The Path Integral approach to Quantum Mechanics Lecture Notes for Quantum Mechanics IV Riccardo Rattazzi May 5, 009
Contents 1 The path integral formalism 5 1.1 Introducing the path integrals.................... 5 1.1.1 The double slit experiment................. 5 1.1. An intuitive approach to the path integral formalism... 6 1.1.3 The path integral formulation................ 8 1.1.4 From the Schröedinger approach to the path integral... 1 1. The properties of the path integrals................ 14 1..1 Path integrals and state evolution............. 14 1.. The path integral computation for a free particle..... 17 1.3 Path integrals as determinants................... 19 1.3.1 Gaussian integrals...................... 19 1.3. Gaussian Path Integrals................... 0 1.3.3 Oħ) correctionstogaussianapproximation........ 1.3.4 Quadratic lagrangians and the harmonic oscillator.... 3 1.4 Operator matrix elements...................... 7 1.4.1 The time-ordered product of operators........... 7 Functional and Euclidean methods 31.1 Functional method.......................... 31. Euclidean Path Integral....................... 3..1 Statistical mechanics..................... 34.3 Perturbation theory......................... 35.3.1 Euclidean n-point correlators................ 35.3. Thermal n-point correlators................. 36.3.3 Euclidean correlators by functional derivatives...... 38.3.4 Computing KE 0 [J] andz0 [J]................ 39.3.5 Free n-point correlators................... 41.3.6 The anharmonic oscillator and Feynman diagrams.... 43 3 The semiclassical approximation 49 3.1 The semiclassical propagator.................... 50 3.1.1 VanVleck-Pauli-Morette formula.............. 54 3.1. Mathematical Appendix 1.................. 56 3.1.3 Mathematical Appendix.................. 56 3. The fixed energy propagator..................... 57 3..1 General properties of the fixed energy propagator..... 57 3.. Semiclassical computation of KE)... 61 3..3 Two applications: reflection and tunneling through a barrier 64 3
4 CONTENTS 3..4 On the phase of the prefactor of Kx f,t f ; x i,t i )..... 68 3..5 On the phase of the prefactor of KE; x f,x i )....... 7 4 Instantons 75 4.1 Introduction.............................. 75 4. Instantons in the double well potential............... 77 4..1 The multi-instanton amplitude............... 81 4.. Summing up the instanton contributions: results..... 84 4..3 Cleaning up details: the zero-mode and the computation of R.............................. 88 5 Interaction with an external electromagnetic field 91 5.1 Gauge freedom............................ 91 5.1.1 Classical physics....................... 91 5.1. Quantum physics....................... 93 5. Particle in a constant magnetic field................ 95 5..1 An exercise on translations................. 96 5.. Motion............................ 97 5..3 Landau levels......................... 99 5.3 The Aharonov-Bohm effect..................... 101 5.3.1 Energy levels......................... 105 5.4 Dirac s magnetic monopole..................... 106 5.4.1 On coupling constants, and why monopoles are different than ordinary charged particles............... 107
Chapter 1 The path integral formalism 1.1 Introducing the path integrals 1.1.1 The double slit experiment One of the important experiments that show the fundamental difference between Quantum and Classical Mechanics is the double slit experiment. It is interesting with respect to the path integral formalism because it leads to a conceptual motivation for introducing it. Consider a source S of approximately monoenergetic particles, electrons for instance, placed at position A. The flux of electrons is measured on a screen C facing the source. Imagine now placing a third screen in between the others, with two slits on it, which can be opened or closed see figure 1.1). When the first slit is open and the second closed we measure a flux F 1,whenthefirstslit is closed and the second open we measure a flux F and when both slits are open we measure a flux F. 1 S A B C Figure 1.1: The double slit experiment: an electron source is located somewhere on A, andadetectorislocatedonthescreenc. AscreenB with two slits 1 and thatcanbeopenorclosedisplacedinbetween,sothattheelectrons have to pass through 1 or to go from A to C. We measure the electron flux on the screen C. In classical physics, the fluxes at C in the three cases are expected to satisfy the relation F = F 1 + F. In reality, one finds in general F = F 1 + F + F int, and the structure of F int precisely corresponds to the interference between two 5
6 CHAPTER 1. THE PATH INTEGRAL FORMALISM waves passing respectively through 1 and : F = Φ 1 +Φ = Φ 1 + Φ +ReΦ }{{}}{{} 1Φ ) 1.1) }{{} F 1 F int How can we interpret this result? What is the electron? More precisely: Does the wave behaviour imply that the electron is a delocalized object? That the electron is passing through both slits? Actually not. When detected, the electron is point-like, and moreremark- ably, if we try to detect where it went through, we find that it either goes through 1 or this is done by setting detectors at 1 and and considering a very weak flux, in order to make the probability of coinciding detections at 1 and arbitrarily small). According to the Copenhagen interpretation, F should be interpreted as a probability density. In practice this means: compute the amplitude Φ as if dealing with waves, and interpret the intensity Φ as a probability density for apoint-likeparticleposition. How is the particle/wave duality not contradictory? The answer is in the indetermination principle. In the case at hand: when we try to detectwhich alternative route the electron took, we also destroy interference. Thus, another formulation of the indetermination principle is: Any determination of the alternative taken by a process capable of following more than one alternative destroys the interference between the alternatives. Resuming: What adds up is the amplitude Φ and not the probability density itself. The difference between classical and quantum composition of probabilities is given by the interference between classically distinct trajectories. In the standard approach to Quantum Mechanics, the probability amplitude is determined by the Schrödinger equation. The Schrödinger viewpoint somehow emphasizes the wave properties of the particles electrons, photons,...). On the other hand, we know that particles, while they aredescribedby a probability wave, are indeed point-like discrete entities. As an example, in the double slit experiment, one can measure where the electron went through at the price of destroying quantum interference). This course will present an alternative, but fully equivalent, method to compute the probability amplitude. In this method, the role of the trajectory of a point-like particle will be formally resurrected, but in awaywhichiscompatible with the indetermination principle. This is the path integral approach to Quantum Mechanics. How can one illustrate the basic idea underlying this approach? 1.1. An intuitive approach to the path integral formalism In the double slit experiment, we get two interfering alternatives for the path of electrons from A to C. TheideabehindthepathintegralapproachtoQuantum Mechanics is to take the implications of the double slit experiment to its extreme consequences. One can imagine adding extra screens and drilling more and more F
