Gauge fixing and BRST formalism in non-Abelian gauge theories

146 Pages·2007·2.52 MB·English

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0 Gauge fixing and BRST formalism
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e in non-Abelian gauge theories
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6

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h Marco Ghiotti
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p
e
h Supervisors: Dr. L. von Smekal and Prof. A. G. Williams
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Special Research Centre for the
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6 Subatomic Structure of Matter
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8 and
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. Department of Physics,
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7 University of Adelaide,
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: Australia
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i
X
r January 2007
a

To the morning star of my life,
my beloved sister Samantha
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Abstract
InthisThesiswepresentacomprehensivestudyofperturbativeandnon-perturbative
non-Abelian gauge theories in the light of gauge-fixing procedures, focusing our
attention on the BRST formalism in Yang-Mills theory. We propose first a
model to re-write the Faddeev-Popov quantisation method in terms of group-
theoreticaltechniques andthenwegiveapossiblewaytosolvetheno-gotheorem
of Neuberger for lattice Yang-Mills theory with double BRST symmetry. In the
final part we present a study of the Batalin-Vilkovisky quantisation method for
non-linear gauges in non-Abelian gauge theories.
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Statement of Originality
This work contains no material which has been accepted for the award of any
other degree or diploma in any university or other tertiary institution and, to
the best of my knowledge and belief, contains no material previously published
or written by another person, except where due reference has been made in the
text.
I give consent to this copy of my thesis, when deposited in the University
Library, being available for loan and photocopying.
Marco Ghiotti
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Acknowledgements
I wish to thank all the people who worked at the CSSM during these three
years. First of all my supervisors, Anthony Williams and Lorenz von Smekal,
who have been not only incredibly competent leading figures in my Ph.D, but
also because they have been good friends. Alongside them, I must also thank
Alex Kalloniatis, who has been my co-supervisor in the first year: I will never
forget our discussions on several issues. I also thank Sara Boffa, Sharon Johnson
andRamonaAdorjan,threeincredibleandwonderfulpersons,whoallowedmeto
go through the necessary bureocracy and the incomprehensible computer world.
Last but not the least, all the other Ph.D students who came and went at the
CSSM in these three years. Thanks to all of you, for the amazing experience
I enjoyed and lived. Yet, my biggest thank must go to my family spread out,
geographically speaking, in the four cornerstones of the world. If the usual
saying affirms that long-distance relations never work, my family is exactly the
perfect countexample. We became more united and we learned how to share our
true deep feelings through our minds. Of course, I will ever thank Mr. Meucci
for his extraordinary invention.
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Contents
1 Introduction 1
2 Symmetries and Constraints in Euclidian Gauge Theories 5
2.1 Classical constrained systems: Hamiltonian and Lagrangian for-
malism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Noether’s theorem and charge algebra . . . . . . . . . . . 9
2.2 Covariant formalism in Abelian gauge-theories: Maxwell theory 11
2.2.1 Dirac quantisation method . . . . . . . . . . . . . . . . . 14
2.2.2 Covariant Quantum Theory of Maxwell Theory . . . . . 14
2.3 Non-Abelian gauge theories: a survey into Yang-Mills theory . . 16
2.3.1 Local gauge invariance . . . . . . . . . . . . . . . . . . . 19
3 Path integrals in Y-M theory 23
3.1 Faddeev-Popov quantisation of non-Abelian gauge theories . . . 26
3.2 The Gribov Ambiguity . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 Gribov pendulum . . . . . . . . . . . . . . . . . . . . . . 35
3.3 The functional spaces and / . . . . . . . . . . . . . . . . . 39
A A G
3.4 The Gribov regions C and the fundamental modular region Λ . 41
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4 BRST formalism in Yang-Mills Theory 45
4.1 Faddeev-Popov ghosts and the birth of a new symmetry . . . . 45
4.2 BRST Noether’s charges and algebra . . . . . . . . . . . . . . . 50
4.3 Kugo-Ojima criterion and Slavnov-Taylor identities . . . . . . . 53
4.4 Another BRST operator . . . . . . . . . . . . . . . . . . . . . . 55
4.5 BRST superalgebra for linear and non-linear gauges . . . . . . . 58
5 Faddeev-Popov Jacobian in non-perturbative Y-M theory 63
5.1 Field theoretic representation for the Jacobian of FP gauge fixing 64
5.1.1 The Nicolai map and Topological Field Theory . . . . . 67
5.2 The Nicolai map in the Faddeev-Popov Jacobian . . . . . . . . . 70
5.3 A new extended BRST . . . . . . . . . . . . . . . . . . . . . . . 71
6 DecontractedDouble Lattice BRST,theCurci-Ferrari Mass and
the Neuberger Problem 75
6.1 Double BRST on the lattice . . . . . . . . . . . . . . . . . . . . 77
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6.2 The Neuberger problem . . . . . . . . . . . . . . . . . . . . . . 85
6.3 The massive Curci-Ferrari model on the lattice . . . . . . . . . . 87
6.4 Parameter Dependences . . . . . . . . . . . . . . . . . . . . . . 93
7 Batalin-Vilkovisky Formalism In Y-M Theory 97
7.1 The Appearance Of Anti-Fields . . . . . . . . . . . . . . . . . . 97
7.2 Non-linear gauges in BV formalism . . . . . . . . . . . . . . . . 99
7.3 Including Double BRST Algebra . . . . . . . . . . . . . . . . . . 103
8 Conclusions 111
9 Appendix A 113
10 Appendix B 120
Bibliography 131