NOTES ON PHYSICS
CONTENTS BOSONS 1 THE HARMONIC CRYSTAL INTRODUCTION CLASSICAL ONE-DIMENSIONAL CRYSTAL DISCRETE LIMIT THERMODYNAMIC LIMIT QUANTUM ONE-DIMENSIONAL CRYSTAL REMINISCENCES ABOUT THE HARMONIC OSCILLATOR HIGHER-DIMENSIONAL GENERALIZATIONS PROBLEMS ABSENCE OF CRYSTALLINE ORDER IN ONE AND TWO DIMENSIONS 2 BOGOLIUBOV THEORY OF A DILUTE BOSE GAS SECOND QUANTIZATION FOR BOSONS BOSE-EINSTEIN CONDENSATION AND SPONTANEOUS SYMMETRY BREAKING DILUTE BOSE GAS: OPERATOR FORMALISM AT VANISHING TEMPERATURE OPERATOR FORMALISM LANDAU CRITERION FOR SUPER?UIDITY DILUTE BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE NON-INTERACTING LIMIT ? = 0 RANDOM-PHASE APPROXIMATION BEYOND THE RANDOM-PHASE APPROXIMATION MAGNONS IN QUANTUM FERROMAGNETS AND ANTIFERROMAGNETS AS EMERGENT BOSONS 3 NON-LINEAR SIGMA MODELS NON-LINEAR SIGMA MODELS (NL?M) DE?NITION OF O(N) NL?M O(N) NL?M AS A ?ELD THEORY ON A RIEMANNIAN MANIFOLD O(N) NL?M AS A ?ELD THEORY ON A SYMMETRIC SPACE OTHER EXAMPLES OF NL?M FIXED POINT THEORIES, ENGINEERING AND SCALING DIMENSIONS, IRRELEVANT, MARGINAL, AND RELEVANT INTERACTIONS FIXED-POINT THEORIES TWO-DIMENSIONAL O(2) NL?M IN THE SPIN-WAVE APPROXIMA- TION GENERAL METHOD OF RENORMALIZATION PERTURBATIVE EXPANSION OF THE TWO-POINT CORRELATION FUNCTION UP TO ONE LOOP FOR THE TWO-DIMENSIONAL O(N) NL?M CALLAN-SYMANZIK EQUATION OBEYED BY THE SPIN-SPIN CORRELATOR FOR THE TWO-DIMENSIONAL O(N) NL?M WITH N > 2 NON-PERTURBATIVE DE?NITIONS OF THE RENORMALIZED COUPLING CONSTANT AND THE WAVE-FUNCTION RENORMALIZATION PHYSICAL INTERPRETATION OF THE CALLAN-SYMANZIK EQUATION PHYSICAL INTERPRETATION OF THE BETA FUNCTION BETA FUNCTION FOR THE D-DIMENSIONAL O(N) NL?M WITH D > 2 AND N > 2 THE MERMIN-WAGNER THEOREM FOR QUANTUM SPIN HAMILTONIANS QUANTUM SPIN COHERENT STATES AND THE O(3) QNL?M CLASSICAL O(N) NL?M WITH N > 2: ONE-LOOP RG USING THE BEREZINSKII-BLANK PARAMETRIZATION OF SPIN WAVES O(N) QNL?M: LARGE-N EXPANSION O(N) QNL?M WITH N > 2: ONE-LOOP RG USING THE BEREZINSKII-BLANK PARAMETRIZATION OF SPIN WAVES 4 KOSTERLITZ-THOULESS TRANSITION CLASSICAL TWO-DIMENSIONAL XY MODEL THE COULOMB-GAS REPRESENTATION OF THE CLASSICAL TWO-DIMENSIONAL XY MODEL EQUIVALENCE BETWEEN THE COULOMB GAS AND SINE-GORDON MODEL DE?NITIONS AND STATEMENT OF RESULTS FORMAL EXPANSION IN POWERS OF THE REDUCED MAGNETIC ?ELD SINE-GORDON REPRESENTATION OF THE SPIN-SPIN CORRELATION FUNCTION IN THE TWO-DIMENSIONAL XY MODEL STABILITY ANALYSIS OF THE LINE OF ?XED POINTS IN THE TWO- DIMENSIONAL SINE-GORDON MODEL FUGACITY EXPANSION OF N-POINT FUNCTIONS IN THE SINE-GORDON MODEL FUGACITY EXPANSION OF THE TWO-POINT FUNCTION TWO-POINT FUNCTION TO LOWEST ORDER IN H/2T TWO-POINT FUNCTION TO SECOND ORDER IN H/2T KOSTERLITZ RENORMALIZATION-GROUP EQUATIONS KOSTERLITZ RG EQUATIONS IN THE VICINITY OF X = Y = 0 CORRELATION LENGTH NEAR X = Y = UNIVERSAL JUMP OF THE SPIN STI?NESS THE CLASSICAL TWO-DIMENSIONAL RANDOM PHASE XY MODEL FERMIONS 5 NON-INTERACTING FERMIONS SECOND QUANTIZATION FOR FERMIONS THE NON-INTERACTING JELLIUM MODEL THERMODYNAMICS WITHOUT MAGNETIC ?ELD SOMMERFELD SEMI-CLASSICAL THEORY OF TRANSPORT PAULI PARAMAGNETISM LANDAU LEVELS IN A MAGNETIC ?ELD TIME-ORDERED GREEN FUNCTIONS DE?NITIONS TIME-ORDERED GREEN FUNCTIONS IN IMAGINARY TIME TIME-ORDERED GREEN FUNCTIONS IN REAL TIME APPLICATION TO THE NON-INTERACTING JELLIUM