identical energy levels and wave functions as conventional quantum mechanics. Notably, we establish a consistent probabilistic framework by introducing real analogues of quantum probabilities via the analytic Born rule. We derive a unified uncertainty principle that coincides with the standard Heisenberg uncertainty bound for complex-valued wave functions, while admitting a vanishing lower bound in the case of real-valued wave functions. The implications of this work are significant: (1) it demonstrates that complex numbers are a mathematical convenience rather than a fundamental requirement in quantum theory, (2) it introduces new tools for analyzing quantum dynamics using signal-processing techniques, (3) it provides insights into non-local quantum evolution, (4) it connects quantum mechanics to real-field theories. Ultimately, RQM reproduces all known quantum predictions while offering a fresh perspective on operator structure, time evolution, and the mathematical foundations of the theory, and (5) it provides a real qubit representation on the modified Bloch sphere for quantum computing.
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.
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.
Quaternionic quantum Hamiltonians describing nonrelativistic spin particles require the ambient physical space to have five dimensions. The quantum dynamics of a spin-21 particle system characterized by a generic Hamiltonian is worked out in detail. It is shown that there exists, within the structure of quaternionic quantum mechanics, a canonical reduction to three spatial dimensions upon which standard quantum theory is retrieved. In this dimensional reduction, three of the five dynamical variables are shown to oscillate around a cylinder, thus behaving in a quasi-one-dimensional manner at large distances. An analogous mechanism is shown to exist in the case of octavic Hamiltonians, where the ambient physical space has nine dimensions. Possible experimental tests in search for the signature of extra dimensions at low energies are briefly discussed.