Table Of ContentLecture Notes
ni Physics
detidE yb .H Araki, Kyoto, .J Ehlers, M0nchen, .K Hepp, hcirJiZ
.R Kippenhahn, ,nehcnQM H.A. ,relIJimnedieW Heidelberg
dna .J Zittartz, Kbln
961
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with Applications to
Yang-Mills Theory, General Relativity,
Classical Spin, Dual String Model
Springer-Verlag
Berlin Heidelberg New York 1982
Author
Kurt Sundermeyer
Freie Universit~t Berlin
Institut flit Theorie der Elementarteilchen
Arnimallee 14, D-1000 Berlin 33
ISBN 3-54041947-7 Springer-Verlag Berlin Heidelberg New York
ISBN 0-38741947-7 Springer-Verlag New York Heidelberg Berlin
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TABLE OF CONTENTS
O. INTRODUCTION I
I. CLASSICAL REGULAR SYSTEMS 9
i. Action, Lagrangian and Hamiltonian for systems with finitely
many degrees of freedom 9
.2 Classical field theory 13
.3 Transformations, symmetries and invariances 16
4. Geometrical formulation of classical dynamics 32
II. CLASSICAL SINGULAR SYSTEMS 38
.i Lagrangian description 38
2. The Dirac-Bergmann algorithm for systems with constraints 45
.3 A simple example 60
4. Field theory with constraints 65
5. "Constrained Dynamics" and presymplectic geometry 72
III. THE REDUCED PHASE-SPACE 79
.i Second class constraints and the Dirac bracket 80
2. First class constraints and gauge transformations 89
.3 Observables and gauge reduction 98
4. The geometrical meaning of the reduction process 106
IV. QUANTIZATION OF CONSTRAINED SYSTEMS 109
.i Canonical quantization 110
.2 Functional integrals for systems with constraints -
Faddeev's method 114
.3 Fradkin's method for relativistic constraints 118
V. THE ELECTROMAGNETIC FIELD 123
i. The singular nature of the Maxwell Lagrangian, and the field
equations 123
2. The Dirac-Bergmann algorithm for the Maxwell theory 126
3. Gauge freedom, the reduced phase-space, and gauge fixing 134
4. The electromagnetic field coupled to a charged boson field 146
.5 The electromagnetic field coupled to a spin-i/2 field 153
VI. YANG-MILLS THEORY 161
.i Lagrangian formalism and local gauge invariance 163
.2 The constraints of Yang-Mills theory and the Coulomb gauge 165
.3 Faddeev-Popov procedure and Gribov ambiguity 170
VI
4. Ghost- and ambiguity-free quantization of Yang-Mills theory 178
5. Yang-Mills theory with massive fields 183
VIII. THE RELATIVISTIC PARTICLE 187
.i Lagrangian and Hamiltonian formulation 187
2. The spinning particle - or - Supergravity in one dimension 194
VIII.THE RELATIVISTIC STRING 205
.i The classical relativistic s£ring 205
.2 Gauge reduction and quantization 215
IX. EINSTEIN'S THEORY OF GRAVITATION 227
1. Different forms of the action - The Lagrangian in terms of
embedding 228
.2 Constraints and the Hamiltonian 237
.3 Towards the true degrees of freedom 244
.4 Miscellaneous remarks 252
APPENDICES 260
App.A: Differentiable manifolds 260
App.B: Feynman path-integrals 271
App.C: Grassmann algebra and Fermi degrees of freedom 280
App. D: Parametrization-invariant theories 287
App.E: Definitions and formulas from Riemannian geometry 297
SYNOP S I S 303
REFERENCES 312
O. INTRODUCTION
For a nonrelativistic system with N degrees of freedom which has a
(~ ~9 )~
Lagrange function L the Euler-Lagrange equations
(z = ...,
are explicitly
with
These can only be solved uniquely for the accelerations if W~detlWik %0.
