Lorentz-invariant Theory Research Articles

Are tachyon causal paradoxes solved?

Tolman's paradox arises in Lorentz-invariant theories of superluminal particles. In this paper we first try to clarify the nature of the paradox and what it means to solve it. We then analyze the various attempts made to either solve or eliminate it. We show that general consequences can be drawn which hold in essentially all paradox-free schemes proposed so far.

On the infrared stability of gauge theories

We consider a class of field theories which contains a Lorentz and gauge invariant theory as a fixed point, but whose generic member possesses none of these symmetries. We show that this fixed point is an infrared repulsor for all non-abelian groups. We also discuss the case of the N = 4 supersymmetric gauge theory whose stability properties are shown to depend on the gauge group.

Classical Lorentz-invariant theory of systems with self-action: Lagrangian formulation

A Lorentz-invariant theory of singular Lagrangian particle systems with selfaction, treated as a dependence of the Lagrangian upon acceleration, is developed. The Lagrangian equations and their exact special solutions are found. The first integrals of 4-momentum and angular momentum are calculated. Particles possessing weak self-action are treated as classical analogs of particles with half-integral spin.

Mach’s principle and the structure of dynamical theories

A structure of dynamical theories is proposed that implements Mach’s ideas by being relational in its treatment of both motion and time. The resulting general dynamics, which is called intrinsic dynamics and by construction treats the evolution of the entire Universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero. The formal structure of Einstein’s general theory of relativity also fits the pattern of intrinsic dynamics and is Machian according to the criteria of this paper provided the so-called thin-sandwich conjecture is generically correct.

Class of Lorentz-invariant theories of gravitation

A class of rank-n tensor theories of gravitation is considered. There is more than one theory for each n≳0. All theories are Lorentz invariant and reduce to Newton’s theory for small velocities and weak gravitational fields. Among these theories is the second-rank tensor theory of Whitehead and the spin-two theory of Fierz and Pauli. The precession of the perihelion of a planet’s orbit is calculated for those theories with n = 0, 1, 2 and compared with the prediction of general relativity.

Independent variables in quantum mechanics

It is conjectured that the particle states of quantum mechanics are represented by functions of independent variables. These functions obey a linear differential equation which has an invariance group homomorphic to the inhomogeneous Lorentz group, thus giving a linear, Lorentz-invariant theory. Simple one-particle examples of equations which lead to a discrete particle spectrum are given, using both space–time variable, xμ, and sets of spinlike variables (pairs of complex numbers). Some of the examples have internal symmetry. No examples of realistic ’’many-body’’ particle theories are given, but we can deduce general characteristics. The differential equation must be of second or higher order to give an interaction. Products of single-particle states will be solutions of the equation and will form a complete set for widely separated particles. But products of one-particle states are not solutions of the equation for strongly interacting particles, and this permits the creation of particles. The origin of antisymmetry in such a theory is not clear.

Spinor theory of general relativity without elementary gravitons

We propose a generally covariant and locally Lorentz invariant theory of a Majorana spinor field ψ μ α . Our theory has no elementary spin-2 quanta, but does reproduce Einstein's general relativity as a classical solution. We compare this situation to the possibility of finding classical monopoles in a gauge theory, even though no such elementary object is introduced at the outset.

Physical foundations of the new theory of gravitation

It is shown that in generalizing simultaneously the special theory of relativity and the Newtonian theory of gravitation into a curved-space theory of gravity one needs to satisfy second-order correspondence conditions as well as the usual first-order ones. It is further shown that the imposition of these second-order conditions leads to a locally Lorentz invariant spin-2 field theory of gravity which is in agreement with all the known facts concerning gravitation. The second-order correspondence requirements endow the curved-space theory of gravitation with experimentally and theoretically novel features which will be discussed in some detail.

On energy-momentum tensors as sourcesof spin-2 fields

The improvement terms in the generalised energy-momentum tensor of Callan, Coleman and Jackiw can be derived from a variational principle if the Lagrangian is generalised to describe coupling between ‘matter’ fields and a spin-2 boson field. The required Lorentz-invariant theory is a linearised version of Kibble-Sciama theory with an additional (generally-covariant) coupling term in the Lagrangian. The improved energy-momentum tensor appears as the source of the spin-2 field, if terms of second order in the coupling constant are neglected.

