場論における経路積分<br>Path Integrals in Field Theory : An Introduction (Advanced Texts in Physics) (2004. XII, 213 p. w. 19 figs. 23,5 cm)

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場論における経路積分
Path Integrals in Field Theory : An Introduction (Advanced Texts in Physics) (2004. XII, 213 p. w. 19 figs. 23,5 cm)

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  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 200 p.
  • 商品コード 9783540403821

基本説明

Concise textbook intended as a primer on path integral formalism both in classical and quantum field theories, although emphasis is on the latter.

Full Description


Concise textbook intended as a primer on path integral formalism both in classical and quantum field theories, although emphasis is on the latter. It is ideally suited as an intensive one-semester course, delivering the basics needed by readers to follow developments in field theory. Path Integrals in Field Theory paves the way for both more rigorous studies in fundamental mathematical issues as well as for applications in hadron, particle and nuclear physics, thus addressing students in mathematical and theoretical physics alike. Assuming some background in relativistic quantum theory (but none in field theory), it complements the authors monograph Fields, Symmetries, and Quarks (Springer, 1999).

Table of Contents

Part I Non-Relativistic Quantum Theory
1 The Path Integral in Quantum Theory 3 (12)
1.1 Propagator of the Schr inger Equation 3 (2)
1.2 Propagator as Path Integral 5 (4)
1.3 Quadratic Hamiltonians 9 (4)
1.3.1 Cartesian Metric 9 (2)
1.3.2 Non-Cartesian Metric 11 (2)
1.4 Classical Interpretation 13 (2)
2 Perturbation Theory 15 (12)
2.1 Free Propagator 15 (2)
2.2 Perturbative Expansion 17 (5)
2.3 Application to Scattering 22 (5)
3 Generating Functionals 27 (12)
3.1 Groundstate-to-Groundstate Transitions 27 (5)
3.1.1 Generating Functional 31 (1)
3.2 Functional Derivatives of Gs-Gs 32 (7)
Transition Amplitudes
Part II Relativistic Quantum Field Theory
4 Relativistic Fields 39 (14)
4.1 Equations of Motion 39 (7)
4.1.1 Examples 41 (5)
4.2 Symmetries and Conservation Laws 46 (7)
4.2.1 Geometrical Space-Time Symmetries
4?
4.2.2 Internal Symmetries 49 (4)
5 Path Integrals for Scalar Fields 53 (6)
5.1 Generating Functional for Fields 53 (6)
5.1.1 Euclidean Representation 56 (3)
6 Evaluation of Path Integrals 59 (16)
6.1 Free Scalar Fields 59 (8)
6.1.1 Generating Functional 59 (2)
6.1.2 Feynman Propagator 61 (3)
6.1.3 Gaussian Integration 64 (3)
6.2 Interacting Scalar Fields 67 (8)
6.2.1 Stationary Phase Approximation 67 (3)
6.2.2 Numerical Evaluation of Path 70 (2)
Integrals
6.2.3 Real Time Formalism 72 (3)
7 Transition Rates and Green's Functions 75 (10)
7.1 Scattering Matrix 75 (2)
7.2 Reduction Theorem 77 (8)
7.2.1 Canonical Field Quantization 77 (1)
7.2.2 Derivation of the Reduction Theorem 78 (7)
8 Green's Functions 85 (12)
8.1 n-point Green's Functions 85 (4)
8.1.1 Momentum Representation 86 (1)
8.1.2 Operator Representations 86 (3)
8.2 Free Scalar Fields 89 (3)
8.2.1 Wick's Theorem 89 (2)
8.2.2 Feynman Rules 91 (1)
8.3 Interacting Scalar Fields 92 (5)
8.3.1 Perturbative Expansion 93 (4)
9 Perturbative   Theory 97 (28)
9.1 Perturbative Expansion of the 97 (4)
Generating Function
9.1.1 Generating Functional up to O(g) 98 (3)
9.2 Two-Point Function 101(5)
9.2.1 Terms up to O(g0) 101(1)
9.2.2 Terms up to O(g) 102(2)
9.2.3 Terms up to O(g2) 104(2)
9.3 Four-Point Function 106(4)
9.3.1 Terms up to O(g) 106(1)
9.3.2 Terms up to O(g2) 107(3)
9.4 Divergences in n-Point Functions 110(15)
9.4.1 Power Counting 110(3)
9.4.2 Dimensional Regularization of   113(6)
Theory
9.4.3 Renormalization 119(6)
10 Green's Functions for Fermions 125(16)
10.1 Grassmann Algebra 125(9)
10.1.1 Derivatives 126(2)
10.1.2 Integration 128(6)
10.2 Green's Functions for Fermions 134(7)
10.2.1 Generating Functional for Fermions 134(4)
10.2.2 Reduction Theorem for Fermions 138(1)
10.2.3 Green's Functions 139(2)
11 Interacting Fields 141(16)
11.1 Feynman Rules 141(4)
11.1.1 Fermion Loops 143(2)
11.2 Wick's Theorem 145(2)
11.3 Bosonization of Yukawa Theory 147(10)
11.3.1 Perturbative Expansion 150(7)
Part III Gauge Field Theory
12 Path Integrals for QED 157(10)
12.1 Gauge Invariance in Abelian Free Field 157(4)
Theories
12.2 Generating Functional 161(1)
12.3 Gauge Invariance in QED 162(2)
12.4 Feynman Rules of QED 164(3)
13 Path Integrals for Gauge Fields 167(22)
13.1 Non-Abelian Gauge Fields 167(4)
13.2 Generating Functional 171(5)
13.3 Gauge Fixing of C 176(2)
13.4 Faddeev Popov Determinant 178(6)
13.4.1 Explicit Forms of the FP 180(2)
Determinant
13.4.2 Ghost Fields 182(2)
13.5 Feynman Rules 184(5)
14 Examples for Gauge Field Theories 189(4)
14.1 Quantum Chromodynamics 189(1)
14.2 Electroweak Interactions 190(3)
Units and Metric 193(4)
A.1 Units 193(1)
A.2 Metric and Notation 194(3)
Functionals 197(6)
B.1 Definition 197(1)
B.2 Functional Integration 197(4)
B.2.1 Gaussian Integrals 198(3)
B.3 Functional Derivatives 201(2)
Renormalization Integrals 203(4)
Gaussian Grassmann Integration 207(2)
References 209(2)
Index 211