Is Minkowski Space-Time Compatible with Quantum Mechanics?

In quantum relativistic Hamiltonian dynamics, the time evolution of interacting particles is described by the Hamiltonian with an interaction-dependent term (potential energy). Boost operators are responsible for (Lorentz) transformations of observables between different moving inertial frames of reference. Relativistic invariance requires that interaction-dependent terms (potential boosts) are present also in the boost operators and therefore Lorentz transformations depend on the interaction acting in the system. This fact is ignored in special relativity, which postulates the universality of Lorentz transformations and their independence of interactions. Taking into account potential boosts in Lorentz transformations allows us to resolve the no-interaction paradox formulated by Currie, Jordan, and Sudarshan [Rev. Mod. Phys. 35, 350 (1963)] and to predict a number of potentially observable effects contradicting special relativity. In particular, we demonstrate that the longitudinal electric field (Coulomb potential) of a moving charge propagates instantaneously. We show that this effect as well as superluminal spreading of localized particle states is in full agreement with causality in all inertial frames of reference. Formulas relating time and position of events in interacting systems reduce to the usual Lorentz transformations only in the classical limit (0) and for weak interactions. Therefore, the concept of Minkowski space-time is just an approximation which should be avoided in rigorous theoretical constructions.     

On the Fizeau Experiment

Transformations of space and time, depending on a synchronisation parameter e ₁, indicate the existence of a privileged inertial system S ₀. The Lorentz transformations are obtained for a particular e ₁0. No classical experiment on relativity is expected to depend on the choice of e ₁. We show, consistently with expectations, that the result of the Fizeau experiment is explained equally well by theories adopting different values of e ₁. In previous papers we showed that if accelerated reference frames are considered only e ₁=0 remains possible.     

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     

Thomas rotation and the parametrization of the Lorentz transformation group

Two successive pure Lorentz transformations are equivalent to a pure Lorentz transformation preceded by a 3×3 space rotation, called a Thomas rotation. When applied to the gyration of the rotation axis of a spinning mass, Thomas rotation gives rise to the well-knownThomas precession. A 3×3 parametric, unimodular, orthogonal matrix that represents the Thomas rotation is presented and studied. This matrix representation enables the Lorentz transformation group to be parametrized by two physical observables: the (3-dimensional) relative velocity and orientation between inertial frames. The resulting parametrization of the Lorentz group, in turn, enables the composition of successive Lorentz transformations to be given by parameter composition. This composition is continuously deformed into a corresponding, well-known Galilean transformation composition by letting the speed of light approach infinity. Finally, as an application the Lorentz transformation with given orientation parameter is uniquely expressed in terms of an initial and a final time-like 4-vector.     

Transformations in special relativity

By using the principle of relativity alone (no assumptions about signals or light) it is shown that a relativisitic group of linear transformations of a spacetime plane is, if infinite, either Galilean, Lorentzian or rotational. The largest such finite group is a Klein 4-group, generated by space-reversal and time-reversal. In the infinite case an invariant of the group, denotedc, appears. Whenc is real, nonzero, noninfinite, then the group is a Lorentz group andc is identified with the speed of light. Lorentz transformations are represented through an algebra ofiterants that provides a link among Clifford algebras, the Pauli algebra and Herman Bondi'sK-calculus.     

On the Ricci Curvature of Compact Spacelike Hypersurfaces in Einstein Conformally Stationary-Closed Spacetimes

In this paper we develop an integral formula involving the Ricci and scalar curvatures of a compact spacelike hypersurface M in a spacetime equipped with a timelike closed conformal vector field K (in short, conformally stationary-closed spacetime), and we apply it, when is Einstein, in order to establish sufficient conditions for M to be a leaf of the foliation determined by K and to obtain some non-existence results. We also get some interesting consequences for the particular case when is a generalized Robertson-Walker spacetime.     

