Relativistic Models of Extended Hadrons Obeying a Mass-Spin Trajectory Constraint: Lectures in Mathematical Physics at the University of Texas at Austin

168 Pages·1982·2.02 MB·English

by  L. C. Biedenharn, H. van Dam, N. Mukunda, A. Böhm, J. D. Dollard (auth.)

Lecture Notes
in Physics
Edited by Ii. Araki, Kyoto, J. Ehlers, Mijnchen, K. Hepp, Ziirich
R. Kippenhahn, MOnchen, H.A. Weidenmijller, Heidelberg
and J. Zittartz. KBln
165
N. Mukunda H.van Dam
L.C. Biedenharn
Relativistic Models
of Extended Hadrons
Obeying a MassSpin
Trajectory Constraint
Lectures in Mathematical Physics at
the University of Texas at Austin
Edited by A. Biihm and J.D. Dollard
1
Springer-Veriag
Berlin Heidelberg New York 1982
Authors
L.C. Biedenharn
Physics Department, Duke University
Durham, NC 27706, USA
H. van Dam
Physics Department, University of North Carolina
Chapel Hill, NC 27514, USA
N. Mukunda
Centre for Theoretical Studies, Indian Institute of Science
Bangalore- 2, India
Editors
A. Bahm
Physics Department, University of Texas
Austin, TX 78712, USA
J.D. Dollard
Mathematics Department, University of Texas
Austin, TX 78712, USA
ISBN O-540-11586-2 Springer-Verlag Berlin Heidelberg New York
ISBN O-387-11586-2 Springer-Verlag New York Heidelberg Berlin
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Preface
The purpose of the Texas lecture notes is to inform graduate
students and "non-specialists" about recent developments in various
areas of mathematics and physics. This volume of notes originated
from two series of lectures which were presented by two of the authors
during the academic year 1979-80 at the University of Texas at Austin.
The subject discussed in this volume is one of the most funda-
mental problems of particle physics - the problem of the hadron mass
spectrum. It is the question of how the continuous parameter mass
depends upon the discrete parameter spin, resulting in the discrete
mass spectrum for the hadrons.
There is no generally accepted solution to this problem nor is
ereh.t a generally accepted proposal on how to attack it. The proposal
discussed here uses the methods of representation theory. It is based
upon the idea that the internal structure of the hadrons is not
explicitely given in terms of elementary constituents but is more
abstract.
Due to the novelty of this approach, the description of a
relativistic quantum mechanics for extended objects proposed in these
notes (in particular Chp. 4), may not be in its final form. Still, it
warrants publication as a review since it will continue to stimulate
discussions and further developments in this important area.
Chapters 5 through 8 are mainly concerned with the application of
constrained Hamiltonian mechanics and give a very clear and beautiful
presentation of classical models for relativistic rotating objects.
If one believes that quantum theory is obtained by the correspondence
principle from classical mechanics, this is a subject that cannot be
avoided.
The Editors
CONTENTS
Page
CHAPTER ONE: Introduction ....................................... 1
CHAPTER TWO: Dirac's New Relativistic Wave Equation
and Its Generalization ............................ 10
§i. Dirac's Presentation of His Ne~; Equation ............... i0
§2. Relationship with the de Sitter Group, S0(3,2) ......... 13
§3. Some Properties of the New Dirac Equation .............. 21
§4. An Alternative, ~re Illuminating, View of the
Structure ............................................. 25
§5. Generalization to Non-Zero Spin States ................. 28
§6. Minimal Electromagnetic Interaction is Forbidden
for the New Dirac Equation ............................ 32
CHAPTER THREE: Unitary Representations of the Poincare Group
in the Thomas Form: Quasi-Newtonian Coordinates 34
§I. Overview ............................................... 34
§2. The Wigner Irreps M,s ................................ 35
§3. Poincar~ Generators for the Wigner Irreps .............. 37
§4. Quasi-Newtonian Coordinates and the Generators in
Thomas Form ........................................... 38
§5. Generalization of the Thomas Form ...................... 40
§6. Application of the Generalized Thomas Form:
Regge Trajectories .................................... 41
§7. Supersymmetry: Relativistic SU(6) .............. ....... 43
CHAPTER FOUR: Explicitly Poincar~ Invariant Formulation,
Relation to Supersymmetry, No-Go Theorems ........ 45
§I. The Transformation from Quasi-Newtonian to
Minkowski Coordinates ................................. 45
§2. Explicit Construction of the Transformation ............ 47
§3. An Algebra which Extends the Poincar~ Algebra and
Contains Operators for Raising and Lowering
Mass and Spin ......................................... 54
§4. Relation with Standard Global Supersymmetry ............ 57
§5. Relativistic SU(6) ..................................... 63