Causality, Lorentz invariance and i) the electromagnetic interaction of a massive spin-one vector field with arbitrary magnetic-dipole moment, ii) the massive Yang-Mills field

In recent years, following the methods first used by VwLo and ZWANZmER (1), several authors have studied the causal na ture of the propagat ion of higher-spin classical fields in external potentials (3). By a consideration of several examples involving a massive spin-one vector field, the author has lent force to the conjecture that , for a field in an external potential , the classical theory is causal (in the sense of VELo and ZWA~ZlGER) if and only if the corresponding quantum theory is Lorentz invar iant (in the sense tha t there exists an operator ordering in terms of which the per turba t ion expansion for the S-operator is normal-independent) (3). I t is of interest to see if the results of ref. (3) can be extended to include interactions between classical fields, and the corresponding fully quantized theories. As a first step in this direction, two theories, whose Lorentz invariance propert ies at the fully quantized level are well known, are here considered at the classical level. The first theory to be discussed is the electromagnetic interact ion of a massive spinone vector field with a rb i t ra ry magnetic-dipole moment. This theory was found b y L ~ and YA~G to be noncovariant a t the fully quantized level (4). Later , NAKAI~UEA (6) and Tzou (e) demonstra ted that , by the inclusion of an appropria te fourth-degree selfinteract ion of the vector field into the Lee-Yang theory, a Lorentzinvar iant theory results. Since i t is the magnetic-dipole interaction, and not the minimal electromagnetic interaction, which gives rise to the covariance difficulties, the la t te r will, for caleulat ional and notat ional simplicity, be omit ted from the following discussion. Thus the

НОВАь тЕОРИь гРАВИтАцИИ

A locally Lorentz-invariant curved-space theory of gravitation, where the local field is a massless, spin-2 field, φΜgn, of Pauli-Fierz type, is presented. In this theory gΜΝ is no longer the gravitational potential itself, but reduces to a functional gΜΝ(φ{α/Β}) of φΜgn. Several rigorous and a general iteration solution of this type are exhibited. The static central-body problem reduces to author’s 1958 theory as a special case φΜgn → φ00 =φ and ∂Μ φΜgn → ∂0φ= 0 exactly, so that as in that theory the three classical tests are satisfied.

Some properties of an ?aesthetic? field theory

We continue our study of the Lorentz-invariant field theory based on the equations Γ jk;l i =0 and gij;k=0. To first order in a perturbation expansion, we find Γ jk;l i =0 reduces to the wave equation. In orders higher than the first, we find that Γ jk;l i =0 cannot be linearized. We also find that the simple wave-type equation gij∂2g/∂xi∂xj=0 is contained in the theory when an appropriate choice is made for the parameters at the origin point.

New Theory of Gravitation

A locally Lorentz-invariant curved-space theory of gravitation where the local field is a massless, spin-2 field ${{\ensuremath{\phi}}_{\ensuremath{\mu}}}^{\ensuremath{\nu}}$ of Pauli-Fierz type is presented. In this theory ${g}_{\ensuremath{\mu}\ensuremath{\nu}}$ is no longer the gravitational potential itself but reduces to a functional ${g}_{\ensuremath{\mu}\ensuremath{\nu}}({{\ensuremath{\phi}}_{\ensuremath{\alpha}}}^{\ensuremath{\beta}})$ of ${{\ensuremath{\phi}}_{\ensuremath{\mu}}}^{\ensuremath{\nu}}$. Several rigorous and a general iteration solution of this type are exhibited. The static central body problem reduces exactly to author's 1958 theory in the special case ${{\ensuremath{\phi}}_{\ensuremath{\mu}}}^{\ensuremath{\nu}}\ensuremath{\rightarrow}{{\ensuremath{\phi}}_{0}}^{0}=\ensuremath{\phi}$ and ${\ensuremath{\partial}}^{\ensuremath{\mu}}{{\ensuremath{\phi}}_{\ensuremath{\mu}}}^{\ensuremath{\nu}}\ensuremath{\rightarrow}{\ensuremath{\partial}}^{0}\ensuremath{\phi}=0$ so that as in that theory the three crucial tests are satisfied.

Gauge invariance and Galilean invariance

The usual method of introducing electromagnetic interaction via a gauge principle is shown to fail for a Galilean-invariant theory. A clarification of the results is obtained by considering the non-relativistic limit of the Lorentz-invariant theory of the electromagnetic interaction of a complex scalar field.

Feldtheoretische Konstruktion der Jordan—Brans—Dicke-Theorie / Fieldtheoretic Construction of the Jordan-Brans-Dicke-Theory

Abstract In the framework of Lorentz invariant theories of gravitation the fieldtheoretic approach of the generally covariant Jordan-Brans-Dicke-theory is investigated. It is shown that a slight restriction of the gauge group of Einstein's linear tensor theory leads to the linearized Jordan-Brans-Dicke-theory. The problem of the inconsistency of the field equations and the equations of motion is solved by introducing the Landau-Lifschitz energy momentum tensor of the gravitational field as an additional source term into the field equations. The second order of the theory together with the corresponding gauge group are calculated explicitly. By means of the structure of the gauge group of the tensor field it is possible to identify the successive orders of the scalar-tensor theory as an expansion of the Jordan-Brans-Dicke-theory in flat space-time. The question of the uniqueness of the procedure is answered by showing that the structure of the gauge group of the tensor field is predetermined by the linear equations of motion. The mathematical proof of this fact confirms formally the meaning of the equations of motion for the geometry of space.

