Constrained Dynamics
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169 Kurt Sundermeyer
Constrained Dynamics with Applications to Yang-Mills Theory, General Relativity, Classical Spin, Dual String Model
Springer-Verlag Berlin Heidelberg New York 1982
Author Kurt S u n d e r m e y e r Freie Universit~t Berlin Institut f l i t T h e o r i e d e r E l e m e n t a r t e i l c h e n A r n i m a l l e e 14, D-1000 Berlin 3 3
I S B N 3 - 5 4 0 4 1 9 4 7 - 7 S p r i n g e r - V e r l a g Berlin H e i d e l b e r g N e w Y o r k I S B N 0 - 3 8 7 4 1 9 4 7 - 7 S p r i n g e r - V e r l a g N e w Y o r k H e i d e l b e r g Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210
TABLE OF CONTENTS
O.
INTRODUCTION
I.
CLASSICAL
I
REGULAR SYSTEMS
9
i. Action, L a g r a n g i a n and H a m i l t o n i a n many degrees of freedom 2. Classical
9
field theory
3. Transformations, 4. Geometrical II.
for systems with finitely 13
symmetries
and invariances
16
formulation of classical dynamics
32
CLASSICAL S I N G U L A R SYSTEMS
38
i. L a g r a n g i a n
38
description
2. The D i r a c - B e r g m a n n
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 2. First class constraints 3. Observables
and gauge transformations
89 98
meaning of the reduction process
QUANTIZATION
OF C O N S T R A I N E D
i. Canonical
quantization
2. Functional
80
and gauge reduction
4. The geometrical IV.
and the Dirac bracket
integrals
106
SYSTEMS
109 110
for systems with constraints
-
Faddeev's method
114
3. Fradkin's method for relativistic V.
THE E L E C T R O M A G N E T I C
constraints
118
FIELD
123
i. The singular nature of the Maxwell Lagrangian,
and the field
equations
123
2. The Dirac-Bergmann 3. Gauge freedom,
VI.
algorithm for the Maxwell
theory
the reduced phase-space, and gauge fixing
126 134
4. The electromagnetic
field
coupled
to a charged boson field 146
5. The electromagnetic
field
coupled
to a spin-i/2
YANG-MILLS
field
THEORY
i. Lagrangian
161
formalism and local gauge invariance
2. The constraints 3. F a d d e e v - P o p o v
153
of Yang-Mills
procedure
theory and the Coulomb gauge
and Gribov ambiguity
163 165 170
IV 4. Ghost- and a m b i g u i t y - f r e e 5. Yang-Mills
theory with massive
VIII. THE R E L A T I V I S T I C i. L a g r a n g i a n
i. The classical
EINSTEIN'S
fields
formulation
- or - S u p e r g r a v i t y
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