Zero-mass Particles Research Articles

Monopoles and instantons from Berry’s phase

The topological phase picked up by an arbitrary spin undergoing a cyclic series of quantum jumps is deduced by geometric considerations. Generalizing the Hamiltonian of a spin in a magnetic field, higher-dimensional Dirac monopoles are derived from the non-Abelian adiabatic phase. These monopoles appear in the momentum space description of zero mass particles in higher dimensions.

Maximal symmetry group of the Hamilton-Jacobi equation: relativistic particle in flat spacetime

Lie's extended group method has been used to obtain the maximal symmetry group of the Hamilton-Jacobi equation for a relativistic particle moving in flat spacetime. For a massive particle it is the 21-parameter inhomogeneous pseudo-orthogonal group IO(4,1) of signature (4,1). For a zero-mass particle like a photon or a neutrino this group is an infinite-parameter Lie group with an infinite-parameter invariant subgroup such that the factor group is isomorphic to the inhomogeneous pseudo-orthogonal group IO(3,1) of signature (3,1).

Classification of gravitational waves in a nonsymmetric gravitational theory

We analyze the polarization properties of weak, plane gravitational waves in theories of gravity containing a nonsymmetric metric. The calculation focuses on the physically observable geodesic-deviation forces that affect gravitational-wave detectors. Considering three different definitions of measurable geodesic deviation, we find that the most general wave may contain up to 12 modes of polarization. For the particular case of the nonsymmetric theory of Moffat, we find that, under two of the three possible definitions of geodesic deviation, the theory admits waves of up to five modes: three transverse, of helicity \ifmmode\pm\else\textpm\fi{}2 and 0, and two longitudinal, of helicity \ifmmode\pm\else\textpm\fi{}1. The presence or absence of the latter modes is found to be Lorentz-frame dependent, meaning that it is impossible to characterize such waves in terms of zero-mass particles of definite helicity. We find that this problem may be overcome, leaving Lorentz-invariant modes of helicity \ifmmode\pm\else\textpm\fi{}2, if the third, ``transposition-invariant,'' definition of measurable geodesic deviation is adopted for this theory. The general formalism developed here may be useful for analyzing other nonsymmetric theories of gravitation.

Machinery for the analysis of field theories with zero-mass particles

We establish a machinery for the analysis of field theory models with zero-mass particles with great generality and in a model independent way. To this end a special class of functions called the class A n -functions, introduced earlier by the author, is quite suitable for such a machinery. We show that Feynman integrands and absolutely convergent Feynman integrals, involving zero-mass particles, belong to such a class as functions of external (and internal) momenta and the non-zero masses of the theory. We show, in particular, how one may find real numbers b r , d r′ , which set 《high-energy》 and 《low-energy》 scales, respectively, with b r >1 and 0< d r′ <1, such that for certain parameters η r′ , λ r′ >0, with η r ⩾ b r and λ r′ ⩽ d r ′, the absolute value of a Feynman integrand may be bounded by introducing suitable high- and low-energy asymptotic coefficients. The analysis provides, in particular, explicit forms for the latter coefficients for the Feynman integrals themselves and hence leads, to powerful results for the study of their asymptotic behaviour.

Galilean-invariant gauge theory.

It is generally characteristic of a field theory with a zero-mass particle that it does not possess a nontrivial Galilean limit. Since all the well-known gauge theories require (at least in the free-field limit) such massless excitations, there are no known examples at this time of Galilean-invariant gauge field theories. However, by making use of a recently formulated gauge theory in two spatial dimensions in which there is no elementary photon, it is shown that there does exist a Galilean theory which incorporates the gauge principle. The general N-particle state for this theory is constructed and subsequently used to obtain the corresponding N-particle Schr\"odinger equation. In the case of two particles the scattering process is considered explicitly, it is being shown that for all partial waves one obtains the same nonzero phase shift.

Elementary proof of ε → +0 limit of renormalised Feynman amplitudes. II. Theories involving zero-mass particles

For pt.I see ibid., vol.16, p.4131 (1983). An elementary proof of the distributional epsilon to +0 limit of renormalised Feynman amplitudes in momentum space, smeared with Schwartz functions and absolutely convergent for epsilon >0, is given for theories involving rigorously zero-mass particles. The proof uses only direct and simple methods and establishes the convergence of such amplitudes in Minkowski space.

