Fractional solutions for the inextensible Heisenberg antiferromagnetic fow and solitonic magnetic flux surfaces in the binormal direction

Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute Lorentz force equations associated with the magnetic b-lines in the binormal direction. Evolution equations of magnetic b-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model.

Dirac Redux - Dirac’s Equation in Physical Spacetime without Internal Degrees of Freedom

The Dirac equation (DE) is one of the cornerstones of quantum physics. We prove in the present contribution that the notion of internal degrees of freedom of the electron represented by Dirac’s matrices is superfluous. One can write down a coordinate-free manifestly covariant vector equation by direct quantization of the 4-momentum vector with modulus m: Pψ=mψ (no slash!), the spinor ψ taking care of the different vector grades at the two sides of the equation. Electron spin and all the standard DEproperties emerge from this equation. In coordinate representation, the four orthonormal time-space frame vectors Xµ appear instead of Dirac’s Yµ-matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x5 that is defining for the C (particle-antiparticle) symmetry and the CPT symmetry of DE, as well as for left- and right-handed rotors and spinors. In 3D it transforms a parity-odd vector x into a parity-even vector σ=x5x and vice versa. STA augmented by the reflector will be referred to as STAR, which operates on a real vector space of same dimension as the equivalent real dimension of Dirac’s complex 4×4 matrices. There are no matrices in STAR and the complex character springs from the signature and dimension of spacetime-reflection. This appears most clearly by first showing that STAR comprises two isomorphic subspaces, one for generators of polar vectors and boosts and the other for generators of axial vectors and rotors, including Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. We prove here that the information from γ-matrices is contained in spacetime-reflection, which makes the matrices redundant. Therefore, it becomes relevant to reexamine those areas of quantum physics that take the γ-matrices and their generalizations as fundamental.

A comparison theorem for cosmological lightcones

Let (M, g) denote a cosmological spacetime describing the evolution of a universe which is isotropic and homogeneous on large scales, but highly inhomogeneous on smaller scales. We consider two past lightcones, the first, $${{\mathcal {C}}_{L}^{-}}(p, g)$$ C L - ( p , g ) , is associated with the physical observer $$p\in \,M$$ p ∈ M who describes the actual physical spacetime geometry of (M, g) at the length scale L, whereas the second, $${\mathcal {C}_{L}^{-}}(p, \hat{g})$$ C L - ( p , g ^ ) , is associated with an idealized version of the observer p who, notwithstanding the presence of local inhomogeneities at the given scale L, wish to model (M, g) with a member $$(M, \hat{g})$$ ( M , g ^ ) of the family of Friedmann–Lemaitre–Robertson–Walker spacetimes. In such a framework, we discuss a number of mathematical results that allows a rigorous comparison between the two lightcones $${\mathcal {C}_{L}^{-}}(p, g)$$ C L - ( p , g ) and $${\mathcal {C}_{L}^{-}}(p, \hat{g})$$ C L - ( p , g ^ ) . In particular, we introduce a scale-dependent (L) lightcone-comparison functional, defined by a harmonic type energy, associated with a natural map between the physical $${\mathcal {C}_{L}^{-}}(p, g)$$ C L - ( p , g ) and the FLRW reference lightcone $${\mathcal {C}_{L}^{-}}(p, \hat{g})$$ C L - ( p , g ^ ) . This functional has a number of remarkable properties, in particular it vanishes iff, at the given length-scale, the corresponding lightcone surface sections (the celestial spheres) are isometric. We discuss in detail its variational analysis and prove the existence of a minimum that characterizes a natural scale-dependent distance functional between the two lightcones. We also indicate how it is possible to extend our results to the case when caustics develop on the physical past lightcone $${\mathcal {C}_{L}^{-}}(p, g)$$ C L - ( p , g ) . Finally, by exploiting causal diamond theory, we show how the distance functional is related (to leading order in the scale L) to spacetime scalar curvature in the causal past of the two lightcones, and briefly illustrate a number of its possible applications.

Dirac Redux - Dirac’s Equation in Physical Spacetime without Internal Degrees of Freedom

The Dirac equation (DE) is one of the cornerstones of quantum physics. We prove in the present contribution that the notion of internal degrees of freedom of the electron represented by Dirac’s matrices1,2 is superfluous. One can write down a coordinate-free manifestly covariant vector equation by direct quantization of the 4-momentum vector with modulus m: Pψ=mψ (no slash!), the spinor ψ taking care of the different vector grades at the two sides of the equation. Electron spin and all the standard DE properties emerge from this equation. In coordinate representation, the four orthonormal time-space frame vectors Xµ appear instead of Dirac’s Yµ-matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x5 that is defining for the C (particle-antiparticle) symmetry and the CPT symmetry of DE, as well as for left- and right-handed rotors and spinors. In 3D it transforms a parity-odd vector x into a parity-even vector σ=x5x and vice versa. STA augmented by the reflector will be referred to as STAR, which operates on a real vector space of same dimension as the equivalent real dimension of Dirac’s complex 4×4 matrices. There are no matrices in STAR and the complex character springs from the signature and dimension of spacetime-reflection. This appears most clearly by first showing that STAR comprises two isomorphic subspaces, one for generators of polar vectors and boosts and the other for generators of axial vectors and rotors, including Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. We prove here that the information from γ-matrices is contained in spacetime-reflection, which makes the matrices redundant. Therefore, it becomes relevant to reexamine those areas of quantum physics that take the γ-matrices and their generalizations as fundamental.

