Relativistic Invariance

The relativistic invariance of the scalar product is also made explicit by Eq. (9-93) since as defined by Eq. (9-92) is a scalar, (k-x is an invariant and d 3k/k0 is the invariant measure element over the hyper-boloid k2 = m2). 

The condition that serves to determine 8(A) is the requirement of relativistic invariance f must satisfy the Dirac equation in the new coordinate system i.e., f(x ) must satisfy the equation... 

This together with the requirements of relativistic invariance (recall discussion in Section 10.1) implies that... 

Consider next the relativistic invariance of quantum electrodynamics. Again, loosely speaking, we say that quantum electrodynamics is relativistically invariant if its observable consequences are the same in all frames connected by an inhomogeneous Lorentz transformation a,A ... 

The previous results become somewhat more transparent when consideration is given to the manner in which matrix elements transform under Lorentz transformations. The matrix elements are c numbers and express the results of measurements. Since relativistic invariance is a statement concerning the observable consequences of the theory, it is perhaps more natural to state the requirements of invariance as a requirement that matrix elements transform properly. If Au(x) is a vector field, call... 

In the present section we shall make this difficulfy apparent in a somewhat different way by showing that it is not possible to satisfy the asymptotic condition when the theory is formulated in terms of an unsubtracted hamiltonian of the form jltAll(x) — JS0JV. We shall work in the Lorentz gauge, where the relativistic invariance of the theory is more obvious. 

Similar remarks apply to the negaton-positon field operator. From spectral assumptions mid relativistic invariance we have previously concluded that... 

Stated differently, relativistic invariance (including space and time inversions) automatically insures that Eq. (11-556) is satisfied. We... 

Rate of change of observables, 477 Ray in Hilbert space, 427 Rayleigh quotient, 69 Reduction from functional to algebraic form, 97 Regula fold method, 80 Reifien, B., 212 Relative motion of particles, 4 Relative velocity coordinate system and gas coordinate system, 10 Relativistic invariance of quantum electrodynamics, 669 Relativistic particle relation between energy and momentum, 496 Relativistic quantum mechanics, 484 Relaxation interval, 385 method of, 62 oscillations, 383 asymptotic theory, 388 discontinuous theory, 385 Reliability, 284... 

The relativistic invariance of the electromagnetic field is conveniently expressed in tensor notation. Factorized in Minkowski space the Maxwell equa-... 

In consequence, the statistical characteristic temperature of relic radiation is fully determined in terms of relativistic invariant spectrum of the cosmic microwave background radiation and the distribution velocity function of radiating particles, i.e., is described with the following expression (compare with the results of reference (Einstein, 1965))... 

Methods for treating relativistic effects in molecular quantum mechanics have always seemed to me, if I may say so without appearing too impertinent to those who work in the field, a complete dog s breakfast. The difficulty is to know to what question they are supposed to be the answer, in the circumstances in which we find ourselves. We do not know what a relativistically invariant theory applicable to molecular behaviour might look like. As was pointed out to us at the last meeting, the Dirac equation certainly will not do to describe interacting electrons and even at the single particle level, where it seems to work, there is an inconsistency in interpreting its solutions in terms... 

The principal aim of the present chapter is twofold. First, we will review the already known ideas, methods, and results centered around the solution techniques that are based on the symmetry reduction method for the Yang-Mills equations (1) in Minkowski space. Second, we will describe the general reduction routine, developed by us in the 1990s, which enables the unified treatment of both the classical and nonclassical symmetry reduction approaches for an arbitrary relativistically invariant system of partial differential equations. As a byproduct, this approach yields exhaustive solution of the problem of... 

