Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics ABSTRACT

325 Pages·2012·1.27 MB·English

Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics
B. D. Keister
Physics Department
Carnegie Mellon University
Pittsburgh, PA 15213
and
W. N. Polyzou
Department of Physics and Astronomy
The University of Iowa
Iowa City, IA 52242
ABSTRACT
This review is intended to provide an introduction to the formulation of rel-
ativistic quantum mechanical models, particularly for use in strong interaction
problems, whose dynamics is given by a unitary representation of the inhomo-
geneous Lorentz group. In the (cid:12)rst portion, an overview is given in which the
properties of these models are de(cid:12)ned and some analytically solvable examples are
given. This is followed by a deductive construction of these models from physical
principles. Particle production, electron scattering, macroscopic locality, and the
relation to local quantum (cid:12)eld theory are discussed in the second half.
RELATIVISTIC HAMILTONIAN DYNAMICS IN
NUCLEAR AND PARTICLE PHYSICS
B. D. Keister and W. N. Polyzou
To be published in:
ADVANCES IN NUCLEAR PHYSICS
Ed. J. W. Negele and Erich Vogt
Plenum Press New York-London
(cid:15)
RELATIVISTIC HAMILTONIAN DYNAMICS IN
NUCLEAR AND PARTICLE PHYSICS
B. D. Keister and W. N. Polyzou
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Relativistic Quantum Mechanics: Principles and Examples . . . . . . . . . . . . 8
2.1 Relativistic Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Con(cid:12)ned Spinless Quarks . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Con(cid:12)ned Relativistic Quarks - With Spin . . . . . . . . . . . . . . . . . 39
2.5 Spinless Two Particle Scattering . . . . . . . . . . . . . . . . . . . . . 45
2.6 Nucleon-Nucleon Scattering - With Spin . . . . . . . . . . . . . . . . . . 61
2.7 Summary of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3. Symmetries in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 71
3.1 Galilean Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2 Special Relativity - The Poincar(cid:19)e Group . . . . . . . . . . . . . . . . . . 79
3.3 Parameterization of Poincar(cid:19)e Transformations . . . . . . . . . . . . . . . 81
3.4 De(cid:12)nition of In(cid:12)nitesimal Generators . . . . . . . . . . . . . . . . . . . 82
3.5 Commutation Relations - Canonical Form . . . . . . . . . . . . . . . . . 82
3.5.1 Commutation Relations - Covariant Form . . . . . . . . . . . . . . . 84
3.5.2 Commutation Relations - Front Form . . . . . . . . . . . . . . . . . 85
3.6 Commuting Self Adjoint Operators . . . . . . . . . . . . . . . . . . . . 86
3.6.1 The Mass Operator . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.6.2 The Pauli-Lubanski Operator . . . . . . . . . . . . . . . . . . . . . 87
3.6.3 Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.4 Spin Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.7 Other Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4. The One-Body Problem - Irreducible Representations . . . . . . . . . . . . . 95
4.1 The Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Unitary Representations . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Position in Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5. The Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1 The Two-Body Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 Relativistic Dynamics of Two Free Particles . . . . . . . . . . . . . . . . 113
5.3 Clebsch-Gordan Coe(cid:14)cients . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Free-Particle Generators and Other Operators . . . . . . . . . . . . . . . 121
5.5 The Bakamjian-Thomas Construction . . . . . . . . . . . . . . . . . . . 123
5.6 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6.1 The Instant Form . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.6.2 The Front Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.6.3 The Point Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6. The 2+1 Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.1 Macroscopic Locality and the 2+1 Body Problem . . . . . . . . . . . . . . 137
6.2 The Three-Body Hilbert Space . . . . . . . . . . . . . . . . . . . . . . 142
6.3 Two 2+1 Body Models . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.4 Packing Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7. The Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1 Three-Body Constructions . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1.1 Poincar(cid:19)e Invariance in the Three-Body Problem . . . . . . . . . . . . . 157
7.1.2 Bakamjian-Thomas Construction . . . . . . . . . . . . . . . . . . . 161
7.1.3 Packing Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.2 Faddeev Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.3 Symmetric Coupling Schemes . . . . . . . . . . . . . . . . . . . . . . . 180
