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Principle of General Q Covariance
Abstract
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 the case of quaternionic quantum theory [29][27] [28], the very notion of 'independent subsystems' is a subject of debate. In fact, quaternion-linear tensor products of quaternionic modules do not exist [63]. ...
... In this thesis, we consider a generalized formulation of quaternionic quantum theory, rather than only considering the restricted class of quantum processes treated in [25] [29][27] [28][1]. We treat generalized quantum measurements as quaternionic positive operator valued measures, and we treat quantum channels as completely positive trace preserving quaternionic maps. ...
... Then i 1 q = (−q 1 + i 3 q 2 ) + i 2 (−q 3 + i 3 q 0 ) → −q 1 + iq 2 −q 3 + iq 0 = 0 i i 0 q 0 + iq 3 q 2 + iq 1 , (A. 28) which shows that left-multiplication by i 1 corresponds to iσ x . Similarly, ...
... The appeal to higher division-algebraic generalizations of quantum theory has a long and distinguished history. Of note, Finkelstein, Jauch, Schiminovich and Speiser provided the first example of quaternionic quantum theory in 1962/1963 [40,41]-the latter of which is remarkable for introducing a mass-generating mechanism for an SU (2)gauge theory by means of a quaternion-imaginary dynamical field playing much the same role as the Higgs field in the later explanation of electroweak symmetry breaking. ...
... A difficulty of this sort was already encountered by Finkelstein, Jauch, Schiminovich and Speiser in their original formulation of quaternionic quantum mechanics [41]. There, it was proposed that the 'special' quaternion imaginary direction should be promoted to a dynamical variable, thus introducing an additional scalar degree of freedom into the theory (a move that was generalized to the octonionic case by Casalbouni, Domokos, and Kövesi-Domokos [68]). ...
Research at the intersection of quantum gravity and quantum information theory has seen significant success in describing the emergence of spacetime and gravity from quantum states whose entanglement entropy approximately obeys an area law. In a different direction, the Kaluza-Klein proposal aims to recover gauge symmetries by means of dimensional reduction of higher-dimensional gravitational theories. Integrating both, gravitational and gauge degrees of freedom in \(3+1\) dimensions may be obtained upon dimensional reduction of higher-dimensional emergent gravity. To this end, we show that entangled systems of two and three qubits can be associated with \(5+1\) and \(9+1\) dimensional spacetimes respectively, which are reduced to \(3+1\) dimensions upon singling out a preferred complex direction. In the latter case, this reduction is invariant under a residual \(SU(3) \times SU(2) \times U(1) /\mathbb{Z}_6\) symmetry, the Standard Model gauge group. This motivates a picture in which spacetime emerges from the area law-contribution to the entanglement entropy, while gauge and matter degrees of freedom are due to area law-violating terms. We remark on a possible natural origin of the chirality of the weak force in the given construction.
... In 1936, Birkhoff and von Neumann [4] introduced the idea of formulating quantum mechanics in quaternion setting. Later several authors continued the study of quaternionic Hilbert spaces in various directions (see [1,2,7,8,9,10,14] for details). There was no suitable notion of spectrum of quaternionic linear operators until the concept of spherical spectrum was proposed, in 2007, by Colombo, Gentile, Sabadini, and Struppa [5]. ...
... It is clear that Z T+ = Z T and Z T+ is bounded C m -linear operator. By Theorem 3.1, there is a measure space (Ω; µ), a unitary operator U + : H Jm + → L 2 (Ω; C m ; µ) and a µ-measurable function φ such that (9) Z T+ = U * + L φ U + . ...
Let H be a right quaternionic Hilbert space and let T be a quaternionic normal operator with domain D(T)⊂H. We prove that there exists a Hilbert basis N of H, a measure space (Ω0,ν), a unitary operator U:H→L2(Ω0;H;ν) and a ν-measurable function η:Ω0→C such thatTx=U⁎MηUx,for allx∈D(T) where Mη is the multiplication operator on L2(Ω0;H;ν) induced by η with U(D(T))⊆D(Mη). We show that every complex Hilbert space can be seen as a slice Hilbert space of some quaternionic Hilbert space and establish the main result by reducing the problem to the complex case then lift it to the quaternion case.
