Quaternionic quantum Hamiltonians describing nonrelativistic spin particles require the ambient physical space to have five dimensions. The quantum dynamics of a spin-1/2 particle system characterised by a generic such Hamiltonian is worked out in detail. It is shown that there exists, within the structure of quaternionic quantum mechanics, a canonical reduction to three spatial dimensions upon which standard quantum theory is retrieved. In this dimensional reduction, three of the five dynamical variables are shown to oscillate around a cylinder, thus behaving in a quasi one-dimensional manner at large distances. An analogous mechanism is shown to exist in the case of octavic Hamiltonians, where the ambient physical space has nine dimensions. Possible experimental tests in search for the signature of extra dimensions at low energies are briefly discussed.
Scattering discussion due to Double Dirac Equation in Quaternionic version of relativistic quantum mechanics has been studied in this paper in details. In such a quantum mechanics Dirac equation in presence vector and scalar potential has been considered. Then a Quaternionic double Dirac delta potential comes to our considered system which causes to scatter the particles. Scattering states of the particles have been derived as well as reflected and transmission coefficients are calculated.
A quaternionic version of Quantum Mechanics is constructed using the Schwinger's formulation based on measurements and a Variational Principle. Commutation relations and evolution equations are provided, and the results are compared with other formulations.
Analysis of the logical foundations of quantum mechanics indicates the possibility of constructing a theory using quaternionic Hilbert spaces. Whether this mathematical structure reflects reality is a matter for experiment to decide. We review the only direct search for quaternionic quantum mechanics yet carried out and outline a recent proposal by the present authors to look for quaternionic effects in correlated multi-particle systems. We set out how such experiments might distinguish between the several quaternionic models proposed in the literature.
We introduce {\em half-whole} dimensions for quaternionic matrices and propose a quaternionic version of the Frobenius-Schur theorem which allows us to obtain the proper quaternionic dimensionality for the representations of the Dirac and Duffin-Kemmer-Petiau (DKP) algebras.
We complete the rules of translation between standard complex quantum mechanics (CQM) and quaternionic quantum mechanics (QQM) with a complex geometry. In particular we describe how to reduce ($2n$+$1$)-dimensional complex matrices to {\em overlapping\/} ($n$+$1$)-dimensional quaternionic matrices with generalized quaternionic elements. This step resolves an outstanding difficulty with reduction of purely complex matrix groups within quaternionic QM and avoids {\em anomalous} eigenstates. As a result we present a more complete translation from CQM to QQM and viceversa.
In this talk I shall first make some brief remarks on quaternionic quantum mechanics, and then describe recent work with A.C. Millard in which we show that standard complex quantum field theory can arise as the statistical mechanics of an underlying noncommutative dynamics.
This study examines Quaternion Dirac solutions for an infinite square well. The quaternion result does not recover the complex result within a particular limit. This raises the possibility that quaternionic quantum mechanics may not be understood as a correction to complex quantum mechanics, but it may also be a structure that can be used to study phenomena that cannot be described through the framework of complex quantum mechanics.
Quaternionic formulation of supersymmetric quantum mechanics has been developed consistently in terms of Hamiltonians, superpartner Hamiltonians, and supercharges for free particle and interacting field in one and three dimensions. Supercharges, superpartner Hamiltonians and energy eigenvalues are discussed and it has been shown that the results are consistent with the results of quantum mechanics.
Quaternion quantum mechanics is examined at the level of unbroken SU(2) gauge symmetry. A general quaternionic phase expression is derived from formal properties of the quaternion algebra.
We discuss the Schrodinger equation in presence of quaternionic potentials. The study is performed analytically as long as it proves possible, when not, we resort to numerical calculations. The results obtained could be useful to investigate an underlying quaternionic quantum dynamics in particle physics. Experimental tests and proposals to observe quaternionic quantum effects by neutron interferometry are briefly reviewed.
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…
We construct N=8 supersymmetric mechanics with four bosonic end eight fermionic physical degrees of freedom. Starting from the most general N=4 superspace action in harmonic superspace for the ({\bf 4,8,4}) supermultiplet we find conditions which make it N=8 invariant. We introduce in the action Fayet-Iliopoulos terms which give rise to potential terms. We present the action in components and give explicit expressions for the Hamiltonian and Poisson brackets. Finally we discuss the possibility of N=9 supersymmetric mechanics.
The algebraic consistency of spin and isospin at the level of an unbroken SU(2) gauge theory suggests the existence of an additional angular momentum besides the spin and isospin and also produces a full quaternionic spinor operator. The latter corresponds to a vector boson in space-time, interpreted as a SU(2) gauge field. The existence of quaternionic spinor fields implies in a quaternionic Hilbert space and its necessary mathematical analysis. It is shown how to obtain a unique representation of a quaternion function by a convergent positive power series.
