Lecture Notes in Physics 864 Statistical Approach to Quantum Field Theory An Introduction Bearbeitet von Andreas Wipf 1. Auflage 2012. Taschenbuch. xviii, 390 S. Paperback ISBN 978 3 642 33104 6 Format (B x L): 15,5 x 23,5 cm Gewicht: 623 g Weitere Fachgebiete > Physik, Astronomie > Physik Allgemein > Theoretische Physik, Mathematische Physik Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Chapter 2 Path Integrals in Quantum and Statistical Mechanics There exist three apparently different formulations of quantum mechanics: HEISEN- BERG’s matrix mechanics, SCHRÖDINGER’s wave mechanics and FEYNMAN’s path integral approach. In contrast to matrix and wave mechanics, which are based on the Hamiltonian approach the latter is based on the Lagrangian approach. 2.1 Summing Over All Paths Already back in 1933 DIRAC asked himself, whether the classical Lagrangian and action are as significant in quantum mechanics as they are in classical mechanics [1, 2]. He observed that the probability amplitude − ˆ K t,q ,q = q e iHt/ |q (2.1) for the propagation of a system from a point with coordinate q to another point with coordinate q in time t is given by [ ] K t,q ,p ∝ eiS qcl / , (2.2) where qcl denotes the classical trajectory from q to q . In the exponent the action of this trajectory enters as a multiple of Planck’s reduced constant . For a free particle with Lagrangian m L = q˙2 (2.3) 0 2 the formula (2.2) is verified easily: A free particle moves with constant velocity (q − q)/t from q to q and the action of the classical trajectory is t m 2 S[qcl]= dsL0 qcl(s) = q − q . 0 2t The factor of proportionality in (2.2) is then uniquely fixed by the condition ˆ e−iHt/ → 1 for t → 0, which in position space reads lim K t,q ,q = δ q ,q . (2.4) t→0 A. Wipf, Statistical Approach to Quantum Field Theory, Lecture Notes in Physics 864, 5 DOI 10.1007/978-3-642-33105-3_2, © Springer-Verlag Berlin Heidelberg 2013 6 2 Path Integrals in Quantum and Statistical Mechanics ˆ ˆ ˆ Alternatively, it is fixed by the property e−iHt/e−iHs/ = e−iH(t+s)/ that takes the form duK t,q ,u K(s,u,q) = K t + s,q ,q (2.5) in position space. Thus, the correct free-particle propagator on a line is given by 1/2 m − 2 K t,q ,q = eim(q q) /2 t . (2.6) 0 2πit Similar results hold for the harmonic oscillator or systems for which ˆq(t) fulfills the classical equation of motion. For such systems V (q)ˆ =V (ˆq) holds true. However, for general systems the simple formula (2.2) must be extended and it was FEYNMAN who discovered this extension back in 1948. He realized that all paths from q to q (and not only the classical path) contribute to the propagator. This means that in quantum mechanics a particle can potentially move on any path q(s) from the initial to the final destination, q(0) = q and q(t) = q . (2.7) The probability amplitude emerges as the superposition of contributions from all trajectories, [ ] K t,q ,q ∼ eiS path / , (2.8) all paths where a single path contributes a term ∼ exp(iS[path]/). In passing we note that already in 1923 WIENER introduced the sum over all paths in his studies of stochastic processes [3]. Thereby a single path was weighted with a real and positive probability and not with a complex amplitude as in (2.8). Wiener’s path integral corresponds to Feynman’s path integral for imaginary time and describes quantum systems in thermal equilibrium with a heat bath at fixed temperature. In this book we will explain this extraordinary result and apply it to in- teresting physical systems. Moreover, the path integral method allows for a uniform treatment of quantum mechanics, quantum field theory and statistical mechanics and can be regarded as a basic tool in modern theoretical physics. It represents an alter- native approach to the canonical quantization of classical systems and earned its first success in the 1950s. The path integral method is very beautifully and intelligibly presented in Feynman’s original work [4] as well as in his book with HIBBS [5]. The latter reference contains many applications and is still recognized as a standard reference. Functional integrals have been developed further by outstanding mathe- maticians and physicists, especially by KAC [9]. An adequate reference for these developments is contained in the review article by GELFAND and YAGLOM [10]. In the present chapter we can only give a short introduction to path integrals. For a deeper understanding the reader should consult more specialized books and review articles. Some of them are listed in the bibliography at the end of this chapter. 