... In 1936, Birkhoff and von Neumann [4] introduced the idea of formulating quantum mechanics in quaternion setting. Later several authors continued the study of quaternionic Hilbert spaces in various directions (see [1,2,7,8,9,10,14] for details). There was no suitable notion of spectrum of quaternionic linear operators until the concept of spherical spectrum was proposed, in 2007, by Colombo, Gentile, Sabadini, and Struppa [5]. ...
... It is clear that Z T+ = Z T and Z T+ is bounded C m -linear operator. By Theorem 3.1, there is a measure space (Ω; µ), a unitary operator U + : H Jm + → L 2 (Ω; C m ; µ) and a µ-measurable function φ such that (9) Z T+ = U * + L φ U + . ...
Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a quaternionic normal operator with the domain $\mathcal{D}(T) \subset \mathcal{H}$. Then for a fixed unit imaginary quaternion $m$, there exists a Hilbert basis $\mathcal{N}_{m}$ of $\mathcal{H}$, a measure space $(\Omega, \mu)$, a unitary operator $U \colon \mathcal{H} \to L^{2}(\Omega; \mathbb{H}; \mu)$ and a $\mu$ - measurable function $\phi \colon \Omega \to \mathbb{C}_m$ (here $\mathbb{C}_{m} = \{\alpha + m \beta; \;\alpha, \beta \in \mathbb{R}\}$) such that
\[
Tx = U^{*}M_{\phi}Ux, \; \mbox{for all}\; x\in \mathcal{D}(T),
\]
where $M_{\phi}$ is the multiplication operator on $L^{2}(\Omega; \mathbb{H}; \mu)$ induced by $\phi$ with $ U(\mathcal{D}(T)) \subseteq \mathcal{D}(M_{\phi})$. In the process, we prove that every complex Hilbert space is a slice Hilbert space.
We establish these results by reducing it to the complex case then lift it to the quaternionic case.
... The product a • a • a is also unambiguously a 3 because of commutativity, but a • a • a • a is potentially ambiguous as it could be (a • a) • (a • a) or a • (a • (a • a). Power associativity ensures that they agree and, more generally, that a n • a m = a m+n (4) for all positive integers m and n. This property allows us to define unambiguously functions of an observable as power series expansions. ...
... The above classification already suggests one generalization of quantum mechanics; i.e. that for which the complex numbers are replaced by the real numbers, quaternions or octonions. It appears that real quantum mechanics is esentially equivalent to complex quantum mechanics, and quaternion complex mechanics suffers from a surplus of imaginary units and does not yield much that is new [4]. The octonionic quantum mechanics based on the exceptional Jordan algebra [5] is the most interesting case because exceptionality implies that no Hilbert space formulation is possible. ...
This is a transcription of a conference proceedings from 1985. It reviews the Jordan algebra formulation of quantum mechanics. A possible novelty is the discussion of time evolution; the associator takes over the role of $i$ times the commutator in the standard density matrix formulation, and for the Jordan algebra of complex Hermitian matrices this implies a Hamiltonian of the form $H= i[x,y] + \lambda \mathbb{I}$ for traceless Hermitian $x,y$ and real number $\lambda$. Other possibilities for time evolution in the Jordan formulation are briefly considered.
... About 100 years later, Finkelstein et al. [2][3][4][5] built the foundations of qQM and gauge theories. Their fundamental works led to a renewed interest in algebrization and geometrization of physical theories by non-commutative fields [6,7]. ...
... Set W = I 4n 1. for r = 1 : n − 1 2. for t = r + 1 : n 3. Generate a generalized symplectic Givens rotation for the (t, r)-th elements of Q s by G 1 = JRSGivens(Q 0 (t, r), Q 1 (t, r), Q 2 (t, r), Q 3 (t, r)). 4. Denote a s = Q s ([r : t − 1, t + 1 : n], t). ...