The present review is based mainly on our publications [33,35-39,49-53]. In Section II we give a detailed description of the general reduction routine for an arbitrary relativistically invariant systems of partial differential equations. The results of Section II are used in Section III to solve the problem of symmetry reduction of Yang-Mills equations (1) by subgroups of the Poincare group P 1,3) and to construct their exact (non-Abelian) solutions. In Section IV we review the techniques for nonclassical reductions of the STJ 2) Yang-Mills equations, which are based on their conditional symmetry. These techniques enable us to obtain the principally new classes of exact solutions of (1), which are not derivable within the framework of the standard symmetry reduction technique. In Section V we give an overview of the known invariant solutions of the Maxwell equations and construct multiparameter families of new ones. 

This form has both potentials propagating away from the source at speed c and is relativistically invariant, and for this reason is often preferred. 

In formulating QED a least-action principle involving a Lagrangian is often used [9,18,20]. This involves the potentials in various forms. Not only is relativistic invariance (Lorenz potentials) desired, but also gauge invariance. At least in the current state of QED, gauge invariance is included as a fundamental part [21,22]. 

In going from the Schrodinger equation to the Klein-Gordon equation, we obtain the necessary symmetry between space and time by having second-order derivatives throughout. It is usually written in a form that brings out its relativistic invariance by using what is called four-vector notation. We define a four-vector x to have components... 

Heitler spoke on the quantum theory of damping, which is a heuristic attempt to eliminate the infinities of quantum field theory in a relativistic invariant manner, Peierls spoke of the problem of self-energy, and Op-penheimer gave an account of the developments of the last years in electrodynamics in which he discussed the problem of the vacuum polarization and charge renormalization with special reference to the recent work of Schwinger and Tomonaga. 

The prediction, and subsequent discovery, of the existence of the positron, e+, constitutes one of the great successes of the theory of relativistic quantum mechanics and of twentieth century physics. When Dirac (1930) developed his theory of the electron, he realized that the negative energy solutions of the relativistically invariant wave equation, in which the total energy E of a particle with rest mass m is related to its linear momentum V by... 

Given the electromagnetic 4-vector field A/( = (A, i(p) and the 4-velocity m/( = (yv, iyc), W is the classical limit of a relativistic invariant y W = A u. This term augments the tree-particle relativistic Lagrangian to give... 

Waves in the moving cavity therefore lock together in a relativistically invariant way. 

In order to reconcile this relativistic invariance with De Broglie s wavelength hypothesis, for waves at rest and in a moving frame,... 

The mysterious phase velocity of the de Broglie wave and the group velocity of the amplitude wave, c2/ > c, refer to the, by now familiar superluminal motion in the interior of the electron. As many authors noted and Molski(1998) recently reviewed [86] an attractive mechanism for construction of dispersion-free wave packets is provided in terms of a free bradyon4 and a free tachyon that trap each other in a relativistically invariant way. It is demonstrated in particular how an electromagnetic spherical cavity may be... 

The bradyon-tachyon coupling was first developed by Corben [87], in terms of trapped masses rather than standing waves. The two objects trap each other in a relativistically invariant way yielding a compound particle of rest mass Ma = yrn% —jjl%, in which m0 and /z0 are respectively the masses of a bradyon and a tachyon. 

The further developments of quantum mechanics, including the discussion of maximal measurements consisting not of the accurate determination of the values of a minimum number of independent dynamical functions but of the approximate measurement of a larger number, the use of the theory of groups, the formulation of a relativistically invariant theory, the quantization of the electromagnetic field, etc., are beyond the scope of this book. 

From 7q = I4 we see immediately that V = cjoVcov- The equation (83) is called the Dirac equation in covariant form. It is best suited for investigations concerning relativistic invariance, because it me is a scalar (which by definition of a scalar is invariant under Lorentz transformations) and the term (7,5) is written in the form of a Minkowski scalar product (if 7 and d were ordinary vectors in Minkowski space, the invariance of this term would be already guaranteed by (81). 

On the other hand, the relativistically invariant scattering amplitude F can be related to the scattering amplitude Tg, which is not Lorentz invariant, according to... 

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