1.1. INTRODUCING THE PATH INTEGRALS 7 holes through them, generalizing the result of the double slit experiment by the superposition principle. This is the procedure illustrated byfeynmaninhis book Quantum Mechanics and Path Integrals. Schematically: With two slits: we know that Φ = Φ 1 +Φ If we open a third slit, the superposition principle still applies: Φ = Φ 1 + Φ +Φ 3 Imagine then adding an intermediate screen D with N holes at positions x 1 D, x D,...,xN D see figure 1.). The possible trajectories are now labelled by x i D and α =1,, 3, that is by the slit they went through at D and by the slit they went through at B. 1 N 1 N 1 3 A D B C Figure 1.: The multi-slit experiment: We have a screen with N slits and a screen with three slits placed between the source and the detector. Applying the superposition principle: Φ= N i=1 α=1,,3 Φ x i D,α) Nothing stops us from taking the ideal limit where N and the holes fill all of D. Thesum i becomes now an integral over x D. Φ= α=1,,3 dx D Φx D,α) D is then a purely fictious device! We can go on and further refine our trajectories by adding more and more fictious screens D 1,D,...,D M Φ= dx D1 dx D dx DM Φx D1,x D,...,x DM ; α) α=1,,3 In the limit in which D i, D i+1 become infinitesimally close, we have specified all possible paths xy) seefigure1.3foraschematicrepresentation,wherethe screen B has been dropped in order to consider the simpler case of propagation in empty space).
8 CHAPTER 1. THE PATH INTEGRAL FORMALISM x xy i ) xy 1 ) xy ) xy 3 ) xy f ) y i y 1 y y 3 y f y Figure 1.3: The paths from x i,y i )tox f,y f )arelabelledwiththefunctions xy) thatsatisfyxy i )=x i and xy f )=x f. In fact, to be more precise, also the way the paths are covered through time is expected to matter t xy),yt))). We then arrive at a formal representation of the probability amplitude as a sum over all possible trajectories: Φ= All trajectories {xt),yt)} Φ{x}) 1.) How is this formula made sense of? How is it normalized? 1.1.3 The path integral formulation Based on the previous section, we will start anew to formulate quantummechanics. We have two guidelines: 1. We want to describe the motion from position x i at time t i to position x f at time t f with a quantum probability amplitude Kx f,t f ; x i,t i )givenby Kx f,t f ; x i,t i )= Φ{γ}) All trajectories where {γ} is the set of all trajectories satisfying xt i )=x i, xt f )=x f.. We want classical trajectories to describe the motion in the formal limit ħ 0. In other words, we want classical physics to be resurrected inthe ħ 0limit. Remarks: ħ has the dimensionality [Energy] [Time]. That is also the dimensionality of the action S which describes the classical trajectories via the principle of least action.
1.1. INTRODUCING THE PATH INTEGRALS 9 One can associate a value of S[γ] toeachtrajectory.theclassicaltrajectories are given by the stationary points of S[γ] δs[γ] = 0). It is thus natural to guess: Φ[γ] =fs[γ]/ħ), with f such that the classical trajectory is selected in the formal limit ħ 0. The specific choice Φ[γ] =e i S[γ] ħ 1.3) implying Kx f,t f ; x i,t i )= {γ} e i S[γ] ħ 1.4) seems promising for two reasons: 1. The guideline.) is heuristically seen to hold. In a macroscopic, classical, situation the gradient δs/δγ is for most trajectories much greater than ħ. Around such trajectories the phase e is/ħ oscillates extremely rapidly and the sum over neighbouring trajectories will tend to cancel. 1 see figure 1.4). γ 1 γ x f x i Figure 1.4: The contributions from the two neighbouring trajectories γ 1 and γ will tend to cancel if their action is big. x i γ cl Figure 1.5: The contributions from the neighbouring trajectories of the classical trajectory will dominate. x f On the other hand, at a classical trajectory γ cl the action S[γ]isstationary. Therefore in the neighbourhood of γ cl, S varies very little, so that all trajectories in a tube centered around γ cl add up coherently see figure 1.5) in the sum over trajectories. More precisely: the tube of trajectories 1 By analogy, think of the integral R + dxeifx) where fx) ax plays the role of S/ħ: the integral vanishes whenever the derivative of the exponent f = a is non-zero
10 CHAPTER 1. THE PATH INTEGRAL FORMALISM in question consists of those for which S S cl ħ and defines the extent to which the classical trajectory is well defined. We cannot expect to define our classical action to better than ħ. However, in normal macroscopic situations S cl ħ. Intheexactlimitħ 0, this effect becomes dramatic and only the classical trajectory survives. Once again, a simple one dimensional analogy is provided by the integral + /h dxeix,whichisdominatedbytheregionx < h around the stationary point x =0.. Eq. 1.3) leads to a crucial composition property. Indeed the action for apathγ 1 obtained by joining two subsequent paths γ 1 and γ,likein fig. 1.6, satisfies the simple additive relation S[γ 1 ]=S[γ 1 ]+S[γ ]. Thanks to eq. 1.3) the additivity of S translates into a factorization property for the amplitude: Φ[γ 1 ]=Φ[γ 1 ]Φ[γ ]whichinturnleadstoacomposition property of K, which we shall now prove. Consider indeed the amplitudes for three consecutive times t i <t int <t f. The amplitude Kx f,t f ; x i,t i )shouldbeobtainablebyevolvingintwo steps: first from t i to t int by Ky, t int ; x i,t i )foranyy ), and second from t int to t f by Kx f,t f ; y, t int ). Now, obviously each path with xt i )=x i and xt f )=x f can be obtained by joining two paths γ 1 y = xt int )) from t i to t int and γ y = xt int )) from t int to t f see figure 1.6). x γ 1 x i y γ x f t i t int t f t Figure 1.6: The path from x i,t i )tox f,t f )canbeobtainedbysummingthe paths γ 1 from x i,t i )toy, t int )withγ from y, t int )tox f,t f ).