In this paper we propose a novel structure-preserving algorithm for solving the right eigenvalue problem of quaternion Hermitian matrices. The algorithm is based on the structure-preserving tridiagonalization of the real counterpart for quaternion Hermitian matrices by applying orthogonal JRSJRS-symplectic matrices. The algorithm is numerically stable because we use orthogonal transformations; the algorithm is very efficient, it costs about a quarter arithmetical operations, and a quarter to one-eighth CPU times, comparing with standard general-purpose algorithms. Numerical experiments are provided to demonstrate the efficiency of the structure-preserving algorithm.
... Quaternions are extensions of real and complex numbers. In recent years, with the development of science and technology, quaternions and quaternion matrices are more and more extensively applied in many fields [10][11][12][13][14][15][16]. Naturally, some scholars turn their interest to the following quaternion equality constrained least squares (QLSE) problem. ...
In this paper, we focus on the quaternion equality constrained weighted least squares problem. First, according to the properties of the Moore-Penrose generalized inverse matrix, the singular value decomposition, the CS decomposition, and the real representation matrices of quaternion matrices, we obtain a real unconstrained equivalent problem of the quaternion equality constrained weighted least squares problem. And then we give the expressions of its general solution and minimal norm solution. Finally, according to the properties of the real representation matrices of quaternion matrices, the special structure of real representation matrices and the real structure-preserving singular value decomposition of quaternion matrix, we propose a real structure-preserving algorithm for the minimal norm solution of the quaternion equality constrained weighted least squares problem, and illustrate the effectiveness of proposed algorithm by a numerical example.
... Some spectral properties of quaternionic matrices have been codeveloped by physicists. In particular, the development of quaternion quantum mechanics and gauge theories by Birkhoff and von Neumann [13], and Finkelstein, Jauch and Speiser [29][30][31][32] inspired much research [1]. More recently, quaternion differential operators [22] and the right quaternionic eigenvalue problem [23,24] have been studied. ...
A quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, which is the inverse of the quaternion unit assigned to the opposite orientation. In this paper we define the adjacency, Laplacian and incidence matrices for a quaternion unit gain graph and study their properties. These properties generalize several fundamental results from spectral graph theory of ordinary graphs, signed graphs and complex unit gain graphs. Bounds for both the left and right eigenvalues of the adjacency and Laplacian matrix are developed, and the right eigenvalues for the cycle and path graphs are explicitly calculated.
... Quaternions were invented by Hamilton in 1843 [1,2]. Starting in the 1960s with the work of Kaneno [3], Finkelstein et al [4,5] and Emch [6], there have been many efforts in formulating quantum mechanics based on the non-commutativity ring of quaternions. Adler has written a comprehensive review of a formulation of quaternionic quantum mechanics (QQM) and quantum field theory based on a theory of anti-Hermitian operators [7]. ...
A thorough approach to quantum mechanics should rely on an algebraic operation of multiplication based on proper algebraic structures, i.e., division algebras. Quaternions, with their commutation rules isomorphic to SU (2), offer such a framework. In this work, a quaternion algebraic approach derived from the finite groups is used to solve the spatially invariant time-dependent Pauli-Schrödinger equation and derive the Rabi formula and the principle of nuclear magnetic resonance and its generalization. The spatial and time-dependent Pauli-Schrödinger equation is then written using left-chirality quaternions, i.e., by expressing the Pauli spinor, its time derivative, and the Hamiltonian as elements of the same division algebra. The quaternion formalism is used to derive the dispersion relation of a free electron and of an electron in a constant magnetic field. Finally, the formalism is applied to calculate the transmission probabilities through a one-dimensional delta scatterer and through a resonant tunneling structure composed of two one-dimensional quaternionic delta-scatterers in series. The results are compared to the transmission probabilities calculated using traditional (complex) quantum mechanics.
... All four algebra's are alternative with totally anti symmetric associators. Quaternions [8,9] were very first example of hyper complex numbers have been widely used [10,11,12] to the various applications of mathematics and physics. Recently, Chanyal et al [13,14,15] studied the role of highest norm division algebra (i.e. ...