7.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8. Particle Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.1 The Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.2 Free-Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.3 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.4 Macroscopic Locality . . . . . . . . . . . . . . . . . . . . . . . . . . 200
9. Electromagnetic Currents and Tensor Operators . . . . . . . . . . . . . . . . 205
9.1 Basic Formulas and Observables . . . . . . . . . . . . . . . . . . . . . . 206
9.2 Matrix Elements and Invariants . . . . . . . . . . . . . . . . . . . . . . 212
9.2.1 Matrix Elements of Tensor Operators . . . . . . . . . . . . . . . . . 213
9.2.2 Front-Form Matrix Elements . . . . . . . . . . . . . . . . . . . . . 221
9.2.3 Example: Matrix Elements of Field Operators . . . . . . . . . . . . . . 221
9.2.4 Four-Vector Current Matrix Elements. . . . . . . . . . . . . . . . . . 222
9.2.5 Symmetries and Constraints . . . . . . . . . . . . . . . . . . . . . . 224
9.2.6 Front-Form Current Matrix Elements . . . . . . . . . . . . . . . . . 225
9.2.7 Example: The (cid:25) (cid:26) Transition Form Factor . . . . . . . . . . . . . . 228
!
9.3 Computation of Composite Form Factors . . . . . . . . . . . . . . . . . . 230
9.3.1 Basic Requirements of Current Operators . . . . . . . . . . . . . . . . 230
9.3.2 Impulse Approximation . . . . . . . . . . . . . . . . . . . . . . . 235
9.3.3 Example: The (cid:25) (cid:26) Transition Form Factor . . . . . . . . . . . . . . 238
!
10. Relation To Covariant Theories . . . . . . . . . . . . . . . . . . . . . . . 241
11. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Appendix A: Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . 258
A.1 The Relation Between S and T . . . . . . . . . . . . . . . . . . . . . . 258
A.2 The Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . 262
A.3 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
A.4 Phenomenological Interactions . . . . . . . . . . . . . . . . . . . . . . 271
Appendix B: Front Form Kinematics . . . . . . . . . . . . . . . . . . . . . . 275
Appendix C: Racah Coe(cid:14)cients . . . . . . . . . . . . . . . . . . . . . . . . 281
Appendix D: Local Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
D.1 Fixed Number of Particles - Galilean Invariance: . . . . . . . . . . . . . . 285
D.2 Fixed Number of Particles - Poincar(cid:19)e Invariance: . . . . . . . . . . . . . . 287
D.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
D.1.1 Quasilocal Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 294
D.4 The One-Body Subspace . . . . . . . . . . . . . . . . . . . . . . . . . 298
D.5 The Two-Body Subspace/Interactions . . . . . . . . . . . . . . . . . . . 303
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
1. Introduction
Strong-interaction problems in nuclear and particle physics are often formulated in terms
of phenomenological models because of the di(cid:14)culties in formulating convergent approximations
in local (cid:12)eld theories such as QCD. Phenomenological models are designed to be simple enough
that they can be solved accurately and, if they are suitably re(cid:12)ned, they can lead to realistic
descriptions of physical systems, as is the case in atomic and molecular physics and low energy
nuclear physics. For many problems of current interest in nuclear physics these models must be
consistent with the principle of special relativity. Relativity is needed to model reactions where
particles are produced, reactions involving energy and momentum transfers that are comparable
to the mass scales of the problem, bound systems where the binding energies are comparable
to the masses of the constituent particles, and coordinate system independent treatments of
problems in lepton-hadron scattering.
Relativistic quantum mechanics began with attempts to construct manifestly covariant ex-
tensions of the Schro(cid:127)dinger equation. Schro(cid:127)dinger (Sc 26) had already discovered and discarded
theKlein-Gordonequation(Kl26,Go26a,Go26b)inhisoriginalpaper. Itwasrealizedearlyon
by Heisenberg, Born and Jordan (Bo 26) that laws of quantum mechanics also should apply to
theelectromagnetic(cid:12)eld,whichtransformscovariantlyunderPoincar(cid:19)etransformations. Thisled
totheintroductionofthequantumtheoryoftheelectromagnetic(cid:12)eld(Di27,He29,Sc58). The
impressive agreement of the predictions of quantum electrodynamics with experiment, coupled
with the realization that a quantized (cid:12)eld provided a means for avoiding the concept of instan-
taneous action at a distance, led to the acceptance of local relativistic (cid:12)eld theory as the correct
way to model the fundamental interactions of nature at accessible energies.