1.1. INTRODUCING THE PATH INTEGRALS 11 Thanks to eq. 1.3) we can then write: dykx f,t f ; y, t int )Ky, t int ; x i,t i ) = γ 1,γ dye i ħ S[γ1y)]+S[γy)]) = γy)=γ y) γ 1y) dye i ħ S[γy)] = Kx f,t f ; x i,t i ). 1.5) Notice that the above composition rule is satisfied in Quantum Mechanics as easily seen in the usual formalism. It is the quantum analogue of the classical composition of probabilities P 1 = α P 1 α P α. 1.6) In quantum mechanics what is composed is not probability P itself but the amplitude K which is related to probability by P = K. It is instructive to appreciate what would go wrong if we modified the choice in eq. 1.3). For instance the alternative choice Φ = e S[γ]/ħ satisfies the composition property but does not in general select the classical trajectories for ħ 0. This alternative choice would select the minima of S but the classical trajectories represent in general only saddle points of S in function space. Another alternative Φ = e is[γ]/ħ),wouldperhapsworkoutinselectingtheclassical trajectories for ħ 0, but it would not even closely reproduce the composition property we know works in Quantum Mechanics. More badly: this particular choice, if S were to represent the action for a system of two particles, would imply that the amplitudes for each individual particle do not factorize even in the limit in which they are very far apart and non-interacting!). One interesting aspect of quantization is that a fundamental unitofaction ħ) isintroduced.inclassicalphysicstheoverallvalueoftheactioninunphysical: if Sq, q) S λ λsq, q) 1.7) the only effect is to multiply all the equations of motion by λ, so that the solutions remain the same the stationary points of S and S λ coincide). Quantum Mechanics sets a natural unit of measure for S. Dependingonthe size of S, thesystemwillbehavedifferently: large S classical regime small S quantum regime In the large S limit, we expect the trajectories close to γ cl to dominate K semiclassical limit). We expect to have: Kx f ; x i ) smooth function) e i S cl ħ. In the S ħ limit, all trajectories are comparably important: we must sum them up in a consistent way; this is not an easy mathematical task.
1 CHAPTER 1. THE PATH INTEGRAL FORMALISM Due to the mathematical difficulty, rather than going on with Feynman s construction and show that it leads to the same results of Schrödinger equation, we will follow the opposite route which is easier: we will derive the path integral formula from Schrödinger s operator approach. 1.1.4 From the Schröedinger approach to the path integral Consider the transition amplitude: Kx f,t f ; x i,t i ) x f e iht f t i ) ħ x i = x f e iht ħ xi 1.8) where we used time translation invariance to set: t i =0,t f t i = t. To write it in the form of a path integral, we divide t into N infinitesimal steps and consider the amplitude for each infinitesimal step see figure 1.7). We label the intermediate times t k = kϵ by the integer k =0,...,N. Notice that t 0 =0andt N = t. 0 ϵ ϵ N 1)ϵ t = Nϵ Figure 1.7: The interval between 0 and t is divided into N steps. We can use the completeness relation x x dx =1ateachstepnϵ: x f e iht ħ xi = N 1 k=1 dx k x f e ihϵ ħ xn 1 x N 1 e ihϵ ħ xn x e ihϵ ħ x1 x 1 e ihϵ ħ xi 1.9) Consider the quantity: x e ihϵ/ħ x. Using p p dp =1: x e ihϵ ħ x = dp x p p e ihϵ ħ x 1.10) If we stick to the simple case H = ˆp m + V ˆx), we can write: p e i ϵ ħ h i h i ˆp m +V ˆx) x = e i ϵ p ħ m +V x) p x + O ϵ ) 1.11) where the Oϵ )termsarisefromthenon-vanishingcommutatorbetweenˆp /m) and V ˆx). We will now assume, which seems fully reasonable, that in the limit ϵ 0thesehigherorertermscanbeneglected. Lateronweshallcome back on this issue and better motivate our neglect of these terms. We then get: x e ihϵ ħ x dp [ e i ϵ ħ ] h p i m +V x) e i ħ px x) πħ 1.1) We can define: x x ϵ ẋ: h dp x e ihϵ ϵ p i ħ x πħ e i ħ m +V x) pẋ 1.13)
1.1. INTRODUCING THE PATH INTEGRALS 13 By performing the change of variables p p mẋ the integral reduces to simple gaussian integral for the variable p: x e ihϵ ħ x d p = h ϵ p πħ e i ħ m m πiħϵ } {{ } 1 A i mẋ +V x) 1.14) e i ϵ ħ[ 1 mẋ V x)] = 1 A eilx,ẋ) ϵ ħ 1.15) At leading order in ϵ we can further identify Lx, ẋ)ϵ with the action Sx,x)= ϵ 0 Lx, ẋ)dt. Byconsideringalltheintervalswethusfinallyget: x f e iht ħ xi =lim ϵ 0 N 1 =lim A ϵ 0 1 k=1 N 1 1 dx k A N e i P N 1 ħ l=0 Sx l+1,x l ) k=1 dx k A }{{} R Dγ e i Sx f,x i ) ħ 1.16) where Dγ should be taken as a definition of the functional measure over the space of the trajectories. We thus have got a path integral formula for the transition amplitude: Kx f,t; x i, 