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E-raamat: Lattice Quantum Field Theory Of The Dirac And Gauge Fields: Selected Topics

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  • Formaat: 316 pages
  • Ilmumisaeg: 30-Jul-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811209710
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Lattice Quantum Field Theory Of The Dirac And Gauge Fields: Selected TopicsSuurem pilt
  • Formaat: 316 pages
  • Ilmumisaeg: 30-Jul-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811209710
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Quantum Chromodynamics is the theory of strong interactions: a quantum field theory of colored gluons (Yang-Mills gauge fields) coupled to quarks (Dirac fermion fields). Lattice gauge theory is defined by discretizing spacetime into a four-dimensional lattice and entails defining gauge fields and Dirac fermions on a lattice. The applications of lattice gauge theory are vast, from the study of high-energy theory and phenomenology to the numerical studies of quantum fields.Lattice Quantum Field Theory of the Dirac and Gauge Fields: Selected Topics examines the mathematical foundations of lattice gauge theory from first principles. It is indispensable for the study of Dirac and lattice gauge fields and lays the foundation for more advanced and specialized studies.
Acknowledgments vii
Preface ix
1 Synopsis
1(6)
1.1 Mathematical Background
2(1)
1.2 Lattice Gauge Field
3(1)
1.3 Lattice Dirac Field
4(1)
1.4 Lattice Gauge Theory Hamiltonian
4(3)
Mathematical Background
7(104)
2 SU (N) Compact Lie Groups
9(20)
2.1 Introduction
9(1)
2.2 Lie Algebras
10(2)
2.3 Vielbein of SU (N)
12(1)
2.4 Metric on SU (N) Group Space
13(2)
2.4.1 SU(2) metric
14(1)
2.5 The Invariant Haar Measure and Delta Function
15(2)
2.5.1 Delta function on group space
16(1)
2.5.2 SU(2) measure
16(1)
2.5.3 SU(2) measure: invariant
17(1)
2.6 Campbell-Baker-Hausdorff (CBH) Formula
17(1)
2.7 Irreducible Representations of SU(TV)
18(2)
2.8 Peter-Weyl Theorem
20(1)
2.9 Lie Algebra Generators: Differential Operators
21(2)
2.10 Vielbein eab and fab for SU(2)
23(1)
2.11 Left and Right Invariant Generators
24(2)
2.11.1 Differential realization of 50(3) generators on state space
26(1)
2.12 SU(Af) Character Function Expansion
26(2)
2.13 Summary
28(1)
3 SU (N) Kac-Moody Algebra
29(26)
3.1 Introduction
29(1)
3.2 Kac-Moody Algebras
30(1)
3.3 Functional Differentiation
31(2)
3.3.1 Chain rule
32(1)
3.4 Kac-Moody Generators
33(5)
3.4.1 Kac-Moody generator and the 2-cocycle
36(2)
3.5 Chiral Field
38(2)
3.6 The WZW Lagrangian
40(3)
3.7 SU{2) Loop Group
43(1)
3.8 SU(2) Kac-Moody Algebra
44(2)
3.9 Kac-Moody Commutation Equations
46(3)
3.10 Virasoro Generator: Point-Split Regularization
49(4)
3.11 Summary
53(1)
3.12 Appendix
54(1)
4 SU (N) Path Integrals
55(26)
4.1 Introduction
55(1)
4.2 Hamiltonian Operator
56(2)
4.2.1 U(1) and SU(2) evolution kernel
57(1)
4.3 Lattice Action for SU(AT)
58(1)
4.4 Classical Paths and Winding Number
59(4)
4.5 U(1) Path Integral
63(1)
4.6 U(1) Path Integral: Classical Paths
64(1)
4.7 St/(AO Continuum Action
65(1)
4.8 SU (2) Path Integration
66(7)
4.8.1 SU (3) path integration
70(2)
4.8.2 SU (M) path integration
72(1)
4.9 Summary
73(1)
4.10 Appendix: Continuum Limit of SU(M) Path Integral
74(7)
5 SU(3) Character Functions
81(6)
5.1 Casimir Operator for SU(3)
81(2)
5.2 Evolution Kernel for SU(3)
83(1)
5.3 Character Functions of SU(3)
84(2)
5.4 Summary
86(1)
6 Fermion Calculus
87(24)
6.1 Fermionic Variables
88(1)
6.2 Fermion Integration
89(1)
6.3 Fermion Hilbert Space
90(4)
6.3.1 Fermionic completeness equation
92(1)
6.3.2 Fermionic momentum operator
93(1)
6.4 Antifermion State Space
94(1)
6.5 Fermion and Antifermion Hilbert Space
95(1)
6.6 Real and Complex Fermions: Gaussian Integration
96(4)
6.6.1 Complex Gaussian fermions
98(2)
6.7 Fermionic Path Integral and Hamiltonian
100(2)
6.8 Fermionic Operators
102(1)
6.9 Fermion-Antifermion Hamiltonians
103(1)
6.10 A Quadratic Hamiltonian
104(1)
6.10.1 Orthogonality and completeness
104(1)
6.11 Fermion-Antifermion Lagrangian
105(2)
6.12 Fermionic Transition Probability Amplitude
107(1)
6.13 Quark Confinement
108(1)
6.14 Summary
109(2)
LATTICE GAUGE FIELD
111(98)
7 Non-Abelian Lattice Gauge Field
