PATH INTEGRALS
CONTENTS INTRODUCTION 1 NOTATIONAL CONVENTIONS 9 1 PATH INTEGRALS IN CLASSICAL THEORY 12 1 1 BROWNIAN MOTION INTRODUCTION TO THE CONCEPT OF PATH INTEGRATION 12 1 1 1 BROWNIAN MOTION OF A FREE PARTICLE DIFFUSION EQUATION AND MARKOV CHAIN 12 1 1 2 WIENER S TREATMENT OF BROWNIAN MOTION WIENER PATH INTEGRALS 22 1 1 3 WIENER S THEOREM AND THE INTEGRATION OF FUNCTIONALS 29 1 1 4 METHODS AND EXAMPLES FOR THE CALCULATION OF PATH INTEGRALS 36 1 1 5 CHANGE OF VARIABLES IN PATH INTEGRALS 45 1 1 6 PROBLEMS 49 1 2 WIENER PATH INTEGRALS AND STOCHASTIC PROCESSES 56 1 2 1 A SHORT EXCURSION INTO THE THEORY OF STOCHASTIC PROCESSES 56 1 2 2 BROWNIAN PARTICLES IN THE ELD OF AN EXTERNAL FORCE TREATMENT BY FUNCTIONAL CHANGE OF VARIABLES IN THE PATH INTEGRAL 63 1 2 3 BROWNIAN PARTICLES WITH INTERACTIONS 66 1 2 4 BROWNIAN PARTICLES WITH INERTIA A WIENER PATH INTEGRAL WITH CONSTRAINT AND IN THE SPACE OF VELOCITIES 69 1 2 5 BROWNIAN MOTION WITH ABSORPTION AND IN THE ELD OF AN EXTERNAL DETERMINISTIC FORCE THE BLOCH EQUATION AND FEYNMAN KAC FORMULA 72 1 2 6 VARIATIONAL METHODS OF PATH INTEGRAL CALCULATIONS SEMICLASSICAL AND QUADRATIC APPROXIMATIONS AND THE METHOD OF HOPPING PATHS 78 1 2 7 MORE TECHNICALITIES FOR PATH INTEGRAL CALCULATIONS NITE DIFFERENCE CALCULUS AND FOURIER DECOMPOSITION 94 1 2 8 GENERATING OR CHARACTERISTIC FUNCTIONALS FOR WIENER INTEGRALS 101 1 2 9 PHYSICS OF MACROMOLECULES AN APPLICATION OF PATH INTEGRATION 108 1 2 10 PROBLEMS 111 2 PATH INTEGRALS IN QUANTUM MECHANICS 122 2 1 FEYNMAN PATH INTEGRALS 123 2 1 1 SOME BASIC FACTS ABOUT QUANTUM MECHANICS AND THE SCHR ODINGER EQUATION 123 2 1 2 FEYNMAN KAC FORMULA IN QUANTUM MECHANICS 137 2 1 3 PROPERTIES OF HAMILTONIAN OPERATORS FROM THE FEYNMAN KAC FORMULA 141 2 1 4 BOHR SOMMERFELD SEMICLASSICAL QUANTIZATION CONDITION FROM PATH INTEGRALS 144 2 1 5 PROBLEMS 149 2 2 PATH INTEGRALS IN THE HAMILTONIAN FORMALISM 153 2 2 1 DERIVATION OF PATH INTEGRALS FROM OPERATOR FORMALISM IN QUANTUM MECHANICS 154 2 2 2 CALCULATION OF PATH INTEGRALS FOR THE SIMPLEST QUANTUM MECHANICAL SYSTEMS A FREE PARTICLE AND A HARMONIC OSCILLATOR 161 2 2 3 SEMICLASSICAL WKB APPROXIMATION IN QUANTUM MECHANICS AND THE STATIONARY PHASE METHOD 169 2 2 4 DERIVATION OF THE BOHR SOMMERFELD CONDITION VIA THE PHASE SPACE PATH INTEGRAL PERIODIC ORBIT THEORY AND QUANTIZATION OF SYSTEMS WITH CHAOTIC CLASSICAL DYNAMICS 176 2 2 5 PARTICLES IN A MAGNETIC ELD THE ITO INTEGRAL MIDPOINT