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Journal of Mathematical Physics

Space−time symmetries and nonunitary representations of the Poincaré semigroup: Description of unstable particles

M. Comi, L. Lanz, Luigi A. Lugiato, G. Ramella

Published Apr 1, 1975• 7 citations
Publisher

Abstract

In this paper we justify the use of nonunitary representations of the space−time symmetry group to describe physical systems, as, e.g., unstable particles. It is shown that such a generalization of the standard theory of symmetries is possible inside the approach to quantum mechanics recently developed by Ludwig. The Poincaré semigroup turns out to be the fundamental symmetry group and only Lorentz transformations must be unitary.

DOI
10.1063/1.522597

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In this paper we justify the use of nonunitary representations of the space−time symmetry group to describe physical systems, as, e.g., unstable particles. It is shown that such a generalization of the standard theory of symmetries is possible inside the approach to quantum mechanics recently developed by Ludwig. The Poincaré semigroup turns out to be the fundamental symmetry group and only Lorentz transformations must be unitary.
This article shows that one can consistently incorporate nonunitary representations of at least one group into the ``ordinary'' nonrelativistic quantum mechanics. This group turns out to be Lorentz group thus giving us an alternative approach to QFT for combining the quantum mechanics and special theory of relativity which keeps the concept of wave function (belonging to some representation of Lorentz group) through the whole theory. Scalar product has been redefined to take into the account the nonunitarity of representations of Lorentz group. Understanding parity symmetry turns out to be the key ingredient throughout the process. Instead of trying to guess an equation or a set of equations for some wave functions or fields (or equivalently trying to guess a Lagrangian for the same), one derives them based only on the superposition principle and properties of wave functions under Lorentz transformations and parity. The resulting model has striking similarities with the standard quantum field theory and yet has no negative energy states, no zitterbewegung effects, symmetric energy momentum tensor and angular momentum density tensor for \emph{all} representations of Lorentz group (unifying the theoretical description of all particles), as well as clear physical interpretation. It also offers a possible interpretation why particles and antiparticles have opposite quantum numbers.
In this talk we discuss how to combine aspects of the theory of stable relativistic particles and of the theory of non-relativistic decaying particles into a relativistic theory of unstable particles and resonances. For the theory of stable relativistic particles we use Wigner’s description by unitary irreducible representations (UIR) of the Poincaré group [ 1 ] which has been the foundation for relativistic quantum mechanics and field theory [ 2 ]. For the non-relativistic decaying particles we use a time asymmetric quantum mechanics in which the Hilbert space axiom, that allows only symmetric (unitary group) time evolution, is replaced by time asymmetric boundary conditions which lead to an irreversible , semigroup time evolution [ 3 ]. In the limit in which certain contributions to the energy continuum are neglected, this new quantum mechanics leads to the Weisskopf-Wigner approximate methods [ 4 ] for decay, and to the Breit-Wigner energy distribution [ 5 ] for the resonance lineshape. We thus generalize and combine two fundamental contributions of Wigner into a description of relativistic resonances and decaying states. The central concept of this description is the relativistic Gamow vector, which is a generalized eigenvector (ket) of the self-adjoint invariant square mass operator https://www.w3.org/1998/Math/MathML" display="inline"> M 2 = P μ P μ P μ = P 1 μ + P 2 μ in resonance scattering 1±2→ R →3±4) with a complex eigenvalue, a relativistic Breit-Wigner energy distribution, and an exponential time evolution governed by a semigroup of continuous operators in the space of kets.
The main subject of the paper is the description of unstable states in quantum mechanics and quantum field theory. Unstable states in quantum field theory can only be introduced as the intermediate states and not as asymptotic states. The absence of the intermediate unstable states from the asymptotic states is compatible with unitarity. Thus the concept of an unstable state is not introduced in quantum field theory despite the fact that an unstable state has well defined linear momentum, angular momentum and other intrinsic quantum numbers. In the rigged Hilbert space quantum mechanics one can define vectors that correspond to the unstable states. These vectors are the generalized eigenvectors (kets in the rigged Hilbert space) with complex eigenvalues of the self-adjoint Hamiltonian. The real part of the eigenvalue corresponds to the mass of an unstable state and the imaginary part is one half of the total width. Such vectors form the minimally complex semigroup representation of the Poincaré transformations into the forward light cone.
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 G5{\mathcal{G}}_{5}G5​ is suggested as the symmetry group of dynamics. G5{\mathcal{G}}_{5}G5​ 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 G5{\mathcal{G}}_{5}G5​ 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.
A Lorentz transformation group SO(m, n) of signature (m, n), m, n ∈ N, in m time and n space dimensions, is the group of pseudo-rotations of a pseudo-Euclidean space of signature (m, n). Accordingly, the Lorentz group SO(1, 3) is the common Lorentz transformation group from which special relativity theory stems. It is widely acknowledged that special relativity and quantum theories are at odds. In particular, it is known that entangled particles involve Lorentz symmetry violation. We, therefore, review studies that led to the discovery that the Lorentz group SO(m, n) forms the symmetry group by which a multi-particle system of m entangled n-dimensional particles can be understood in an extended sense of relativistic settings. Consequently, we enrich special relativity by incorporating the Lorentz transformation groups of signature (m, 3) for all m ≥ 2. The resulting enriched special relativity provides the common symmetry group SO(1, 3) of the (1 + 3)-dimensional spacetime of individual particles, along with the symmetry group SO(m, 3) of the (m + 3)-dimensional spacetime of multi-particle systems of m entangled 3-dimensional particles, for all m ≥ 2. A unified parametrization of the Lorentz groups SO(m, n) for all m, n ∈ N, shakes down the underlying matrix algebra into elegant and transparent results. The special case when (m, n) = (1, 3) is supported experimentally by special relativity. It is hoped that this review article will stimulate the search for experimental support when (m, n) = (m, 3) for all m ≥ 2.