The three-nucleon continuum: achievements, challenges and applications

This paper is devoted to a detailed study of nonrelativistic particles and their properties, as described by Galilei invariant wave equations, in order to obtain a precise distinction between the specifically relativistic properties of elementary quantum mechanical systems and those which are also shared by nonrelativistic systems. After having emphasized that spin, for instance, is not such a specifically relativistic effect, we construct wave equations for nonrelativistic particles with any spin. Our derivation is based upon the theory of representations of the Galilei group, which define nonrelativistic particles. We particularly study the spin 1/2 case where we introduce a four-component wave equation, the nonrelativistic analogue of the Dirac equation. It leads to the conclusion that the spin magnetic moment, with its Lande factorg=2, is not a relativistic property. More generally, nonrelativistic particles seem to possess intrinsic moments with the same values as their relativistic counterparts, but are found to possess no higher electromagnetic multipole moments. Studying “galilean electromagnetism” (i.e. the theory of spin 1 massless particles), we show that only the displacement current is responsible for the breakdown of galilean invariance in Maxwell equations, and we make some comments about such a “nonrelativistic electromagnetism”. Comparing the connection between wave equations and the invariance group in both the relativistic and the nonrelativistic case, we are finally led to some vexing questions about the very concept of wave equations.

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Nonrelativistic particles and wave equations

We ask whether Cohen and Glashow's very special relativity model for Lorentz violation might be modified, perhaps by quantum corrections, possibly producing a curved space-time with a cosmological constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family of continuous deformations, but none of these give rise to noncommutative translations analogous to those of the de Sitter deformation of the Poincar\'e group: space-time remains flat. Only a 1-parameter family ${\mathrm{DISIM}}_{b}(2)$ of deformations of SIM(2) is physically acceptable. Since this could arise through quantum corrections, its implications for tests of Lorentz violations via the Cohen-Glashow proposal should be taken into account. The Lorentz-violating point-particle action invariant under ${\mathrm{DISIM}}_{b}(2)$ is of Finsler type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive ${\mathrm{DISIM}}_{b}(2)$-invariant wave equations for particles of spins 0, $\frac{1}{2}$, and 1. The experimental bound, $|b|l{10}^{\ensuremath{-}26}$, raises the question ``Why is the dimensionless constant $b$ so small in very special relativity?''

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General very special relativity is Finsler geometry

Publisher Summary Galileo Galilei explicitly introduced the principle of relativity in physics. He was the first one to recognize the existence of inertial transformations, connecting various frames of reference in which the laws of physics take the same form. Starting with the recognition that the Galilean transformation laws were but an approximation to the more exact Lorentz formulas, one could distinguish the abstract principle of relativity from its concrete expressions, as various possible theories of relativity. At the same time, one could begin to see the basic role played by the relevant theory of relativity in structuring a given physical theory—Galilean relativity for classical mechanics, Einstein relativity for “relativistic” mechanics, and electromagnetism. Applications of the Galilei group are described for classical physics and for quantum physics. A “super-Galilei group” has been defined in relation to non-relativistic cosmology. This is a sixteen-parameter group acting on the five-dimensional manifold of space-time and gravitational potential.

Galilei Group and Galilean Invariance

After a pedagogical review of elementary cosmology, I go on to discuss some obstacles to obtaining inflationary or accelerating universes in M/string theory. In particular, I give an account of an old 'no-go theorem' to this effect. I then describe some recent ideas about the possible role of the tachyon in cosmology. I stress that there are many objections to a naive inflationary model based on the tachyon, but there remains the possibility that the tachyon was important in a possible pre-inflationary 'open string era' preceding our present 'closed string era'.

http://iopscience.iop.org/article/10.1088/0264-9381/20/12/301/pdf

Thoughts on tachyon cosmology

This paper deals with the theory of deformation of Lie algebras. A connection is established with the usual contraction theory, which leads to some ``more singular'' contractions. As a consequence it is shown that the only groups which can be contracted in the Poincare group are SO(4, 1) and SO(3, 2).

Deformation and Contraction of Lie Algebras

We introduce here a new coupling scheme for three angular momenta. It relies on the properties of an operator which depends ``democratically'' upon the three individual angular momenta; this is in fact their mixed product. This operator, the total angular momentum and one of its components form a complete set of commuting observables. In the case where the three individual angular momenta are equal, the eigenstates of this set possess remarkable symmetry properties with respect to the permutation group S3.

Symmetrical Coupling of Three Angular Momenta

The kinematics of a nonrelativistic three‐particle system is studied with the help of the general method devised by Levy‐Leblond and Lurcat. Basis states are constructed which are eigenstates, in addition to the total momentum‐energy, angular momentum, etc., of new observables; among these, the ``togetherness tensor'' describes the simultaneous localization of the three particles and therefore is of great physical interest. All of these observables arise as Casimir operators of a ``great group'' acting on the three‐particle phase‐space manifold in a transitive way, and of some of its subgroups. In the present case, by trying to keep all the particles on the same footing (``democracy'' arguments), we are led to choose the SU3 group as a particularly convenient ``great group''. We thus recover completely the Dragt classification of non‐relativistic three‐particle states. The explicit calculation of the basis functions is done in a new way, by analytical methods, solving partial derivative equations. This en...

Three‐Particle Nonrelativistic Kinematics and Phase Space

We study the finite‐dimensional representations of non‐semisimple Lie algebras. We give some general properties, and apply them to the case of the motion group and of the inhomogeneous Lorentz group.

Monique Lévy‐Nahas

Papers

Deformation and Contraction of Lie Algebras

Symmetrical Coupling of Three Angular Momenta

Three‐Particle Nonrelativistic Kinematics and Phase Space

Finite‐Dimensional Representations of Some Non‐Semisimple Lie Algebras