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.
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.
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.
The conventional correspondence basis for quantum dynamics is here replaced by a self-contained quantum dynamical principle from which the equations of motion and the commutation relations can be deduced. The theory is developed in terms of the model supplied by localizable fields. A short review is first presented of the general quantum-mechanical scheme of operators and eigenvectors, in which emphasis is placed on the differential characterization of representatives and transformation functions by means of infinitesimal unitary transformations. The fundamental dynamical principle is stated as a variational equation for the transformation function connecting eigenvectors associated with different spacelike surfaces, which describes the temporal development of the system. The generator of the infinitesimal transformation is the variation of the action integral operator, the spacetime volume integral of the invariant lagrange function operator. The invariance of the lagrange function preserves the form of the dynamical principle under coordinate transformations, with the exception of those transformations which include a reversal in the positive sense of time, where a separate discussion is necessary. It will be shown in Sec. III that the requirement of invariance under time reflection imposes a restriction upon the operator properties of fields, which is simply the connection between the spin and statistics of particles. For a given dynamical system, changes in the transformation function arise only from alterations of the eigenvectors associated with the two surfaces, as generated by operators constructed from field variables attached to those surfaces. This yields the operator principle of stationary action, from which the equations of motion are obtained. Commutation relations are derived from the generating operator associated with a given surface. In particular, canonical commutation relations are obtained for those field components that are not restricted by equations of constraint. The surface generating operator also leads to generalized Schr\"odinger equations for the representative of an arbitrary state. Action integral variations which correspond to changing the dynamical system are discussed briefly. A method for constructing the transformation function is described, in a form appropriate to an integral spin field, which involves solving Hamilton-Jacobi equations for ordered operators. In Sec. III, the exceptional nature of time reflection is indicated by the remark that the charge and the energy-momentum vector behave as a pseudoscalar and pseudovector, respectively, for time reflection transformations. This shows, incidentally, that positive and negative charge must occur symmetrically in a completely covariant theory. The contrast between the pseudo energy-momentum vector and the proper displacement vector then indicates that time reflection cannot be described within the unitary transformation framework. This appears most fundamentally in the basic dynamical principle. It is important to recognize here that the contributions to the lagrange function of half-integral spin fields behave like pseudoscalars with respect to time reflection. The non-unitary transformation required to represent time reflection is found to be the replacement of a state vector by its dual, or complex conjugate vector, together with the transposition of all operators. The fundamental dynamical principle is then invariant under time reflection if inverting the order of all operators in the lagrange function leaves an integral spin contribution unaltered, and reverses the sign of a half-integral spin contribution. This implies the essential commutativity, or anti-commutativity, of integral and half-integral field components, respectively, which is the connection between spin and statistics.
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 the absence of external relata, internal quantum reference frames (QRFs) appear widely in the literature on quantum gravity, gauge theories and quantum foundations. Here, we extend the perspective-neutral approach to QRF covariance to general unimodular Lie groups. This is a framework that links internal QRF perspectives via a manifestly gauge-invariant Hilbert space in the form of "quantum coordinate transformations", and we clarify how it is a quantum extension of special covariance. We model the QRF orientations as coherent states which give rise to a covariant POVM, furnishing a consistent probability interpretation and encompassing non-ideal QRFs whose orientations are not perfectly distinguishable. We generalize the construction of relational observables, establish a variety of their algebraic properties and equip them with a transparent conditional probability interpretation. We import the distinction between gauge transformations and physical symmetries from gauge theories and identify the latter as QRF reorientations. The "quantum coordinate maps" into an internal QRF perspective are constructed via a conditioning on the QRF's orientation, generalizing the Page-Wootters formalism and a symmetry reduction procedure. We find two types of QRF transformations: gauge induced "quantum coordinate transformations" as passive unitary changes of description and symmetry induced active changes of relational observables from one QRF to another. We reveal new effects: (i) QRFs with non-trivial orientation isotropy groups can only resolve isotropy-group-invariant properties of other subsystems; (ii) in the absence of symmetries, the internal perspective Hilbert space "rotates" through the kinematical subsystem Hilbert space as the QRF changes orientation. Finally, we invoke the symmetries to generalize the quantum relativity of subsystems before comparing with other approaches. [Abridged]
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
An isomorphism is shown to exist between the group of point transformations in classical mechanics and a certain subgroup of the group of all unitary transformations in quantum mechanics. This isomorphism is used to indicate that the quantum analogs of physically significant classical expressions can be constructed uniquely in any coordinate system. There is no ambiguity in the ordering of noncommuting quantum operators, and the method of constructing the quantum analogs is covariant under general coordinate transformations. The method is actually only applicable to systems having Lagrangians which are at most quadratic in the velocities, but this includes all systems which are presently of interest in physics. The method is applied to two intrinsically nonlinear examples, one of which is the gravitational field. The correct Hamiltonian operator for a quantized version of Einstein's gravitational theory is constructed.