PRESCRIPTION AND GAUGE INVARIANCE 183 2 2 6 APPLICATIONS OF PATH INTEGRALS TO OPTICAL PROBLEMS BASED ON A FORMAL ANALOGY WITH QUANTUM MECHANICS 187 2 2 7 PROBLEMS 190 2 3 QUANTIZATION THE OPERATOR ORDERING PROBLEM AND PATH INTEGRALS 200 2 3 1 SYMBOLS OF OPERATORS AND QUANTIZATION 200 2 3 2 GENERAL CONCEPT OF PATH INTEGRALS OVER TRAJECTORIES IN PHASE SPACE 209 2 3 3 NORMAL SYMBOL FOR THE EVOLUTION OPERATOR COHERENT STATE PATH INTEGRALS PERTURBATION EXPANSION AND SCATTERING OPERATOR 216 2 3 4 PROBLEMS 226 2 4 PATH INTEGRALS AND QUANTIZATION IN SPACES WITH TOPOLOGICAL CONSTRAINTS 230 2 4 1 POINT PARTICLES IN A BOX AND ON A HALF LINE 231 2 4 2 POINT PARTICLES ON A CIRCLE AND WITH A TORUS SHAPED PHASE SPACE 238 2 4 3 PROBLEMS 243 2 5 PATH INTEGRALS IN CURVED SPACES SPACETIME TRANSFORMATIONS AND THE COULOMB PROBLEM 245 2 5 1 PATH INTEGRALS IN CURVED SPACES AND THE ORDERING PROBLEM 245 2 5 2 SPACETIME TRANSFORMATIONS OF HAMILTONIANS 251 2 5 3 PATH INTEGRALS IN POLAR COORDINATES 258 2 5 4 PATH INTEGRAL FOR THE HYDROGEN ATOM THE COULOMB PROBLEM 266 2 5 5 PATH INTEGRALS ON GROUP MANIFOLDS 272 2 5 6 PROBLEMS 282 2 6 PATH INTEGRALS OVER ANTICOMMUTING VARIABLES FOR FERMIONS AND GENERALIZATIONS 286 2 6 1 PATH INTEGRALS OVER ANTICOMMUTING GRASSMANN VARIABLES FOR FERMIONIC SYSTEMS 286 2 6 2 PATH INTEGRALS WITH GENERALIZED GRASSMANN VARIABLES 298 2 6 3 LOCALIZATION TECHNIQUES FOR THE CALCULATION OF A CERTAIN CLASS OF PATH INTEGRALS 304 2 6 4 PROBLEMS 315 INTRODUCTION THE AIM OF THIS BOOK IS TO PRESENT AND EXPLAIN THE CONCEPT OF THE PATH INTEGRAL WHICH IS INTENSIVELY USED NOWADAYS IN ALMOST ALL THE BRANCHES OF THEORETICAL PHYSICS THE NOTION OF PATH INTEGRAL SOMETIMES ALSO CALLED FUNCTIONAL INTEGRAL OR INTEGRAL OVER TRAJECTORIES OR INTEGRAL OVER HISTORIES OR CONTINUOUS INTEGRAL WAS INTRODUCED FOR THE RST TIME IN THE 1920S BY NORBERT WIENER 1921 1923 1924 1930 AS A METHOD TO SOLVE PROBLEMS IN THE THEORY OF DIFFUSION AND BROWNIAN MOTION THIS INTEGRAL WHICH IS NOW ALSO CALLED THE WIENER INTEGRAL HAS PLAYED A CENTRAL ROLE IN THE FURTHER DEVELOPMENT OF THE SUBJECT OF PATH INTEGRATION IT WAS REINVENTED IN A DIFFERENT FORM BY RICHARD FEYNMAN 1942 1948 IN 1942 FOR THE REFORMULATION OF QUANTUM MECHANICS THE SO CALLED THIRD FORMULATION OF QUANTUM MECHANICS BESIDES THE SCHR ODINGER AND HEISENBERG ONES THE FEYNMAN APPROACH WAS INSPIRED BY DIRAC S PAPER 1933 ON THE ROLE OF THE LAGRANGIAN AND THE LEAST ACTION PRINCIPLE IN