Energy-momentum Density Research Articles

Energy and frames of reference in general relativity

When relativistic theory amalgamates space and time into “event”, other magnitudes of classical mechanics merge or superimpose. In special relativity it is still easy to find them again by referring to the inertial frame of reference, but general relativity presents them to us as already identified in the curvatures of the Riemannian universe, which admits of no inertial frames, except local ones. It is a search for particular magnitudes and relations (such as momentum and conservation laws), physically interpretablein the classical sense, which specifies a privileged frame. Starting from this frame of reference, the present Note sets out to define the energy-momentum density and super-potentials that give rise, in the same frame, to the known integral results due to the Einstein and Freud psendotensors. Thus the snags arising out of the transformation laws of these pseudotensors are eliminated at the same time.

Exchange Energy, Stress, and Momentum in a Rigid Ferrimagnet

A self-consistent formulation of the exchange energy density, effective exchange field, and Poynting vector is carried out, for a general ferrimagnet, in terms of the spin and orbital angular momentum densities. When there is more than one magnetic lattice or when the gyromagnetic ratios are tensor quantities, this formulation leads to an exchange energy density and Poynting vector that differ from those commonly employed. The associated boundary conditions at a surface of passive discontinuity are also discussed; power flow considerations lead to the conclusion that one of these is a dynamic boundary condition. With the help of the correspondence principle and a quasi particle interpretation of spin waves, a small-signal energy density, momentum density, and stress tensor are postulated for a ferromagnet. The associated small-signal force density (postulated to equal the divergence of the stress tensor minus the time rate of change of the momentum density) is found to be the negative of the force density that accelerates the envelope of the spin wave packet and is interpreted as a reaction force density acting upon the lattice. A prediction is made that a magnon incident upon a surface of discontinuity such as a material—air interface, will exert an exchange-radiation pressure on it.

A gauge-invariant model of quantum electrodynamics without spinor fields

A model of quantum electrodynamics is given exclusively in terms of gauge-invariant observable fields. The electromagnetic fields are obeying the conventional Maxwell equations. The rest of the fields, called the current fields and the gauge-invariant Dirac energy-momentum density fields, obey a system of algebraic and differential equations which are, in the ordinary quantum-mechanical case, equivalent to the system of Dirac equations. The theory is not formulated in terms of Hilbert-space concepts but rather in the language of the ℒ-formalism. Thus the direct construction of a theory of interacting fields is enabled, without the necessity of resorting to creation and annihilation operators or any Fock-space methods.

Note on variational principle in bimetric relativity

In bimetric relativity, involving a Riemannian metric tensor g μν and a flat-space metric tensor γ μν , one can write down a variational principle which gives the field equations and also leads to a gravitational energy-momentum density tensor.

An Attempt to Derive the Gravitational Potential

A model theory is proposed to introduce both the mass of the matter field and the repulsive gravitational potential between matters within the framework of Minkowski's space-time. In consideration of the expansion of universe, the conformal invariant equations for two coup led scalar fields are taken as starting equations. By two procedures breaking conformal invariancy, it is shown that one field is the negative energy field and the self­ consistent mass equations, while they allow zero for the negative energy field, allow a non­ zero value for the mass of the other field. The negative energy field, in the energy density, is replaced by the repulsive adiabatic potential with a retardation effect. The transformed energy density docs not satisfy the usual. equal-time commutator of energy density which is valid for the ordinary system of fields with spin;:--::;:1. The additive potential is introduced so that the total energy density satisfies the usual energy density commutator. The total Hamiltonian is shown to be nothing but the one which gives the post Newtonian equation of Einstein, Infelcl and Hoffmann for the two-body problem in the general relativity. § L Introduction In spite of many approaches to the quantization of the gravitational field in the theory of the general relativity, there are neither necessary reasons for the quantization of the general relativity nor experimental evidence for the ex­ istence of the gravitational wave. On the other hand, since the general rela­ tivity has little to do with the ordinary theory of fields in Minkowski's space­ time, there are attempts to yield the same result as that of the general relativi­ ty standing on the usual theory of fields in Minkowski's space-time. As an example, there has been the approach of Gupta.l) l--Ie introduced a graviton with spin 2 and mass 0 and obtained the Lagrangian density for the graviton coupled with the matter field by the successive approximation method so as to bring about the energy-momentum density of the system on the right-hand side of the equation for the graviton as the source term. However, unless the experi­ mental evidence of the graviton exists, it may be no more than the expansion of the general relativity on Minkowski's space-time, though the equivalence prin­ ciple may be derived from the existence of the spin 2 particle with mass 0 and the Lorentz invariancy of S-matrix as shown by Weinberg. 2 l

