Group Lorentz

We shall only mention the fact, that a unitary representation of the inhomogeneous proper Lorentz group is exhibited in this Hilbert space through the following identification of the generators of the... 

The requirement of Postulate 3 that the equations of motion be form-invariant (i.e., that fix ) and A u(x ) satisfy the same equation of motion with respect to a as did fx) and Au x) with respect to x demands that the field variables transform under such transformations according to a finite dimensional representation of the Lorentz group. In other words it demands that transform like a spinor... 

These are the structure relations for the Poincar4 group and are consequences of the multiplication law (11-171) and (11-172) for the inhomogeneous Lorentz group. In terms of the three vectors... 

Lorentz approximation, 46 Lorentz condition, 551 Lorentz gauge, 657,664 Lorentz group homogeneous, 490... 

Lorentz group, inhomogeneous proper, unitary representation in Hilbert space, 497... 

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. 

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 history of the study of symmetry properties of Eq. (3) goes back to the beginning of the twentieth century. Invariance properties of Maxwell equations have been studied by Lorentz [40] and Poincare [41,42]. They have proved that Eq. (3) are invariant with respect to the transformation group named by the Poincare suggestion the Lorentz group. Furthermore, Larmor [43] and Rainich [44] have found that equations (3) are invariant with respect the singleparameter transformation group... 

As discussed in the earlier part of this review, Eq. (511) is an identity between generators of the Poincare group, which differs from the Lorentz group because the former contains the generator of spacetime translations... 

J. P. Vigier, and P. Hillion, Elementary particle waves and irreductible representations of the Lorentz group, Nucl. Phys. 16, 361 (1960). 

The formalism has 8 components and is therefore an irreducible representation of the Lorentz group. This implies that... 

M. Gel Fand, R. Minlos, Z. Shapiro, Representation of the rotations and Lorentz groups (Pergamon Oxford, 1963)... 

More explicit representations for M are described below. Here we note that the matrices M are, in general, not unitary. The Lorentz group is non-compact and hence has no finite-dimensional unitary representations. An exception is the subgroup of rotations, which is compact, and matrices M representing rotations are indeed unitary. 

As the boosts constitute the non-compact part of the Lorentz group, the corresponding matrices M = " which act on the spinor-components are... 

Levy-Leblond [16] has realized that not only the Lorentz group (or rather the homomorphic group SL(2) [32, 7]), but also the Galilei group has spinor-field representations. While the simplest possible spinor field with s = I and m 0 in the Lorentz framework is described by the Dirac equation, the corresponding field in a Galilei-invariant theory satisfies the Levy-Leblond equation (LLE)... 

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