Zinn-Justin: Path Integrals in Quantum Mechanics
ジン－ジュスタン: 量子力学における経路積分
・本体状態良好
・書き込み線引き押印無し
1 Gaussian integrals
1.1 Generating function
1.2 Gaussian expectation values. Wick's theorem
1.3 Perturbed gaussian measure.
Connected contributions
1.4 Expectation values.
Generating function. Cumulants
1.5 Steepest descent method
1.6 Steepest descent method:
Several variables, generating functions
1.7 Gaussian integrals: Complex matrices
Exercises
2 Path integrals in quantum mechanics
2.1 Local markovian processes
2.2 Solution of the evolution equation
for short times
2.3 Path integral representation
2.4 Explicit calculation: gaussian path integrals
2.5 Correlation functions: generating functional
2.6 General gaussian path integral and
correlation functions
2.7 Harmonic oscillator: the partition function
2.8 Perturbed harmonic oscillator
2.9 Perturbative expansion in powers of h
2.10 Semi-classical expansion
Exercises
3 Partition function and spectrum
3.1 Perturbative calculation
3.2 Semi-classical or WKB expansion
3.3 Path integral and variational principle
3.4 O(N) symmetric quartic potential for N → ∞
3.5 Operator determinants
3.6 Hamiltonian: structure of the ground state
Exercises
4 Classical and quantum statistical physics
4.1 Classical partition function. Transfer matrix
4.2 Correlation functions
4.3 Classical model at low temperature:
an example
4.4 Continuum limit and path integral
4.5 The two-point function:
perturbative expansion,
spectral representatin.
4.6 Operator formalism. Time-ordered products
Exercises
5 Path integrals and quantization
5.1 Gauge transformations
5.2 Coupling to a magnetic field:
gauge symmetry
5.3 Quantization and path integrals
5.4 Magnetic field: direct calculation
5.5 Diffusion, random walk,
Fokker-Planck equation
5.6 The spectrum of the O(2) rigid rotator
Exercises
6 Path integrals and holomorphic formalisn
6.1 Complex integrals and Wick's theorem
6.2 Holomorphic representation
6.3 Kernel of operators
6.4 Path integral: the harmonic oscillator
6.5 Path integral: general hamiltonians
6.6 Bosons: second quantization
6.7 Partition function
6.8 Bose Einstein condensation
6.9 Generalized path integrals:
the quantum Bose gas
Exercises
7 Path integrals: fermions
7.1 Grassmann algebras
7.2 Differentiation in Grassmann algebras
7.3 Integration in Grassmann algebras
7.4 Gaussian integrals and
perturbative expansion
7.5 Fermion vector space and operators
7.6 One-state hamiltonian
7.7 Many-particle states. Partition function
7.8 Path integral: one-state problem
7.9 Path integrals: Generalization
7.10 Quantum Fermi gas
7.11 Real gaussian integrals. Wick's theorem
7.12 Mixed change of variables:
Berezinian and supertrace
Exercises
8 Barrier penetration: semi-classical approximation
8.1 Quartic double-well potential and instantons
8.2 Degenerate minima:
semi-classical approximation
8.3 Collective coordinates and
gaussian integration
8.4 Instantons and metastable states
8.5 Collective coordinates: alternative method
8.6 The jacobian
8.7 Instantons: the quartic anharmonic oscillator
Exercises
9 Quantum evolution and scattering matrix
9.1 Evolution of the free particle and S-matrix
9.2 Perturbative expansion of the S-matrix
9.3 S-matrix: bosons and fermions
9.4 S-matrix in the semi-classical limit
9.5 Semi-classical approximation: one dimension
9.6 Eikonal approximation
9.7 Perturbation theory and operators
Exercises
10 Path integrals in phase space
10.1 A few elements of classical mechanics
10.2 The path integral in phase space
10.3 Harmonic oscillator.
Perturbative calculations
10.4 Lagrangians quadratic in the velocities
10.5 Free motion on the sphere or
rigid rotator
Exercises
Appendix Quantum mechanics: minimal background
A1 Hilbert space and operators
A2 Quantum evolution, symmetries and
density matrix
A3 Position and momentum.
Schrdinger equation
Bibliography
Index
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