The Lagrangian is called singular if the Hessian W vanishes, and re-
gular otherwise. - If one defines as usual the momenta
9k
one finds again the condition W • O to be necessary for solving these
equations uniquely for the q~ in terms of Pi and q~, ,t which is
a must in order to perform the Legendre transformation from the Lagrange
to the Hamilton function of the system. In the singular case the coordi-
nates and momenta are not independent, there are constraints among them.
Systems with this property are the subject of these lecture notes.
They are equivalently called "singular Lagrangian systems", "constrained
Hamiltonian systems", sometimes also "degenerate systems". Although the
property of being singular is a property of the Lagrangian, I will be
sloppy and call a system itself regular (singular) if its dynamics is
derived from a regular (singular) Lagrangian.
Hardly any textbook on Classical Mechanics cares about W being zero
or not. And there are good reasons for this negligence. It is true that
in a formal sense also "normal" (holonomic and non-holonomic) constraints
render a Lagrangian singular if Lagrange multipliers are considered as
dynamical variables. But this point of view is artificial and so are
all singular Lagrange functions which you might construct. - So why
lectures on "Constrained Dynamics"? - Well, go one step ahead from non-
relativistic to relativistic mechanics. Then you have to take into
account the mass-shell condition p2+ m 2 = O (in "my" metric and units)
demonstrating that not all momenta are independent; and this is a con-
straint! - But that's not the final peal of wisdom. Extend the express-
ions above to infinitely many degrees of freedom, that is to field
theory. Now take for instance electrodynamics. In its covariant formu-
lation where the Lagrange density is a functional of the vector potent-
ials ~ , the momentum canonically conjugate to T o vanishes; again
a constraint. (To be correct these are even~ constraints, one for
each space point.) - As will be explained in detail in these lectures
the singular nature of the Maxwell Lagrangian is rooted in its local
gauge invariance. For reasons of local invariance also Einstein's
theory of gravitation will unavoidably have constraints. But not only
theories with this kind of invariance will be systems with constraints.
For instance any field theory in light-cone description is singular.
And since Lagrangians for Fermi fields are linear in derivatives they are
singular too. - This collection of only few and respectable examples
should be motivation enough toget you interested in "Constrained
Dynamics".
Attempts to handle singular systems date back thirty years. It is due
to P.G.Bergmann and collaborators to clarify the connection between
constraints and invariance properties of a theory. This is especially
interesting for field theories. However the methods to investigate
constrained systems can already be studied for a finite number of degrees
of freedom. The credit for this goes to P.A.M. Dirac who set up a forma-
lism to treat the constraints selfconsistently and which allowS to
entangle their meaning. - Right from the early days one had in mind the
quantization of constrained Hamiltonian systems. Although from the work
of Bergmann, Dirac and others a pathway to canonical quantization was
opened up, the methods were hard to apply to theories of physical inte-
rest. It was a big step ahead when during the sixties constraints were
incorporated into Feynman path-integral quantization - connected today
with the names of Faddeev and Popov. - With the advent of supersymmetry
there was growing interest in the generalization of the methods to
fermionic degrees of freedom. In addition there has been considerable
progress in formulating singular systems in the language of symplectic
geometry and global analysis.
You notice the big range of "Constrained Dynamics". It leads from
elementary notions in point mechanics to present day problems in
theoretical physics. I will try to cover these broad aspects in these
lecture notes since a complete presentation does not exist to my know-
ledge. Besides the by now classic lectures of Dirac (1964) there are
excellent reviews either on the treatment of constraints in systems
with a finite number of degrees of freedom only, Sudarshan and Mukunda
(1974), or mainly on examples from field theory, Hanson, Regge and
Teitelboim (1976). Since the above three references will be cited
frequently they get the shorthand notation
D - Dirac (1964)
~ Sudarshan and Mukunda (1974)
-
~R~ - Hanson, Regge and Teitelboim (1976).