CHAPTER FIVE: Constrained Hamiltonian Mechanics ................. 64
§I. Theory ........ , ........................................ 64
§2. Examples ............................................... 74
IV
Page
CHAPTER SIX: Vector Lagrangian Model ............................ 79
§i. Choice of Variables and Lagrangian ..................... 79
§2. Phase Space Structure and Conservation Laws ............ 82
§3. Two-Variable Models: Model A ........................... 86
§4. Two-Variable Model B ................................... 91
§5. Two-Variable Model C ................................... 94
§6. The General Three-Variable Case ........................ 96
§7. Quantization ........................................... 98
§8. Electromagnetic Interaction ............................ 105
CHAPTER SEVEN: Lagrangian Spinor Model, Electromagnetic
Interaction ..................................... 106
§i. Choice of Variables and Lagrangian ........ • ............. 106
§2. Phase Space, Conservation Laws and Forms of Constraint.109
§3. Equations of Motion and Space-Time Trajectory .......... ii0
§4. Quantization of the Free Spinor Model .................. 113
§5. Relation to the Models of Chapters 3 and 4 ............. 115
§6. Electromagnetic Interactions ........................... 116
§7. Calculation of the Magnetic Moments of the States ...... 118
§8. Relativistic SU(6) ~del, String Model ................. 122
CHAPTER EIGHT: Further Analysis of the Classical Motion
of the Spinor Model ............................. 124
§i. Dynamics in Terms of Primary Variables ................. 124
§2. Free Particle Kinematics, Introduction of
Secondary Coordinates and Spin ........................ 125
§3. Free Particle Motion ................................... 127
§4. Spin for Particle in an External Electromagnetic
Field, Gyromagnetic Ratio ............................. 132
§5. Energy-Momentum Tensor ................................. 135
§6. Classical SU(6) Model with Three Pairs of
Oscillators ........................................... 136
APPENDIX A: The Relativistic Spherical Top ...................... 138
APPENDIX B: Galilean Subdynamics ............ . ................... 153
CHAPTER ONE
INTRODUCTION
One of the significan£ early developments in hadron spectroscopy
was the empirical realization that excited hadronic states--having the
same SU3 labels and parity--could be organized into families called
"Regge sequence" or "Regge trajectories". The resulting (Mass) 2 versus
spin plots, developed by Chew and and Frautschi, were expected (from
potential model calculations) to show strong curvature, but as the data
were assembled straight line fits, M 2 = C s, with a universal slope
o
C( o ~ .i GeV )2 seemed indicated. Such linearly rising Regge trajec-
tories became an acceptable empirical model for hadronic structure,
which could be considered to agree with unitarity constraints as a sort
of 'zero-width first approximation' MAN i. Coupled with the concept
of duality, this viewpoint--as is well known--led first to the Veneziano
model, then to dual resonance models, and eventually to the technically
very impressive dual string models FRA i.
An alternative line of development of the trajectory concept came
from the discovery of SU(6) symmetry DYS i, combining Gell-Mann SU(3)
with the in£risinic spin SU2, in a way familiar from Wigner's introduc-
tion of (non-relativistic) SU4 symmetry in nuclear structure physics.
In hadronic physics, SU6 symmetry is problematic, since it appeared
justifiable only non-relativistically, but was applied in a relativistic
context. ~re exotic models were developed in great profusion, most--
unfortunately--wholly incorrect (.such as U(!2)) and in obvious contra-
diction to basic principles.
This line of development was brought to a halt by the McGlinn i
O'Raifeartaigh theorem which asserts (in essence) that: If one has a
finitely generated Lie symmetry group G which contains the Poincar~
group P as a subgroup (and G is not a direct product 'G x )P , then in
any UIR of G either the mass spectrum is continuous, or consists of a
single discrete point. Any hope of producing the empirical Regge
trajectories via the representations of a symmetry group seemed doomed.
The reaction to this theorem was almost as extreme as the U(12)
fiasco itself, which the theorem had eliminated. This was especially
unfortunate, since the use of group structure as a substitute for
dynamics is a valid--and very useful--theoretical technique for model
building, and moreover, some of the symmetry models being developed
*S. Coleman, Proc. Midwest Conference on Theoretical Physics, Indiana
University, Bloomington, Ind., 1966.
at this time NAM l TAK i BOH i were actually valid ways to circum-
vent this "no-go' theorem. Nevertheless, the belief is still widespread
that only the use of supersymmetry (boson-fermion multiplets) avoids the
no-go theorem.