MODEL EQUAL-TIME NON-INTERACTING TWO-POINT GREEN FUNCTION FOR A FERMI GAS APPLICATION OF THE KUBO FORMULA TO THE HALL CONDUCTIVITY IN THE INTEGER QUANTUM HALL E?ECT THE HALL CONDUCTIVITY AND GAUGE INVARIANCE 6 JELLIUM MODEL FOR ELECTRONS IN A SOLID DE?NITION OF THE COULOMB GAS IN THE SCHR¨ODINGER PICTURE THE CLASSICAL THREE-DIMENSIONAL COULOMB GAS THE QUANTUM THREE-DIMENSIONAL COULOMB GAS PATH-INTEGRAL REPRESENTATION OF THE COULOMB GAS THE RANDOM-PHASE APPROXIMATION HUBBARD-STRATONOVICH TRANSFORMATION INTEGRATION OF THE ELECTRONS GAUSSIAN EXPANSION OF THE FERMIONIC DETERMINANT DIAGRAMMATIC INTERPRETATION OF THE RANDOM-PHASE APPROXIMATION GROUND-STATE ENERGY IN THE RANDOM-PHASE APPROXIMATION LINDHARD RESPONSE FUNCTION LONG-WAVELENGTH AND QUASI-STATIC LIMIT AT T = 0 LONG-WAVELENGTH AND DYNAMIC LIMIT AT T = 0 RANDOM-PHASE APPROXIMATION FOR A SHORT-RANGE INTERACTION FEEDBACK E?ECT ON AND BY PHONONS STATIC LINDHARD FUNCTION IN ONE-DIMENSIONAL POSITION SPACE LUTTINGER THEOREM REVISITED: ADIABATIC ?UX INSERTION FERMIONIC SLAVE PARTICLES 7 SUPERCONDUCTIVITY IN THE MEAN-?ELD AND RANDOM-PHASE APPROXIMATIONS PAIRING-ORDER PARAMETER PHASE OPERATOR CENTER-OF-MASS AND RELATIVE COORDINATES SCALING OF ELECTRONIC INTERACTIONS CASE OF A REPULSIVE INTERACTION CASE OF AN ATTRACTIVE INTERACTION TIME- AND SPACE-INDEPENDENT LANDAU-GINZBURG ACTION E?ECTIVE POTENTIAL AT T = 0 E?ECTIVE FREE ENERGY IN THE VICINITY OF TC MEAN-?ELD THEORY OF SUPERCONDUCTIVITY NAMBU-GORKOV REPRESENTATION E?ECTIVE ACTION FOR THE PAIRING-ORDER PARAMETER E?ECTIVE THEORY IN THE VICINITY OF T = SPATIAL TWIST AROUND |?| TIME TWIST AROUND |?| POLARIZATION TENSOR FOR A BCS SUPERCONDUCTOR E?ECTIVE THEORY IN THE VICINITY OF T = TC BCS VARIATIONAL METHOD TO SUPERCONDUCTIVITY FLUX QUANTIZATION IN A SUPERCONDUCTOR COLLECTIVE EXCITATIONS WITHIN THE RPA APPROXIMATION THE HALL CONDUCTIVITY IN A SUPERCONDUCTOR AND GAUGE INVARIANCE 8 A SINGLE DISSIPATIVE JOSEPHSON JUNCTION PHENOMENOLOGICAL MODEL OF A JOSEPHSON JUNCTION DC-JOSEPHSON E?ECT AC-JOSEPHSON E?ECT DISSIPATIVE JOSEPHSON JUNCTION CLASSICAL CALDEIRA-LEGGETT MODEL INSTANTONS IN QUANTUM MECHANICS SEMI-CLASSICAL APPROXIMATION WITHIN THE EUCLIDEAN-PATH- INTEGRAL REPRESENTATION OF QUANTUM MECHANICS APPLICATION TO A PARABOLIC POTENTIAL WELL APPLICATION TO THE DOUBLE WELL POTENTIAL APPLICATION TO THE PERIODIC POTENTIAL THE CASE OF AN UNBOUNDED POTENTIAL OF THE CUBIC TYPE THE QUANTUM-DISSIPATIVE JOSEPHSON JUNCTION DUALITY IN A DISSIPATIVE JOSEPHSON JUNCTION REGIME M Y ?1 AND M Y ?? REGIME M Y ?1 AND M Y ?? DUALITY RENORMALIZATION-GROUP METHODS DI?USIVE REGIME WHEN M Y ?1 AND M Y ?? DI?USIVE REGIME WHEN M Y ?1 AND M Y ?? CONJECTURED PHASE DIAGRAM FOR A DISSIPATIVE JOSEPHSON JUNCTION THE KONDO E?ECT: A PERTURBATIVE APPROACH THE KONDO E?ECT: A NON-PERTURBATIVE APPROACH 9 ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME ABELIAN BOSONIZATION OF THE THIRRING MODEL FREE-?ELD ?XED POINT IN THE MASSIVE THIRRING MODEL THE U(1) AXIAL-GAUGE ANOMALY ABELIAN BOSONIZATION OF THE MASSLESS THIRRING MODEL ABELIAN BOSONIZATION OF THE MASSIVE THIRRING MODEL APPLICATIONS SPINLESS FERMIONS WITH E?ECTIVE LORENTZ AND GLOBAL U(1) GAUGE SYMMETRIES QUANTUM XXZ SPIN-1/2 CHAIN SINGLE IMPURITY OF THE MASS TYPE QUANTUM CHIRAL EDGE THEORY TWO-POINT CORRELATION FUNCTION IN THE MASSLESS THIRRING MODEL