But even if you take these together some pieces are missing which I
would like to supply. At the one end of the spectrum the connection
between constraints and invariances will be added, this being so impor-
tant because gauge invariance is a leitmotiv in current theoretical
research. And at the other end both the methods with functional inte-
grals and the treatment of Fermi systems shall be included since they
became of outstanding interest during the last years. -Most of the
material has been published. My aim is to tie the available information
on "Constrained Dynamics" together and actualize the older reviews.
From the Table of Contents you can see that the book roughly divides
into two parts: I prefer to present the techniques for the investiga-
tions of singular systems (both classical as well quantized) first
(chap. I-IV) before they are applied to theories or models in the last
five chapters. In this way for each example the full machinery will be
at our disposal, although in some cases we do not have to turn it all
on. Someone else might take the view that a more pedagogical way would
be the development of methods in the course of the analysis of examples.
Then one would see why recipes worked out up to a certain level need
refinement. In turn this could invoke the danger that general features
of the exhibited methods get lost within the discussion of specific
models.
Someone who works in any of the fields treated as examples in the
last chapters will be aware of their character as singular systems, and
he may read the lecture notes backwards to understand the formal basis
of the methods, or crosswise to see the relations to other fields.
Since in writing this book I had in mind a physicist who is interested
in doing "practical" calculations within a specific example, most of
these notes are written in a "conventional" language. The only exception
are the last sections of each of the first three chapters where the
geometrical aspects and formal structures are described. Depending on
your education you may in a first reading leave out these parts. - It
is true that he who gets confronted with "Constrained Dynamics" for the
first time will see the flesh on the bones of the formalisms only very
late. Nevertheless any student who attended a course in theoretical
physics should be able to follow the first five chapters. For his benefit
I added an artificial example to spice the somewhat dry calculations.
With some patience he will be guided through the other chapters to Some
current problems. Of course he should not expect to find in this book
the motivation to deal with, say, Yang-Mills theory or the Dual String
model. They would be topics of an extra series of lectures. But he has
a chance of studying them under the heading of "Constrained Dynamics"
And since I took the name "lecture notes'! serious they are redundant
on purpose. I also tried to make them as self-contained as reasonable.
But the reader should become aware that some of the chapters and appen-
dices like for instance "Classical Regular Systems" or "Differentiable
Manifolds" do not pretend to give a resum~ of their heading. Only those
topics are included which will be used in the context of singular systems.
In this sense, in chapt. I out of Lagrange and Hamilton theory for
classical regular systems (not only for point mechanical systems but in
part also for field theories) those items are dealt with which are picked
up in later chapters and get another interpretation for singular systems.
An important part is the section on symmetries and invariances. There
both Noether theorems are derived. Logically the second of these theorems
would belong to the next chapter since it characteristically applies to
singular systems. - After the treatment of Lagrange constraints in
sect. II.l, the phase-space formulation of a singular system is described
in sect. 2. Here the terminology of the pioneers in this field, terms
like "weak and strong equality", "primary and secondary constraints",
"first and second class quantity" is presented. A simple example (with-
out any physical background) is chosen in such a manner that it shows
characteristics of topics discussed up to this stage and typical features
exploited later on. Indeed the methods worked out up to that level are
sufficient for nonrelativistic classical point mechanical systems for
which they are of hardly any relevance. It needs more efforts to trans-
cribe the methods to field theories. There peculiarities can arise
originating from the presence of non-algebraic constraints and the
occurence of surface integrals, as is explained in sect.4. - Chapt. III
contrasts three different spaces, the full phase space ~ , the constraint
space 7~ , a subspace in ~ (at least locally) defined by the con-
straints, and the reduced phase space ~ spanned by the true degrees of
freedom. First and second class constraints have different properties
as generators of transformations. Whereas first class constraints
mediate canonical transformations that stay on q , second class ones
do not. With the help of the Dirac bracket the second class constraints
can be taken care of. This is a generalized Poisson bracket and some of
its properties and its meaning are discussed in the first section. Due
to folklore first class constraints are called "generators of gauge
transformations", but I rather prefer to give another name to them since
in principle not all first class constraints generate transformations
that leave the system unchanged and since they do in general not direct-
ly correspond to local gauge transformations in gauge theories of Yang-
Mills type or to coordinate transformations in generally covariant theo-
ries; this dubiousness is brought up in sect.2. In only few cases and un-
luckily in most of the theories under current investigation, one is able
to isolate and identify the true degrees of freedom. In these instance
one can try to reduce the dynamics from~ by choosing specific gauges.