The models we shall discuss in detail in these lectures follow the
line of development of symmetry structures; these models were clearly
foreshadowed, more or less explicitly, in a brief survey by Takabayashi
in 1967 TAK i. These same models may also be considered in terms of
the other line of development since--in the limit--they fit into string
model structures.
In order to treat, relativistically, structures which are no% ele-
mentary--as for example an entire Regge trajectory--, one is forced to
make a choice right at the beginning: )a( is the structure to be made
up of other (possibly elementary) particles? or )b( is the "internal
structure" to be more abstract (for example not realizable as particles?)
If one chooses )a( then one faces serious, as yet not fully solved,
problems: cluster decomposition, existence of a continuum,.., and
essentially the only way to proceed is via a fully fledged field theory.
(Cf. however, the recent work of Komar KOM i, F. Rohrlich ROH i and
I. T. TodorovTOD i.)
The fact that quarks are not (as yet) seen, leads one to hope that
alternative )b( is a valid approach, for which, in particular, there is
no continuum and the indefinitely rising discrete Regge trajectory is
an acceptable first approximation.
The essential point in this use of alternative )b( is that the in-
ternal coordinates, while varying under Lorentz transformations (and
hence carrying spin angular momentum) do not carry any linear momentum.*
(This is not quite the same as removing the center-of-mass coordinate,
having translation invariant internal coordinates, since (by reversing
the process) one could recover the linear momentum carried by the
separate constituents.) The internal (spin) angular momentum is defined
by spin operators acting on the abstract internal coordinates and these
do not arise from differences of particle positions. (This is discussed
in Chapter 2.)
Such a viewpoint is a literal generalization of the concepts used
in the four-component Dirac wave function. There is one, and only one,
position coordinate (x) and the spin is carried by the indices. Gen-
eralizing to a denumerable infinity of indices is equivalent to using:
* This circumstance is less surprising if one recalls that phonons (of
non-zero frequency) also do not carry linear momentum, as Peierls has
discussed PEI i. This analogy would suggest that the internal varia-
bles used here are Fourier modes of the internal structure.
~n(X ) + ~(x ;~), where we have introduced in ~ one (or more) continuous
internal variables. (Hence the term: infinite component wave functions.)
Such a generalization was suggested very early (950's) by Yukawa
YUK i as bi-local (later quadri-local) fields. Yukawa used as inter-
nal variables four-vectors, and this leads at once to an insuperable
problem for interactions. (Constraints are required to remove redundant
timelike variables and the associated negative energy timelike oscilla-
tions, and interactions are incompatible with the constraints, see
Chapters 5 and 6.)
The proper way to proceed--or at least a way which is successful in
allowing interactions--is to use spinorial internal variables. This is,
once again, an idea that appears in the early literature--nost notably
in the Majorana equation MAJ I,FRD 1,GEL 1--but the bad Spectral
properties (accumulation points in the mass spectrum at zero-energy,...)
led to an equally early discard. For another approach see DRE i.
Our interest in this already well-studied field was stimulated by
a paper by Dirac in 1970 DIR i. In this paper, Dirac gave a new wave
equation with spinor internal variables that superficially was similar
in appearance to the famous Dirac electron equation. The equation was,
in fact, a very ingenious way to avoid the spacelike (and light-like)
solutions of the Majorana spectrum, and it had an inherently positive
energy spectrum, a conserved current, but forbade any (minimal) coupling
ot electromagnetism. An especially intriguing point was that Dirac
claimed that in a moving system the "particle" appeared to have all
spins. This is an error*, resolved later by Dirac himself DIR 2 and
is due to a misidentification of the spin. Alternatively, one may say
that the orbital angular momentum is misidentified by using the wrong
position coordinate.
This latter is in itself a very interesting idea, for it suggests
that we are dealing in Dirac's new model with two positions: the posi-
tion of the "charge" (the Minkowski position, x ) as contrasted with the
position of the center-of-mass (the coordinate defining, through the
orbital angular momentum, the proper spin). These ideas are physically
appealing and quite suggestive, for they are just the old Zitterbewegung
concepts introduced by Schr6dinger SCH i, DIR 3, but now realized
in the context of a (positive energy) extended structure. (We discuss
this in detail in Chapter 4 and, more physically, in Chapter 8.)
We may summarize now the essential ideas on which the models of
these lectures are based and then outline the contents of the successive
chapters:
* It is curious to note that in Dirac's 1949 paper on expansors the
same (erroneous) claim appears DIR 4.
)a( a discrete, rising, unbounded mass spectrum (versus spin) is
taken for the Regge trajectory constraint;
)b( internal variables are abstract (unobservable), translation-
independent, spinorial coordinates;
)c( the internal variables are oriented by the state of motion
(Lorentz frame) of the object (see Chapter 2, §4.);
)d( the structure is a generalization of Dirac's new equation to
encompass all spins~ (Dirac's structure had only spin 0.)