Use of the Lorentz Transformations in Noninertial Reference Frames

The Lorentz transformations are used within the model of a noninertial reference frame without infinitely high accelerations arising at instantaneous jumps of an accelerated observer between different inertial reference frames. It is demonstrated that the twin paradox can be explained within this model with the help of the Lorentz transformations. Based on the model of a noninertial reference frame, the acceleration a measured in the noninertial reference frame is related to the acceleration a measured in an inertial reference frame.     

Electrodynamics Under a Possible Alternative to the Lorentz Transformation

A generalization of the classical electrodynamics for systems in absolute motion in presented using a possible alternative to the Lorentz transformation. The main hypothesis assumed in this work are: a) The inertial transformations relate two inertial frames: the privileged frame S and the moving frame S with velocity v with respect to S. b) The transformation of the fields from S to the moving frame S is given by H = a(H – v × D) and E = a(E + v × B), where a is a matrix whose elements depend of the absolute velocity of the system. c) The constitutive relations in the moving frame S are given by D = E, B = H and J = E. It is found that Maxwell's equations, which are transformed to the moving frame, take a new form depending on the absolute velocity of the system. Moreover, differing from classical electrodynamics, it is proven that the electrodynamics proposed explains satisfactorily the Wilson effect.     

The mathematical structure of Newtonian spacetime: Classical dynamics and gravitation

We give a precise and modern mathematical characterization of the Newtonian spacetime structure (). Our formulation clarifies the concepts of absolute space, Newton's relative spaces, and absolute time. The concept of reference frames (which are timelike vector fields on ) plays a fundamental role in our approach, and the classification of all possible reference frames on is investigated in detail. We succeed in identifying a Lorentzian structure on and we study the classical electrodynamics of Maxwell and Lorentz relative to this structure, obtaining the important result that there exists only one intrinsic generalization of the Lorentz force law which is compatible with Maxwell equations. This is at variance with other proposed intrinsic generalizations of the Lorentz force law appearing in the literature. We present also a formulation of Newtonian gravitational theory as a curve spacetime theory and discuss its meaning.     

The introduction of Superluminal Lorentz transformations: A revisitation

We revisit the introduction of the Superluminal Lorentz transformations which carry from bradyonic inertial frames to tachyonic inertial frames, i.e., which transform time-like objects into space-like objects, andvice versa. It has long been known that special relativity can be extended to Superluminal observers only by increasing the number of dimensions of the space-time or—which is in a sense equivalent—by releasing the reality condition (i.e., introducing also imaginary quantities). In the past we always adopted the latter procedure. Here we show the connection between that procedure and the former one. In other words, in order to clarify the physical meaning of the imaginary units entering the classical theory of tachyons, we have temporarily to call into play anauxiliary six-dimensional space-time M(3, 3); however, we are eventually able to go back to the four-dimensional Minkowski space-time. We revisit the introduction of the Superluminal Lorentz transformations also under another aspect. In fact, the generalized Lorentz transformations had been previously written down in a form suited only for the simple case of collinear boosts (e.g., they formed a group just for collinear boosts). We express now the Superluminal Lorentz transformations in a more general form, so that they constitute a group together with the ordinary—orthochronousand antichronous—Lorentz transformations, and reduce to the previous form in the case of collinear boosts. Our approach introduces either real or imaginary quantities, with exclusion of (generic) complex quantities. In the present context, a procedure—in two steps—for interpreting the imaginary quantities is put forth and discussed. In the case of a chain of generalized Lorentz transformations, such a procedure (when necessary) is to be applied only at the end of the chain. Finally, we justify why we call transformations also the Superluminal ones.     