One way of doing so is an implicit gauge choice in which additional con-
straints are introduced. - As mentioned previously, the curiosity for
looking into the structure of systems with constraints was provoked by
the interest in quantizing theories of that type. One can try to apply
and extend the "rules" of canonical quantization to singular systems.
There is however no general precept, and the only justification for a
quantization method is its success. Another such method is Faddeev's
path-integral quantization for constrained systems. However it is still
not the full story if one wants to retain relativistic covariance of a
theory, and the more recent progress achieved in this respect by Fradkin
and coworkers is displayed in the last section of chapt. IV.
After these chapters on the skeleton of "Constrained Dynamics" it
will come to life in examples. The selection is guided by their impor-
tance as physical models and by their pedagogical usefulness. Common
features of these examples shall be stressed without wiping away intrin-
sic differences. Most of them are treated also in the review of HRT.
So I cannot avoid duplications of certain points discussed by these
authors. But I add topics either because they became relevant in the
meantime or serve as clarification of the methods exhibited in foregoing
chapters.
In chapt.V on electrodynamics different formulations (first order
instant form, light-cone and second order form) are compared (sects.
1,2) to illustrate how the nature and number of constraints depend on
the formalism chosen. Electrodynamics has the virtue of being a non-
trivial example in that a clean application of the general methods of
chapts. I-IV is feasible. So for instance the structure of the reduced
phase-space (sect. )3 can be made transparent either by choosing gauge
constraints or without doing so. Since free theories are not a picture
of Nature but only an exercise ground for theorists, the coupling of
the electromagnetic field to other fields within the context of
"Constrained Dynamics" is described, too. Especially the coupling to
a spin-i/2 field ("classical quantum electrodynamics") is a nice and
instructive exercise for dealing with Grassmann variables in a res-
pectable theory.- The first two sections of chapt.VI on Yang-Mills
theory superficially look very similar to Maxwell's theory. But the for-
mer differs from the Abelian theory in gauge specification as is demon-
strated by the existence of Gribov a~biguities. And whereas Maxwell's
theory can be quantized canonically this procedure encounters obstacles
for the Yang-Mills theory. Indeed it is this case where Faddeev and Popov
overcame some difficulties using functional integral methods. Alternative
proposals for the quantization of Yang-Mills field are discussed in sect.
4. Massive Yang-Mills theory (the last section of this chapter) is much
simpler than the massless version; it constitutes an example with second
class constraints only.
The examples of the last three chapters have in common that they are
generally covariant (or parametrization-invariant). In these, geometrical
parameters are raised to the status of dynamical variables. - The sim-
plest one is the relativistic particle, chapt.VII. At the spinning par~ m
cle I want to introduce some basic notions from supersymmetry and super-
gravity since these themselves are not treated in this book. Also I think
it is quite intriguing to see the Dirac equation for spin-i/2 particles
arising as a constraint. - In chapt.VIII the relativistic string is
dealt with, a model for strong interactions which currently is under dis-
cussion in approximations to quantum chromodynamics. The classical string
does not present major difficulties, however the question of the exist-
ence of a covariant quantized relativistic string in four dimensions
is still disputed.- The last chapter is concerned with Einstein's
theory of gravitation. Only because of its complexity it is situated
at the end, while historically it has been the starting-point for the