The fact that center-of-mass and charge positions are not the same
(in the models we treat) introduces another basic theme of these lec-
tures: the relation between the quasi-Newtonian (or Newton-Wigner)
NEW i coordinates and the Minkowski position coordinates. The general
concept stems from the Foldy-Wouthuysen transformation for the Dirac
electron equation FOL i, PRY i, BEC I : the center-of-mass coordi-
nates are the "mean position"coordinates, the related orbital angular
momentum and spin, the "mean orbital angular momentum" and "mean spin"
respectively. These latter angular momenta are each separately con-
served. Although fully relativistic, these quasi-Newtonian position
coordinates are not manifestly relativistic; in the particular they do
not constitute the space components of some four-vector. Moreover,
nothing couple8 ot the quasi-Newtonian coordinates. The troubles of
"relativistic SU(6)" are, in fact, precisely the troubles of quasi-
Newtonian coordinates. We discuss in Chapter 3 the general subject of
quasi-Newtonian coordinates, developing these ideas from the Wigner form
of the Poincare irreps for massive particles with arbitrary spin WIG i.
The problem of transforming from quasi-Newtonian to Minkowski
coordinates is discussed in Chapter .4 Here we develop the "inverse
Foldy-Wouthuysen" transformation, but in our case for all spins at once.
The successful, explicit, realization of this "invers~ F-W" transforma-
tion (for the entire set of ~(s),s) Wigner irreps) should be an instruc-
tive example for more complicated--but closely related--situations
(such as the Melosh transformation between "current quarks" and
"constituent quarks" MEL i.
The end result achieved in Chapter 4 is a constrained Lagrangian
for an extended object in Minkowski space. To study this structure
in the required detail we develop in Chapter 5, the powerful techniques
of Dirac for analyzing constrained Lagrangian-Hamiltonian mechanics.
In this, our work again has predecessors: in this case the Hanson-Regge
HAN i, ~/~H 2 treatment of the relativistic top. Their discussion is ~
based on internal coordinates built from vectors (a Lorentz tensor)
and--unlike the spinor model we develop--fails in both interactions
(constraints) and in the use of non-commuting Minkowski coordinates for
the structure to be quantized. (The Hanson-Regge model is dicussed in
Appendix A.) Using the techniques of Chapter 5 we discuss ni( Chapter 6)
vectorial models and ni( Chapter 7) the spinorial model, which--as we
show--permits of electromagnetic interactions. In this chapter, we show
that minimal electromagnetic coupling results in anomalous magnetic
moments.
In the concluding chapter, Chapter 8, we exploit the fact that
the sDinorial model has a well-defined classical structure (classical
relativistic mechanics) to develop a clearer physical picture under-
lying the free particle motion. For example, it is by no means clear,
a priori, that (unobservable) abstract coordinates, not realized in
Minkowski space, can give the particle "spatial extension" in Minkowski
space. Just this is, in fact, the case: for it is the spin motion
that causes the charge position to differ from the center-of-mass
position and this separation--the spatial extension--is proportional
to the spin (and hence via the trajectory to the mass). In this
structure spin is literally a type of 'orbital' motion and this fact
underlies the existence of an anomalous magnetic moment.
Equations of motion describing a classical relativistic particle
with charge and with a magnetic moment in interaction with an external
field--such as the structure discussed in Chapter 8--have been the sub-
ject of very many papers in the literature. There were early argu-
ments by Bohr which cast doubt on the validity of such a classical
description of spinning particles MOT I. In particular, Bohr
showed that such a description can only be correct if the magnetic
moment of the particle is large compared to eh/mc. Similarly it was
argued that the Stern-Gerlach effect cannot be observed for electrons
as distinct from atoms. That such equations may be quite useful was
demonstrated by the experimental use of the (Thomas)-Bargmann-Michel-
Telegdi ("BMT") equation BAR i in determining the g-factor of elec-
tron and muon and has been defended further DIX i.
The B.M.T. equations are incomplete in that they neglect deri-
vatives of the electromagnetic field and do not include the Stern-
Gerlach effect. The attempt by Good GO0 i to complete these
equations to include field gradients was unsuccessful as his equa-
tions conflict with the conservation laws. Suttorp and de Groot
SUT i gave the correct equations. Van Dam and Ruygrok VAN i
recently derived the correct equations using the source terms by which
the particles act back on the field and using the i0 conservation
laws of linear and angular momentum for field plus particle.
Historically Frenkel FRE i was the first one to attempt to
write relativistic equations for spinning particles in an electro-