Dirac Redux - Dirac’s Equation in Physical Spacetime without Internal Degrees of Freedom

The Dirac equation (DE) is one of the cornerstones of quantum physics. We prove in the present contribution that the notion of internal degrees of freedom of the electron represented by Dirac’s matrices is superfluous. One can write down a coordinate-free manifestly covariant equation by direct quantization of the energy-momentum 4-vector P with modulus m: P(psi) = m(psi) (no slash!), the spinor (psi) taking care of the different vector grades at the two sides of the equation. Electron spin and all the standard DE properties emerge from this equation. In coordinate representation, the four orthonormal time-space frame vectors x0, x1, x2, x3 formally substitute Dirac’s gamma-matrices, the two sets obeying to the same Clifford algebra. The present formalism expands Hestenes’ spacetime algebra (STA) by adding a reflector vector x5, which in 3D transforms a parity-odd vector x into a parity-even vector x5x and vice versa. STA augmented by the reflector will be referred to as STAR, which operates on a real vector space of same dimension as the equivalent real dimension of Dirac’s complex 4 x 4 matrices. There are no matrices in STAR and the complex character springs from the signature and dimension of spacetime-reflection. This appears most clearly by first showing that STAR comprises two isomorphic subspaces, one for the generators of polar vectors and boosts and the other for the generators of axial vectors and rotors, comprising Pauli spin vectors. These then help to discuss the symmetries, probability current, transformation properties and nonrelativistic approximation of STAR DE. By proving that Dirac’s matrices are redundant, because all the information from them is contained in spacetime-reflection, it becomes relevant to reexamine those areas of modern physics that take Dirac matrices and their generalizations as fundamental.

Peeling property and asymptotic symmetries with a cosmological constant

This paper establishes two things in an asymptotically (anti-)de Sitter spacetime, by direct computations in the physical spacetime (i.e. with no involvement of spacetime compactification): (1) The peeling property of the Weyl spinor is guaranteed. In the case where there are Maxwell fields present, the peeling properties of both Weyl and Maxwell spinors similarly hold, if the leading order term of the spin coefficient [Formula: see text] when expanded as inverse powers of [Formula: see text] (where [Formula: see text] is the usual spherical radial coordinate, and [Formula: see text] is null infinity, [Formula: see text]) has coefficient [Formula: see text]. (2) In the absence of gravitational radiation (a conformally flat [Formula: see text]), the group of asymptotic symmetries is trivial, with no room for supertranslations.

The Now and the Passage of Time

The notion of the “present moment” has intrigued philosophers, physicists, and psychologists alike. Here we review the literature in the physics and the neuropsychology of the “now” in order to connect those two yet unrelated fields. Such a unitary perspective helps us to explain why there cannot be an objective and absolute “now” and why we naïvely tend to believe in a cosmically extended present. In particular, invoking the recent identification in the Cognitive Neurosciences of various temporal integration windows underlying an individual’s temporal experience within physical spacetime enables us to qualify in a more precise way in what sense the now, as frequently claimed by philosophers, is mind-dependent.

WESS–ZUMINO–WITTEN MODEL FOR GALILEAN CONFORMAL ALGEBRA

In this note, we construct a Wess–Zumino–Witten model based on the Galilean conformal algebra in two-spacetime dimensions, which is a nonrelativistic analogue of the relativistic conformal algebra. We obtain exact background corresponding to σ-models in six dimensions (the dimension of the group manifold) and a central charge c = 6. We carry out a Sugawara type construction to verify the conformal invariance of the model. Further, we discuss the feasibility of the background obtained as a physical spacetime metric.

DE SITTER SPACETIME WITH TORSION AS PHYSICAL SPACETIME IN THE VACUUM AND ISOTROPIC COSMOLOGY

Homogeneous isotropic models with two torsion functions built in the framework of the Poincaré gauge theory of gravity (PGTG) based on general expression of gravitational Lagrangian without cosmological constant are analyzed. It is shown that the physical spacetime in the vacuum in the framework of PGTG can have the structure of flat de Sitter spacetime with torsion. Some physical consequences of obtained conclusion are discussed.

THE INCOMPLETENESS OF KRUSKAL–SZEKERES SPACETIME

Novikov has developed a reference system built around times measured by radially moving geodesic clocks that is equivalent to Kruskal–Szekeres coordinates. From analysis of the construction of Novikov's reference system, I give arguments showing that the reference system is not maximally extended, as is commonly reported in the literature. On both Novikov and Kruskal–Szekeres spacetime diagrams, the left-hand side, corresponding to negative values of the spatial coordinate, should not be included when describing a physical spacetime. In turn, this means we have to rethink widely-accepted concepts such as black and white holes that arise from the usual picture of a maximally-extended Kruskal–Szekeres spacetime.