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In this paper, we prove the sum and product uncertainty relations conjectured by V Dodonov for multiple observables. The uncertainty relations for linear combinations of position and momentum recently obtained by Kechrimparis and Weigert are recovered. Furthermore, the entropic uncertainty relations conjectured by the latter authors are proved for specific cases. At last, we revisit the uncertainty relation for triple canonical operators and obtain a tighter bound on real Hilbert space. A quantitative stability result is given as well. [ABSTRACT FROM AUTHOR]

We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced by Alexeeva et al (Ann. Phys., NY 403 198) (ABS model), which admits an exact explicit solitary-wave (soliton for short) solution. The charge, the momentum, and the energy of this solution are conserved. We investigate the dynamics of the soliton subjected to several potentials: a ramp, a harmonic, and a periodic potential. We develop a collective coordinates (CCs) theory by making an ansatz for a moving soliton where the position, rapidity, and momentum, are functions of time. We insert the ansatz into the Lagrangian density of the model, integrate over space and obtain a Lagrangian as a function of the CCs. This Lagrangian differs only in the charge and mass with the Lagrangian of a CCs theory for the Gross–Neveu equation. Thus the soliton dynamics in the Alexeeva–Barashenkov–Saxena (ABS) spinor model is qualitatively the same as in the Gross–Neveu equation, but quantitatively it differs. These results of the CCs theory are confirmed by simulations, i.e. by numerical solutions for solitons of the ABS spinor model, subjected to the above potentials. [ABSTRACT FROM AUTHOR]

We investigate the relations between spinors and null vectors in Clifford algebra of any dimension with particular emphasis on the conditions that a spinor must satisfy to be simple (also: pure). In particular, we prove: (i) a new property for null vectors: each of them bisects spinor space into two subspaces of equal size; (ii) that simple spinors form one-dimensional subspaces of spinor space; (iii) a necessary and sufficient condition for a spinor to be simple that generalizes a theorem of Cartan and Chevalley which becomes a corollary of this result. We also show how to write down easily the most general spinor with a given associated totally null plane. [ABSTRACT FROM AUTHOR]

We revisit the higher spin extensions of the anti de Sitter algebra in four dimensions that incorporate internal symmetries and admit representations that contain fermions, classified long ago by Konstein and Vasiliev. We construct the dS4, Euclidean and Kleinian version of these algebras, as well as the corresponding fully nonlinear Vasiliev type higher spin theories, in which the reality conditions we impose on the master fields play a crucial role. The N = 2 supersymmetric higher spin theory in dS4, on which we elaborate further, is included in this class of models. A subset of the Konstein-Vasiliev algebras are the minimal higher spin extensions of the AdS4 superalgebra osp(4∣N) with N = 1, 2, 4 mod 4, whose R-symmetry can be realized using fermionic oscillators. We tensor these algebras with appropriate internal symmetry algebras, namely u(n) for N = 2 mod 4 and so(n) or usp(n) for N = 1, 4 mod 4. We show that the N = 3 mod 4 higher spin algebras are isomorphic to those with N = 4 mod 4. We describe the fully nonlinear higher spin theories based on these algebras, including the coupling between the adjoint and twisted-adjoint master fields. We elaborate further on the N = 6 model in AdS4, and provide two equivalent descriptions one of which exhibits manifestly its relation to the N = 8 model. [ABSTRACT FROM AUTHOR]

We present the full list of conserved currents built of two massless spinor fields in Minkowski space and their derivatives multiplied by Clifford algebra elements. The currents have particular mixed-symmetry type described by Young diagrams with one row and one column of arbitrary lengths and heights. Along with Yukawa-like totally antisymmetric currents, the complete set of constructed currents exactly matches the spectrum of AdS mixedsymmetry fields arising in the generalized Flato-Frønsdal theorem for two spinor singletons. As a by-product, we formulate and study general properties of primary fields and conserved currents of mixed-symmetry type. [ABSTRACT FROM AUTHOR]