The quaternions are first hyper-complex numbers, having four-dimensional structure, which may be useful to express the 4-dimensional theory of dyons carrying both electric and magnetic charges. Keeping in mind t'Hooft's monopole solutions and the fact that despite the potential importance of massive monopole, we discuss a connection between quaternionic complex field, to the generalized electromagnetic field equations of massive dyons. Starting with the Euclidean space-time structure and two four-components theory of dyons, we represent the generalized charge, potential, field and current source in quaternion form with real and imaginary part of electric and magnetic constituents of dyons. We have established the quaternionic formulation of generalized complex-electromagnetic fields equations, generalized Proca-Maxwell's (GPM) equations and potential wave equations for massive dyons. Thus, the quaternion formulation be adopted in a better way to understand the explanation of complex-field equations as the candidate for the existence of massive monopoles and dyons where the complex parameters be described as the constituents of quaternion.
... They are suitable options to express some classical phenomena. The idea of using quaternions for quantum mechanics yielded some works on its foundations, such as papers by Kaneno [1], Finkelstein et al. [2,3], and Emch [4]. They tried to make a comprehensive fundamental discussion for quaternionic quantum mechanics. ...
In this paper, the Schrödinger equation for quaternionic quantum mechanics with a Dirac delta potential has been investigated. The derivative discontinuity condition for the quaternionic wave function has been derived and the boundary conditions for the quaternionic wave function have been applied. Probability current densities for different regions of the problem have been determined along with reflection and transmission coefficients. Résumé : Nous étudions ici l'équation de Schrödinger pour la mécanique quantique des quaternions avec un potentiel en fonction delta. Nous dérivons la condition de discontinuité dérivative pour la fonction d'onde de quaternion et appliquons les conditions limites a ` cette fonction d'onde. Les densités de courant de probabilité sont évaluées pour différentes régions du problème et de même pour les coefficients de réflexion et de transmission. [Traduit par la Rédaction] Mots-clés : mécanique quantique des quaternions, potentiel en fonction delta, équation de Schrödinger, coefficients de réflexion et de transmission.
... Physical applications of the widest normed algebra of octonions are relatively rare (see reviews [4][5][6][7][8]). One can point to the possible impact of octonions on color symmetry [9][10][11][12][13]; GUTs [14][15][16][17]; representation of Clifford algebras [18][19][20]; quantum mechanics [21][22][23][24][25][26][27][28][29]; space-time symmetries [30,31]; field theory [32][33][34][35]; quantum Hall effect [36]; Kaluza-Klein program without extra dimensions [37][38][39][40][41][42]; Strings and Mtheory [43][44][45][46][47][48]; SUSY [49]; and so forth. ...
Physical signals and space-time intervals are described in terms of split
octonions. Geometrical symmetries are represented by the automorphism group of
the algebra - the real non-compact form of Cartan's smallest exceptional group
G2. This group generates specific rotations of (3+4)-vector parts of split
octonions with three extra time-like coordinates and in certain limits reduces
to standard Lorentz group. In this picture several physical characteristics of
ordinary (3+1)-dimensional theory (such as: number of spatial dimensions,
existence of maximal velocities, the uncertainty principle, some quantum
numbers) are naturally emerge from the properties of the algebra.
... Over the last decade, after fundamental works of Finkelstein et al. [2,3] on foundation of quaternionic quantum mechanics and gauge theories, we have witnessed a renewed interest in algebrization and geometrization of physical theories by non-commutative fields [1]. Among the numerous references on this subject, we recall the important paper of Horwitz and Biedenharn [5], where the authors showed that the assumption of a complex projection of the scalar product that permits the OSWALDO GONZÁLEZ-GAXIOLA 2 definition of a suitable tensor product between single-particle quaternionic wavefunctions. ...
The purpose of this note is in addition to establishing Fréchet derivativesand Gâteuax, considering the basic different implications between them. Also be considered a counterexample of a Lipschitzian real-valuedfunction Gâteaux differentiable but not Fréchet differentiable.
... All 4 algebras are alternative with totally antisymmetric associators. Quaternions [2,3] were the very first example of hypercomplex numbers and have been widely used [4,5,6,7,8,9] in various applications of mathematics and physics. Since octonions [10,11] share with complex numbers and quaternions many attractive mathematical properties, one might expect that they would be equally as useful. ...
