Model Tests of Cluster Separability In Relativistic Quantum Mechanics

B. D. Keister

Physics Division, National Science Foundation, Arlington, VA 22230

W. N. Polyzou

Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242 (Dated: May 31, 2012)

A relativistically invariant quantum theory first advanced by Bakamjian and Thomas has proven very useful in modeling few-body systems. For three particles or more, this approach is known formally to fail the constraint of cluster separability, whereby symmetries and conservation laws that hold for a system of particles also hold for isolated subsystems. Cluster separability can be restored by means of a recursive construction using unitary transformations, but implementation is difficult in practice, and the quantitative extent to which the Bakamjian-Thomas approach violates cluster separability has never been tested. This paper provides such a test by means of a model of a scalar probe in a three-particle system for which (1) it is simple enough that there is a straightforward solution that satisfies Poincaré invariance and cluster separability, and (2) one can also apply the Bakamjian-Thomas approach. The difference between these calculations provides a measure of the size of the corrections from the Sokolov construction that are needed to restore cluster properties. Our estimates suggest that, in models based on nucleon degrees of freedom, the corrections that restore cluster properties are too small to effect calculations of observables.

PACS numbers: 21.45+v

I. INTRODUCTION

There are two distinct requirements for describing quantum mechanical systems of particles under the requirements of special relativity. The first requirement is Poincaré invariance: probabilities, expectation values and ensemble averages for equivalent experiments performed in different inertial frames are identical. A necessary and sufficient condition for a quantum theory to be Poincaré invariant is that the dynamics is described by a unitary representation of the Poincaré group [1]. The second requirement is cluster separability: isolated subsystems must have the same observable properties as they would in a framework in which the other “spectator” particles are absent entirely. This requirement justifies tests of special relativity on isolated subsystems. It applies both to systems of particles interacting among themselves (e.g. via the strong interaction) and to the current operators that allow them to interact with external fields.

There is more than one way to implement the requirements of Poincaré invariance in a quantum system, and cluster separability is not necessarily an automatic consequence of the implementation. In local quantum field theory, Poincaré invariance and cluster separability are satisfied formally as a consequence of the covariance, spectral properties and the local commutation relations of the field operators. For systems amenable to perturbation methods, these properties can still hold. For non-perturbative systems (such as those based on the strong interaction) controlled approximations are replaced by a truncation or an expansion scheme, which replaces an infinite hierarchy of coupled non-linear field equations with a more tractable finite subset, and which may be valid only in restricted parameter domains. These schemes may still exhibit Lorentz covariance, but cluster separability does not automatically follow, and must be demonstrated on a case-by-case basis that depends on the scheme.

An alternative implementation of Poincaré invariance in a few-body system involves direct construction of the dynamical representations of the Poincaré group in the presence interactions with a finite number of particle (rather than field) degrees of freedom. Such a construction was originally studied by Dirac [2], and Bakamjian and Thomas [3] provided an explicit construction of the generators for a system of two interacting particles.

Many realistic, Poincaré invariant, quantum mechanical models of strongly interacting few-body systems are based on a generalization of the Bakamjian-Thomas construction [4]. This construction can be used to construct models of systems of arbitrary numbers of particles and systems that do not conserve particle number. A representative sample of few-body applications of Poincaré invariant quantum mechanics based on the Bakamjian-Thomas construction includes relativistic constituent quark models [5–9], relativistic few-nucleon models [10–14], relativistic models involving electromagnetic probes [15–17], and relativistic models with particle production [18, 19].

The virtue of the Bakamjian-Thomas construction is that it provides a means for constructing Poincaré generators for systems of interacting particles, and where the problems have a finite number of degrees of freedom and are in principle solvable. The limitation of this construction is that for systems of more than two particles, the dynamical representation of the Poincaré group does not become a tensor product on states representing asymptotically separated subsystems. While this limitation does not lead to observable consequences in the three-body S-matrix or bound-state observables, there are observable consequences when the three-body system is embedded in the four-particle Hilbert space, as it is in four-body problems.

The most ambitious applications of the Bakamjian-Thomas construction are relativistic Faddeev calculations [10– 14] of three-nucleon scattering, which have been performed using realistic nucleon-nucleon forces and three-nucleon forces. The next step in developing a relativistic few-body theory would be to model four-nucleon systems or electron scattering from a three-nucleon system. In both of these applications a three-nucleon subsystem is embedded in a four-particle Hilbert space, and, for the first time, there is the possibility of observable consequences of violations of cluster properties.

The restoration of cluster properties can be achieved through a recursive construction due to Sokolov [20, 21]. The Sokolov construction generates a dynamical representation of the Poincaré group satisfying cluster properties using of a hierarchy of unitary transformations, each of which preserves the S matrix while transforming tensor products of subsystem representations of the Poincaré group to representations where the interactions can be added in a manner that preserves the underlying Poincaré symmetry. The Bakamjian-Thomas representation is retrieved by setting the relevant unitary transformations to the identity operator. The Sokolov construction is non-trivial and the unitary transformations are generated by the interactions. An isolated three-body system is a special case where the cluster property can be achieved by means of a single overall unitary transformation that preserves the S matrix. For systems of more than three particles there are observable differences between the S matrices in the two representations. The interested reader can find a complete discussion in [21].

The construction of the Sokolov hierarchy of unitary transformations is sufficiently complicated that the technique has never been used in realistic calculations. Before undertaking computationally intensive four-body calculations, one would like to know the importance of the corrections required by cluster properties. While Ref. [21] argued that these corrections should be small in nuclear physics applications, this was never quantified in any model calculations. An additional investigation is needed to determine if these corrections can be ignored, can be treated perturbatively, or must be treated exactly. To address this question we construct a model involving matrix elements of a scalar probe for a system of three particles. The model is simple enough that Poincaré invariance and cluster separability are easily satisfied; this unusual pair of features is due to the simplicity of the test model. At the same time the model is sufficiently rich that it permits an equivalent Bakamjian-Thomas treatment of the three-body system that illustrates the quantitative impact of the breakdown of cluster separability in this four-body problem. The conclusion of our preliminary analysis suggests that the corrections required by cluster properties are too small to be observable in nuclear physics applications. The corrections are more important for models based on sub-nucleon degrees of freedom.

II. CLUSTER PROPERTIES

In a few-body quantum mechanical model the requirement of cluster separability means that the Poincaré generators {H, P, J, K} that generate time translations, translations, rotations, and rotationless Lorentz transformations and current operators I^{μ}(x) have cluster expansions of the form

H = ∑_{i} H_{i} + ∑_{ij} H_{ij} + ∑_{jk} H_{ijk} + ⋯  (2.1)

P = ∑_{i} P_{i} + ∑_{ij} P_{ij} + ∑_{ijk} P_{ijk} + ⋯  (2.2)

J = ∑_{i} J_{i} + ∑_{ij} J_{ij} + ∑_{ijk} J_{ijk} + ⋯  (2.3)

K = ∑_{i} K_{i} + ∑_{ij} K_{ij} + ∑_{ijk} K_{ijk} + ⋯  (2.4)

I^{μ}(x) = ∑_{i} I_{i}^{μ}(x) + ∑_{ij} I_{ij}^{μ}(x) + ∑_{jk} I_{ijk}^{μ}(x) + ⋯  (2.5)