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.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.

Journal of Mathematical PhysicsJun 1, 1963241 citations

Nonrelativistic particles with spin

Springer eBooksApr 6, 20080 citations

Galilean Symmetry of Non-Relativistic Quantum Mechanics

Domenico Giulini

Springer eBooksJan 1, 20030 citations

New Dynamical Group for the Relativistic Quantum Mechanics of Elementary Particles

J. J. Aghassi, P. Román, Ruggero María Santilli

Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group ${\mathcal{G}}_{5}$ is suggested as the symmetry group of dynamics. ${\mathcal{G}}_{5}$ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of ${\mathcal{G}}_{5}$ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group ${\mathcal{G}}_{5}$ is suggested as the symmetry group of dynamics. ${\mathcal{G}}_{5}$ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of ${\mathcal{G}}_{5}$ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMay 15, 197043 citations

Representations of the Galilei Group

Erdal İnönü, E. P. Wigner

Springer eBooksJan 1, 19930 citations

{\mathcal{G}}_{5}

is suggested as the symmetry group of dynamics.

{\mathcal{G}}_{5}

contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of

{\mathcal{G}}_{5}

by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.