We know that the octonion algebra is the largest division algebra. Therefore, we have discussed the octonion 8-dimensional space as the combination of 2 (external and internal) 4-dimensional spaces. The octonion wave equations in terms of 8 components has been written in terms of an 8 × 8 matrix. Octonion forms of potential as well as fields equations of dyons in terms of an 8 × 8 matrix are discussed in a consistent manner. At last, we have obtained the generalized Dirac-Maxwell equations of dyons in two 4-dimensional spaces, from the 8 × 8 matrix representation of octonion wave equations. Generalized Dirac-Maxwell equations are fully symmetric Maxwell's equations and allow for the possibility of magnetic charges and currents, i.e. analogous to electric charges and currents..
... All four algebra's are alternative with totally anti symmetric associators. Quaternions [17, 18] were very first example of hyper complex numbers have been widely used [19][25] to the various applications of mathematics and physics. Since octonions [26] share with complex numbers and quaternions, many attractive mathematical properties, one might except that they would be equally as useful as others. ...
In this paper, we have made an attempt to discuss the role of octonions in superstring theory (i.e. a theory of everything to describe the unification of all four types of forces namely gravitational, electromagnetic, weak and strong) where, we have described the octonion representation of the super-string (SS) formulation as the combination of four complex (C) spaces namely associated with the gravitational (G-space), electromagnetic (EM-space), weak (W-space) and strong (S-space) interactions. We have discussed the octonionic differential operator, octonionic valued potential wave equation, octonionic field equation and other various quantum equations of superstring formulation in simpler, compact and consistent manner. Consequently, the generalized Dirac-Maxwell's equations are studied with the preview of superstring theory by means of octonions.
This article delves into the Ramsauer-Townsend effect regarding the realm of quaternionic relativistic quantum mechanics. The investigation centers on studying the Dirac equation within this framework. The article presents a thorough examination of various facets of the subject matter, including a comprehensive analysis of quaternionic potential wells, wave functions, probability currents, reflection coefficients, and transmission coefficients.
Keeping in view the advantages of quaternion, an attempt is made to explain supersymmetry breaking in the interacting field in terms of quaternion superpartner Hamiltonian. A complete theory for supersymmetry breaking in the thermal field in terms of bosonic and fermionic operators has been obtained along with certain restrictions due to the non-commutativity of quaternions that can hint at some hidden results. Witten index in terms of quaternion units has been obtained, and we find that energy is different along different axes. It has been observed that supersymmetry remains good symmetry at zero temperature but breaks down with temperature.
In this paper, we study error estimates of the quaternion least squares problem and the quaternion weighted least squares problem. For different cases, we define different condition numbers and effective condition numbers. We propose upper bounds of absolute and relative errors of the minimal norm LS solution and the general LS solution of the quaternion least squares problem and the quaternion weighted least squares problem, respectively, by using different condition numbers and effective condition numbers to make error estimates sharper.
We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.
In this paper, we study the power method of the right eigenvalue problem of a quaternion matrix A. If A is Hermitian, we propose the power method that is a direct generalization of that of complex Hermitian matrix. When A is non-Hermitian, by applying the properties of quaternion right eigenvalues, we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector. We also briefly discuss the inverse power method and shift inverse power method for the both cases. The real structure-preserving algorithm of the power method in the two cases are also proposed, and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method.
We argue that quaternions form a natural language for the description of quantum-mechanical wavefunctions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the non-relativistic limit to find the Schr\"odinger's equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the non-interacting non-relativistic limit the quaternionic description effectively reduces to the regular complex wavefunction language. We provide an easy to use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell's equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.
In this article, one of the well-known effects in quantum mechanics is addressed and also the extended form of quantum mechanics which is based on quaternions is presented. In the presence of this version of quantum mechanics the Ramsauer–Townsend effect has been investigated and the existence of this phenomenon is studied according to quaternionic calculations; results are presented by graphs.
