The orthodox formulation of quantum theory invokes the mathematical apparatus of complex Hilbert space. In this thesis, we consider a quaternionic quantum formalism for the description of quantum states, quantum channels, and quantum measurements. We prove that probabilities for outcomes of quaternionic quantum measurements arise from canonical inner products of the corresponding quaternionic quantum effects and a unique quaternionic quantum state. We embed quaternionic quantum theory into the framework of usual complex quantum information theory. We prove that quaternionic quantum measurements can be simulated by usual complex positive operator valued measures. Furthermore, we prove that quaternionic quantum channels can be simulated by completely positive trace preserving maps on complex quantum states. We also derive a lower bound on an orthonormality measure for sets of positive semi-definite quaternionic linear operators. We prove that sets of operators saturating the aforementioned lower bound facilitate a reconciliation of quaternionic quantum theory with a generalized Quantum Bayesian framework for reconstructing quantum state spaces.
The orthodox formulation of quantum theory invokes the mathematical apparatus of complex Hilbert space. In this thesis, we consider a quaternionic quantum formalism for the description of quantum states, quantum channels, and quantum measurements. We prove that probabilities for outcomes of quaternionic quantum measurements arise from canonical inner products of the corresponding quaternionic quantum effects and a unique quaternionic quantum state. We embed quaternionic quantum theory into the framework of usual complex quantum information theory. We prove that quaternionic quantum measurements can be simulated by usual complex positive operator valued measures. Furthermore, we prove that quaternionic quantum channels can be simulated by completely positive trace preserving maps on complex quantum states. We also derive a lower bound on an orthonormality measure for sets of positive semi-definite quaternionic linear operators. We prove that sets of operators saturating the aforementioned lower bound facilitate a reconciliation of quaternionic quantum theory with a generalized Quantum Bayesian framework for reconstructing quantum state spaces.
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.
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 paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space X with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of X under the semi-inner product, and then the state space S(X) of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space X equipped with a fixed semi-inner product, and then define a physical event at a quantum state ω∈S(X) to be a projection P (bounded operator such that P2=P) in X satisfying the positivity condition 0≤ω(P)≤1, and a physical quantity at a quantum state ω to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at ω. The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schrödinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.
A systematic study of additive isometries on a quaternionic Hilbert space is presented. A number of new results describing the properties of such operators are proved. The work culminates in the first mathematical proof of Wigner’s theorem for quaternionic Hilbert spaces of dimension other than 2 which asserts that any operator which preserves the absolute value of the inner product on a quaternionic Hilbert space is equivalent, in the sense of differing pointwise by a mere phase factor, to a linear isometry. A complete and concise description of the exceptional situation in a two-dimensional quaternionic Hilbert space is given.
We present an axiomatic construction of quaternionic probability, extending Kolmogorov’s classical framework to the noncommutative algebra of quaternions. The theory introduces quaternionic probability spaces, conditional probabilities, Bayes’ rules, independence, random variables, expectations, and transport equations, all formulated in a consistent manner. Classical probability is recovered through scalar projection, while restriction to complex subalgebras reproduces the standard quantum formalism. Uniquely quaternionic structures arise, including noncommutative conditional probabilities, inequivalent forms of independence, and quaternionic transport laws. The framework further develops quaternionic Markov chains, entropy, and divergence measures that separate scalar uncertainty from vectorial coherence. Several illustrative examples are provided to show how quaternionic probability captures order effects, hidden correlations, and orthogonal divergences—features invisible to both classical and complex approaches. These results establish quaternionic probability as a rigorous generalization of Kolmogorov’s axioms and as a potential foundation for future studies in noncommutative probability, integrable structures, and quaternionic extensions of mathematical physics.
Quantum theory combines density matrices, Born probabilities, tensor-product composites, positive-operator-valued measures (POVMs), and quantum channels. In a finite-dimensional causal operational theory, we prove that two postulates suffice: local input-output statistics identify channels, and every state admits an equivalent-system purification, unique up to reversible dynamics. The full complex quantum formalism follows; every consistent probability rule is realized as a POVM, so measurement no-restriction is derived rather than assumed.