An empirical,purely spatial criterion for the planes ofF-simultaneity

The claim that distant simultaneity with respect to an inertial observer is conventional arose in the context of a space-and-time rather than a spacetime ontology. Reformulating this problem in terms of a spacetime ontology merely trivializes it. In the context of flat space, flat time, and a linear inertial structure (a purely space-and-time formalism), we prove that the hyperplanes of space for a given inertial observer are determined by a purely spatial criterion that depends for its validity only on the two-way light principle, which is universally regarded as empirically verified. All (empirically determined) spacetime entities, such as the conformal structure or light surface equation, are used in a purely mathematical manner that is independent of and hence isneutral with respect to the ontological status that is ascribed to them. In this regard, our criterion is significantly stronger than thespacetime criterion recently advanced by D. Malament, which appeals explicitly to the conformal orthogonality of spacetime vectors and to the invariance of the conformal-orthogonal structure of spacetime under the causal automorphisms of spacetime. Once the hyperplanes of space for a given inertial observer have been determined by our empirical and purely spatial criterion, the following holds: there exists one and only one -synchronization procedure, namely the standard procedure proposed by Einstein, such that the planes of common time are thesame as the nonconventional hyperplanes of space for the inertial observer. It follows that our criterion provides an empirical even if indirect method for determining that the one-way speed of light is the same as the average two-way speed of light. In addition, two inertial observers that are not at rest with respect to each other necessarily havedifferent hyperplanes of space, and consequently their respective spatial views cannot be encompassed in a single three-dimensional space. Hence, our purely spatial criterion provides an empirical motivation for adopting the more comprehensive spacetime ontology.     

States and operators in the spacetime algebra

The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum - and -matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed.Supported by a SERC studentship.     

Coulomb scattering of an electron by a monopole

The Coulomb scattering of an electron by a magnetic monopole is analyzed using a lowest-order quantum perturbation approximation suggested by a two-potential Lagrangian form for classical electromagnetism, generalized through the use of spacetime algebra to include magnetic monopoles. Good agreement with existing conventional analyses of this problem is demonstrated.1. Work supported by Department of Energy contract DE-AC03-76SF00515.2. The idea to employ spacetime algebra (sometimes called Dirac algebra) to incorporate magnetic monopoles into classical electromagnetic theory was proposed by de Faria-Rosaet al. [3].3. This is a factori difference between the definition of 5 by Eq. (3) and that by Bjorken and Drell [6]. Since a cross section (without interference terms) is being calculated, we can ignore this distinction.     

Geometric Equations of State in Friedmann-Lemaître Universes Admitting Matter and Ricci Collineations

As a rule in General Relativity the spacetime metric fixes the Einstein tensor and through the Field Equations (FE) the energy-momentum tensor. However one cannot write the FE explicitly until a class of observers has been considered. Every class of observers defines a decomposition of the energy-momentum tensor in terms of the dynamical variables energy density (), the isotropic pressure (p), the heat flux q ^a and the traceless anisotropic pressure tensor _ab . The solution of the FE requires additional assumptions among the dynamical variables known with the generic name equations of state. These imply that the properties of the matter for a given class of observers depends not only on the energy-momentum tensor but on extra a priori assumptions which are relevant to that particular class of observers. This makes difficult the comparison of the physics observed by different classes of observers for the same spacetime metric. One way to overcome this unsatisfactory situation is to define the extra condition required among the dynamical variables by a geometric condition, which will be based on the metric and not to the observers. Among the possible and multiple conditions one could use the consideration of collineations. We examine this possibility for the Friedmann-Lemaître-Robertson-Walker models admitting matter and Ricci collineations and determine the equations of state for the comoving observers. We find linear and non-linear equations of state, which lead to solutions satisfying the energy conditions, therefore describing physically viable cosmological models.     

Extension of the Virasoro and Neveu-Schwarz Algebras and Generalized Sturm-Liouville Operators

We consider the universal central extension of the Lie algebra Vect(S ¹) C(S ¹). The coadjoint representation of thisLie algebra has a natural geometric interpretation by matrix analogues ofthe Sturm –Liouville operators. This approach leads to new Liesuperalgebras generalizing the well-known Neveu –Schwarz algebra.     

Relativistic Linear Spacetime Transformations Based on Symmetry

Usually the Lorentz transformations are derived from the conservation of the spacetime interval. We propose here a way of obtaining spacetime transformations between two inertial frames directly from symmetry, the isotropy of the space and principle of relativity. The transformation is uniquely defined except for a constant e, that depends only on the process of synchronization of clocks inside each system. Relativistic velocity addition is obtained, and it is shown that the set of velocities is a bounded symmetric domain. If e=0, Galilean transformations are obtained. If e>0, the speed 1/e and a spacetime interval are conserved. By assuming constancy of the speed of light, we get e=1/c ² and the transformation between the frames becomes the Lorentz transformation. If e<0, a proper speed and a Hilbertian norm are conserved.     