We solve Klein-Gordon equation (KGE) in the framework of the real Hilbert space approach to quaternionic quantum mechanics ($\mathbbm H$QM). The presented solution is the simplest ever obtained for quaternionic quantum theories, and the closest to the complex solution. The scattering of a quaternionic charged scalar particle from an electric field is also obtained. A remarkable feature of quaternionic scalar particles is the existence of massive light cone particles.
The second quantization of the quaternionic fermionic field is undertaken using the real Hilbert space approach to quaternionic quantum mechanics ($\mathbbm H$QM). The solution responds to an open problem of quaternionic quantum theory, and launches the basis to the development of the quaternionic interaction theory.
A pseudoclassical model, reproducing, upon quantization, the dynamics of the chiral sectors of the massless spin-1/2 field theory is proposed. The discrete symmetries of the action are studied in details. In order to reproduce the positive and negative chiral sectors of the particle and antiparticle, we promote the algebra of functions on the phase space to a bimodule over the complexified quaternions - biquaternions. The quantization is performed by means of strictly canonical methods (Dirac brackets formalism) reproducing the Dirac equation in the Foldy-Wouthuysen representation in the parti…
We perform a one-dimensional complexified quaternionic version of the Dirac equation based on $i$-complex geometry. The problem of the missing complex parameters in Quaternionic Quantum Mechanics with $i$-complex geometry is overcome by a nice ``trick'' which allows to avoid the Dirac algebra constraints in formulating our relativistic equation. A brief comparison with other quaternionic formulations is also presented.
We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})\cong \mathfrak{o}(3;\mathbb{C})\oplus \mathfrak{o}(3;\mathbb{C})$, describing quantum rotational symmetry of four-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\ma…
In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with $SL(2;\mathbb{H})$, which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects b…
The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann's foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has been started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigrou…
Introducing a non-compact version of the Hopf map, we demonstrate remarkable close relations between quantum Hall effect and twistor theory. We first construct quantum Hall effect on a hyperboloid based on the noncompact 2nd Hopf map of split-quaternions. We analyze a hyperbolic one-particle mechanics, and explore many-body problem, where a many-body groundstate wavefunction and membrane-like excitations are derived explicitly. In the lowest Landau level, the symmetry is enhanced from $SO(3, 2)$ to the $SU(2, 2)$ conformal symmetry. We point out that the quantum Hall effect naturally realizes …
While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and coquaternionic (split-signature analogue of quaternions) extensions of Hamiltonian mechanics are introduced, and are shown to offer a unifying framework for complexified classical and quantum mechanics. In particular, quantum theories characterised by complex Hamiltonians invariant under space-time reflection are shown to be equivalent to certain coquaternionic exte…
The Kerr theorem is revisited as part of the twistor program in six dimensions. The relationship between pure spinors and integrable 3-planes is investigated. The real condition for Lorentzian spacetimes is seen to induce a projective property in the space of solutions, reminiscent of the quaternionic structure of the 6-dimensional Lorentz group. The twistor equation (or Killing spinor equations generically) also has an interpretation as integrable null planes and a family of Einstein spacetimes with this property are presented in the Kerr-Schild fashion.
We describe the couplings of six-dimensional supergravity, which contain a self-dual tensor multiplet, to $n_T$ anti-self-dual tensor matter multiplets, $n_V$ vector multiplets and $n_H$ hypermultiplets. The scalar fields of the tensor multiplets form a coset $SO(n_T,1)/SO(n_T)$, while the scalars in the hypermultiplets form quaternionic Kähler symmetric spaces, the generic example being $Sp(n_H,1)/Sp(n_H)\otimes Sp(1)$. The gauging of the compact subgroup $Sp(n_H) \times Sp(1)$ is also described. These results generalize previous ones in the literature on matter couplings of $N=1$ supergravit…
One-dimensional sigma-models with N supersymmetries are considered. For conventional supersymmetries there must be N-1 complex structures satisfying a Clifford algebra and the constraints on the target space geometry can be formulated in terms of these. In the cases in which the complex structures are simultaneously integrable, a conventional extended superspace formulation is given, with the geometry determined by a 2-form potential for N=2, by a 1-form potential for N=3 and a scalar potential for N=4; for N>4 it is given by a scalar potential satisfying differential constraints. This gives e…
We present the first Monte Carlo results for supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature. The recently proposed non-lattice simulation enables us to include the effects of fermionic matrices in a transparent and reliable manner. The internal energy nicely interpolates the weak coupling behavior obtained by the high temperature expansion, and the strong coupling behavior predicted from the dual black hole geometry. The Polyakov line takes large values even at low temperature suggesting the absence of a phase transition in sharp contrast to the bosonic…
We use a local scale invariance of a classical Hamiltonian and describe how to construct six different formulations of quantum mechanics in spaces with two time-like dimensions. All these six formulations have the same classical limit described by the same Hamiltonian. One of these formulations is used as a basis for a complementation of the usual quantum mechanics when in the presence of gravity.