Hamiltonians and Wave Equations for Particles of Spin 0 and Spin 12 with Nonzero Mass

The formal mathematical structure of the truly Hamiltonian formulations of nonrelativistic and relativistic quantum mechanics is critically examined for both spin 0 and spin 12 particles with nonzero mass. It is shown that the relativistic quantum mechanics can, in principle, at least be fitted into a truly covariant Hamiltonian procedure where the Hamiltonian is a Lorentz-invariant world scalar (and is the negative of 12 the rest energy of the particle, regardless of its spin). In particular, a truly covariant Hamiltonian theory of the electron is presented. It will be seen that the Lorentz-invariant Dirac theory of the electron does not fall within the framework of a truly covariant Hamiltonian procedure. Further, the dynamical variables associated with the translational degree of freedom of the particle (such as the position, the canonical momentum, and the velocity four-vectors of the relativistic quantum mechanics) correspond precisely to the same variables of the truly covariant formulation of relativistic classical mechanics. The mathematical structure of relativistic quantum mechanics is seen to be rather similar to that of nonrelativistic quantum mechanics for both spin 0 and spin 12 particles with nonzero mass.

Lorentz-Invariant Gravitational Perturbations and the Evaluation of Generalized Green's Functions

The Lorentz-invariant perturbation theory of classical nonlinear field theories, notably the fast-motion approximation for a system of gravitationally interacting particles, is used to illustrate the appearance of generalized Green's functions (GGF's) in physics. The (retarded) GGF's are defined as the convolution of the usual (retarded) Green's function with certain other linear functionals. A manifestly Lorentz-invariant technique for the evaluation of the convolution integral is described and applied to two important classes of GGF's. Time-symmetric GGF's are also discussed briefly. The technique exhibited provides the tools for the calculation of the higher-order approximations in a Lorentz-invariant perturbational approach to the classical n-body problem.

Relativistic Theories of Gravitation

THE results which general relativity predicts for the three well known tests (the motion of the perihelion, the deflexion of light rays and the gravitational red-shift) and for new tests which we have proposed1,2 in the one-body problem are also obtained in certain Lorentz-invariant theories of gravitation. Whitrow and Morduch3,4 have examined the Lorentz-invariant theories of gravitation developed earlier and have calculated most of the known empirical tests for these theories in order to analyse the possibility of deciding experimentally between them and Einstein's general theory of relativity. We have continued the work of Whitrow and Morduch by classifying all possible Lorentz-invariant theories of gravitation that satisfy certain general axioms and by analysing the one-body problem in all these theories.

Possibility of the Speed of Sound Exceeding the Speed of Light in Ultradense Matter

We show that, in the classical physics of very dense matter, Lorentz invariance imposes no restriction on the speed of sound or on the ratio of pressure to energy density. Indeed, the simplest and most reasonable classical many-particle theory can manifest such apparently noncausal behavior whenever the calculated self-energy of a particle exceeds its observed (renormalized) rest energy. This comes about because ordinary mass renormalization subtracts out part of a particle's self-interaction energy without altering the interaction with other particles that contributes to pressure. Two types of models are exhibited which, at low densities, show normal behavior and, at high densities, become superluminal (speed of sound greater than speed of light in vacuum) and ultrabaric (pressure greater than energy density). One is a system of classical particles which, when stationary, repel each other by a short-range repulsive Yukawa interaction. Although the particles interact through ordinary retarded neutral vector fields, after mass renormalization there must always be a domain of sufficiently high densities where this matter becomes superluminal and ultrabaric. The second group of models is a class of classical Lorentz-invariant nonlinear field theories which, in the limit of low densities, reduces to a noninteracting Klein-Gordon field. If matter deep inside superdense stars could be ultrabaric, then the limiting gravitational red shift from the star's surface would be slightly under 2. This is perhaps suggestive of the observed clustering of quasar absorption-line red shifts at 1.95.

Kosmologische Lösungen der lorentzinvarianten Gravitationstheorie

In the framework of the Lorentz invariant theory of gravitation a cosmology in the flat space-time is investigated. As in the Newtonian cosmology we start from an infinitely extended system of incoherent matter under the influence of its own gravitational field. The field equations, the equations of motion and the world postulate of homogenity and isotropy for geodetic observes lead then to the Friedman equation. In order to handle the coupled system of equations for the gravitational field and the matter a conveniant approximation method is developed. The calculations are carried out in the second order of this method. The Einstein theory, which is in some respect equivalent to the Lorentz invariant theory of gravitation, serves as a guiding principle for our formal developements. On the other hand the flat space-time cosmology presented here, gives rise to a new interpretation of the Einstein Cosmology.