On non-homogeneous Cauchy–Fueter equations and Hartogs’ phenomenon in several quaternionic variables

The Cauchy–Fueter complex is the counterpart of the Dolbeault complex in the theory of several quaternionic variables. By using the fundamental solution to the Laplacian operators of fourth order associated to this differential complex on Hⁿ${\mathbb{H}}^n$, we can solve the system of non-homogeneous Cauchy–Fueter equations and prove the Hartogs’ extension phenomenon for quaternionic regular functions on any domain. The quaternionic version of Bochner–Martinelli integral representation formula for H$\mathbb{H}$-valued functions is also given.     

Quaternionic electroweak theory and Cabibbo-Kobayashi-Maskawa matrix

We find in our quaternionic version of the electroweak theory an apparently hopeless problem: In going from complex numbers to quaternions, the calculation of the real-valued parameters of the Cabibbo-Kobayashi-Maskawa matrix drastically changes. We aim to explain thisquaternionic puzzle.     

Schwinger algebra for quaternionic quantum mechanics

It is shown that the measurement algebra of Schwinger, a characterization of the properties of Pauli measurements of the first and second kinds, forming the foundation of his formulation of quantum mechanics over the complex field, has a quaternionic generalization. In this quaternionic measurement algebra some of the notions of quaternionic quantum mechanics are clarified. The conditions imposed on the form of the corresponding quantum field theory are studied, and the quantum fields are constructed. It is shown that the resulting quantum fields coincide with the fermion or boson annihilation-creation operators obtained by Razon and Horwitz in the limit in which the number of particles in physical states N→∞.     

One representation of the proper Lorentz group,and quaternionic analyticity

A representation of the proper Lorentz group G is considered, with operators realized as transformations of complex-valued functions of real variables x, y, z, and t. These transformations leave invariant the set of solutions of the Moisil-Fueter equation, defining quaternionic analytic functions. A subset of quaternionic analytic functions is found, which realizes irreducible representations of the group G. A slightly different representation is also considered, in the form of a transformation of functions which leaves invariant the set of solutions of the Dirac equation for a massless particle in the quaternionic form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 73–79, October, 1982.     

Computational aspects of the continuum quaternionic wave functions for hydrogen

Over the past few years considerable attention has been given to the role played by the Hydrogen Continuum Wave Functions (HCWFs) in quantum theory. The HCWFs arise via the method of separation of variables for the time-independent Schrödinger equation in spherical coordinates. The HCWFs are composed of products of a radial part involving associated Laguerre polynomials multiplied by exponential factors and an angular part that is the spherical harmonics. In the present paper we introduce the continuum wave functions for hydrogen within quaternionic analysis ((R)QHCWFs), a result which is not available in the existing literature. In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with R³${\mathbb{R}}^3$ and R⁴${\mathbb{R}}^4$). We prove that the (R)QHCWFs are orthonormal to one another. The representation of these functions in terms of the HCWFs are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of fundamental properties and further computation of the hydrogen-like atom transforms of the (R)QHCWFs are also discussed. We address all the above and explore some basic facts of the arising quaternionic function theory. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach.     

Quaternionic Lorentz Group and Dirac Equation

We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component spinors, we derive a quaternionic chiral representation for the space-time algebra. Finally, we obtain a quaternionic bi-dimensional version of the Dirac equation.     

Global effects in quaternionic quantum field theory

We present some striking global consequences of a model quaternionic quantum field theory which is locally complex. We show how making the quaternionic structure a dynamical quantity naturally leads to the prediction of cosmic strings and nonbaryonic hot dark matter candidates.     

Quantum mechanics: From complex to complexified quaternions

This paper is an attempt to simplify and clarify the mathematical language used to express quaternionic quantum mechanics (QQM). In our quaternionic approach the choice of “complex” geometries allows an appropriate definition of momentum operator and gives the possibility to obtain consistent formulations of standard theories. Barred operators represent the key to realizing a set of translation rules between quaternionic and complex quantum mechanics (QM). These translations enable us to obtain a rapid quaternionic counterpart of standard quantum mechanical results.     