On invariant electromagnetic and energy-momentum tensors under Weyl transformations

Invariant electromagnetic and energy-momentum tensors are studied under the Weyl symmetry. Theirkinematical groups are discussed and related to the corresponding study in the Poincare context. Their physical interest is pointed out in connection with zero-mass particle descriptions.

Harmonic analysis on the space-time gauge continuum

The classical Kaluza-Klein unified field theory has previously been extended to unify and geometrize gravitational and gauge fields, through a study of the geometry of a bundle spaceP over space-time. Here, we examine the physical relevance of the Laplace operator on the complex-valued functions onP. The spectrum and eigenspaces are shown (via the Peter-Weyl theorem) to determine the possible masses of any type of particle field. In the Euclidean case, we prove that zero-mass particles necessarily come in infinite families. Also, lower bounds on masses of particles of a given type are obtained in terms of the curvature ofP.

Generalized decoupling theorem in quantum field theory

Previous analyses of the decoupling theorem, in Euclidean space, considered only one mass scale in the theory becoming large and had the stringent constraint of allowing no zero mass particles in the theory. We generalize the decoupling theorem in both respects. We prove the vanishing property of renormalized Feynman amplitudes, with subtractions, when any subset of the masses in the theory become large and, in general, at different rates, thus providing different large mass scales in the theory. This theorem is then extended and we give sufficiency conditions for the validity of the decoupling theorem when any subset of the remaining nonasymptotic masses are scaled to zero and, in general, at different rates. All the subtractions of renormalization are carried out at the origin of momentum space. The proof applies for theories with derivative couplings and with higher spin fields as well.

Massless particles of high spin

The existence of a Lorentz-covariant energy-momentum tensor in a quantum field theory has certain strong implications about the matrix elements of this Lorentz tensor between one-particle states of high-spin ($j\ensuremath{\ge}\frac{3}{2}$) zero-mass particles, composite or elementary. A similar result obtains for theories with a Lorentz-covariant current for zero-mass particles, composite or elementary with spin $j\ensuremath{\ge}1$. If these results are taken together with a continuity requirement on the matrix elements of the energy-momentum tensor and the current, respectively, one ends with a contradiction which can be removed only by assuming that such particles do not exist. We reexamine this conclusion and recognize that theories of higher-spin free particles are consistent but the continuity of the matrix elements do not obtain. The reason is traced to the abrupt change in the sign of the polarization when the momentum transfer changes from spacelike to null values. In the general case with particles of $j\ensuremath{\ge}\frac{3}{2}$ and $j\ensuremath{\ge}1$, respectively, we prove that the divergence of any tensor and any vector vanish between one-particle states.

Necessity of many-particle forces in relativistic Newtonian mechanics for massive and massless particles

The Lorentz-invariance conditions for Newtonian equations of motion for three particles are assumed to be satisfied by sums of two-particle forces that satisfy the Lorentz-invariance conditions for two particles. Then it is shown that a particle can be accelerated only by forces from particles that do not accelerate, provided every particle has positive mass. There are exceptional cases when one or more of the particles has zero mass. Relativistic Newtonian mechanics for zero-mass particles is formulated two different ways. When the equations of motion specify the accelerations as functions of the positions and velocities, the result is the same as for positive-mass particles. When the time derivatives of the momenta are specified as functions of the positions and momenta, the result is that a particle can be accelerated only by forces from particles that do not accelerate continuously. However, there are forces that change the magnitude of the momentum without changing the velocity, for a particle with zero mass. They produce discontinuous acceleration when the velocity abruptly changes direction as the momentum reaches zero and changes sign. For a particle accelerated by a force from a massless particle that accelerates in this discontinuous way, there are two-particle forces with acceleration of both particles.

Low-energy analysis of subtracted Feynman amplitudes

A low-energy asymptotic analysis of subtracted Feynman amplitudes is carried out, in Euclidean space, for theories which on experimental grounds may contain zero-mass particles. Initially all the propagators are defined to have nonzero masses in their denominators. The study is general enough and deals with the asymptotic low-energy behaviour of the amplitudes when some (or all) of the external (non-exceptional) momenta become small, in general, atdifferent rates such that i) the asymptotic momenta are led to approach their asymptotic values atslower rates than those masses in the theory which are led to vanish, also ii) the vanishing masses are led to vanish, in general, atdifferent rates as required on experimental grounds. Rules are given for applications and examples are worked out as illustrations of the rules.