QUANTUM MECHANICS THIS EVENTUALLY LED FEYNMAN TO REPRESENT THE PROPAGATOR OF THE SCHR ODINGER EQUATION BY THE COMPLEX VALUED PATH INTEGRAL WHICH NOW BEARS HIS NAME AT THE END OF THE 1940S FEYNMAN 1950 1951 WORKED OUT ON THE BASIS OF THE PATH INTEGRALS A NEW FORMULATION OF QUANTUM ELECTRODYNAMICS AND DEVELOPED THE WELL KNOWN DIAGRAM TECHNIQUE FOR PERTURBATION THEORY IN THE 1950S PATH INTEGRALS WERE STUDIED INTENSIVELY FOR SOLVING FUNCTIONAL EQUATIONS IN QUANTUM ELD THEORY SCHWINGER EQUATIONS THE FUNCTIONAL FORMULATION OF QUANTUM ELD THEORY WAS CONSIDERED IN THE WORKS OF BOGOLIUBOV 1954 GELFAND AND MINLOS 1954 KHALATNIKOV 1952 1955 MATHEWS AND SALAM 1954 EDWARDS AND PEIERLS 1954 SYMANZIK 1954 FRADKIN 1954 AND OTHERS OTHER AREAS OF APPLICATIONS OF PATH INTEGRALS IN THEORETICAL PHYSICS DISCOVERED IN THIS DECADE WERE THE STUDY OF BROWNIAN MOTION IN AN ABSORBING MEDIUM SEE KAC 1959 WIEGEL 1975 1986 AND REFERENCES THEREIN AND THE DEVELOPMENT OF THE THEORY OF SUPER UIDITY FEYNMAN 1953 1954 TER HAAR 1954 KIKUCHI 1954 1955 STARTING FROM THESE PIONEERING WORKS MANY IMPORTANT APPLICATIONS OF PATH INTEGRALS HAVE BEEN FOUND IN STATISTICAL PHYSICS IN THE THEORY OF PHASE TRANSITIONS SUPER UIDITY SUPERCONDUCTIVITY THE ISING MODEL QUANTUM OPTICS PLASMA PHYSICS IN 1955 FEYNMAN USED THE PATH INTEGRAL TECHNIQUE FOR INVESTIGATING THE POLARON PROBLEM FEYNMAN 1955 AND INVENTED HIS VARIATIONAL PRINCIPLE FOR QUANTUM MECHANICS THIS WORK HAD AN IMPORTANT IMPACT ON FURTHER APPLICATIONS OF PATH INTEGRALS IN STATISTICAL AND SOLID STATE PHYSICS AS WELL AS IN QUANTUM ELD THEORY IN GENERAL AT THE SAME TIME ATTEMPTS WERE INITIATED TO WIDEN THE CLASS OF EXACTLY SOLVABLE PATH INTEGRALS I E TO EXPAND IT BEYOND THE CLASS OF GAUSSIAN LIKE INTEGRALS IN THE EARLY 1950S OZAKI IN UNPUBLISHED LECTURE NOTES KYUSHU UNIVERSITY 1955 STARTED WITH A SHORT TIME ACTION FOR A FREE PARTICLE WRITTEN IN CARTESIAN COORDINATES AND TRANSFORMED IT INTO THE POLAR FORM LATER PEAK AND INOMATA 1969 CALCULATED EXPLICITLY THE RADIAL PATH INTEGRAL FOR THE HARMONIC OSCILLATOR THIS OPENED THE WAY FOR AN ESSENTIAL BROADENING OF THE CLASS OF PATH INTEGRABLE MODELS FURTHER IMPORTANT STEPS IN THIS DIRECTION WERE STUDIES OF SYSTEMS ON MULTIPLY CONNECTED SPACES IN PARTICULAR ON LIE GROUP MANIFOLDS SCHULMAN 1968 DOWKER 1972 AND THE TREATMENT OF THE QUANTUM MECHANICAL COULOMB PROBLEM BY DURU AND KLEINERT 1979 WHO APPLIED THE SO CALLED KUSTAANHEIMO STIEFEL SPACETIME