Geometric quantization and internal symmetry

As a part of an attempt to geometrize physics, internal symmetries in the covariant classification of matter by itsT type are considered in relation to phase transformations generated by complex and quaternionic structures on space-time. The Rainich theory of electromagnetism and neutrinos is compared with the theory ofU(1) ×SO(1, 3) torsional gauge fields, and extended to the quaternionic case. It is shown by the Kostant technique of geometric quantization that complex and quaternionic phase transformations for an Einstein space are associated with one-dimensional and three-dimensional harmonic oscillators.     

Division Algebras and Quantum Theory

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure.     

Quantum mechanics of the quaternionic Hilbert spaces based upon the imprimitivity theorem

We survey the realization of quantum mechanics in quaternionic Hilbert spaces following the methods of Mackey, who examined the complex and real cases exploiting the imprimitivity theorem. We show that there exists a unique unitary skew-adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic imaginary unit. Difficulties in the definition of time reversal, however, arise because of the properties of the quaternionic field. The introduction of an extra imaginary unit, commuting with the others, is suggested in order to implement time reversal properly. In the Appendix we give the proof of the imprimitivity theorem, in the quaternionic case, that we use in the paper.     

Models of Local Relativistic Quantum Fields with Indefinite Metric (in All Dimensions)

A condition on a set of truncated Wightman functions is formulated and shown to permit the construction of the Hilbert space structure included in the Morchio--Strocchi modified Wightman axioms. The truncated Wightman functions which are obtained by analytic continuation of the (truncated) Schwinger functions of Euclidean scalar random fields and covariant vector (quaternionic) random fields constructed via convoluted generalized white noise, are then shown to satisfy this condition. As a consequence such random fields provide relativistic models for indefinite metric quantum field theory, in dimension 4 (vector case), respectively in all dimensions (scalar case). Received: 25 April 1996 / Accepted: 29 July 1996     

Field Extension of Real Values of Physical Observables in Classical Theory can Help Attain Quantum Results

Physical quantities are assumed to take real values, which stems from the fact that an usual measuring instrument that measures a physical observable always yields a real number. Here we consider the question of what would happen if physical observables are allowed to assume complex values. In this paper, we show that by allowing observables in the Bell inequality to take complex values, a classical physical theory can actually get the same upper bound of the Bell expression as quantum theory. Also, by extending the real field to the quaternionic field, we can puzzle out the GHZ problem using local hidden variable model. Furthermore, we try to build a new type of hidden-variable theory of a single qubit based on the result.     

Star-product in the presence of a monopole

We present a deformed ?-product for a particle in the presence of a magnetic monopole. The product is obtained within a self-dual quantization-dequantization scheme, with the correspondence between classical observables and operators defined with the help of a quaternionic Hilbert space, following work by Emch and Jadczyk. The resulting product is well defined for a large class of complex functions and reproduces (at first order in ?) the Poisson structure of the particle in the monopole field. The product is associative only for quantized monopole charges, thus incorporating Dirac's quantization requirement.     

Quaternionic and hyper-Kähler metrics from generalized sigma models

The problem of finding new metrics of interest, in the context of SUGRA, is reduced to two stages: first, solving a generalized BPS sigma model with full quaternionic structure proposed by the authors and, second, constructing the hyper-Kähler metric, or suitable deformations of this condition, taking advantage of the correspondence between the quaternionic left-regular potential and the hyper-Kähler metric of the target space. As illustration, new solutions are obtained using generalized Q-sigma model for Wess–Zumino type superpotentials. Explicit solutions analog to the Berger?s sphere and Abraham–Townsend type are given and generalizations of 4-dimensional quaternionic metrics, product of complex ones, are shown and discussed.     

Quaternionic complexes

Each regular or semi-regular integral affine orbit of the Weyl group of gl(2n + 2, ) invariantly determines a locally exact differential complex on a 4n dimensional quaternionic manifold. This gives quaternionic analogues of Dolbeault cohomology on complex manifolds. We compute the index of such complexes in the hyper-Kähler case, showing that quaternionic cohomology is not trivial.