0) = D[xt)]e i S cl x,ẋ) ħ 1.17) We see that we actually somebody else before us!) had guessedwelltheform of the transition amplitude. The path integral approach to QM wasdeveloped by Richard Feyman in his PhD Thesis in the mid 40 s, following ahintfrom an earlier paper by Dirac. Dirac s motivation was apparently toformulateqm starting from the lagrangian rather than from the hamiltonian formulation of classical mechanics. Let us now come back to the neglected terms in eq. 1.11). To simplify the notation let us denote the kinetic operator as T = i p mħ and the potential U = iv/ħ; wecanthenwrite p e ϵt +U) x = p e ϵt e ϵt e ϵt +U) e ϵu e ϵu x = p e ϵt e ϵc e ϵu x 1.18) h = e i ϵ p i ħ m +V x) p e ϵc x 1.19) where C is given, by using the Campbell-Baker-Haussdorf formula twice, as a series of commutators between T and U C = 1 [T,U]+ ϵ {[T,[T,U]] + [U, [U, T ]]} +... 1.0) 6 By iterating the basic commutation relation [ˆp, V ˆx)]] = iv ˆx)andexpanding the exponent in a series one can then write p e ϵc x =1+ϵ n= ϵ n p r s P n,s,r x) 1.1) r=n s=r n=0 r=1 s=0
14 CHAPTER 1. THE PATH INTEGRAL FORMALISM where P n,s,r x) isahomogenouspolynomialofdegreen +1 r in V and its derivatives, with each term involving exactly r + s derivatives. For instance P 1,0,1 x) =V x). We can now test our result under some simple assumption. For instance, if the derivatives of V are all bounded, the only potential problem to concentrate on in the ϵ 0limitisrepresentedbythepowersofp. This is because the leading contribution to the p integral in eq. 1.15) comes from the region p 1/ ϵ,showingthatp diverges in the small ϵ limit. By using p 1/ ϵ the right hand side of eq. 1.1) is 1+Oϵ 3/ )sothat,eventaking into account that there are N 1/ϵ such terms one for each step), the final result is still convergent to 1 ) lim 1+aϵ 3 1 ϵ =1. 1.) ϵ 0 Before proceeding with technical developments, it is worth assessing the rôle of the path integral P.I.) in quantum mechanics. As it was hopefully highlighted in the discussion above, the path integral formulation is conceptually advantageous over the standard operatorial formulation of Quantum Mechanics, in that the good old particle trajectories retain some rôle. The P.I. is however technically more involved. When working on simple quantum systems like the hydrogen atom, no technical profit is really given by path integrals. Nonetheless, after overcoming a few technical difficulties, the path integral offers a much more direct viewpoint on the semiclassical limit. Similarly, for issues involving topology like the origin of Bose and Fermi statistics, the Aharonov-Bohm effect, charge quantization in the presence of a magnetic monopole, etc...path integrals offer a much better viewpoint. Finally, for advanced issues like the quantization of gauge theories and for effects like instantons in quantum field theory it would be hard to think how to proceed without path integrals!...butthatisforanother course. 1. The properties of the path integrals 1..1 Path integrals and state evolution To get an estimate of the dependence of the amplitude Kx f,t f ; x i,t i )onits arguments, let us first look at the properties of the solutions totheclassical equations of motion with boundary conditions x c t i )=x i, x c t f )=x f. Let us compute tf S cl.wheres cl is defined S cl S[x c ]= tf t i Lx c, ẋ c )dt 1.3) with x c asolution,satisfyingtheeuler-lagrangeequation: [ L t ẋ L ] =0. 1.4) x x=x c We can think of x c as a function x c fx i,x f,t i,t f,t)
1.. THE PROPERTIES OF THE PATH INTEGRALS 15 such that fx i,x f,t i,t f,t= t i ) x i = const fx i,x f,t i,t f,t= t f ) x f = const Differentiating the last relation we deduce: [ tf x c + t x c ] t=t f =0 = tf x c t=tf = ẋ c t=tf 1.5) Similarly, differentiating the relation at t = t i we obtain And the following properties are straightforward: Using these properties, we can now compute: tf S = Lx, ẋ) tf t=tf + t i = Lx, ẋ) tf t=tf + dt t i tf x c t=ti =0 1.6) xf x c t=tf =1 1.7) [ L dt All this evaluated at x = x c gives: xf x c t=ti =0 1.8) x t f x + L [ L t ẋ t f x ] ẋ t f ẋ )] tf t i L dt t ẋ L ) tf x x 1.9) tf S cl = Lx c, ẋ c ) L t=t f t=tf + ẋ tf x c) x=xc t=t i = Lx c, ẋ c ) L ẋ ẋ c) = E 1.30) x=xc t=tf We recognize here the definition of the hamiltonian of the system, evaluated at x = x c and t = t f. We can also compute: xf S = tf t i [ )] L tf dt t ẋ L x f x dt t t i ẋ L ) xf x 1.31) x Evaluated at x = x c,thisgives: L xf S cl x c )= ẋ xf x c) x=xc t=t f t=t i = L ẋ = P 1.3) x=xc,t=t f We recognize here the definition of the momentum of the system,evaluated at x = x c and t = t f.