Intensive Observables in Quantum Theory

The notion of strictly intensive observables is introduced in a theory of local observables (such as that of R. Haag). If O1, O2 are two disjoint regions, then the value of a strictly intensive observable in O1 ∪ O2 is the sum of its values in O1 and O2. It is shown that energy-momentum can never be strictly intensive. This result is used to prove that the algebras of observables is not of Type I for some regions. By analogy with the energy-momentum tensor density for the free field, the definition is weakened, so that an intensive observable in a region O is only ``approximately'' in a(O). This leads to the introduction of germs of intensive observables. It is proved that the unitary intensive operators form a sheaf 𝔉 of groups, and the Hermitian intensive operators form a sheaf 𝔊 of Abelian groups with operators in 𝔉, and on which the inner derivative Y → i(XY − YX) is defined.

Flat-space metric in general relativity theory

The main points in the formalism involving a flat-space metric tensor γ μν together with the Riemannian metric tensor g μν are summarized and discussed. The advantages of this formalism in distinguishing between an inertial field and a permanent gravitational field and in providing an energy-momentum density tensor for the latter are pointed out. Many different field equations are possible, but an invariance principle is proposed which essentially singles out the Einstein field equations.

Asymptotic Covariant Conservation Laws for Gravitational Radiation

The usual conserved quantities of Lorentz-covariant theories are associated with the descriptors of coordinate transformations in Killing directions. By suitably defining the concept of asymptotic-Killing vector fields, it is possible to extend the definition of the usual conserved quantities to situations in which, properly speaking there are no Killing vector fields, but in which, nevertheless, it is still meaningful to speak of the energy, momentum and perhaps angular momentum radiated by the gravitational field. Among the results obtained ln the course of the investigation are: (a) an understanding of the circumstance which singles out the energy as a positive-definite quantity; (b) a possible global consequence that vanishing energy implies that the space is flat; (c) an understanding of the position of the Fock harmonic coordinate conditions in Trautman's treatment of gravitational radiation; (d) an extension of the validity of the Moller pseudotensor to the Trautman radiative solutions. Also a brief indication is given of how this work might be extended to give a proper treatment of energy, momentum, and angular momentum densities for general metrics. (auth)

Energy-Momentum Tensor for Plane Waves

A general form is established for the energy momentum tensor for plane waves propagating in a homogeneous medium, the field equations of which are derivable from a quadratic Lagrangian function. Energy density and momentum density are proportional to frequency and the wave vector, the coefficient of proportionality being density. Energy flow and momentum flow are related to energy density and momentum density by the group velocity. The relation between momentum density and the wave vector is valid even in a nonlinear system. For a wave packet, one finds that the total energy is related to frequency and the total momentum to the wave vector by the total action of the packet, in close analogy with corresponding relations of quantum mechanics.

