Galilean-Covariant Clifford Algebras in the Phase-Space Representation

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrödinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner–Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.     

Quantum Marginal Inequalities and the Conjectured Entropic Inequalities

A conjecture – the modified super-additivity inequality of relative entropy – was proposed in Zhang et al. (Phys. Lett. A 377:1794–1796, 2013): There exist three unitary operators \(U_{A}\in \mathrm {U}(\mathcal {H}_{A}), U_{B}\in \mathrm {U}(\mathcal {H}_{B})\) , and \(U_{AB}\in \mathrm {U}(\mathcal {H}_{A}\otimes \mathcal {H}_{B})\) such that $$\mathrm{S}\left(U_{AB}\rho_{AB}U^{\dagger}_{AB}||\sigma_{AB}\right)\geqslant \mathrm{S}\left(U_{A}\rho_{A}U^{\dagger}_{A}||\sigma_{A}\right) + \mathrm{S}\left(U_{B}\rho_{B}U^{\dagger}_{B}||\sigma_{B}\right), $$ where the reference state σ is required to be full-ranked. A numerical study on the conjectured inequality is conducted in this note. The results obtained indicate that the modified super-additivity inequality of relative entropy seems to hold for all qubit pairs.     

Linear operators in Clifford algebras

We consider the real vector space structure of the algebra of linear endomorphisms of a finite-dimensional real Clifford algebra (2, 4, 5, 6, 7, 8). A basis of that space is constructed in terms of the operators M _{eI, eJ} defined by xe _I · x · e _J , where the e _I are the generators of the Clifford algebra and I is a multi-index (3, 7).In particular, it is shown that the family (M _{eI, eJ} ) is exactly a basis in the even case.     

Idempotents of Clifford Algebras

A classification of idempotents of Clifford algebras C ^p,q is presented. It is shown that using isomorphisms between Clifford algebras C ^p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one-sided ideals in Clifford algebras. Some low-dimensional examples are discussed.     

All the Bell Inequalities

Bell inequalities are derived for any number of observers, any number of alternative setups for each one of them and any number of distinct outcomes for each experiment. It is shown that if a physical system consists of several distant subsystems, and if the results of tests performed on the latter are determined by local variables with objective values, then the joint probabilities for triggering any given set of distant detectors are convex combinations of a finite number of Boolean arrays, whose components are either 0 or 1 according to a simple rule. This convexity property is both necessary and sufficient for the existence of local objective variables. It leads to a simple graphical method which produces a large number of generalized Clauser-Horne inequalities corresponding to the faces of a convex polytope. It is plausible that quantum systems whose density matrix has a positive partial transposition satisfy all these inequalities, and therefore are compatible with local objective variables, even if their quantum properties are essentially non-local.     

Clifford Fields and the Relativistic Equation of the Nucleon

A relativistic equation for free fields whichtake their values in the Clifford algebra associatedwith the Minkowski metric is shown to be interpretableas the equation of the nucleon. The internal symmetry group SU(2) arises naturally from theassociative algebra structure of the representationspace. The latter structure can be used to constructcoupling terms consistent with the transformationproperties of the interacting fields; in particular, itallows the familiar couplings of the nucleon field withthe electromagnetic field and with the -mesonfield.     

Free states and automorphisms of the Clifford algebra

We study automorphisms of the Clifford algebra which map the set of quasi-free states onto itself. We show that they are quasi-free if the one-particle space is infinite dimensional, and give counter examples in finite dimensions.     

Clifford Space as the Arena for Physics

A new theory is considered according to which extended objects in n-dimensional space are described in terms of multivector coordinates which are interpreted as generalizing the concept of center of mass coordinates. While the usual center of mass is a point, by generalizing the latter concept, we associate with every extended object a set of r-loops, r=0,1,...,n–1, enclosing oriented (r+1)-dimensional surfaces represented by Clifford numbers called (r+1)-vectors or multivectors. Superpositions of multivectors are called polyvectors or Clifford aggregates and they are elements of Clifford algebra. The set of all possible polyvectors forms a manifold, called C-space. We assume that the arena in which physics takes place is in fact not Minkowski space, but C-space. This has many far reaching physical implications, some of which are discussed in this paper. The most notable is the finding that although we start from the constrained relativity in C-space we arrive at the unconstrained Stueckelberg relativistic dynamics in Minkowski space which is a subspace of C-space.     

Dirac and Maxwell equations in the Clifford and spin-Clifford bundles

We show how to write the Dirac and the generalized Maxwell equations (including monopoles) in the Clifford and spin-Clifford bundles (of differential forms) over space-time (either of signaturep=1,q=3 orp=3,q=1). In our approach Dirac and Maxwell fields are represented by objects of the same mathematical nature and the Dirac and Maxwell equations can then be directly compared. We show also that all presentations of the Maxwell equations in (matrix) Dirac-like spinor form appearing in the literature can be obtained by choosing particular global idempotents in the bundles referred to above. We investigate also the transformation laws under the action of the Lorentz group of Dirac and Maxwell fields (defined as algebraic spinor sections of the Clifford or spin-Clifford bundles), clearing up several misunderstandings and misconceptions found in the literature. Among the many new results, we exhibit a factorization of the Maxwell field into two-component spinor fields (Weyl spinors), which is important.     