Letter: Spherically Symmetric Metrics with Shear Controlling Expansion

An isotropic spatially inhomogeneous spacetime with the stress tensor satisfying the limiting case of the strong energy condition [T₀₀ + 1/2)T] = 0 in the locally inertial coordinates where the observer's four-velocity is u ^a = ₀ ^a satisfying the constraint u ^a u_a = –1 is studied. Special metrics with accelerating expansions of the inhomogeneous spacetime merely controlled by the shear are presented as an alternative model.     

Nonlinear Transformation Group of CAR Fermion Algebra

Based on our previous work on the recursive fermion system in the Cuntz algebra, it is shown that a nonlinear transformation group of the CAR fermion algebra is induced from a U(2 ^p ) action on the Cuntz algebra ₂ ^p with an arbitrary positive integer p. In general, these nonlinear transformations are expressed in terms of finite polynomials in generators. Some Bogoliubov transformations are involved as special cases.     

Noninvariant one-way velocity of light

After discussing in the first five sections the meaning and the difficulties of the principle of relativity we present a new sel of spacetime transformations between inertial systems (inertial transformations), based on three assumptions: (1) The two-way velocity of light is c in all inertial systems and in all directions; (2) Time dilation effects take place with the usual relativistic factor; (3) Clocks are synchronized in the way chosen by nature itself, e.g., in the Sagnac effect. We show that our new transformation laws can explain the available experimental evidence in spite of the implied noninvariance of the one-way velocity of light.     

Fusion Rules for the Continuum Sectors of the Virasoro Algebra with c=1

The Virasoro algebra with c = 1 has a continuum of superselection sectors characterized by the ground state energy h 0. Only the discrete subset of sectors with h = s ², s ₀, arises by restriction of representations of the SU(2) current algebra at level k=1. The remaining continuum of sectors is obtained with the help of (localized) homomorphisms into the current algebra. The fusion product of continuum sectors with discrete sectors is computed. A new method of determining the sector of a state is used.     

New q-Deformed Oscillator Algebra and Thermodynamic Model

In this paper we propose the new q-oscillator algebra. We discuss the coherent state and the deformed su(2) algebra for this algebra when q is real. As is different from Arik–Coon algebra (J. Math. Phys. 17:524, 1976), this algebra is invariant under the hermitian conjugation for complex q. When q is a root of unity, we obtain the finite dimensional Fock space. Finally we discuss the thermodynamics of particle obeying this algebra when q is a root of unity.     

An Analytical Treatment of the Clock Paradox in the Framework of the Special and General Theories of Relativity

No Heading In this paper we treat the so called clock paradox in an analytical way by assuming that a constant and uniform force F of finite magnitude acts continuously on the moving clock along the direction of its motion assumed to be rectilinear (in space). No inertial motion steps are considered. The rest clock is denoted as (1), the to and fro moving clock is (2), the inertial frame in which (1) is at rest in its origin and (2) is seen moving is I and, finally, the accelerated frame in which (2) is at rest in its origin and (1) moves forward and backward is A. We deal with the following questions: (1) What is the effect of the finite force acting on (2) on the proper time interval ⁽²⁾ measured by the two clocks when they reunite? Does a differential aging between the two clocks occur, as it happens when inertial motion and infinite values of the accelerating force is considered? The special theory of relativity is used in order to describe the hyperbolic (in spacetime) motion of (2) in the frame I. (II) Is this effect an absolute one, i.e., does the accelerated observer A comoving with (2) obtain the same results as that obtained by the observer in I, both qualitatively and quantitatively, as it is expected? We use the general theory of relativity in order to answer this question. It turns out that _I = _A for both the clocks, ⁽²⁾ does depend on g = F/m, and = ⁽²⁾/⁽¹⁾ = (1 – ²atanhj)/ < 1. In it ; = V/c and V is the velocity acquired by (2) when the force is inverted.     

“True Transformations Relativity” and Electrodynamics

Different approaches to special relativity (SR) are discussed. The first approach is an invariant approach, which we call the true transformations (TT) relativity. In this approach a physical quantity in the four-dimensional spacetime is mathematically represented either by a true tensor (when no basis has been introduced) or equivalently by a coordinate-based geometric quantity comprising both components and a basis (when some basis has been introduced). This invariant approach is compared with the usual covariant approach, which mainly deals with the basis components of tensors in a specific, i.e., Einstein's coordinatization of the chosen inertial frame of reference. The third approach is the usual noncovariant approach to SR in which some quantities are not tensor quantities, but rather quantities from 3+1 space and time, e.g., the synchronously determined spatial length. This formulation is called the apparent transformations (AT) relativity. It is shown that the principal difference between these approaches arises from the difference in the concept of sameness of a physical quantity for different observers. This difference is investigated considering the spacetime length in the TT relativity and spatial and temporal distances in the AT relativity. It is also found that the usual transformations of the three-vectors (3-vectors) of the electric and magnetic fields E and B are the AT. Furthermore it is proved that the Maxwell equations with the electromagnetic field tensor F^ab and the usual Maxwell equations with E and B are not equivalent, and that the Maxwell equations with E and B do not remain unchanged in form when the Lorentz transformations of the ordinary derivative operators and the AT of E and B are used. The Maxwell equations with F^ab are written in terms of the 4-vectors of the electric E^a and magnetic B^a fields. The covariant Majorana electromagnetic field 4-vector ^a is constructed by means of 4-vectors E^a and B^a and the covariant Majorana formulation of electrodynamics is presented. A Dirac like relativistic wave equation for the free photon is obtained.     