16 CHAPTER 1. THE PATH INTEGRAL FORMALISM At this point, we can compare these results with those of the path integral computation in the limit ħ 0. Kx f,t f ; x i,t i )= D[xt)]e i S ħ 1.33) In the limit ħ we expect the path integral to be dominated by the classical trajectory and thus to take the form and we shall technically demostrate this expectation later on) Kx f,t f ; x i,t i )=Fx f,t f,x i,t i ) e i S cl ħ 1.34) }{{} smooth function where F is a smooth function in the ħ 0limit.Wethenhave: so that we obtain t,x e i S cl 1 ħ ħ large 1.35) t,x F O1) comparatively negligible 1.36) iħ xf K = xf S cl K + O ħ) =P cl K + O ħ) 1.37) iħ tf K = tf S cl K + O ħ) =E cl K + O ħ) 1.38) Momentum and energy, when going to the path integral description of particle propagation, become associated to respectively the wave number i x ln K) 1 ) and oscillation frequency i t ln K)). The last two equations make contact with usual quantum mechanical relations, if we interpret K has the wave function and iħ x and iħ t as respectively the momentum and energy operators iħ x K P K 1.39) iħ t K E K 1.40) The function Kx,t ; x, t) = x e iht t) ħ x is also known as the propagator, as it describes the probability amplitude for propagating a particle from x to x in a time t t. From the point of view of the Schrödinger picture, Kx,t; x, 0) represents the wave function at time t for a particle that was in the state x at time 0: ψt, x) x e iht ħ ψ0 x ψt) 1.41) By the superposition principle, K can then be used to write the most general solution. Indeed: 1. Kx,t ; x, t) solvestheschrödingerequationfort, x as variables: iħ t K = dx Hx,x )Kx,t ; x, t) 1.4) where Hx,x )= x H x.
1.. THE PROPERTIES OF THE PATH INTEGRALS 17. lim t t Kx,t ; x, t) = x x = δx x) Thus, given Ψ 0 x) thewavefunctionattimet 0,thesolutionatt>t 0 is: Ψx, t) = x e iht t 0 ) ħ Ψ 0 = x e iht t 0 ) ħ y y Ψ 0 dy = Kx, t; y, t 0 )Ψ 0 y)dy 1.43) Having Kx,t ; x, t), the solution to the Schrödinger equation is found by performing an integral. Kx,t ; x, t) containsalltherelevantinformationonthedynamicsofthe system. 1.. The path integral computation for a free particle Let us compute the propagator of a free particle, decribed by the lagrangian Lx, ẋ) =mẋ /, using Feyman s time slicing procedure. Using the result of section 1.1.4 we can write where: N 1 1 dx k Kx f,t f ; x i,t i )=lim ϵ 0 A A e is ħ 1.44) k=1 ϵ = t f t i N N 1 1 S = mx k+1 x k ) ϵ x 0 = x i k=0 x N = x f Let us compute eq. 1.44) by integrating first over x 1.Thiscoordinateonly appears only in the first and the second time slice so we need just to concentrate on the integral = dx 1 e i m ϵħ [x 1 x 0) +x x 1) ] dx 1 e i m ϵħ hx 1 1 x0+x)) 1 + x x0)i πiϵħ 1 = e i m 1 ϵħ x x0) 1.45) m }{{} A 1 Notice that x 1 has disappeared: we integrated it out in the physics jargon. Putting this result back into eq. 1.44) we notice that we have anexpression
18 CHAPTER 1. THE PATH INTEGRAL FORMALISM similar to the original one but with N 1 instead of N time slices. Moreover the first slice has now a width ϵ instead of ϵ: the factors of 1/ inboththe exponent and prefactor of eq. 1.45) are checked to match this interpratation. It is now easy to go on. Let us do the integral on x to gain more confidence. The relevant terms are = dx e i m ϵħ [ 1 x x0) +x 3 x ) ] h dx e i m 3 ϵħ x 1 3 x0 3 x3) + 1 3 x3 x0)i πiϵħ = e i m 1 ϵħ 3 x3 x0) 1.46) m 3 }{{} A 3 It is now clear how to proceed by induction. At the n-th step the integral over x n will be: dx n e i ϵħ[ m 1 n xn x0) +x n+1 x n) ] = h dx n e i m n+1 ϵħ n x n 1 n+1 x0+ n n+1 xn+1)) + 1 n+1 xn+1 x0)i πiϵħ n = m n +1 }{{} A n n+1 e i m 1 ϵħ n+1 xn+1 x0) 1.47) Thus, putting all together, we find the expression of the propagator for a free particle: 1 1 Kx f,t f ; x i,t i )=lim ϵ 0 A 3...N 1 N ei m ħnϵ x f x i) 1 =lim ϵ 0 A N ei m m x f x i ) = πiħt f t i ) ei ħ t f t i 1.48) m ħnϵ x f x i) where we used the fact that Nϵ = t f t i. We can check this result by computing directly the propagatorofaone- dimensional free particle using the standard Hilbert space representation of the propagator: Kx f,t; x i, 0) = x f p p e iht ħ xi dp = 1 πħ = 1 πħ = m πiħt ei e i p ħ x f x i) e i p t mħ dp e it mħ mx f x i ) ħt p m x f x i t +i m x f x i ) ħ t dp = F t)e i S cl x f,t;x i,0) ħ 1.49)
