S>The implications of group theory for the foundations of quaternionic quantum mechanics are discussed. It is proved that even in this case a relativistic description is possible. It is shown that the elementary systems for the Poincare group, as they appear in this realization, are in one-to-one correspondence with those of the usual complex quantum mechanics. The occurrence of charge conjugation related to space reflexion is discussed and some consequences on a TCP-relativistic theory are explored. The framework of a general relativistic quaternionic quantum mechanics is developed and the particular ties that exist between this theory and its complex analog are emphasized. The possible adequacy of a mixed complex-quaternionic description for isospin is discussed. (auth)
S>The implications of group theory for the foundations of quaternionic quantum mechanics are discussed. It is proved that even in this case a relativistic description is possible. It is shown that the elementary systems for the Poincare group, as they appear in this realization, are in one-to-one correspondence with those of the usual complex quantum mechanics. The occurrence of charge conjugation related to space reflexion is discussed and some consequences on a TCP-relativistic theory are explored. The framework of a general relativistic quaternionic quantum mechanics is developed and the particular ties that exist between this theory and its complex analog are emphasized. The possible adequacy of a mixed complex-quaternionic description for isospin is discussed. (auth)
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.
The possibility of a new quantum mechanical description in which the observables are realized as self-adjoint operators in quaternionic Hilbert spaces is investigated from the point of view of relativistic invariance. The problem is first formulated in the natural language of quantum mechanics, i.e. the one of proposition systems. This is followed by the theory of projective group representation and their reduction to unitary group representations in quaternionic Hilbert spaces. The general theory is then applied to the group of special relativity which leads to the classification of elementary systems in this theory.
In this paper the physical implications of quaternion quantum mechanics are further explored. In a quanternionic Hilbert space HQ, the lattice of subspaces has a symmetry group which is isomorphic to the group of all co-unitary transformations in HQ. In contrast to the complex space HC (ordinary Hilbert space), this group is connected, while for HC it consists of two disconnected pieces. The subgroup of transformations in HQ which associates with every quaternion q of magnitude 1, the correspondence ψ → qψq−1 for all ψ∈HQ (called Q conjugations), is isomorphic to the three-dimensional rotation group. We postulate the principle of Q covariance: The physical laws are invariant under Q conjugations. The full significance of this postulate is brought to light in localizable systems where it can be generalized to the principle of general Q covariance: Physical laws are invariant under general Q conjugations. Under the latter we understand conjugation transformations which vary continuously from point to point. The implementation of this principle forces us to construct a theory of parallel transport of quaternions. The notions of Q-covariant derivative and Q curvature are natural consequences thereof. There is a further new structure built into the quaternionic frame through the equations of motion. These equations single out a purely imaginary quaternion η(x) which may be a continuous function of the space—time coordinates. It corresponds to the i in the Schrödinger equation of ordinary quantum mechanics. We consider η(x) as a fundamental field, much like the tensor gμν in the general theory of relativity. We give here a classical theory of this field by assuming the simplest invariant Lagrangian which can be constructed out of η and the covariant Q connection. It is shown that this theory describes three vector fields, two of them with mass and charge, and one massless and neutral. The latter is identifiable with the classical electromagnetic field.
An attempt is made to enlarge the number field underlying quantum mechanics from complex to quaternion. It is not impossible to construct quaternion quantum mechanics in quite a similar manner to the usual complex quantum mechanics, though some limitations are necessary for the position of factors. Further, as a special case, the charge properties of the spin-0 particles are considered. This example may show some new way to generalized the concept of the charge of the elementary particles without adoption of the iso-space.
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.
New insight to the principles of the quantum theory construction is given. It is based on the symmetry study of main differential equations of mechanics and electrodynamics. It has been shown, that differential equations, which are invariant under transformations of groups, which are symmetry groups of mathematical numbers (considered within the frames of the number theory) determine the mathematical nature of the quantities, incoming in given equations. The substantial consequence of the given consideration is the proof of the main postulate of quantum mechanics, that is, the proof of the statement, that to any quantum mechanical quantity can be set up into the correspondence the Hermitian matrix. It is shown, that a non-abelian character of the multiplicative group of the quaternion ring leads to the nonapplicability of the quaternion calculus for the construction of new versions of quantum mechanics directly. The given conclusion seems to be actual, since there is a number of modern publications with the development of the quantum mechanics theory using the quaternions with the standard basis {e, i, j, k}. The correct way for the construction of new versions of quantum mechanics on the quaternion base is discussed in the paper presented. It is realized by means of the representation of the quaternions through the basis of the linear space of complex numbers over the field of real numbers, under the multiplicative group of which the equations of the dynamics of mechanical systems are invariant. At the same time the quaternion calculus is applicable in electrodynamics, at that the new versions of quantum electrodynamics can be constructed by an infinite number of the ways corresponding to an infinite number of the matrix representations of the standard quaternion basis {e, i, j, k}. The given conclusion is the consequence of the high symmetry of Maxwell equations.