We investigate the Schrödinger equation in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. We propose an analytical method to solve bicomplex-version of Schrödinger equation corresponds to the systems of Hamiltonians of both hermitian (self-adjoint) and non-hermitian symmetric type. In our approach we extend the existing mathematical formulation of quantum system searching for the exact or quasi-exact solution for ground state energy eigenvalues and associated wave functions acting in bicomplex Hilbert space. The model concerning hermitian Hamiltonians is then applied to the problems of two bicomplex valued polynomial oscillators one involving and another of isotonic type. The ground states and associated energy values for both the oscillators are found to be hyperbolic in nature. The model in connection to the unbroken symmetric Hamiltonians is then applied to illustrate the problems of complex and bicomplex valued shifted oscillators.
In this paper, we survey three different forms of Householder based transformations for quaternion matrices in the literature, and propose a new form of quaternion Householder based transformation. We propose real structure-preserving algorithms of these Householder based transformations, which make the procedure computationally more flexible and efficient. We compare the computation counts and assignment numbers of these algorithms. We also compare the effectiveness of these real structure-preserving algorithms applying to the quaternion QRD and the quaternion SVD.
All these four real structure-preserving algorithms are more efficient, comparing to the algorithms which apply Quaternion Toolbox using quaternion arithmetics, or algorithms which directly performs real Householder transformations on the real representation of a quaternion matrix. Among these four real structure-preserving algorithms, the most efficient ones are based on quaternion Householder reflection, and new proposed Householder based transformation.
The use of complex geometry allows us to obtain a consistent formulation of octonionic quantum mechanics (OQM). In our octonionic formulation we solve the hermiticity problem and define an appropriate momentum operator within OQM. The nonextendability of the completeness relation and the norm conservation is also discussed in detail.
In this article we prove the existence of the polar decomposition for densely
defined closed right linear operators in quaternionic Hilbert spaces: If $T$ is
a densely defined closed right linear operator in a quaternionic Hilbert space
$H$, then there exists a partial isometry $U_{0}$ such that $T = U_{0}|T|$. In
fact $U_{0}$ is unique if $N(U_{0}) = N(T)$. In particular, if $H$ is separable
and $U$ is a partial isometry with $T = U|T|$, then we prove that $U = U_{0}$
if and only if either $N(T) = \{0\}$ or $R(T)^{\bot} = \{0\}$.
We prove the Weyl-von Neumann-Berg theorem for quaternionic right linear
operators (not necessarily bounded) in a quaternionic Hilbert space: Let $N$ be
a right linear normal (need not be bounded) operator in a quaternionic
separable Hilbert space $H$. Then for a given $\epsilon>0$ there exists a
compact operator $K$ with $\|K\|<\epsilon$ and a diagonal operator $D$ on $H$
such that $N=D+K$.
A quaternionic wavefunction consisting of real and scalar functions is found to satisfy the quaternionic
momentum eigen value equation. Each of these components is found to satisfy a generalized damped wave
equation. This reduces to the massless Klein-Gordon equation for certain cases. For a plane wave solution the
angular frequency is complex. This equation is in agreement with the Einstein energy-momentum formula. The
spin of the particle is obtained from the interaction of the particle with the photon field.
Since non-semisimple groups are unstable, every non-semisimple group in physics will probably be the locus of a major conceptual reformation. We carry out such reformations of the canonical commutation relations, the space-time continuum, and the interface between experimenter and system. The canonical reformation introduces a large orthogonal group and its Clifford algebra and spinors. The space-time reformation eliminates the differential calculus from basic physical laws, introducing a generic elementary quantum process and a fundamental time %. The interface reformation leads to a non-operational cosmic quantum theory that treats the physical experimenter as a quantum system, recovering the operational quantum theory by a systematic quotient process. We propose a real form of Fermi statistics for the elementary processes, in a Clifford algebra generated from the real line (which represents the empty set) by iterated quantification. We use this quantum set language to program the spin and space-time variables and equations of motion of a Dirac particle.
In this Chapter we briefly recall the fundamentals of quantum physics that we shall need in what follows. In section 2.1 we give the rules of the game, in section 2.2 we discuss one of the more disconcerting issues of quantum theory, namely, the whole and its parts, and in section 2.3 we discuss the fundamental (yet open) problem of the classical limit of the theory and the phenomenon of decoherence. In section 2.4 we study some proposed proofs of the frequency interpretation.