1.3. PATH INTEGRALS AS DETERMINANTS 19 These results trivially generalize to the motion in higher dimensional spaces, as for the free particle the result factorizes. As we already saw, we can perform the same integral à la Feynman by dividing t into N infinitesimal steps. It is instructive interpret the result for the probability distribution: dp dx = K = m πħt 1.50) Let us interpret this result in terms of a flux of particles that startedat x i =0att =0withadistributionofmomenta dnp) =fp)dp 1.51) and find what fp) is. Aparticlewithmomentump at time t will have travelled to x =p/m)t. Thus the particles in the interval dp will be at time t in the coordinate interval dx =t/m)dp. Therefore we have dnx) =fxm/t)m/t)dx. Comparing to 1.50), we find f =1/πħ) = const. Thus,dnp) =1/πħ)dp dn/dp [Length] 1 [Momentum] 1. We recover the result that the wave function ψx, t) =Kx, t;0, 0) satisfies ψx, 0) = δx) whichcorrespondsinmomentumspacetoanexactlyflat distribution of momenta, with normalization dnp)/dp =1/πħ). 1.3 Path integrals as determinants We will now introduce a systematic small ħ expansion for path integrals and show that the computation of the leading term in the expansion corresponds to the computation of the determinant of an operator. Before going to that, let us remind ourselves some basic results about Gaussian integrals in several variables. 1.3.1 Gaussian integrals Let us give a look at gaussian integrals: dx i e P i,j λij xixj 1.5) i With one variable, and with λ real and positive we have: 1 π dxe λx = dye y = λ λ 1.53) Consider now a Gaussian integral with an arbitrary number of real variables dx i e P i,j λijxixj 1.54) where λ ij is real. i
0 CHAPTER 1. THE PATH INTEGRAL FORMALISM We can rotate to the eigenbasis: x i = O ij x j,witho T λo =diagλ 1,...,λ n ), O T O = OO T =1. Then, i dx i = det O i d x i = i d x i.thus: dx i e P i,j λijxixj = d x i e P i λi x i = i i i π 1 = λ i det 1 ˆλ) π 1.55) where again the result makes sense only when all eigenvalues λ n are positive. Along the same line we can consider integrals with imaginary exponents, which we indeed already encountered before I = dxe iλx = e i sgnλ) π π 4 1.56) λ This integral is performed by deforming the contour into the imaginary plane as discussed in homework 1. Depending on the sign of λ we must choose different contours to ensure convergence at infinity. Notice that the phase depends on the sign of λ. For the case of an integral with several variables I = dx i e i P j,k λ jkx jx k 1.57) the result is generalized to i I = e in+ n ) π 4 1 det 1 π ˆλ) 1.58) where n + and n are respectively the number of positive and negative eigenvalues of the matrix λ jk. 1.3. Gaussian Path Integrals Let us now make contact with the path integral: Kx f,t f ; x i,t i )= D[xt)]e i S[xt)] ħ 1.59) with S[xt)] = tf t i Lxt), ẋt))dt 1.60) To perform the integral, we can choose the integration variables that suit us best. The simplest change of variables is to shift xt) to center itaroundthe classical solution x c t): xt) =x c t)+yt) D[xt)] = D[yt)] the Jacobian for this change of variables is obviously trivial, as we indicated).
1.3. PATH INTEGRALS AS DETERMINANTS 1 We can Taylor expand S[x c t)+yt)] in yt) notethatyt i )=yt f )= 0, since the classical solution already satisfies the boundary conditions x c t i )=x i and x c t f )=x f ). By the usual definition of functional derivative, the Taylor expansion of a functional reads in general δs S[x c t)+yt)] = S[x c ]+ dt 1 δxt 1 ) yt 1 ) 1.61) x=xc + 1 δ S dt 1 dt δxt 1 )δxt ) yt 1 )yt )+O y 3) x=xc with obvious generalization to all orders in y. InourcaseS = dtlx, ẋ) so that each term in the above general expansion can be written asasingle dt integral { δs dt 1 L δxt 1 ) yt 1 ) dt x=xc x y + L } x=xc ẋ ẏ 1.6) x=xc 1 δ S dt 1 dt δxt 1 )δxt ) yt 1 )yt ) x=xc { } L dt x y + L x=xc x ẋ yẏ + L x=xc ẋ ẏ x=xc 1.63) and so on. The linear term in the expansion vanishes by the equations of motion: δs δx yt)dt = x=xc L x d ) L yt)dt + L dt ẋ x=x c ẋ yt) t=t f t=t i =0 1.64) Thus, we obtain the result: Kx f,t f ; x i,t i )= D[yt)]e i ħ h S[x ct)]+ 1 δ S δx y +Oy 3 ) i 1.65) where δ S/δx is just short hand for the second variation of S shown in eq. 1.63). We can rewrite it rescaling our variables y = ħ ỹ: Kx f,t f ; x i,t i )=N e i ħ S[xct)] D[ỹt)]e i δ S δx ỹ +O ħ ỹ 3 ) 1.66) where the overall constant N is just the Jacobian of the rescaling. The interpretation of the above expression is that the quantum propagation of a particle can be decomposed into two pieces: 1. the classical trajectory x c t), which gives rise to the exponent e is[xc]/ħ. Given that N is a constant its role is only to fix the right overall normalization of the propagator, and does not matter in assessing the relative importance of each trajectory. Therefore it does not play a crucial role in the following discussion.