Fefferman and Graham showed some time ago that four dimensional conformal geometries could be analyzed in terms of six dimensional, ambient, Riemannian geometries admitting a closed homothety. Recently it was shown how conformal geometry provides a description of physics manifestly invariant under local choices of unit systems. Strikingly, Einstein's equations are then equivalent to the existence of a parallel scale tractor (a six component vector subject to a certain first order covariant constancy condition at every point in four dimensional spacetime). These results suggest a six dimensiona…
This monograph is derived from a series of six lectures I gave at the Centre de Recherches Mathematiques in Montreal, in March and June 2000, while titulary of the Aisenstadt Chair.
In order to establish an explicit connection between four-dimensional Hall effect on $S^4$ and six-dimensional Hall effect on $\DC P^3$, we perform the Hamiltonian reduction of a particle moving on $\DC P^3$ in a constant magnetic field to the four-dimensional Hall mechanics (i.e. a particle on $S^4$ in a SU(2) instanton field). This reduction corresponds to fixing the isospin of the latter system.
In classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz group O(3,3) applicable to a six-dimensional space consisting of three space-like and three time-like coordinates, or SL(4,r) in the four-dimensional phase space consisting of two position and two momentum variables. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r) respectively. It is shown that among the six Sp(4)-like subgroups, only one possesses the symmetry which can be translated into the group of unitary …
We study the (2,0) superconformal theories in six dimensions, which arise from the low-energy limit of k coincident 5-branes, using their discrete light-cone formulation as a superconformal quantum mechanical sigma model. We analyze the realization of the superconformal symmetry in the quantum mechanics, and the realization of primary operators. As an example we compute the spectrum of chiral primary states in symmetric Spin(5)_R representations. To facilitate the analysis we introduce and briefly discuss a new class of Lorentz non-invariant theories, which flow in the IR to the (2,0) supercon…
The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows to recover quantum mechanics as mechanics on a non-differentiable (fractal) space-time. The Schrödinger and Klein-Gordon equations have already been demonstrated as geodesic equations in this framework. We propose here a new development of the intrinsic properties of this theory to obtain, using the mathematical tool of Hamilton's bi-quaternions, a derivation of the Dirac equation, which, in standard physics, is merely postulated. The bi-quaternionic natu…
We propose descriptions of interacting (2,0) supersymmetric theories without gravity in six dimensions in the infinite momentum frame. They are based on the large $N$ limit of quantum mechanics or 1+1 dimensional field theories on the moduli space of $N$ instantons in $\IR^4$.
Via supersymmetry argument, we determine the effective action of the SU(2) supersymmetric Yang-Mills quantum mechanics up to two constants, which results from the full supersymmetric completion of the F^4 term. The effective action, consisting of zero, two, four, six and eight fermion terms, agrees with the known perturbative one-loop calculations from the type II string theory and the matrix theory. Our derivation thus demonstrates its non-renormalization properties, namely, the one-loop exactness of the aforementioned action and the absence of the non-perturbative corrections. We briefly dis…
We find the most general low energy dynamics of 1/2 BPS monopoles in the N=4 supersymmetric Yang-Mills theories (SYM) when all six adjoint Higgs expectation values are turned on. When only one Higgs is turned on, the Lagrangian is purely kinetic. When all six are turned on, however, this moduli space dynamics is augmented by five independent potential terms, each in the form of half the squared norm of a Killing vector field on the moduli space. A generic stationary configuration of the monopoles can be interpreted as stable non BPS dyons, previously found as non-planar string webs connecting …
We use the recently developed dimensional regularization (DR) scheme for quantum mechanical path integrals in curved space and with a finite time interval to compute the trace anomalies for a scalar field in six dimensions. This application provides a further test of the DR method applied to quantum mechanics. It shows the efficiency in higher loop computations of having to deal with covariant counterterms only, as required by the DR scheme.
We propose descriptions of interacting (1,0) supersymmetric theories without gravity in six dimensions in the infinite momentum frame. They are based on the large N limit of quantum mechanics or 1+1 dimensional field theories with SO(N) gauge group and four supercharges. We argue that this formulation allows for a concrete description of the chirality-changing phase transitions which connect (1,0) theories with different numbers of tensor multiplets.