TRANSFORMATION TO THE PATH INTEGRAL 1 IN THE 1960S A NEW ELD OF PATH INTEGRAL APPLICATIONS APPEARED NAMELY THE QUANTIZATION OF GAUGE ELDS EXAMPLES OF WHICH ARE THE ELECTROMAGNETIC GRAVITATIONAL AND YANG MILLS ELDS THE SPECI C PROPERTIES OF THE ACTION FUNCTIONALS FOR GAUGE ELDS THEIR INVARIANCE WITH RESPECT TO GAUGE TRANSFORMATIONS SHOULD BE TAKEN INTO ACCOUNT WHEN QUANTIZING OTHERWISE WRONG RESULTS EMERGE THIS WAS RST NOTICED BY FEYNMAN 1963 USING THE EXAMPLE OF YANG MILLS AND GRAVITATIONAL ELDS HE SHOWED THAT QUANTIZATION BY STRAIGHTFORWARD USE OF THE FERMI METHOD IN ANALOGY WITH QUANTUM ELECTRODYNAMICS VIOLATES THE UNITARITY CONDITION LATER AS A RESULT OF WORKS BY DE WITT 1967 FADDEEV AND POPOV 1967 MANDELSTAM 1968 FRADKIN AND TYUTIN 1969 AND T HOOFT 1971 THE PROBLEM WAS SOLVED AND THE PATH INTEGRAL METHOD TURNED OUT TO BE THE MOST SUITABLE ONE FOR THIS AIM IN ADDITION IN THE MID 1960S BEREZIN 1966 TOOK A CRUCIAL STEP WHICH ALLOWED THE COMPREHENSIVE USE OF PATH INTEGRATION HE INTRODUCED INTEGRATION OVER GRASSMANN VARIABLES TO DESCRIBE FERMIONS ALTHOUGH THIS MAY BE CONSIDERED TO BE A FORMAL TRICK IT OPENED THE WAY FOR A UNI ED TREATMENT OF BOSONS AND FERMIONS IN THE PATH INTEGRAL APPROACH IN THE 1970S WILSON 1974 FORMULATED THE ELD THEORY OF QUARKS AND GLUONS I E QUANTUM CHROMODYNAMICS ON A EUCLIDEAN SPACETIME LATTICE THIS MAY BE CONSIDERED AS THE DISCRETE FORM OF THE ELD THEORETICAL PATH INTEGRAL THE LATTICE SERVES AS BOTH AN ULTRAVIOLET AND INFRARED CUT OFF WHICH MAKES THE THEORY WELL DE NED AT LOW ENERGIES IT IS THE MOST FRUITFUL METHOD TO TREAT THE THEORY OF STRONG INTERACTIONS FOR EXAMPLE MAKING USE OF COMPUTER SIMULATIONS A FEW YEARS LATER FUJIKAWA 1979 SHOWED HOW THE QUANTUM ANOMALIES EMERGE FROM THE PATH INTEGRAL HE REALIZED THAT IT IS THE MEASURE IN THE PATH INTEGRAL WHICH IS NOT INVARIANT UNDER A CERTAIN CLASS OF SYMMETRY TRANSFORMATIONS AND THIS MAKES THE LATTER ANOMALOUS ALL THESE ACHIEVEMENTS LED TO THE FACT THAT THE PATH INTEGRAL METHODS HAVE BECOME AN INDISPENSABLE PART OF ANY CONSTRUCTION AND STUDY OF ELD THEORETICAL MODELS INCLUDING THE REALISTIC THEORIES OF UNI ED ELECTROMAGNETIC AND WEAK INTERACTIONS GLASHOW 1961 WEINBERG 1967 SALAM 1968 AND QUANTUM CHROMODYNAMICS THE THEORY OF STRONG INTERACTIONS GROSS AND WILCZEK 1973 POLITZER 1973 AMONG OTHER APPLICATIONS OF PATH INTEGRALS IN QUANTUM ELD THEORY AND ELEMENTARY PARTICLE PHYSICS IT IS WORTH MENTIONING THE DERIVATION OF ASYMPTOTIC