2.2 Recalling Quantum Mechanics 7 2.2 Recalling Quantum Mechanics There are two well-established ways to quantize a classical system: canonical quan- tization and path integral quantization. For completeness and later use we recall the main steps of canonical quantization both in Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics. i A classical system is described by its coordinates {q } and momenta {pi} on phase space Γ . An observable O is a real-valued function on Γ . Examples are the coordinates on phase space and the energy H(q,p) of the system under considera- tion. We assume that phase space comes along with a symplectic structure and has local coordinates with Poisson brackets i = i q ,pj δj . (2.9) The brackets are extended to observables through antisymmetry and the derivation rule {OP,Q}=O{P,Q}+{O,Q}P . The evolution in time of an observable is determined by ˙ i i O ={O,H}, e.g. q˙ = q ,H and p˙i ={pi,H}. (2.10) In the canonical quantization the function on phase space are mapped to operators and the Poisson brackets of two functions become commutators of the associated operators: 1 O(q,p) → O(ˆ q,ˆ p)ˆ and {O,P}→ [O,ˆ Pˆ]. (2.11) i The time evolution of an (not explicitly time-dependent) observable is determined by Heisenberg’s equation dOˆ i = [H,ˆ Oˆ ]. (2.12) dt i In particular the phase space coordinates (q ,pi) become operators with commuta- [ˆi ˆ ]= i tion relations q , pj i δj and time evolution given by ˆi ˆ dq i ˆ i dpi i ˆ = H,qˆ and = [H,pˆi]. dt dt For a system of non-relativistic and spinless particles the Hamiltonian reads 1 Hˆ = Hˆ + Vˆ with Hˆ = pˆ2, (2.13) 0 0 2m i and one arrives at Heisenberg’s equations of motion, ˆi ˆ ˆ dq pi dpi ˆ = and =−V,i . (2.14) dt 2m dt Observables are represented by hermitian operators on a Hilbert space H , whose elements characterize the states of the system: O(ˆ q,ˆ p)ˆ : H → H . (2.15) 8 2 Path Integrals in Quantum and Statistical Mechanics Consider a particle confined to an endless wire. Its Hilbert space is H = L2(R) and its position and momentum operator are represented in position space as (qψ)(q)ˆ = qψ(q) and (pψ)(q)ˆ = ∂ ψ(q). (2.16) i q In experiments we can measure matrix elements of observables, represented by her- mitian operators, and in particular expectation values of hermitian operators in a state of the system. The time dependence of an expectation value ψ|O(t)ˆ |ψ is determined by the Heisenberg equation (2.12). The transition from the Heisenberg picture to the Schrödinger picture involves a time-dependent similarity transformation, ˆ −itH/ˆ ˆ itH/ˆ −itH/ˆ Os = e Oe and |ψs=e |ψ, (2.17) and leads to time-independent observables in the Schrödinger picture, d − ˆ i d ˆ Oˆ = e itH/ − [H,ˆ Oˆ ]+ Oˆ eitH/ = 0. dt s dt ˆ ˆ Note that the Hamiltonian operator is the same in both pictures, Hs = H and that all expectation values are left invariant by the similarity transformation, ˆ ˆ ψ|O(t)|ψ= ψs(t) Os ψs(t) . (2.18) A state vector in the Schrödinger picture |ψs(t) fulfills the Schrödinger equation d − ˆ i |ψ =Hˆ |ψ ⇐⇒ψ (t) = e itH/ ψ (0) . (2.19) dt s s s s In position space this formal solution of the evolution equation has the form −itH/ˆ ψs t,q ≡ q |ψs(t) = q e |q q|ψs(0) dq ≡ K t,q ,q ψs(0,q)dq, (2.20) where we inserted the resolution of the identity with qˆ-eigenstates, dq|qq|=1, (2.21) and introduced the kernel of the unitary time evolution operator − ˆ K t,q ,q = q K(t)ˆ |q, K(t)ˆ = e itH/ . (2.22) The propagator K(t,q,q) is interpreted as the probability amplitude for the prop- agation from q at time 0 to q at time t. This is emphasized by the notation K t,q ,q ≡ q ,t|q,0 . (2.23) The amplitude solves the time-dependent Schrödinger equation d i K t,q ,q = HKˆ t,q ,q , (2.24) dt 2.3 Feynman–Kac Formula 9 where Hˆ acts on q, and fulfills the initial condition lim K t,q ,q = δ q − q .