Zero Mass Case Research Articles

Calculation of dimensionally regularized box graphs in the zero mass case

We evaluate the non trivial one loop graphs for massless particles which are relevant for the QCD description ofe+e− → three jets. Especially we present a method which allows a quick calculation of the imaginary part of box graphs.

Remarks on some nonunitary representations of the Poincaré group

We analyse a class of reducible but indecomposable representations of the Poincare group, characterized by real masses and momenta. The nonunitary character of these representations is reflected in the fact that the momentum eigenvectors no longer form a generalized basis for the representation space. A formalism is set up, which, through the introduction of a basis consisting of eigenvectors and eigenghosts of the momentum operators, allows us to analyse and discuss these representations in a way very close to the standard. We then apply this formalism to the discussion of field theories defined by weak equations of motion of the form (□+m2) n ,m2⩾0,n⩾1, and obtain the most general bilinear forms (commutators, two-point functions, etc.) consistent with the representation. Some peculiarities of the zero-mass case are also exhibited.

On the Poincaré noninvariance of a recent alternative to the Dirac equation

Explicit construction of the infinitesimal generators of the Poincare group, demonstrating that the algebra of commutators closes only for the case of zero mass. Hence, the so-called Stigma equation proposed by Biedenharn et al. (1971) as an alternative to the Dirac equation (1928) for spin one-half, finite mass particles is not Poincare invariant except when the leptonic mass is zero.

Lorentz-Invariant Localization for Elementary Systems. III. Zero-Mass Systems

The present series of papers is devoted to the localization problem in relativistic quantum mechanics. The consequences of imposing the condition that the descriptions of a localized state seen from two different frames of reference should be physically consistent were investigated in Paper II. Only instantaneous localization was studied. Considering elementary systems of nonzero mass and spin 0 and \textonehalf{}, the following results were found. (a) As regards the localization problem, one cannot avoid accepting at least one of the following strong departures from the usual ideas: (i) Position has no meaning; (ii) it violates the physical equivalence of inertial frames; (iii) it is the only quantum variable which cannot be represented by an operator; (iv) it is non-Hermitian; (v) some unusual interaction effects do not disappear when the interaction is switched off. (b) If a component of position is to be precisely measurable as a point, then for spin 0 there is only one possible position operator; for spin \textonehalf{} the operator is unique up to a parameter; in both cases the operators are non-Hermitian with respect to the relativistically invariant scalar product, but in spite of this the eigenvalues of all components are real. The components of position are compatible with each other for spin 0 and incompatible for spin \textonehalf{}. The communication relations of position with linear momentum are the standard ones. The velocity operator, which is Hermitian, has the expected form. Several authors have stated that the localization problem can have a solution for mass greater than zero, but for the zero-mass case a solution does not exist. In this paper the localization problem is studied for zero-mass elementary systems by only imposing, as in Paper II, the above requirement of physical consistency. We consider systems with spin 0, \textonehalf{}, and 1. We prove that the consequences are essentially the same as those obtained for systems of mass greater than zero. The parameter is no longer arbitrary.

Momentum-Space Behavior of Integrals in Nonpolynomial Lagrangian Theories

Methods are developed for constructing momentum-space amplitudes corresponding to nonpolynomial nonderivative interactions of a real scalar field. The methods give rise to a supergraph technique and rules for writing down matrix elements very similar to Feynman techniques. The methods are not established rigorously; at several points the argument requires certain analytic properties of Feynman integrands which, though plausible, can only be demonstrated rigorously for the zero-mass case. Asymptotic behavior, both in spacelike and timelike directions, is discussed. Rough arguments are given that indicate that the singularity structure of the amplitudes is likely to be consistent with unitarity.