The concept of electric and magnetic field lines is intrinsically non-relativistic. Nonetheless, for certain types of fields satisfying certain geometric properties, field lines can be defined covariantly. More precisely, two Lorentz-invariant 2D surfaces in spacetime can be defined such that magnetic and electric field lines are determined, for any observer, by the intersection of those surfaces with spacelike hyperplanes. An instance of this type of field is constituted by the so-called Hopf-Rañada solutions of the source-free Maxwell equations, which have been studied because of their interesting topological properties, namely, linkage of their field lines. In order to describe both geometric and topological properties in a succinct manner, we employ the tools of geometric algebra (aka Clifford algebra) and use the Clebsch representation for the vector potential as well as the Euler representation for both magnetic and electric fields. This description is easily made covariant, thus allowing us to define electric and magnetic field lines covariantly in a compact geometric language. The definitions of field lines can be phrased in terms of 2D surfaces in space. We display those surfaces in different reference frames, showing how those surfaces change under Lorentz transformations while keeping their topological properties. As a byproduct we also obtain relations between optical helicity, optical chirality and generalizations thereof, and their conservation laws. [ABSTRACT FROM AUTHOR]

We investigate two seemingly disjoint definitions of helicity, one commonly used in particle physics, the other one used when studying bilinear covariants of Clifford algebras. We can prove that the 'mathematical' definition of helicity implies its 'physical' counterpart. As an unexpected application of our result, we show that the Hamiltonian describing the one-layer superconductor graphene is proportional to the trace of an operator that is used in the 'mathematical' definition of helicity. [ABSTRACT FROM AUTHOR]

In this paper, the classical theory of spherical harmonics in is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk-Hecke theorem to superspace. Finally, this integration over the supersphere is used to define an integral over the whole superspace, and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral. [ABSTRACT FROM AUTHOR]

For over a decade, there has been intensive work on the numerical and analytic construction of SICs (d2 equiangular lines in ) as an orbit of the Heisenberg group. The Clifford group, which consists of the unitary matrices which normalise the Heisenberg group, plays a key role in these constructions. All of the known fiducial (generating) vectors for such SICs are eigenvectors of symplectic operations in the Clifford group with canonical order 3. Here we describe the Clifford group and the subgroup of symplectic operations in terms of a natural set of generators. From this, we classify all its elements of canonical order three. In particular, we show (contrary to prior claims) that there are symplectic operations of canonical order 3 for , which are not conjugate to the Zauner matrix. It is as yet unknown whether these give rise to SICs. [ABSTRACT FROM AUTHOR]

A finite Hilbert space can be associated to a periodic phase space, that is, a torus. Then a subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving transition Wigner or chord functions of transition operators . Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator. [ABSTRACT FROM AUTHOR]

We show that the Atiyah–Singer index theorem of Dirac operator can be directly proved in the canonical formulation of quantum mechanics, without using the path-integral technique. This proof takes advantage of an algebraic isomorphism between Clifford algebra and exterior algebra in small τ (high temperature) limit, together with simple properties of quantum harmonic oscillator. Compared to the proofs given by full symbol calculus and heat kernel, we try to prove this theorem in a more quantum mechanical way. [ABSTRACT FROM AUTHOR]

Graph states are quantum states that can be described by a stabilizer formalism and play an important role in quantum information processing. We consider the action of local unitary operations on graph states and hypergraph states. We focus on non-Clifford operations and find for certain transformations a graphical description in terms of weighted hypergraphs. This leads to the indentification of hypergraph states that are locally equivalent to graph states. Moreover, we present a systematic way to construct pairs of graph states which are equivalent under local unitary operations, but not equivalent under local Clifford operations. This generates counterexamples to a conjecture known as LU-LC conjecture. So far, the only counterexamples to this conjecture were found by random search. Our method reproduces the smallest known counterexample as a special case and provides a physical interpretation. [ABSTRACT FROM AUTHOR]

Stabilizer states are eigenvectors of maximal commuting sets of operators in a finite Heisenberg group. States that are far from being stabilizer states include magic states in quantum computation, MUB-balanced states, and SIC vectors. In prime dimensions the latter two fall under the umbrella of minimum uncertainty states (MUSs) in the sense of Wootters and Sussman. We study the correlation between two ways in which the notion of ‘far from being a stabilizer state’ can be quantified. Two theorems valid for all prime dimensions are given, as well as detailed results for low dimensions. In dimension 7 we identify the MUB-balanced states as being antipodal to the SIC vectors within the set of MUS, in a sense that we make definite. In dimension 4 we show that the states that come closest to being MUS with respect to all of the six stabilizer MUBs are the fiducial vectors for Alltop MUBs. [ABSTRACT FROM AUTHOR]

