Massless Particles and Fields ”

Free fields of massless particles transforming covariantly under the Poincare group are constructed. The allowed infinite and finite dimensional representations of the Lorentz group are obtained. The wave functions are calculated in these representations in various bases u The commutation rules are computed, and turn out to be non-local for any infinite dimens ional fields. The transformation law of a certain irreducible infinite dimensional represen+&tion is shown to coincide, for its lowest spin component., with the usual, radiation gauge, vector potential transforma tion law, as already discovered by Bender. * -Work supported by the U. S. Atomic Energy Commission. >:: +,c Cm lea\-e from the Weizmann Institute, Rchovoth, Israel. *** On leave from Service de Physique Theorique, Saclay, France, I. INTRODUC+ION This paper is devoted to a general treatment of free zero-mass fields, transforming covariantly under the Poincare’ group. The requirement of covariante is shown to impose restrictions on the transformation law for a free massless field. For a field transforming according to an allowed representation we construct the wave functions in various bases and study their properties. We also compute the explicit expression for a commutator or anti-commutator of two fields. It follows that locality can be obta.ined only in the finite dimensional case, and here only with the usual connection between spin and statistics. It is _ also demonstrated that in the spherical jabasis, the j f 1 components of the field in momentum space can be expressed in terms of the jth with coefficients linear in the components of the unit vector $ = ii z along the three momentum F. This imIPI plies that the result of a Lorentz transformatiom on a j component can be expressed in terms of the jth components itself. In particular, an infinitesimal Lorentz transformation can be so expressed, with coefficients linear in I;,. As a special case, the transformation laws of the j =l components for helicity + 1 fields in special representations turn out to be those of the free electromsgnetic vector potential in the radiation gauge. This result was obtained, using a somewhat less direct method, by Bender. 1 .~ As is shown in this paper, a free massless field can be incorporated in irreducible representations of the Lorentz group for which the lowest spin equals the absolute value of the helicity. This is no more true when interactions are introduced. A study of the electromagnetic potentials in the radiation gauge shows that a direct sum of a finite number of irreducible representations is not sufficient to describe the transformation law of these potentials. These facts and a study of the interaction case deserve further attention, -fIt was shown by Weinberg& that the requirement of covariance singles out, among the finite dimensional representations of the Lorentz group, those for which the minimal spin equals to the helicity X of the considered massless particle. The sign of A is determined by the representation, In the representation [ 1 ja’jb $ with l/4 (?-iz)2 = ja (ja + 1) and l/4 (T-iiK’)2 = jb (jb + 1) candzare the generators of rotations and Lorentz transformations) only helicity A = j,-j, can be incorporated. We show that this result is general, and applies to infinite representations as well. The allowed representations are those for which the lowest spin equals the absolute value of the h&city. In section II we summarize the properties of physical states for massless particles and establish our notation. In section IIT we discuss the allowed representations for free massless fields and the appropriate wave functions in these representations. We also show there that,starting from a massive field and letting the mass go to zero, the only non-vanishing terms are those for which the absolute value ~of the helicity equals the minimal spin, as expected. We compute, in the same section, the various components and recursion relations (mentioned above) among them, in the jcrbasis and in a Cartesian basis. Finally we give expressions for the irreducible massless fields and the relations among their various components which correspond to the relations found for the wave functions. In section IV we express the Lorentz. transformed lowest spin component in terms of the various components of the same spin, and discover, for h = f 1 , the connection with clectroma@etism mentioned above. In section V ne discuss the commutation relations among the various massless fields. The computations of the wave functions are performed in two ways. One uses generators and their matrix elements, and the other global methods. The first is summarized in appendix A, and the second in aypendix B. The reader