eBook ISBN:  9781470401856 
Product Code:  MEMO/126/600.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $27.00 
eBook ISBN:  9781470401856 
Product Code:  MEMO/126/600.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $27.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 126; 1997; 85 ppMSC: Primary 81
This work presents a rigorous account of quantum gauge field theory for bundles (both trivial and nontrivial) over compact surfaces. The Euclidean quantum field measure describing this theory is constructed and loop expectation values for a broad class of Wilson loop configurations are computed explicitly. Both the topology of the surface and the topology of the bundle are encoded in these loop expectation values. The effect of wellbehaved areapreserving homeomorphisms of the surface is to take these loop expectation values into those for the pullback bundle. The quantum gauge field measure is constructed by conditioning an infinitedimensional Gaussian measure to satisfy constraints imposed by the topologies of the surface and of the bundle. Holonomies, in this setting, are defined by interpreting the usual paralleltransport equation as a stochastic differential equation.
ReadershipGraduate students, research mathematicians and physicists interested in quantum field theory, gauge theory or stochastic geometry.

Table of Contents

Chapters

1. Introduction

2. Terminology and basic facts

3. The structure of bundles and connections over compact surfaces

4. Quantum gauge theory on the disk

5. A conditional probability measure

6. The YangMills measure

7. Invariants of systems of curves

8. Loop expectation values I

9. Some tools for the abelian case

10. Loop expectation values II

Appendix


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This work presents a rigorous account of quantum gauge field theory for bundles (both trivial and nontrivial) over compact surfaces. The Euclidean quantum field measure describing this theory is constructed and loop expectation values for a broad class of Wilson loop configurations are computed explicitly. Both the topology of the surface and the topology of the bundle are encoded in these loop expectation values. The effect of wellbehaved areapreserving homeomorphisms of the surface is to take these loop expectation values into those for the pullback bundle. The quantum gauge field measure is constructed by conditioning an infinitedimensional Gaussian measure to satisfy constraints imposed by the topologies of the surface and of the bundle. Holonomies, in this setting, are defined by interpreting the usual paralleltransport equation as a stochastic differential equation.
Graduate students, research mathematicians and physicists interested in quantum field theory, gauge theory or stochastic geometry.

Chapters

1. Introduction

2. Terminology and basic facts

3. The structure of bundles and connections over compact surfaces

4. Quantum gauge theory on the disk

5. A conditional probability measure

6. The YangMills measure

7. Invariants of systems of curves

8. Loop expectation values I

9. Some tools for the abelian case

10. Loop expectation values II

Appendix