We investigate the action of the first three levels of the Clifford hierarchy on sets of mutually unbiased bases comprising the Ivanovic mutually unbiased base (MUB) and the Alltop MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop vectors splits into three Clifford orbits. These vectors form configurations with so-called Zauner subspaces, eigenspaces of order 3 elements of the Clifford group highly relevant to the SIC problem. We identify Alltop vectors as the magic states that appear in the context of fault-tolerant universal quantum computing, wherein the appearance of distinct Clifford orbits implies a surprising inequivalence between some magic states. [ABSTRACT FROM AUTHOR]

Gauged \mathcal {P}\mathcal {T} quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie-triple structure is found and an interpretation as \mathcal {P}\mathcal {T}-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space-related J-self-adjoint extensions for PTQM setups with ultra-localized potentials. [ABSTRACT FROM AUTHOR]

By using a recently developed method, we report five different families of isospectral 2 x 2 matrix Hamiltonians defined on a four-dimensional (4D) phase space. The employed method is based on a realization of the supersymmetry idea on the phase space whose complexified Clifford algebra structure is deformed with the Moyal star-product. Each reported family comprises many physically relevant special models. 2D Pauli Hamiltonians, systems involving spin-orbit interactions such as Aharonov-Casher-type systems, a supermembrane toy model and models describing motion in noncentral electromagnetic fields as well as Rashba- and Dresselhaus-type systems from semiconductor physics are obtained, together with their super-partners, as special cases. A large family of isospectral systems characterized by the whole set of analytic functions is also presented. [ABSTRACT FROM AUTHOR]

In order to realize supersymmetric quantum mechanics methods on a four-dimensional classical phase space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase-space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jaynes-Cummings-type models of quantum optics are presented as special cases of the method. Their spectra, eigenspinors and Wigner functions as well as their constants of motion are also obtained within the autonomous framework of deformation quantization. [ABSTRACT FROM AUTHOR]

Prompted by a recent demonstration that the structure of a single quark-lepton generation may be understood via a Dirac-like linearization of the form p2 + x2, we analyse the corresponding Clifford algebra in some detail. After classifying all elements of this algebra according to their U(1) [?] SU(3) and SU(2) transformation properties, we identify the element which might be associated with the concept of lepton mass. This element is then transformed into a corresponding element for a single coloured quark. It is shown that--although none of the three thus obtained individual quark mass elements is rotationally invariant--the rotational invariance of the quark mass term is restored when the sum over quark colours is performed. [ABSTRACT FROM AUTHOR]

We consider a straightforward extension of the four-dimensional spacetime M4 to the space of extended events associated with strings/branes, corresponding to points, lines, areas, 3-volumes and 4-volumes in M4. All those objects can be elegantly represented by the Clifford numbers X\equiv x^A \gamma_A \equiv x^{a_1 \ldots a_r} \gamma_{a_1 \ldots a_r}, r=0,1,2,3,4 . This leads to the concept of the so-called Clifford space {\cal C} , a 16-dimensional manifold whose tangent space at every point is the Clifford algebra {\cal {C} \ell }(1,3) . The latter space besides an algebra is also a vector space whose elements can be rotated into each other in two ways: (i) either by the action of the rotation matrices of SO(8, 8) on the components xA or (ii) by the left and right action of the Clifford numbers R = exp [aAgA] and S = exp [bAgA] on X. In the latter case, one does not recover all possible rotations of the group SO(8, 8). This discrepancy between the transformations (i) and (ii) suggests that one should replace the tangent space {\cal {C} \ell }(1,3) with a vector space V8,8 whose basis elements are generators of the Clifford algebra {\cal {C} \ell }(8,8) , which contains the Lie algebra of the exceptional group E8 as a subspace. E8 thus arises from the fact that, just as in the spacetime M4 there are r-volumes generated by the tangent vectors of the spacetime, there are R-volumes, R = 0, 1, 2, 3, ..., 16, in the Clifford space {\cal C} , generated by the tangent vectors of {\cal C} . [ABSTRACT FROM AUTHOR]

We give an eight-dimensional realization of the Clifford algebra in the five-dimensional Galilean covariant spacetime by using a dimensional reduction from the (5 + 1) Minkowski spacetime to the (4 + 1) Minkowski spacetime which encompasses the Galilean covariant spacetime. A set of solutions of the Dirac-type equation in the five-dimensional Galilean covariant spacetime is obtained, based on the Pauli representation of 8 × 8 gamma matrices. In order to find an explicit solution, we diagonalize the Klein-Gordon divisor by using the Galilean boost. [ABSTRACT FROM AUTHOR]