F(4) supergravity, the gauge theory of the exceptional six-dimensional Anti-de Sitter superalgebra, is coupled to an arbitrary number of vector multiplets whose scalar components parametrize the quaternionic manifold SO(4,n)/SO(4)\times SO(n). By gauging the compact subgroup SU(2)_d \otimes G, where SU(2)_d is the diagonal subgroup of SO(4)= SU(2)_L\otimes SU(2)_R (the R-symmetry group of six-dimensional Poincare' supergravity) and G is a compact group such that dim G = n. The potential admits an AdS background for g=3m, as the pure F(4)-supergravity. The boundary F(4) superconformal fields ar…
We show that the resolution of moduli space of ideal instantons parameterizes the instantons on non-commutative $\IR^{4}$. This moduli space appears as a Higgs branch of the theory of $k$ $D0$-branes bound to $N$ $D4$-branes by the expectation value of the $B$ field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of $(2,0)$ superconformal theories in six dimensions.
Using a quaternionic formulation of the moduli space $\boldsymbol{M}({\small IIA/K3}) $ of 10D type IIA superstring on a generic K3 complex surface with volume $\boldsymbol{V}_{0}$, we study extremal $\mathcal{N}=2$ black attractors in 6D space-time and their uplifting to 7D. For the 6D theory, we exhibit the role played by 6D $\mathcal{N}=1$ hypermultiplets and the $Z^{m}$ central charges isotriplet of the 6D $\mathcal{N}=2$ superalgebra. We construct explicitly the special hyperKahler geometry of $% \boldsymbol{M}({\small IIA/K3}) $ and show that the $SO(4) \times SO(20) $ invariant hyperKah…
Massive spinning particle in $6d$-Minkowski space is described as a mechanical system with the configuration space $R^{5,1} \times CP^3$. The action functional of the model is unambiguously determined by the requirement of identical (off-shell) conservation for the phase-space counterparts of three Casimir operators of Poincaré group. The model proves to be completely solvable. Its generalization to the constant curvature background is presented. Canonical quantization of the theory leads to the relativistic wave equations for the irreducible $6d$ fields.
We classify the supersymmetric mass deformations of all the super Yang-Mills quantum mechanics, which are obtained by dimensional reductions of minimal super Yang-Mills in spacetime dimensions: ten, six, four, three and two. The resulting actions can be viewed as the matrix descriptions of supermembranes in nontrivial backgrounds of one higher dimensional supergravity theories. We also discuss the utmost generalization of the light-cone formulation of the Nambu-Goto action for a p-brane, including time dependent backgrounds.
In this paper we propose a quiver model of matrix quantum mechanics with 8 supercharges which, on a Higgs branch, deconstructs the worldsheet of Matrix String Theory. This discrete model evades the fermion doubling problem and, in the continuum limit, enhances the number of supersymmetries to sixteen. Our model is motivated by orbifolding the Matrix Model, and the deconstruction {\it ansatz} exhibits a duality between target space compactification and worldsheet deconstruction.
An action for a superconformal particle is constructed using the non linear realization method for the group PSU(1,1|2), without introducing superfields. The connection between PSU(1,1|2) and black hole physics is discussed. The lagrangian contains six arbitrary constants and describes a non-BPS superconformal particle. The BPS case is obtained if a precise relation between the constants in the lagrangian is verified, which implies that the action becomes kappa-symmetric.
F(4) supergravity, the gauge theory of the exceptional six-dimensional Anti-de Sitter superalgebra, is coupled to an arbitrary number of vector multiplets whose scalar components parametrize the quaternionic manifold $SO(4,n)/SO(4)\times SO(n)$. By gauging the compact subgroup $SU(2)_d \otimes \cG$, where SU(2)_d is the diagonal subgroup of $SO(4)\simeq SU(2)_L\otimes SU(2)_R$ (the R-symmetry group of six-dimensional Poincaré supergravity) and $\cG$ is a compact group such that $dim\cG = n$, we compute the scalar potential which, besides the gauge coupling constants, also depends in non trivia…
We calculate one-loop radiative correction to the mass of Higgs identified with the extra space components of the gauge field in a six dimensional massive scalar QED compactified on a two-sphere. The radiatively induced Higgs mass is explicitly shown to be finite for arbitrary bulk scalar mass M. Furthermore, the remaining finite part also turns out to vanish, at least for the case of small M, thus suggesting that the radiatively induced Higgs mass exactly vanishes, in general. The non-zero "Kaluza-Klein" modes in the gauge sector are argued to have a Higgs-like mechanism and quantum mechanica…