On the Quantization of Physical Space-time Operators

§ 1. IntrDduction In the conventional field theory, space-time coordinates are treated as c-number parameters, and field operators (consequently energy-momentum density, current density and such quantities are expressed as the products of field operators also) are represented as their functions. On the other hand, many authorsl) investigated the possibility to regard space-time coordinates as operators, or to quantize them, in order to get rid of the so-called divergence difficulties. However, no promising guiding principle to quantize space-time coordinates has been found. Among many attempts, a series of investigations 2 )-4) which was begun by the famous Dirac's work 5 ) has an exceptional attraction for the present author, although these works were developed regardless of the divergence difficulties. For their quantization of the physical coordinate operators seems to express the uncertainty principle in the measurements, as will be seen in the following discussions. The purpose of this article is to make the physical meaning of the quantization clear and to analyse the nature of the extended Hilbert space. The meaning of the uncertainty principle in quantum mechanics is as follows. As long as the particle, for which the uncertainty relation between its position and momentum holds, is used in the observation of the motion of some particle, the position and the momentum of the object also cannot be measured simultaneously and exactly. If any particle which is free from the uncertainty principle does exist, we would be able to use this particle to make the simultaneous measurement of the position and the momentum of the object, whether they are elementary or not. It may in fact be possible to take the standpoint that the uncertainty relation is derived from such a theory that introduces some hidden variable free from the uncertainty principle, as Bohm

The Self-Stress of the Electron

The diagonal elements of the symmetrical energy-momentum tensor density of an electron at rest are calculated. The covariant formalism of Tomonaga, Schwinger, Feynman, and Dyson is used, and it is shown that it is necessary to use a relativistic cut-off in addition to the covariant separation of the infinite renormalizations. Therefore "formalistic regulators" are used in the form of additional neutral vector fields. The integrations are carried out with the Feynman method. The resultant vanishing value for the self-stress constitutes a proof of the consistency of the relativistic formalism. It is also shown how the Feynman-Dyson method can be used for the calculation of expectation values of operators, of which the self-stress calculation constitutes an example.

The impulse-energy tensor of material particles

The following is a direct and general construction, from the field quantities of the electrons and mesons and their derivatives, of a real, symmetrical and gauge-invariant tensor T kl whose divergence to l is — 1 times the four-force f k experienced by the matter. Interpreting T kl as the impulse-energy tensor of the material particles, for the case of no electromagnetic field the energy-momentum density obtained from T k4 is compared with that obtained from treating (ℏ/ i ) (∂/∂x k ) as the energy-momentum operator. Further, in the general case the Hamiltonian of the matter is compared with ∫ T 44 dxdydz . Both times, the two quantities compared agree apart from small modifications.

Statistical Mechanics of Fields and the 'Apeiron'

IN view of the difficulties encountered in the quantum theory of fields, we have developed a new approach to this subject which bears a much closer resemblance to ordinary quantum mechanics of particles than the existing theories (Heisenberg and Pauli), We consider the field in a finite volume Ω as a mechanical system described by its total energy and momentum. The latter are obtained from the classical expressions for the energy-momentum densities simply by multiplication with Ω. Each field component is considered as an operator (for the whole volume Ω) and is not regarded as a function of the co-ordinates at all. Between the field components we have established very simple commutation laws, analogous to the ordinary law pq—qp = ℏ/i which depend on the transformation character of the field considered. For example, for the scalar meson field (potential v, its gradient f, its time derivative g; * means Hermitean adjoint; [a,b] means ab – ba):

Über freie teilchen mit nichtverschwindender masse und beliebiger spinquantenzahl

The properties are investigated of the solutions of a wave equation for an undor of rank N (Eq. (1)), which is a simple generalization of the Dirac equation for free electrons and by which particles with arbitrary spin and non-vanishing mass might eventually be described. The solutions can be classified in groups satisfying Klein-Gordon equations with (for N > 2) different values of the mass-constant. In the cases N = 1 (Dirac electron) and N = 2 (the meson theory as given by Møller and Rosenfeld and by Belinfante) the mass-constant can take only one single value. Those solutions of lowest mass-constant, which are symmetrical in all undor indices of the wave function, correspond to Dirac's particles of arbitrary spin. From the variational principle, from which the wave equation can be derived, unambiguous expressions are established for the vector of the charge current-density and for the tensor of the energy-momentum density. In the case N > 2, the direct applicability to the description of actual elementary particles is seriously impeded by the circumstances that the expression for the energy obtained in this way is not positive definite. This difficulty cannot be overcome by a “hole theory” as it can for N = 1 (Dirac electron).