For the strong interaction, however, ab initio calculations based on local (cid:12)eld theories are
di(cid:14)cultbecausethein(cid:12)nitenumberofdegreesoffreedomandthelargecoupling constantsmake
it di(cid:14)cult to control the size of the error in any calculation. Field theoretic calculations involve
manipulations of a (cid:12)nite number of renormalized Feynman diagrams, using ladder sums (Sa 51)
or other techniques. These calculations ignore an in(cid:12)nite number of graphs with large coupling
constants and theyfail toaddress the extenttowhich theterms in theperturbationseries de(cid:12)ne
1
thedynamics. Inaddition,mostapplicationsinnuclearphysicsinvolvecompositesystems,either
of nuclei composed of nucleons, or of nucleons composed of quarks and gluons. The treatment
of composite systems in quantum (cid:12)eld theories is nonperturbative at the outset. For the case of
nucleonsascompositesofquarksandgluons,theproblemismoredi(cid:14)cultbecausethequarkand
gluon (cid:12)elds do not correspond to observable particles. Atpresent there areno known algorithms
for constructing approximate solutions of dynamical problems in strongly interacting quantum
(cid:12)eld theories with arbitrary precision.
Integralformulationsof(cid:12)eldtheory,such aslatticeapproximations, (Wi74)mayultimately
leadtocomputationalmethodswhereerrorscanbecontrolled. Thesemethodsarenotdeveloped
to the point where they provide su(cid:14)cient control of computational error to make many useful
quantitative statements about nuclear or hadronic dynamics of realistic systems.
In spite of the acceptance of (cid:12)eld theories as a matter of principle, most realistic dynam-
ical calculations in nuclear physics, and many in particle physics, utilize the nonrelativistic
Schro(cid:127)dinger equation. Nonrelativistic models can be solved using well de(cid:12)ned computational
algorithms (Fa 65, Ya 67) in which errors can be made as small as desired. In a nonrelativistic
approach, onebeginswithalargeclassofmodels,mostofwhichcanbediscardeduponcompari-
sonwithexperiment. Ina(cid:12)eldtheoreticapproach,onebeginswithasmallerclassofmodels,but
in most cases, it is impossible to perform a calculation with errors small enough to discard the
model if it is not in agreement with experiment. Although the problem of putting error bounds
on (cid:12)eld theoretic calculations can be justi(cid:12)ably considered a technical problem, it has resisted
solution for over 50 years.
Subsequenttothedevelopmentofquantum(cid:12)eldtheory,Wigner(Wi39)analyzedthemath-
ematical formulation of the physical requirement of special relativity in quantum mechanics.
Physical states in quantum mechanics are in one-to-one correspondence with one-dimensional
subspaces, or \rays," of the Hilbert space. Wigner showed that a necessary and su(cid:14)cient condi-
tion for quantum mechanical probabilities to have values that are independent of the choice of
inertial coordinate system is the existence of a unitary ray representation of the inhomogeneous
Lorentzgroup(Poincar(cid:19)egroup)onthequantummechanicalHilbertspace. Wigner’sanalysisap-
pliesbothtoquantum(cid:12)eldtheoriesandtoquantumtheoriesofparticles,althoughitsapplication
2
to theories of particles was not vigorously pursued at the time.
Most of what follows in this review is motivated by (cid:12)ve seminal papers that took the work
of Wigner to its logical conclusion for systems of interacting particles. First, Dirac (Di 49)
formulated the problem of including interactions in relativistic classical mechanics. This was
done in Hamiltonian form, which has a natural canonical quantization. Although Dirac did
not solve the classical problem, he simpli(cid:12)ed it to one of several simpler problems. These three
di(cid:11)erenttypesofsolutionstothisproblemarenowcalledthe\point,"\instant"and\front"forms
of the dynamics. Bakamjian and Thomas (Ba 53) successfully constructed the (cid:12)rst relativistic
quantum mechanical model of two interacting particles in Dirac’s \instant" form of dynamics.
