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Physical Review Letters

QUATERNIONIC QUANTUM FIELD THEORY

Stephen L. Adler

Published Aug 19, 1985• 70 citations
Publisher

Abstract

We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions, as well as the bosons, obey a second-order wave equation. The theory is initially defined in terms of a quaternion-imaginary Lagrangean using the Feynman sum over histories. A Schr\"odinger equation can be derived from the functional integral, which identifies the quaternion-imaginary quantum Hamiltonian. Conversely, the transformation theory based on this Hamiltonian can be used to rederive the functional-integral formulation.

DOI
10.1103/physrevlett.55.783

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We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions, as well as the bosons, obey a second-order wave equation. The theory is initially defined in terms of a quaternion-imaginary Lagrangean using the Feynman sum over histories. A Schr"odinger equation can be derived from the functional integral, which identifies the quaternion-imaginary quantum Hamiltonian. Conversely, the transformation theory based on this Hamiltonian can be used to rederive the functional-integral formulation.
The correspondence between classical and quaternion quantum equations, permits considering the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase the energy density, consequently generate a gravity forcing and affect the wave speed. Consequently the gravity could be described by an index of refraction. Theory was created by combining the Cauchy model of the elastic continuum with the Planck-Kleinert crystal hypothesis. The quaternion-imaginary Lagrange’an, the quaternion motion equation and the quaternionic oscillator allowed deriving: - The Schrödinger equation from the functional integral, which identifies the quaternion-imaginary quantum Hamiltonian. - The 2nd order wave equation system describing both the bosons and the gravity in terms of quaternionic Poisson equation. - The first order quaternionic wave equation system. - The family of the second order wave equation systems describing both the particles and the generated quaternionic force-fields (four-potentials). - The fundamental Planck and gravity constants. - The quaternionic continuity equation in an ideal elastic solid.
We present a formulation of quantum field theory based entirely on real numbers, which we call real quantum field theory (RQFT). The construction is obtained from the standard complex-number formulation by replacing the imaginary unit i with a matrix J throughout all formulas. Here J is the real 2x2 matrix satisfying the condition J^2=-1. This extends the recently developed formulation of real quantum mechanics based on the real Kähler space to systems with infinitely many degrees of freedom. As the basic example, we construct the RQFT for a scalar field. We define a field operator acting on the real Kähler analogue of the bosonic Fock space. This operator satisfies the Klein-Gordon equation and the J-form of canonical commutation relation. To construct the RQFT we develop the corresponding J-calculus. In particular, we introduce the direct and inverse J-Fourier transforms and the associated J-valued distributions. In RQFT, the usual unitarity condition for the complex S-matrix is replaced by the statement that the real scattering operator is both orthogonal and symplectic. The physical observables in RQFT coincide with those of ordinary QFT. Thus RQFT does not change the physical predictions of scalar QFT, but provides an equivalent formulation. We explore the possibility of purely real formulations of the Standard Model of elementary particles. We show that it does admit this formulation and the resulting theory has ortho-symplectic symmetry. The choice of the Standard Model as the testbed for exploring the possibility of purely real formulations is related with the fact it provides a realistic description of all known fundamental particles. The real formulation of the Standard Model naturally suggests a possible exit beyond the Standard Model. In particular, we consider the implications of breaking the J-symmetry as a marker for new physics.
Abstract: 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 → ∞.
We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.
Solving the Schrödinger equation in quaternionic quantum mechanics (QQM) has been a nut to crack. In complex quantum mechanics (CQM), approximation methods have been used to find the solutions of the Schrödinger equation when exact solutions are difficult to obtain. This study challenges the earlier Eikonal (or WKB) approximation formalism [S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995)] developed for an imaginary quaternion potential. In this paper, the correct form of coupled equations, obeying the noncommutative structure and approximated solution for quaternionic time-independent Schrödinger equation (QTISE), has been obtained.
A new formulation of quantum mechanics based on differential commutator brackets is developed. We have found a wave equation representing the fermionic particle. In this formalism, the continuity equation mixes the Klein-Gordon and Schrodinger probability density while keeping the Klein-Gordon and Schrodinger current unaltered. We have found time and space transformations under which Dirac’s equation is invariant. The invariance of Maxwell’s equations under these transformations shows that the electric and magnetic fields of a moving charged particle are perpendicular to the velocity of the propagating particle. This formulation agrees with the quaternionic formulation recently developed by Arbab.