Poincare group

In this section, we extend consideration from the Lorentz to the Poincare group within the structure of 0(3) electrodynamics, by introducing the generator of spacetime translations along the axis of propagation in the normalized (unit 12-vector) form ... 

The unit 12-vector acts essentially as a normalized spacetime translation on the classical level. The concept of spacetime translation operator was introduced by Wigner, thus extending [100] the Lorentz group to the Poincare group. The PL vector is essential for a self-consistent description of particle spin. 

In Eq. (715), GM is dual to the third rank GCTpsv in four dimensions and normal to it with the same magnitude. In the received view, there is nothing normal to the purely transverse GCTp on the U(l) level, and therefore cannot be consistently dual with GCTpsv. This result is inconsistent with the four-dimensional algebra of the Poincare group. If we adopt the notation Gv — Bv, we obtain... 

The PL vector was originally constructed for particles from the generators of the Poincare group. The PL vector corresponding to the photon s angular momentum corresponds in free space and in = 1 units to... 

In the Poincare group, therefore, the fundamental spin of the electromagnetic field is represented ineluctably by the PL vector ... 

The Lie algebra of the PL vector within the Poincare group is not well known and is given here for convenience. The PL vector is defined by... 

In electromagnetic theory, we replace W 1 by GM the relativistic helicity of the field. Therefore, Eq. (770) forms a fundamental Lie algebra of classical electrodynamics within the Poincare group. From first principles of the Lie algebra of the Poincare group, the field B is nonzero. 

The first of these equations is an equation of the cyclic theorem, which therefore emerges from the symmetry of the Poincare group in free space. Similarly, Eq. (772c) gives ... 

The structure of the 0(3) equations in condensed form [i.e., Eqs. (612)] emerges from the symmetry of the Poincare group. Consider, for example, the three equations ... 

Consideration of the symmetry of the Poincare group also shows that the cyclic theorem is independent of Lorentz boosts in any direction, and also reveals the physical meaning of the E(2) little group of Wigner. This group is unphysical for a photon without mass, but is physical for a photon with mass. This proves that Poincare symmetry leads to a photon with identically nonzero mass. The proof is as follows. Consider in the particle interpretation the PL vector... 

We have just seen that the symmetry of the Poincare group leads to vacuum charge and current as proposed by Panofsky and Phillips [86], Lehnert and Roy... 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

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. 

Thus, to completely solve the problem of symmetry reduction within the framework of the formulated algorithm above, we need to be able to perform steps 3-5 listed above. However, solving these problems for a system of partial differential equations requires enormous amount of computations moreover, these computations cannot be fully automatized with the aid of symbolic computation routines. On the other hand, it is possible to simplify drastically the computations, if one notes that for the majority of physically important realizations of the Euclid, Galileo, and Poincare groups and their extensions, the corresponding invariant solutions admit linear representation. It was this very idea that enabled us to construct broad classes of invariant solutions of a number of nonlinear spinor equations [31-33]. 

In the previous section we gave a complete list of P( 1,3)-inequivalent ansatzes for the Yang-Mills held, which are invariant under the three-parameter subgroups of the Poincare group P(l, 3). These ansatzes can be represented in the unified form (53), where Bv(co) are new unknown vector functions, a> - co(x) is the new independent variable, and the functions (x) are given by (54). 

Consequently, to describe all the ansatzes of the form (53),(54) reducing the Yang-Mills equations to a system of ordinary differential equations, one has to construct the general solution of the overdetermined system of partial differential equations (54),(86). Let us emphasize that system (54),(86) is compatible since the ansatzes for the Yang-Mills field ( ) invariant under the three-parameter subgroups of the Poincare group satisfy equations (54),(86) with some specific choice of the functions F, F2, , 7Mv, [35]. 

W. I. Fushchych, L. F. Barannik, and A. F. Barannik, Subgroup Analysis of the Galilei and Poincare Groups and Reduction of Nonlinear Equations, Naukova Dumka, Kiev, 1991 (in Russian). 

Representations of the Poincare group and their relation to mass and spin. 

In this section, we suggest a resolution of this > 70-year-old paradox using 0(3) electrodynamics [44]. The new method is based on the use of covariant derivatives combined with the first Casimir invariant of the Poincare group. The latter is usually written in operator notation [42,46] as the invariant P P 1, where P1 is the generator of spacetime translation ... 

Therefore Eq. (18) has been shown to be an invariant of the Poincare group, Eq. (12), and a product of two Poincare covariant derivatives. In momentum space, this operator is equivalent to the Einstein equation under any condition. The conclusion is reached that the factor g is nonzero in the vacuum. 

The field equations of electrodynamics for any gauge group are obtained from the Jacobi identity of Poincare group generators [42,46] ... 

On the 0(3) level, the clearest insight into the meaning of the Jacobi identity (37) is obtained by writing the covariant derivative in terms of translation (P) and rotation (J) generators of the Poincare group ... 

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