Foldy (Fo 61) recognized the importance of macroscopic locality as an additional constraint on
thesemodels. This condition replaces theconceptof Einstein causality ormicroscopic localityin
local(cid:12)eldtheories. Coester(Co65)thenextendedtheworkofBakamjianandThomastosystems
ofthreeparticles,withascatteringoperatorconsistentwiththeprincipleofmacroscopiclocality.
Finally, Sokolov(So77)providedthegeneralconstructionforN particlesinamanner consistent
withmacroscopiclocality. These(cid:12)vepapersde(cid:12)nethescopeofthisreview. Relativisticquantum
mechanical models of directly interacting particles have the following features:
consistency with requirements of relativity and quantum mechanics
(cid:15)
connection between few-body dynamics and the many-body problem
(cid:15)
possibility of composite particles
(cid:15)
large class of permissible interactions
(cid:15)
tractable few-body calculation
(cid:15)
We believe that such models are very attractive for a wide variety of applications in nuclear and
particle physics.
Relativistic direct interaction theories of particles lie between local (cid:12)eld theoretic models
and nonrelativistic quantum mechanical models. They are applicable to situations involving
larger momentum transfers and binding energies than nonrelativistic models, and they permit
theformulationofinvariantcalculationsinvolvingparticleproduction,electromagneticandweak
probes (in the one-boson exchange approximation); none of the latter applications is possible
3
in nonrelativistic models. They replace the microscopic locality of (cid:12)eld theories with a weaker
condition, called macroscopic locality, but, unlike (cid:12)eld theories, lead to mathematically well
de(cid:12)nedmodelswherecomputationalerrorcanbecontrolled. Becauseofthis,theyshouldprovide
a useful framework for the construction of mathematical models of the dynamics of hadrons and
nuclei at intermediate energies.
In comparing the contents of this review to other formulations of relativistic quantum me-
chanics, itisusefultokeepinmindthatthereare(atleast)twowaystoconsidertheformulation
of this problem. The most common is to begin with a local relativistic (cid:12)eld theory, and trun-
cate the dynamics in such a way that what remains is a closed system of dynamical equations
involving a (cid:12)nite number of important degrees of freedom. A second approach is to assume that
the system is governed by a (cid:12)nite number of degrees of freedom, and then to construct the most
general class of dynamical models with these degrees of freedom, consistent with a set of general
principles that include relativistic invariance. In some cases, equations obtained by these two
di(cid:11)erent approaches may be identical, but the emphasis and subsequent application is usually
di(cid:11)erent.
Inthe(cid:12)rstapproach, theconnectionto(cid:12)eldtheoryisemphasized. Ingeneral,relationssuch
as the Schwinger-Dyson equations involve an in(cid:12)nite number of coupled amplitudes. Models
are constructed by retaining the coupling only among a (cid:12)nite number of amplitudes, or by
replacingan unknownamplitude withaphenomenological amplitude. Ladderapproximations to
the Bethe-Salpeter equation (Sa 51) and approaches based upon mean (cid:12)eld theory (Se 86) are
typical examples. We refer to these procedures as truncations. In general, truncations are not
controlledapproximations,andthephysicalpropertiesofthe(cid:12)eldtheory(i.e.,theaxiomsof(cid:12)eld
theory)arenotnecessarilypreservedontruncation. ThemodelsmayviolatePoincar(cid:19)einvariance,
currentcovariance,orothersymmetries; onemustthenbelievethatforasensibletruncation,the
corrections needed to restore these symmetries are small.
In the second approach, basic principles are emphasized. The connection to (cid:12)eld theory
is of secondary concern. In this case, some principles have to be given up in passing from
a local (cid:12)eld theory to a particle theory, but this is accomplished directly by weakening speci(cid:12)c
axioms. Inthispapertheaxiomofmicroscopiclocalityisreplacedbyaweakerrequirementcalled
4