Path Integrals in Quantum Physics Lectures given at ETH Zurich R. Rosenfelder Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland Abstract These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many- body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin & color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions. arXiv:1209.1315v4 [nucl-th] 30 Jul 2017 2 Content 0. Contents First, an overview over the planned topics. The subsections marked by ⋆ are optional and may be left out if there is no time available whereas the chapters printed in blue deal with basic concepts. Problems from the optional chapters or referring to ”Details” are marked by a ⋆ as well. 1. Path Integrals in Non-Relativistic Quantum Mechanics of a Single Particle 1. Action Principle and Sum over all Paths ...................................................................... 6 2. Lagrange, Hamilton and other Path-Integral Formulations ........................................ 10 3. Quadratic Lagrangians ................................................................................................ 22 4. Perturbation Theory ⋆ ................................................................................................ 26 5. Semiclassical Expansions .............................................................................................. 29 6. Potential Scattering and Eikonal Approximation ⋆ ..................................................... 33 7. Green Functions as Path Integrals .............................................................................. 44 8. Symmetries and Conservation Laws ⋆ .......................................................................... 53 9. Numerical Treatment of Path Integrals ⋆ ..................................................................... 56 10. Tunneling and Instanton Solutions ⋆ ............................................................................ 63 2. Path Integrals in Statistical Mechanics and Many-Body Physics 1. Partition Function ......................................................................................................... 72 2. The Polaron ................................................................................................................. 74 3. Dissipative Quantum Systems ⋆ ................................................................................... 79 4. Particle Number Representation and Path Integrals over Coherent States ................. 88 5. Description of Fermions: Grassmann Variables ........................................................... 95 6. Perturbation Theory and Diagrams ⋆ .......................................................................... 98 7. Auxiliary Fields and Hartree Approximation ⋆ .......................................................... 105 8. Asymptotic Expansion of a Class of Path Integrals ⋆ ................................................ 109 3. Path Integrals in Field Theory 1. Generating Functionals and Perturbation Theory ................................................... 120 2. Effective Action ⋆ .................................................................................................... 135 3. Quantization of Gauge Theories ............................................................................... 139 4. Worldlines and Spin in the Path Integral ⋆ .............................................................. 149 5. Anomalies ⋆ .............................................................................................................. 157 6. Lattice Field Theories ⋆ ............................................................................................ 162 Additional Literature ................................................................................................................ 175 Original Publications ............................................................................................................... 176 Problems ................................................................................................................................... 180 Solutions ................................................................................................................................... 195 R. Rosenfelder : Path Integrals in Quantum Physics 3 Literature: There is a large number of text books which are dealing with path integrals either as a tool for specific problems or as basis for an unified treatment. In the following a small selection is presented in which my favorite books (whom I mostly follow 1) are marked by the symbol . ♣ Section 1 : R. P. Feynman and A. R. Hibbs: Quantum Mechanics and Path Integrals, McGraw-Hill (1965). • L. S. Schulman: Techniques and Applications of Path Integration, John Wiley (1981). • ♣ J. Glimm and A. Jaffe: Quantum Physics: A Functional Point of View, Springer (1987). • G. Roepstorff: Path Integral Approach to Quantum Physics, Springer (1994). • G. W. Johnson and M. L. Lapidus: The Feynman Integral and Feynman’s Operational Calculus, • Oxford University Press (2000). J. Zinn-Justin: Path Integrals in Quantum Mechanics, Oxford University Press (2006). • Section 2 : R. P. Feynman: Statistical Mechanics, Benjamin (1976). • V. N. Popov: Functional Integrals in Quantum Field Theory and Statistical Physics, Reidel (1983). • J. W. Negele and H. Orland: Quantum Many-Particle Systems, Addison-Wesley (1987). • ♣ J. Zinn-Justin: Quantum Field Theory and Critical Phenomena, 4th ed., Oxford U. Press (2002). • H. Kleinert: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, • 3rd ed., World Scientific (2004). Section 3 : C. Itzykson and J.-B. Zuber: Quantum Field Theory, McGraw-Hill (1980), ch. 9. • T.-P. Cheng and L.-F. Li: Gauge Theory of Elementary Particle Physics, Clarendon (1988). • R. J. Rivers: Path Integral Methods in Quantum Field Theory, Cambridge University Press (1990). • M. E. Peskin and D. V. Schroeder: An Introduction to Quantum Field Theory, Addison-Wesley • (1995), ch. 9. ♣ V. Parameswaran Nair: Quantum Field Theory. A Modern Perspective, Springer (2005). • ♣ H. J. Rothe: Lattice Gauge Theories. An Introduction, 3rd ed., World Scientific Lecture Notes in • Physics, Vol. 74, World Scientific (2005). A. Das: Field Theory. A Path Integral Approach, 2nd ed., World Scientific Lecture Notes in Physics, • Vol. 75, World Scientific (2006). ♣ 1”(My father used to say: ’If you steal from one book, you are condemned as a plagiarist, but if you steal from ten books, you are considered a scholar, and if you steal from thirty or fourty books, a distinguished scholar.’)” {Oz}, p.129. 4 Content Prerequisites: Quantum Mechanics: A two-semester course, a little bit of Statistical Mechanics, basic concepts of Field Theory. Practice Lessons: Participation in the Practice Sessions is strongly recommended – according to the motto “I hear – and I forget, I see – and I remember, I do – and I understand !” (Chinese proverb 2) Remarks: In order to find some orientation in the jungle of (over 1300 listed) equations, the most important equations are framed according to the following scheme fundamental ( 9 times) very important ( 24 times) important ( 101 times) , where, of course, my rating may be debatable ... The “Details” in small print give more in-depth information or detailed derivations and can be omitted in a first study. Additional literature is referenced in the text in curly brackets, e.g. {Weiss} and is listed at the end in aphabetical order. Original publications are cited by consecutive numbers, e.g. [11], which are collected at the end of the text just before the Problems. 2Attributed to Xun Kuang (c. 310 - c. 235 BC) known as Xunzi (”Master Xun”), a Chinese Realist Confucian philosopher (see https://en.wikipedia.org/wiki/Xun−Kuang). R. Rosenfelder : Path Integrals in Quantum Physics 5 The following abbreviations are frequently used in the text: w.r.t. : with respect to i.e. : (Latin: id est) that is viz. : (Latin: videlicet) namely l.h.s. : left-hand side r.h.s. : right-hand side e.g. : (Latin: exemplum gratum) for example cf. : (Latin: confer) compare q.e.d. : (Latin: quod erat demonstrandum) which was to be proved. Despite the word ”Integral” appearing in the title no knowledge of particular integrals is needed – the only integral which appears again and again in various (multi-dimensional) generalizations is the Gaussian integral + ∞ ax2 π (0.1) G(a) = dx e− = , Re a> 0 . a Z−∞ r Proof: Evaluate + + G2(a) = ∞ ∞ dx dy exp a(x2 + y2) (0.2a) − Z−∞ Z−∞ h i in polar coordinates x = r cos φ,y = r sin φ, dxdy = rdrdφ . Then one obtains 2π 1 d π G2(a) = dφ ∞ dr r exp( ar2) = 2π ∞ dr exp( ar2) = . (0.2b) − − 2a dr − a Z0 Z0 Z0 The constraint for the complex parameter a is needed for convergence of the integral or for vanishing of the integrated term at infinity,