Antiparticles from special Relativity with ortho-chronous and antichronous Lorentz transformations

Special Relativity can be based on the whole proper group of both ortho- and antichronous Lorentz transformations, and a clear physical meaning can be given also to antichronous (i.e., nonorthochronous) Lorentz transformations. From the active point of view, the latter requires existence, for any particle, of its antiparticle within a purely relativistic, classical context. From the passive point of view, they give rise to frames dual to the ordinary ones, whose properties—here briefly discussed—are linked with the fact that in relativity it is impossible to teach another, far observer (by transmitting only instructions, and no physical objects) our own conventions about the choices right/left, matter/antimatter, and positive/negative time direction. Interesting considerations follow, in particular, by considering—as it is the case—theCPT operation as an actual (even if antichronous) Lorentz transformation.Work partially supported by FAPESP and CNPq (Brazil).     

The structure of space-time transformations

LetT be a one-to-one mapping ofn-dimensional space-timeM onto itself. IfT maps light cones onto light cones and dimM3, it is shown thatT is, up to a scale factor, an inhomogeneous Lorentz transformation. Thus constancy of light velocity alone implies the Lorentz group (up to dilatations). The same holds ifT andT ^–1 preserve (x–y)²>0. This generalizes Zeeman's Theorem. It is then shown that ifT maps lightlike lines onto (arbitrary) straight lines and if dimM3, thenT is linear. The last result can be applied to transformations connecting different reference frames in a relativistic or non-relativistic theory.     

Euclidean Special Relativity

New four coordinates are introduced which are related to the usual space-time coordinates. For these coordinates, the Euclidean four-dimensional length squared is equal to the interval squared of the Minkowski space. The Lorentz transformation, for the new coordinates, becomes an SO(4) rotation. New scalars (invariants) are derived. A second approach to the Lorentz transformation is presented. A mixed space is generated by interchanging the notion of time and proper time in inertial frames. Within this approach the Lorentz transformation is a 4-dimensional rotation in an Euclidean space, leading to new possibilities and applications.     

Structure of supermanifolds and supersymmetry transformations

After giving a global, constraint-free Lagrangian formulation of theN=1 superspace supergravity in terms of super fibre bundles and differential forms over a supermanifold, we show that the concept of body manifold of a supermanifold provides a natural manner to reduce the theory to spacetime. This reduction, however, is not canonical, and the various ways in which it can be done give rise to transformations of the field variables which generalise the known invariances of theN=1 spacetime supergravity under supersymmetry transformations and spacetime diffeomorphisms.Research partly supported by the Gruppo Nazionale per la Fisica Matematica of the Italian Research Council and by the Italian Ministry of Public Education through the research project Geometria e Fisica     

Differential Groupoids and Their Application to the Theory of Spacetime Singularities

The transformation groupoid = × G, where is the total space of the generalized frame G-bundle over spacetime with a singular boundary, is not a Lie groupoid but a differential groupoid, i.e., a smooth groupoid in the category of structured spaces. We define this concept and use it to investigate spacetimes with various kinds of singularities. Any differential transformation groupoid can be represented by an algebra of operators on a bundle of Hilbert spaces defined on the groupoid fibers. This algebra reflects the structure of a given fiber even if it is a fiber over a singularity. It is also shown that any spacetime with singularities can be regarded as a noncommutative space. Its geometry is done in terms of a noncommutative algebra defined on the corresponding differential transformation groupoid. We focus on the structure of malicious singularities such as the ones appearing in the beginning and in the end of the closed Friedman universe.     

Covariant Formulation of Electromagnetic 4-Momentum in Terms of 4-Vectors E α and B α

The fundamental difference between the true transformations (TT) and the apparent transformations (AT) is explained. The TT refer to the same quantity, while the AT refer, e.g., to the same measurement in different inertial frames of reference. It is shown that the usual transformations of the three-vectors E and B are - the AT. The covariant electrodynamics with the four-vectors E and B of the electric and magnetic field is constructed. It is also shown that the conventional synchronous definitions of the electromagnetic energy and momentum contain both, the AT of the volume, i.e., the Lorentz contraction, and the AT of E and B, while Rohrlich's expressions contain only the AT of E and B. A manifestly covariant expression for the energy-momentum density tensor and the electromagnetic 4-momentum is constructed using E and B . The 4/3 problem is discussed and it is shown that all previous treatments either contain the AT of the volume, or the AT of E and B, or both of them. In our approach all quantities are four-dimensional spacetime tensors whose transformations are the TT.     