FORMULAS FOR INFRARED AND ULTRAVIOLET BEHAVIOUR OF GREEN FUNCTIONS THE SEMICLASSICAL APPROXIMATION REARRANGEMENT AND PARTIAL SUMMATION OF PERTURBATION SERIES CALCULATIONS IN THE PRESENCE OF TOPOLOGICALLY NON TRIVIAL ELD CON GURATIONS AND EXTENDED OBJECTS SOLITONS AND INSTANTONS THE STUDY OF COSMOLOGICAL MODELS AND BLACK HOLES AND SUCH AN ADVANCED APPLICATION AS THE FORMULATION OF THE RST QUANTIZED THEORY OF SUPER STRINGS AND BRANES IN ADDITION THE PATH INTEGRAL TECHNIQUE NDS NEWER AND NEWER APPLICATIONS IN STATISTICAL PHYSICS AND NON RELATIVISTIC QUANTUM MECHANICS IN PARTICULAR IN SOLID BODY PHYSICS AND THE DESCRIPTION OF CRITICAL PHENOMENA PHASE TRANSITIONS POLYMER PHYSICS AND QUANTUM OPTICS AND IN MANY OTHER BRANCHES OF PHYSICS DURING THE TWO LAST DECADES OF THE MILLENNIUM MOST WORKS IN THEORETICAL AND MATHEMATICAL PHYSICS CONTAINED SOME ELEMENTS OF THE PATH INTEGRAL TECHNIQUE WE SHALL THEREFORE NOT PURSUE THE HISTORY OF THE SUBJECT PAST THE 1970S EVEN BRIE Y FUNCTIONAL INTEGRATION HAS PROVED TO BE ESPECIALLY USEFUL FOR THE DESCRIPTION OF COLLECTIVE EXCITATIONS FOR EXAMPLE QUANTUM VORTICES IN THE THEORY OF CRITICAL PHENOMENA AND FOR SYSTEMS ON TOPOLOGICALLY NON TRIVIAL SPACES IN SOME CASES THIS TECHNIQUE ALLOWS US TO PROVIDE SOLID FOUNDATIONS FOR THE RESULTS OBTAINED BY OTHER METHODS TO CLARIFY THE LIMITS OF THEIR APPLICABILITY AND INDICATE THE WAY OF CALCULATING THE CORRECTIONS IF AN EXACT SOLUTION IS POSSIBLE THEN THE PATH INTEGRAL TECHNIQUE GIVES A SIMPLE WAY TO OBTAIN IT IN THE CASE OF PHYSICALLY REALISTIC PROBLEMS WHICH NORMALLY ARE FAR FROM BEING EXACTLY SOLVABLE THE USE OF PATH INTEGRALS HELPS TO BUILD UP THE QUALITATIVE PICTURE OF THE CORRESPONDING PHENOMENON AND TO DEVELOP APPROXIMATE METHODS OF CALCULATION THEY REPRESENT A SUF CIENTLY EXIBLE MATHEMATICAL APPARATUS WHICH CAN BE SUITABLY ADJUSTED FOR THE EXTRACTION OF THE ESSENTIAL INGREDIENTS OF A COMPLICATED MODEL FOR ITS FURTHER PHYSICAL ANALYSIS ALSO SUGGESTING THE METHOD FOR A CONCRETE REALIZATION OF SUCH AN ANALYSIS ONE CAN JUSTLY SAY THAT PATH INTEGRATION IS AN INTEGRAL CALCULUS ADJUSTED TO THE NEEDS OF CONTEMPORARY PHYSICS PATH INTEGRALS IN PHYSICS 2 UNIVERSALITY OF THE PATH INTEGRAL FORMALISM THE MOST CAPTIVATING FEATURE OF THE PATH INTEGRAL TECHNIQUE IS THAT IT PROVIDES A UNI ED APPROACH TO SOLVING PROBLEMS IN DIFFERENT BRANCHES OF THEORETICAL PHYSICS SUCH AS THE THEORY OF STOCHASTIC PROCESSES QUANTUM MECHANICS QUANTUM