CHAPTER 1. THE PATH INTEGRAL FORMALISM. the fluctuation yt)overwhichwemustintegrate;becauseofthequadratic ħ ) term in the exponent, the path integral is dominated by y O that is ỹ O1)) so that yt) representsthe quantum fluctuation around the classical trajectory. In those physical situations where ħ can be treated as a small quantity, one can treat Oħ) intheexponentofthepathintegralasaperturbation Kx f,t f ; x i,t i ) = N e i ħ S[xct)] D[ỹt)]e i δ S δx ỹ [1 + O ħ)] = 1.67) = F x f,t f ; x i,t i )e i ħ S[xct)] [1 + O ħ)] 1.68) Where the prefactor F is just the gaussian integral around the classical trajectory. The semiclassical limit should thus correspond to the possibility to reduce the path integral to a gaussian integral. We will study this in moredetaillater on. In the small ħ limit, the classical part of the propagator, expis[x c ]/ħ) oscillates rapidly when the classical action changes, while theprefactorf and the terms in square brakets in eq. 1.68)) depend smoothly on ħ in this limit. As an example, we can compute the value of S cl for a simple macroscopic system: a ball weighing one gram moves freely along one meter in one second. We will use the following unity for energy: [erg] = [g] [cm]/[s]). S cl = 1 mx f x i ) t f t i = 1 mv x =5000erg s ħ =1.0546 10 7 erg s S cl /ħ 10 30 1.3.3 Oħ) corrections to Gaussian approximation It is a good exercise to write the expression for the Oħ) correctiontothe propagator K in eq. 1.68). In order to do so we must expand the action to order ỹ 4 around the classical solution using the same short hand notation as before for the higher order variation of S) S[x c + ħỹ] ħ = S[x c] ħ + 1 δ S δx ỹ + ħ δ 3 S 3! δx 3 ỹ3 + ħ δ 4 S 4! δx 4 ỹ4 + Oħ 3/ ) 1.69) and then expand the exponent in the path integral to order ħ. Theleadingħ 1/ correction vanishes because the integrand is odd under ỹ ỹ ħ δ 3 S D[ỹt)] 3! δx 3 ỹ3 e i δ S δx ỹ =0 1.70) and for the Oħ) correctiontok we find that two terms contribute { K = iħe i ħ S[xct)] 1 δ 4 S N D[ỹt)] 4! δx 4 ỹ4 + i 1 [ 1 δ 3 ] } S 3! δx 3 ỹ3 e i δ S δx ỹ 1.71)
1.3. PATH INTEGRALS AS DETERMINANTS 3 1.3.4 Quadratic lagrangians and the harmonic oscillator Consider now a special case: the quadratic lagrangians, defined by the property: δ n) S =0 forn> 1.7) δxn For these lagrangians, the gaussian integral coresponds to the exact result: Kx f,t f ; x i,t i )=e i S[xc] ħ D[y]e i 1 δ S ħ δx y = const e i S[xc] ħ ) 1 det δ S δx 1.73) To make sense of the above we must define this strange beast : the determinant of an operator. It is best illustrated with explicit examples, and usually computed indiretly using some trick. This is a curious thing about path integrals: one never really ends up by computing them directly. Let us compute the propagator for the harmonic oscillator using the determinant method. We have the lagrangian: Lx, ẋ) = 1 mẋ 1 mω x 1.74) It is left as an exercise to show that: S cl = tf t i Lx c, ẋ c )dt = mω [ ) x f + x i cotωt) x ] f x i sinωt) 1.75) where T = t f t i. Then, tf t i Lx c + y, ẋ c +ẏ)dt = tf 1 t i m ẋc ω xc And thus, the propagator is: Kx f,t f ; x i,t i )=e i S cl ħ D[y]e i ħ ) dt + tf R t f t i t i 1 m ẏ ω y ) dt m ẏ ω y )dt 1.76) = e i S cl x f,x i,t f t i ) ħ Jt f t i ) 1.77) This is because yt) satisfiesyt i )=yt f )=0andanyreferencetox f and x i has disappeared from the integral over y. Indeed J is just the propagator from x i =0tox f =0: Kx f,t f ; x i,t i )=e i S cl x f,x i,t f t i ) ħ K0,t f ;0,t i ) }{{} Jt f t i) 1.78) We already have got a non trivial information by simple manipulations. The question now is how to compute Jt f t i ).
4 CHAPTER 1. THE PATH INTEGRAL FORMALISM 1. Trick number 1: use the composition property of the transition amplitude: Kx f,t f ; x i,t i )= dx Kx f,t f ; x, t)kx, t; x i,t i ) Applying this to Jt f t i ), we get: Jt f t i )=K0,t f ;0,t i )= Jt f t)jt t i ) e i ħ S cl0,x,t f t)+s cl x,0,t t i)) dx 1.79) We can put the expression of the classical action in the integral, and we get T = t f t i, T 1 = t f t, T = t t i ): JT ) JT 1 )JT ) = dxe i mω ħ [x cotωt 1)+cotωT ))] = dxe iλx = iπ λ = [ iπħ mω ] 1 sinωt 1 )sinωt ) sinωt) 1.80) The general solution to the above equation is mω JT )= πiħ sinωt) eat 1.81) with a an arbitrary constant. So this trick is not enough to fully fix the propagator, but it already tells us a good deal about it. We will now compute J directly and find a =0. Noticethatinthelimitω 0, the result goes back to that for a free particle.. Now, let us compute the same quantity using the determinant method. What we want to compute is: R JT )= D[y]e i t f m ħ t i ẏ ω y ) 1.8) with the boundary conditions: yt i )=yt f )=0 1.83) We can note the property: tf m i ẏ ω y ) tf ) m d dt = i t i ħ t i ħ y dt + ω ydt= i tf yôy dt t i 1.84) where Ô = m d ħ dt + ω ). Formally, we have to perform a gaussian integral, and thus thefinalresult Ô) 1/. is proportional to det To make it more explicit, let us work in Fourier space for y: yt) = n a n y n t) 1.85)
1.3. PATH INTEGRALS AS DETERMINANTS 5 where y n t) aretheothonormaleigenfunctionsofô: y n = T sinnπ T t),n N 1.86) Ôy n t) =λ n y n t) 1.87) y n y m dt = δ nm 1.88) The eigenvalues are λ n = m ħ [ nπ ) ω ] T 1.89) Notice that the number of negative eigenvalues is finite and given by n = int Tω π ), that is the number of half periods of oscillation contained int. The y n form a complete basis of the Hilbert space L [t i,t f ]) mod yt i )= yt f )=0. Thus,thea n form a discrete set of integration variables, and we can write: D[yt)] = da n Ñ n πi }{{} 1.90) Jacobian where the 1/ πi factors are singled out for later convenience and also to mimick the 1/A factors in our definition of the measure in eq. 1.16)). We get: yôy dt = a m a n y m Ôy n dt = λ n an 1.91) m,n n And thus, JT ) = Ñ da n e i Pn λna n = πi = n n λ n ) 1 ill defined ) 1 {}}{ λ n Ñ n }{{} well defined e in+ n ) π 4 e in++n ) π 4 Ñe in π 1.9) Notice that in the continuum limit ϵ 0inthedefinitionofthemeasure eq. 1.16), we would have to consider all the Fourier modes with arbitrarily large n. InthislimitboththeproductofeigenvaluesandtheJacobian Ñ are ill defined. Their product however, the only physically relevant quantity, is well defined. We should not get into any trouble if weavoid computing these two quantities separately. The strategy is to work only with ratios that are well defined. That way we shall keep at large from mathematical difficulties and confusion!), as we will now show. Notice that Ô depends on ω, whileñ obviously does not: the modes y n do not depend on ω 3. For ω =0,ourcomputationmustgivethefree 3 Indeed a stronger result for the independence of the Jacobian onthequadraticaction holds, as we shall discuss in section 3.1.