113(18)
7.1 Introduction
113(3)
7.2 The Weak Coupling Approximation
116(2)
7.3 Gauge-Fixing the Lagrangian
118(1)
7.4 Zero Mode
118(1)
7.5 Gauge-Fixed Path Integral
119(1)
7.6 The Faddeev-Popov Counter-Term
120(1)
7.7 Abelian Gauge-Fixed Path Integral
121(2)
7.8 Lattice Faddeev-Popov Non-Abelian Ghost Action
123(3)
7.9 Lattice BRST Symmetry
126(4)
7.10 Summary
130(1)
8 Abelian Lattice Gauge Field in d = 3
131(12)
8.1 Introduction
131(1)
8.2 Strong Coupling Representation
132(2)
8.3 Weak Coupling Representation
134(1)
8.4 Gauge Invariance
135(2)
8.5 Wilson Loop
137(1)
8.6 Phase Transition
137(3)
8.6.1 Mean field approximation
139(1)
8.7 Summary
140(1)
8.8 Appendix A
141(1)
8.9 Appendix B
142(1)
9 Lattice Gauge Field Mass Renormalization
143(14)
9.1 Introduction
143(1)
9.2 The Propagator and Mass Renormalization
144(2)
9.3 The Computational Scheme
146(1)
9.4 Determination of mα and mμ
147(1)
9.5 Determination of mc for Sc [ θ + B]
148(1)
9.6 Expansion of the Gauge Field Action
149(2)
9.7 Determination of mA for S (θ + B) + Sα[ B]
151(2)
9.8 One-Loop Mass Renormalization
153(1)
9.9 Slavnov-Taylor Identity
154(1)
9.10 Summary
155(2)
10 Gauge Field Block-Spin Renormalization
157(30)
10.1 Introduction
157(1)
10.2 Two-Dimensional Lattice Gauge Field
158(1)
10.3 The Renormalization Group Transformation
159(6)
10.3.1 Renormalization group and fixed points
165(1)
10.4 The Recursion Equation for d Dimensions
165(4)
10.5 Strong Coupling Approximation for SU(2)
169(1)
10.6 Weak Coupling Expansion for SU(2)
170(4)
10.7 Weak Coupling Approximation for d = 4 + e
174(1)
10.8 Weak Coupling 517(2) Gauge Field: β-Function
175(1)
10.9 Numerical Solution of the Recursion Equation
176(6)
10.9.1 Change of integration variable
176(2)
10.9.2 Numerical algorithm
178(1)
10.9.3 Total grid size S(I)
179(3)
10.10 Numerical Results
182(2)
10.11 Summary: Confinement and Asymptotic Freedom
184(1)
10.12 Appendix
185(2)
11 Lattice Gauge Field Hamiltonian
187(22)
11.1 Lattice Gauge Field Hamiltonian
188(3)
11.2 Gauge-Fixed Chromoelectric Operator
191(3)
11.3 Gauss's Law
194(3)
11.4 Gauge-Fixed Lattice Gauge Field Hamiltonian
197(1)
11.5 Hamiltonian and Covariant Gauge: Faddeev-Popov Quantization
198(1)
11.6 Ghost State Space and Hamiltonian
199(3)
11.6.1 BRST cohomology: state space
201(1)
11.7 BRST Charge Qb
202(2)
11.8 QB and State Space
204(4)
11.8.1 Gupta-Bleuler condition
206(2)
11.9 Summary
208(1)
LATTICE DIRAC FIELD
209(60)
12 Dirac Lattice Path Integral
211(32)
12.1 Introduction
211(1)
12.2 Dirac Field Coordinates
212(1)
12.3 Dirac Lattice Lagrangian
213(3)
12.4 Lattice Fermions and Chiral Symmetry
216(3)
12.5 Dirac Field: Boundary Conditions
219(1)
12.6 Dirac Fermionic State Space
220(3)
12.7 Hilbert Space Metric and Transfer Matrix
223(4)
12.8 Dirac Lattice Hamiltonian
227(2)
12.9 Lattice Path Integral
229(4)
12.9.1 Normalization constant
232(1)
12.10 Evolution Kernel
233(2)
12.11 Energy Eigenfunctions
235(3)
12.12 Propagator
238(3)
12.13 Summary
241(2)
13 Dirac Hamiltonian
243(26)
13.1 Fermionic Operators
244(1)
13.2 Lattice Dirac Hamiltonian
245(2)
13.3 Continuum Hilbert Space
247(1)
13.4 Continuum Hamiltonian
248(6)
13.5 Dirac Field's Energy Eigenfunctionals
254(2)
13.6 Dirac Charge Operator
256(3)
13.6.1 Momentum and spin operators
258(1)
13.7 Finite Time Dirac Action
259(2)
13.8 Continuum Evolution Kernel
261(2)
13.9 Evolution Kernel: General Quadratic Case
263(2)
13.10 Evolution Kernel: Dirac Hamiltonian
265(2)
13.10.1 Chiral charge operator
267(1)
13.11 Summary
267(2)
Lattice Gauge Theory Hamiltonian
269(24)
14 Lattice Gauge Theory Hamiltonian
271(22)
14.1 Introduction
271(1)
14.2 Finite Time Action and Transfer Matrix
272(3)
14.3 Axial Gauge: Un0=1
275(1)
14.4 Noncanonical Fermion Anticommutation Equations
276(1)
14.5 Lattice Gauge Theory Hamiltonian
277(2)
14.6 Canonical Fermions
279(1)
14.7 Color Charge Operator and Gauss's Law
280(4)
14.8 Lattice Action from Lattice Hamiltonian
284(2)
14.9 Summary
286(1)
14.10 Appendix A: Fermion Calculus with Gauge Field
286(2)
14.11 Appendix B: Matrix M
288(1)
14.12 Appendix C: Lagrangian for an Asymmetric Lattice
289(2)
14.13 Appendix D: Classical Continuum Limit
291(2)
Bibliography 293(4)
Index 297
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