A group-theoretical approach to relativistic eikonal physics

A contraction of the Poincare group is performed leading to the eikonal approximation. Invariants, one-particle states, spinning particles and some interaction problems are studied with the following results: momenta of ultrarelativistic particles behave as lightlike, the little group beingE2, spin behaves as that of zero-mass particles, helicity being conserved in the presence of interactions. The full eikonal results are rederived for Green’s functions, wave functions, etc. The way for computing corrections due to transverse momenta and spin-dependent interactions is outlined. A parallel analysis is made for the infinitemomentum frame, the similarities and differences between this formalism and the eikonal approach being disclosed.

Infrared behavior of zero-mass Green's functions and the absence of quark confinement in perturbation theory

We present a simple proof that all Green's functions for nonexceptional Euclidean momenta in any renormalizable (but not superrenormalizable) field theory containing zero-mass particles are infrared finite. We discuss the relationship of a corollary result, that any physical discontinuity of Green's functions is also infrared finite, to the question of quark confinement in non-Abelian gauge theories.

Induced representations, reproducing kernels and the conformal group

The properties of “induced” (or multiplier) representations of groups which act in Hilbert spaces with a reproducing kernel are investigated. A resume of earlier work is followed by a discussion of criteria for the irreducibility of such representations. The notions of reproducing kernel and positive definite spherical function are found to overlap. As a result, functional equations (analogous to those of Godement for spherical functions) are found for the reproducing kernel. The abstract theory is illustrated by certain discrete series representations of the conformal group and by their “limit points”. In particular the so-called ladder representations (which give rise to the conformal symmetry of zero mass particles) are analysed from this viewpoint.

Study of a two-dimensional model with axial-current-pseudoscalar derivative interaction

A detailed study is made of a massive pseudoscalar field interacting via derivative coupling with massless fermions in two-dimensional space-time. The model provides an example of a soluble renormalizable theory with an anomalous axial-vector current and a zero-mass particle interpretation for the fermion. Except for a finite mass and wavefunction renormalization, the boson remains free in the presence of the interaction. The canonical fermion field exhibits an anomalous dimension that is found to be in agreement with the asymptotic Callan-Symanzik equation. The connection between the Wilson expansion for defining operator products in this model and the Dyson equations of renormalized perturbation theory is discussed, and agreement with second-order perturbation theory is verified by explicit calculation.

A geometrical model for nonhadrons and its implications for hadrons

With the aid of the ideas suggested by the string model, the possibility of describing nonhadrons by a geometrical approach is studied. It is shown how it is possible to construct a Lagrangian which contains a fundamental length, by selecting a class of motions of the string. The quantization of the model gives rise to a spin tower of zeromass particles. The interaction among these states is realized in analogy with the theory of Mandelstam for the interacting strings. The vertices for the lower spin states are studied in some detail. It is shown that in order to get the correct electromagnetic and gravitational interactions the fundamental length √α′ must be of the order of 10−33 cm. This choice of √α′ gives a very great value for the three-scalar coupling constant. This fact allows some speculation on the possibility to interpret these scalars as the constituents of the hadrons.

Stimulated Emission from Nuclei

The feasibility of obtaining stimulated emission of both zero and finite mass particles from nuclei is considered, and the limitations imposed on the relevant nuclear parameters determined.

Generalized Pauli-Villars regularization in the presence of zero-mass particles

Generalized Pauli-Villars regularization in the presence of zero-mass particles

On the Localizability of Massless Particles

Wightman's reformulation of the Wigner-Newton position operators and localizability of a massive particle is generalized so as to include zero mass particles. Different a priori procedures of generalizations are examined: we first prove some negative results based upon a projective induction technique. We then present some positive results which state that in some sense zero-mass particles are localizable and that their corresponding position operators are candidates for observables. Our treatment–which is group theoretical and kinematical in the same sense as that of Wightman–ends with suggestions for further possible developments.