From a number of qualitative conjectures, the constantsm e ,c, ħ, and a spin(8) gauge field theory, I derive the following particle masses (quark masses are constituent masses) and force constants: up quark mass=312.7542 MeV; down quark mass=312.7542 MeV; proton mass=938.2626 MeV; neutrino masses (all types)=0; muon mass=104.76 MeV; strange quark mass=523 MeV; charmed quark mass=1989 MeV; tauon mass=1877 MeV; bottom quark mass=5631 MeV; top quark mass=129.5 GeV;W + mass=80.87 GeV;W − mass=80.87 GeV;W 0 mass=99.04 GeV; fine structure constantα= 1/137.036082; weak constant times the proton mass squared φM p2=0.97×10−5; color constant=0.6286. From the pion mass in addition, I derive the Planck mass ≈(1–1.6)×1019 GeV, so that the gravitational constant times the proton mass squared GM p2 ≈ (3.6–8.8)×10−39.
A new approach to reconciling General Relativity with Quantum Field Theory is Relativization, the act of making a physical model which obeys the principles of special relativity and General Relativity. This approach immediately yields results that no other approach has. I have established the foundations and fundamentals of relativization via a set of axioms. Expressions such as appear P μ | p > = k μ | p > P^{\mu}\left|p\right\rangle\,=k^{\mu}\left|p\right\rangle , or < p | P μ | p > = < p | k μ | p > = k μ < p | p > = k μ \left\langle p\right|P^{\mu}\left|p\right\rangle\,=\,\left\langle p\right|k^{% \mu}\left|p\right\rangle\,=k^{\mu}\left\langle p\right.\left|p\right\rangle\,=% k^{\mu} . Such expressions appear in textbooks and papers but they are given a clearer interpretation in this model. Using this new approach to the problem, I will formulate the standard model as a relativized model in a curved space time with a locally valid graviton-Higgs interaction. This interaction will lead to a renormalized perturbation theory that can be summed up exactly to give the amplitude of graviton-graviton interaction as approximately C o s h ( p ) Cosh(p) . I will solve a Schrodinger equation for a gravitationally bound system and get theoretical predictions relating to the thermodynamics of Planck scale black holes. Relativization has already provided a finite quantitative prediction for the quantum corrections to the local gravitational field, and gives results compatible with established black hole thermodynamics. Further research will certainly yield new insights.
It is well known that the Schrödinger equation is only suitable for the particle in common potential V (⃗r, t). In this paper, a general Quantum Mechanics is proposed, where the Lagrangian is the general form. The new quantum wave equation can describe the particle which is in general potential V (⃗r, ˙ ⃗r, t). We think these new quantum wave equations can be applied in many fields. PACS numbers: 03.65.Ta; 03.65.-w
We investigate bicomplex Hamiltonian systems in the framework of an analogous version
of the Schr¨odinger equation. Since in such a setting different types of conjugates of bicomplex
numbers appear, each defines a separate class of time reversal operator. We are thus in
a position to explore the corresponding extensions of parity (P)-time (T)-symmetric models
by generalizing the concept of an extended phase space. Applications to the problems of
harmonic oscillator, inverted oscillator and isotonic oscillator are considered and many new
interesting properties are uncovered for the new types of PT symmetries.
A theoretical foundation for spin(8) gauge field theory is proposed to explain the numerical results announced in a previous paper.
Motivated in the mathematical fundamentation of the quaternionic quantum mechanics, in the present work we introduce the concept of imaginary operator J to decompose any normal operator on a quaternionic Hilbert space as an integral of operators of the form a + Jb (J2 = -I) through establishment of a spectral theorem for normal operators on a quaternionic Hilbert space.