Observer-Dependence of Chaos Under Lorentz and Rindler Transformations

The behavior of Lyapunov exponents and dynamical entropies h, whose positivity characterizes chaotic motion, under Lorentz and Rindler transformations is studied. Under Lorentz transformations, and h are changed, but their positivity is preserved for chaotic systems. Under Rindler transformations, and h are changed in such a way that systems, which are chaotic for an accelerated Rindler observer, can be nonchaotic for an inertial Minkowski observer. Therefore, the concept of chaos is observer-dependent.     

Use of Lorentz Transformations for Noninertial Frames of Reference

By analogy with the calculation of the path of a mass point in terms of the integral of the point velocity with respect to time, such that the point has a constant velocity V(t _i) within a time interval dt _i, then changes this velocity stepwise by V(t _i+1), moves with this velocity within a time interval dt _i+1, etc., an accelerated motion of an observer with a clock is represented by alternating states of rest in a sequence of inertial frames of reference and instantaneous jumps from one frame of reference into another. Lorentz transformations are used to calculate the readings of a resting clock observed from a noninertial frame of reference represented in this manner, during the rest of a noninertial observer in a next-in-turn inertial frame of reference belonging to the mentioned sequence, and upon a jump. For the observation from a noninertial frame of reference, the relation of the time interval counted by the resting clock to the time interval counted by the accelerated clock and to the acceleration has been obtained.     

Derivation of Dirac's Equation from the Evans Wave Equation

The Evans wave equation [1] of general relativity is expressed in spinor form, thus producing the Dirac equation in general relativity. The Dirac equation in special relativity is recovered in the limit of Euclidean or flat spacetime. By deriving the Dirac equation from the Evans equation it is demonstrated that the former originates in a novel metric compatibility condition, a geometrical constraint on the metric vector qused to define the Einstein metric tensor. Contrary to some claims by Ryder, it is shown that the Dirac equation cannot be deduced unequivocally from a Lorentz boost in special relativity. It is shown that the usually accepted method in Clifford algebra and special relativity of equating the outer product of two Pauli spinors to a three-vector in the Pauli basis leads to the paradoxical result X = Y = Z = 0. The method devised in this paper for deriving the Dirac equation from the Evans equation does not use this paradoxical result.     

Revised Robertson's test theory of special relativity: Space-time structure and dynamics

The experimental testing of the Lorentz transformations is based on a family of sets of coordinate transformations that do not comply in general with the principle of equivalence of the inertial frames. The Lorentz and Galilean sets of transformations are the only member sets of the family that satisfy this principle. In the neighborhood of regular points of space-time, all members in the family are assumed to comply with local homogeneity of space-time and isotropy of space in at least one free-falling elevator, to be denoted as Robertson'sab initio rest frame [H. P. Robertson,Rev. Mod. Phys. 21, 378 (1949)].Without any further assumptions, it is shown that Robertson's rest frame becomes a preferred frame for all member sets of the Robertson family except for, again, Galilean and Einstein's relativities. If one now assumes the validity of Maxwell-Lorentz electrodynamics in the preferred frame, a different electrodynamics spontaneously emerges for each set of transformations. The flat space-time of relativity retains its relevance, which permits an obvious generalization, in a Robertson context, of Dirac's theory of the electron and Einstein's gravitation. The family of theories thus obtained constitutes a covering theory of relativistic physics.A technique is developed to move back and forth between Einstein's relativity and the different members of the family of theories. It permits great simplifications in the analysis of relativistic experiments with relevant Robertson's subfamilies. It is shown how to adapt the Clifford algebra version of standard physics for use with the covering theory and, in particular, with the covering Dirac theory.Part of this work was done at the Department of Physics, Utah State University, Logan, Utah 84322.