Inequalities in Entanglement Percolation

We present proofs of two results concerning entanglement in three-dimensional bond percolation. Firstly, the critical probability for entanglement with free boundary conditions is strictly less than the critical probability for connectivity percolation. (The proof presented here is a detailed justification of the ideas sketched in Aizenman and Grimmett.) Secondly, under the hypothesis that the critical probabilities for entanglement with free and wired boundary conditions are different, for p between the two critical probabilities, the size of the entangled cluster at the origin with free boundary conditions does not have exponentially decaying tails.     

The Bertlmann-Martin Inequalities in the Three-Body Problem

The Bertlmann-Martin inequalities (BMI) are studied in the three-body case. We consider distinguishable particles and scalar interactions. The systems are described by Schr?dinger equations with local potentials. Under these conditions we show that the lower bound character of the BMI is preserved. We discuss the question of the correction factors transforming the inequalities into approximate or exact relationships. As illustrative example, a simple model in the (D = 1)-dimensional space is considered, for which the exact solution is known. We also study what can be learned from the Hartree approximation and the hyperspherical method in the central potential approximation with respect to the BMI inequalities and correction factors.     

The Violation of Bell Inequalities in the Macroworld

We show that Bell inequalities can be violated in the macroscopic world. The macroworld violation is illustrated using an example involving connected vessels of water. We show that whether the violation of inequalities occurs in the microworld or the macroworld, it is the identification of nonidentical events that plays a crucial role. Specifically, we prove that if nonidentical events are consistently differentiated, Bell-type Pitowsky inequalities are no longer violated, even for Bohm's example of two entangled spin 1/2 quantum particles. We show how Bell inequalities can be violated in cognition, specifically in the relationship between abstract concepts and specific instances of these concepts. This supports the hypothesis that genuine quantum structure exists in the mind. We introduce a model where the amount of nonlocality and the degree of quantum uncertainty are parameterized, and demonstrate that increasing nonlocality increases the degree of violation, while increasing quantum uncertainty decreases the degree of violation.     

Charged Representations of the Infinite Fermi and Clifford Algebras

The real and quaternionic charge conjugation operators invariant under the infinite-dimensional Clifford algebra, or compatible with the Fermi algebra, are determined. There results a maze of inequivalent irreducible charged representations, all of which are non-Fock. The representation vectors and their charges admit two interpretations besides those of spinors or states of quantum fields: as wavelets on the circle, with charge conjugations acting via ordinary complex conjugation; and as infinite-dimensional numbers, with charge conjugations acting by automorphisms.     

Pauli-Dirac matrix generators of Clifford Algebras

This article presents a Pauli-Dirac matrix approach to Clifford Algebras. It is shown that the algebra C₂ is generated by two Pauli matrices i₂ and i₃; C₃ is generated by the three Pauli matrices ₁, ₂, ₃; C₄ is generated by four Dirac matrices ₀, ₁, ₂, ₃ and C₅ is generated by five Dirac matrices i₀, i₁, i₂, i₃, i₅. The higher dimensional anticommuting matrices which generate arbitrarily high order Clifford algebras are given in closed form. The results obtained with this Clifford algebra approach are compared with the vector product method which was described in a recent article [Found. Phys. 10, 531–553 (1980) by Poole, Farach and Aharonov] and with the Dirac, Rashevskii and Ramakrishnan methods of matrix generation.Supported by the National Science Foundation under Grant ISP-80-11451.     

Connections on Clifford bundles and the Dirac operator

It is shown, how, in the setting of Clifford bundles, the spin connection (or Dirac operator) may be obtained by averaging the Levi-Civita connection (or Kähler-Dirac operator) over the finite group generated by an orthonormal frame of the base manifold.The familiar covariance of the Dirac equation under a simultaneous transformation of spinors and matrix representations emerges very naturally in this scheme, which can also be applied when the manifold does not possess a spin structure.     

Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation

In this paper we show how the dynamics of the Schr?dinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, C_1,3, C_3,0{\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}, and C_0,1{\mathcal{C}}_{0,1}. Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can be completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time.     

The Clifford bundle and the nature of the gravitational field

In this paper we formulate Einstein's gravitational theory with the Clifford bundle formalism. The formalism suggests interpreting the gravitational field in the sense of Faraday, i.e., with the field residing in Minkowski spacetime. We succeeded in discovering the condition for this interpretation to hold. For the variables that play the role of the gravitational field in our theory, the Lagrangian density turns out to be of the Yang-Mills type (with an auto-interaction plus gauge-fixing terms). We give a brief comparison of our theory with other field theories of the gravitational field in the flat Minkowski spacetime.     

On the relationship between Twistors and Clifford algebras

Basis p-forms of a complexified Minkowski spacetime can be used to realize a Clifford algebra isomorphic to the Dirac algebra of matrices. Twistor space is then constructed as a spin space of this abstract algebra through a Witt decomposition of the Minkowski space. We derive explicit formulas relating the basis p-forms to index one twistors. Using an isomorphism between the Clifford algebra and a space of index two twistors, we expand a suitably defined antisymmetric index two twistor basis on p-forms of ranks zero, one, and four. Together with the inverse formulas they provide a complete passage between twistors and p-forms.