The Poincaré group and the Λ-representation

A basis where one has tried to take into account the special azimuthal symmetry in scattering experiments, and which is of importance in crossing symmetry, is defined. The basis is chosen such that the energy, the third components of the momentum and the angular momentum are diagonal together with the helicity. The Poincaré algebra is represented in this basis. A particular spin algebra, defined in a previous article is utilized in the construction of the representation. The discrete elements of the Poincaré group are also represented. The basis is very well suited for discussing the singularities which come from the definition of the helicity itself. The connection between our basis and that defined by Jacob and Wick is given. The zero-mass case is discussed and it is proven that in our representation the discrete case of zero mass is identical with the limit of the continuous positive-mass case.

Analytic Functionals in Quantum Field Theory

A class of Lorentz invariant generalized functions can be defined as analytic functionals, i.e., as continuous linear functionals which are contour integrals over a suitable space of test functions. These generalized functions include in particular the invariant functions of quantum field theory, but also include ``propagators with higher order poles.'' The analysis shows exactly which of these are well defined, especially in the important special case of zero mass. Various applications to quantum field theory are indicated.

Integration of the infinitesimal generators of the inhomogeneous Lorentz group and application to the transformation of the wave function

We integrate the infinitesimal generators of the proper, orthochronous, inhomogeneous Lorentz group for the nonzero mass case when the infinitesimal generators are given in the Foldy-Shirokov form. We also integrate the infinitesimal generators for the zero mass case when they are given in the Lomont-Moses form. The results are essentially the same as those given by Ritus, but we use a notation which we believe is more convenient for our purposes. The formulas of the present paper will be used to obtain properties of generalized spherical harmonics and to obtain the relativistic quantization of the electromagnetic vector potential in an angular momentum basis in papers which we shall write shortly.

On Some Unitary Representations of the Galilei Group I. Irreducible Representations

The true irreducible unitary representations of central extensions GM of the Galilei universal covering group G and hence the physical representations of G are constructed by Mackey's method of induced representations. The elements of the representation space ℋ are obtained from functions defined on GM and restricted to their values at one representative of each left coset of GM modulo K where K is the induction subgroup. The physical interpretation of these functions is in terms of wave functions and comes from the definition of a basis in ℋ. This interpretation depends on the choice of a fundamental frame of reference in space-time and on the physical meaning given to a fundamental state. To a change of the representatives corresponds a change of basis in ℋ. By a suitable choice of these representatives, we obtain in particular the momentum-spin representation and the momentum-helicity representation. The zero mass case named class II by Inönü and Wigner is then obtained by the limit process M → 0 applied to the helicity representation.

Galilei Group and Nonrelativistic Quantum Mechanics

This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.

Wave Equations for Scalar and Vector Particles in Gravitational Fields

An earlier representation of the Kemmer wave equation in Riemann space is modified so as to remove the matrices, which in general have to be added to the differential operators of the equation. It is pointed out that the possibility of this removal is equivalent to the fact, known from Maxwell's equations, that if only scalars, vectors, and antisymmetrical tensors are involved, the field equations can be written with ordinary derivatives without explicit use of the affine connection. The wave equation is written in component form, and the photon zero mass case is obtained by means of the most general matrix mass term, without any questionable limiting process.

Covariant description of polarization

A description is presented of a particle of mass m and spin j with a momentam p. A density matrix is used to describe a system of pure states and mixtures of polarization states. In the zero mass case only two polarization states are available and these have opposite helicities. An application of the Dirac theory was used to explicitly construct the density matrix of the pure polarization state. Any field theoretical computation of polarization effects can be made covariantly and reduces to trace computation; it is possible to describe the computation in terms of two by two density matrices. Application is made to the Bargmann-Wigner theory of particles of arbitrary spin in the polarization states.

Unitary Irreducible Representations of the Lorentz Group

The unitary irreducible representations of the inhomogeneous, proper Lorentz group are determined, using a prescription given by Wigner, with special emphasis on the case of zero rest mass. The principal results are: (a) the construction of one-component representations for the case of zero mass and discrete spin; (b) the existence of a Foldy-Wouthuysen transformation for zero mass and spin \textonehalf{}; (c) the construction of "position operators" for zero mass and spins \textonehalf{}, 1; (d) the complete synthesis of the Dirac, Majorana, and Maxwell one-particle theories.