In this paper, we represent 16-component sedenions, the generalization of octonions, which is noncommutative space–time algebra. The sedenions is neither a composition algebra nor a division algebra because it has zero divisors. Here we have formulated the sedenionic unified potential equations, unified fields equations and unified current equations of dyons and gravito-dyons. We have developed the sedenionic unified theory of dyons and gravito-dyons in terms of two eight-potentials leading to the structural symmetry between generalized electromagnetic fields of dyons and generalized gravito-Heavisidian fields of gravito-dyons. Thus we have obtained the sedenionic form of generalized Dirac–Maxwell’s equations, unified work–energy theorem (Poynting theorem), generalized unified gravi-electromagnetic force and other quantum equations of dyons and gravito-dyons in simple, compact and consistent way incorporating the non-associativity and non-commutativity of sedenion variables.
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A generalization of quantum mechanics is proposed, where the Lagrangian is the general form. The new quantum wave equation can describe the particle which is in general potential $V({\bf r}, \dot{\bf r}, t)$, and the Schrödinger equation is only suited for the particle in common potential V(r, t). We think these new quantum wave equations can be used in some fields.
It is shown that a simple continuity condition in the algebra of split octonions suffices to formulate a system of differential equations that are equivalent to the standard Dirac equations. In our approach the particle mass and electromagnetic potentials are part of an octonionic gradient function together with the space–time derivatives. As distinct from previous attempts to translate the Dirac equations into different number systems here the wave functions are real split octonions and not bi-spinors. To formulate positively defined probability amplitudes four different split octonions (transforming into each other by discrete transformations) are necessary, rather then two complex wave functions which correspond to particles and antiparticles in usual Dirac theory.
In this paper we propose a fast structure-preserving algorithm for computing the singular value decomposition of quaternion matrices. The algorithm is based on the structure-preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal JRS-symplectic matrices. The algorithm is efficient and numerically stable.
In this paper, we have made an attempt to reformulate the generalized field equation and various quantum equations of massive dyons in terms of octonion eight dimensional space as the combination of two (external and internal) four dimensional spaces. The octonion forms of generalized potential and current equations of massive dyons are discussed in consistent manner. It has been shown that due to the non associativity of octonion variables it is necessary to impose certain constraints to describe generalized octonion massive electrodynamics in manifestly covariant and consistent manner.
Starting with the usual definitions of octonions and split octonions in terms of Zorn vector matrix realization, we have made an attempt to write the continuity equation and other wave equations of dyons in split octonions. Accordingly, we have investigated the work energy theorem or "Poynting Theorem," Maxwell stress tensor and Lorentz invariant for generalized fields of dyons in split octonion electrodynamics. Our theory of dyons in split octonion formulations is discussed in term of simple and compact notations. This theory reproduces the dynamic of electric (magnetic) in the absence of magnetic (electric) charges.
It is shown 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
Schroedinger 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.
Gauge fields interacting with source fields in a gauge‐noninvariant way are examined in a purely classical framework. The existence of conserved currents for the solutions of such theories follows from the properties of the free gauge fields rather than from the usual requirement of global gauge covariance. One of the simplest gauge‐noninvariant theories of electromagnetism is shown to be an extension of Dirac's theory of classical clouds of charge. In this extension the clouds exhibit some quantum features. Gauge‐noninvariant interactions for Einstein's theory of gravitation are presented in which the properties of the sources are carried by the metric field. In the case of the Yang‐Mills field, internal symmetries for a real scalar source field are a consequence of the properties of the gauge field.
By using the representation of its complex-conjugate pairs, we have investigated the diagonalization of a bounded linear operator on separable infinite-dimensional right quaternionic Hilbert space. The sufficient condition for diagonalizability of quaternionic operators is derived. The result is applied to anti-Hermitian operators, which is essential for solving Schrdinger equation in quaternionic quantum mechanics.
A new kind of quantum mechanics using inner products, matrix elements,
and coefficients assuming values that are quaternionic (and thus noncommutative)
instead of complex is developed. This is the most general kind of quantum
mechanics possessing the same kind of calculus of assertions as conventional
quantum mechanics. The role played by the new imaginaries is studied. The
principal conceptual difficulty concerns the theory of composite systems where
the ordinary tensor product fails due to noncommutativity. It is shown that the
natural resolution of this difficulty introduces new degrees of freedom similar
to isospin and hypercharge. The problem of the Schrodinger equation, which i
should appear?'' is studied and a generalization of Stone's theorem is used to
resolve this problem. (auth)
One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).