ELD THEORY THE THEORY OF SUPERSTRINGS AND STATISTICAL BOTH CLASSICAL AND QUANTUM MECHANICS INDEED THE GENERAL FORM OF THE BASIC OBJECT NAMELY THE TRANSITION PROBABILITY W X F T F X0 T0 IN THE THEORY OF STOCHASTIC PROCESSES READS W X F T F X0 T0 ALL TRAJECTORIES FROM X0 TO X F EXP 1 4D F X 0 0 1 WHERE X0 DENOTES THE SET OF COORDINATES OF THE STOCHASTIC SYSTEM UNDER CONSIDERATION AT THE INITIAL TIME T0 AND W X F T F X0 T0 GIVES THE PROBABILITY OF THE SYSTEM TO HAVE THE COORDINATES X F AT THE NAL TIME T F THE EXPLICIT FORM OF THE FUNCTIONAL F X T0 T F AS WELL AS THE VALUE AND PHYSICAL MEANING OF THE CONSTANT D DEPEND ON THE SPECI C PROPERTIES OF THE SYSTEM AND SURROUNDING MEDIUM SEE CHAPTER 1 THE SUMMATION SIGN SYMBOLICALLY DENOTES SUMMATION OVER ALL TRAJECTORIES OF THE SYSTEM OF COURSE THIS OPERATION REQUIRES FURTHER CLARI CATION AND THIS IS ONE OF THE GOALS OF THIS BOOK IN QUANTUM MECHANICS THE BASIC OBJECT IS THE TRANSITION AMPLITUDE K X F T F X0 T0 NOT A PROBABILITY BUT THE PATH INTEGRAL EXPRESSION FOR IT HAS A FORM WHICH IS QUITE SIMILAR TO 0 0 1 K X F T F X0 T0 ALL TRAJECTORIES FROM X0 TO X F EXP I S X 0 0 2 OR IN A MORE GENERAL CASE K X F T F X0 T0 ALL TRAJECTORIES IN PHASE SPACE WITH XED X0 AND X F EXP I S X P 0 0 3 HERE S X IS THE ACTION OF THE SYSTEM IN TERMS OF THE CON GURATION SPACE VARIABLES WHILE S X P IS THE ACTION IN TERMS OF THE PHASE SPACE VARIABLES COORDINATES AND MOMENTA THOUGH NOW WE HAVE PURELY IMAGINARY EXPONENTS IN CONTRAST WITH THE CASE OF STOCHASTIC PROCESSES THE GENERAL FORMAL STRUCTURE OF EXPRESSIONS 0 0 1 0 0 3 IS TOTALLY ANALOGOUS MOREOVER AS WE SHALL SEE LATER THE PATH INTEGRALS 0 0 2 0 0 3 CAN BE CONVERTED INTO THE FORM 0 0 1 I E WITH A PURELY REAL EXPONENT BY A TRANSITION TO PURELY IMAGINARY TIME VARIABLES T IT AND IN MANY CASES THIS TRANSFORMATION CAN BE MATHEMATICALLY JUSTI ED IN THE CASE OF SYSTEMS WITH AN IN NITE NUMBER OF DEGREES OF FREEDOM IT WAS ALSO REALIZED EVEN IN THE 1960S THAT AN ESSENTIAL SIMILARITY BETWEEN QUANTUM ELD THEORY AND CLASSICAL OR QUANTUM STATISTICAL PHYSICS EXISTS IN PARTICULAR THE VACUUM EXPECTATIONS GREEN FUNCTIONS IN QUANTUM ELD THEORY ARE GIVEN BY EXPRESSIONS OF THE TYPE 0 A 0 ALL ELD CON GURATIONS A EXP I S 0 0 4 WHERE ON THE LEFT HAND SIDE A IS AN OPERATOR MADE OF THE ELD OPERATORS AND ON THE RIGHT HAND SIDE A IS THE CORRESPONDING CLASSICAL QUANTITY AFTER THE TRANSITION TO PURELY IMAGINARY TIME T IT PATH INTEGRALS IN PHYSICS 3