Clifford代数中的双曲相位变换群及其在四维相对论时空中的应用

将Clifford代数所定义的双曲复空间R_H和作用在双曲复空间R_H上的双曲相位变换群U₄(H)赋予了明确的物理意义. 双曲复空间R_H同构于四维Minkowski时空，而其上的双曲相位变换群U₄(H)就是四维相对论时空中的洛仑兹(Lorentz)变换群. 进一步，利用U₄(H)群的复合变换性质，自然导出了四维Minkowski时空中Lorentz变换和速度变换的一般表达式. 由此，将狭义相对论中的特殊Lorentz变换作为特例包含其中. 关键词： 双曲复数 双曲相位变换 Minkowski时空 Clifford代数     

Quantum fields in 1+1-dimension carrying a true ray representation of the Poincaré group

In 1+1 spacetime dimensions there are genuine ray representations of the Poincaré group P ₊ ^sup . We shall construct a free relativistic quantum field theory, such that the fields _d are covariant under P ₊ ^sup and transform with a nontrivial infinitesimal exponent d.     

Implication of Spatial and Temporal Variations of the Fine-Structure Constant

Temporal and spatial variations of fine-structure constant \(\alpha \equiv e^{2}/\hbar c\) in cosmology have been reported in analysis of combination Keck and VLT data. This paper studies the variations based on consideration of basic spacetime symmetry in physics. Both laboratory α ₀ and distant α _z are deduced from relativistic spectrum equations of atoms (e.g., hydrogen atom) defined in inertial reference systems. When Einstein’s Λ≠0, the metric of local inertial reference systems in SM of cosmology is Beltrami metric instead of Minkowski, and the basic spacetime symmetry has to be de Sitter (dS) group. The corresponding special relativity (SR) is dS-SR. A model based on dS-SR is suggested. Comparing the predictions on α-varying with the data, the parameters are determined. The best-fit dipole mode in α’s spatial varying is reproduced by this dS-SR model. α-varyings in whole sky are also studied. The results are generally in agreement with the estimations of observations. The main conclusion is that the phenomenon of α-varying cosmologically with dipole mode dominating is due to the de Sitter (or anti de Sitter) spacetime symmetry with a Minkowski point in an extended special relativity called de Sitter invariant special relativity (dS-SR) developed by Dirac-Inönü-Wigner-Gürsey-Lee-Lu-Zou-Guo.     

Quantum Kinematic Theory of the Poincare Group in Two-Dimensional Spacetime

Non-Abelian quantum kinematics is applied to thePoincare group P ₊ (1, 1),as an example of the quantization-through-the-symmetryapproach to quantum mechanics. Upon quantizing thegroup, generalized Heisenberg commutation relations are obtained, and aclosed Heisenberg–Weyl algebra follows. Then,according to the general theory, the three basicquantum-kinematic invariant operators are calculated;these afford the superselection rules for diagonalizing theincoherent rigged Hilbert space H(P ₊ ) of the regularrepresentation. This paper examines only one of thesediagonalization schemes, while introducing a irreducible spacetime representation carried by isotopicplane-wave eigenvectors of two compatible superselectionoperators (which define a Poincare-invariant linear2-momentum). Thereafter, the principle of microcausality produces massive 2-spinor isotopic states in 1+ 1 Minkowski space. The Dirac equation is thus deducedwithin the quantum kinematic formalism, and the familiarJordan–Pauli propagation kernel in 2-dimensional spacetime is also obtained as a Hurwitzinvariant integral over the group manifold. The maininterest of this approach lies in the adoptedgroup-quantization technique, which is a strictlydeductive method and uses exclusively the assumed Poincaresymmetry.     

Essay: Why Gravity Has No Choice: Bulk Spacetime Dynamics Is Dictated by Information Entanglement Across Horizons

The principle of equivalence implies that gravity affects the light cone (causal) structure of the space-time. It follows that there will exist observers (in any space-time) who do not have access to regions of space-time bounded by horizons. Since physical theories in a given coordinate system must be formulated entirely in terms of variables which an observer using that coordinate system can access, gravitational action functional must contain a foliation dependent surface term which encodes the information inaccessible to the particular observer. I show that: (i) It is possible to determine the nature of this surface term from general symmetry considerations and prove that the entropy of any horizon is proportional to its area. (ii) The gravitational action can be determined using a differential geometric identity related to this surface term. The dynamics of spacetime is dictated by the nature of quantum entanglements across the horizons and the flow of information, making gravity inherently quantum mechanical at all scales. (iii) In static space-times, the action for gravity can be given a purely thermodynamic interpretation and the Einstein equations have a formal similarity to laws of thermodynamics. (iv) The horizon area must be quantized with A _horizon = (8 G /c ³)m with m = 1, 2, in the semi-classical limit.