6 CHAPTER 1. THE PATH INTEGRAL FORMALISM particle result, which we already know. So we have ) 1 J ω T )=Ñ det Ôω e in π 1.93) ) 1 J 0 T )=Ñ det Ô0 1.94) We can thus consider J ω T ) π det Ô0 J 0 T ) ein = det Ôω ) 1 The eigenvalues of the operator Oω ˆ = m ħ functions yt) havebeengivenbefore: n=1 λ n 0) λ n ω) = nπ T λ n 0) λ n ω) = n=1 nπ ) T n = λ ) 1 nω =0) n λ = ) 1 λ n 0) nω) λ n n ω) 1.95) ) d dt + ω over the space of ) ω = 1 1 1 a nπ 1 ωt nπ ) = a sin a ) = 1 1 ) a 1.96) nπ 1.97) By eq. 1.95) we get our final result: ωt π mω π J ω T )=J 0 T ) sinωt) e in = πiħ sinωt) e in 1.98) So that by using eq. 1.75) the propagator is assume for simplicity ωt < π) h mω mω Kx f,t; x i, 0) = πiħ sinωt) ei ħ x f +x i ) cotωt) x f x i i sinωt ) 1.99) At this point, we can make contact with the solution of the eigenvalue problem and recover the well know result for the energy levels of the harmonic oscillator. Consider {Ψ n } acompleteorthonormalbasisofeigenvectorsofthe hamiltonian such that H Ψ n = E n Ψ n.thepropagatorcanbewrittenas: Kx f,t f ; x i,t i )= m,n x f Ψ n Ψ n e iht f t i ) ħ Ψ m Ψ m x i = Let us now define the partition function: Zt) = Kx, T ; x, 0)dx = ) Ψ n x) dx n n Ψ nx i )Ψ n x f )e ient f t i ) ħ 1.100) e ient ħ For the harmonic oscillator, using eq. 1.100) we find: 1 Kx, T ; x, 0)dx = i sin ) = e i 1 ωt = ωt 1 e i ωt n=0 = n e ient ħ e in+1) ωt 1.101) 1.10)
1.4. OPERATOR MATRIX ELEMENTS 7 E n = ħω n + 1 ) 1.103) 1.4 Operator matrix elements We will here derive the matrix elements of operators in the path integral formalism. 1.4.1 The time-ordered product of operators Let us first recall the basic definition of quantities in the Heisenberg picture. Given an operator Ô in the Schroedinger picture, the time evolved Heisenberg picture operator is Ôt) e iht Ôe iht. 1.104) In particular for the position operator we have ˆxt) e ihtˆxe iht. 1.105) Given a time the independent position eigenstates x, the vectors x, t e iht x 1.106) represent the basis vectors in the Heisenberg picture notice the + in the exponent as opposed to the intheevolutionofthestatevectorinthe Schroedinger picture!), being eigenstates of ˆxt) ˆxt) x, t = x x, t. 1.107) We have the relation: x f,t f x i,t i = D[xt)]e i S[xt)] ħ. 1.108) By working out precisely the same algebra which we used in section Consider now a function Ax). It defines an operator on the Hilbert space: ˆx) x x Ax)dx 1.109) Using the factorization property of the path integral we can derive the following identity: D[x]Axt 1 ))e i S[x] ħ = D[x a ]D[x b ]dx 1 Ax 1 )e i S[xa]+S[x b ] ħ = dx 1 x f e iht f t 1 ) ħ x 1 x 1 e iht 1 t i ) ħ x i Ax 1 )= x f e iht f t 1 ) ħ ˆx)e iht 1 t i ) ħ x i = x f e iht f ħ ˆxt 1 ))e iht i ħ x i = x f,t f ˆxt 1)) x i,t i 1.110)
8 CHAPTER 1. THE PATH INTEGRAL FORMALISM Let us now consider two functions O 1 xt)) and O xt)) of xt) andletus study the meaning of: D[x]O 1 xt 1 ))O xt ))e i S[x] ħ 1.111) By the composition property, assuming that t 1 <t,wecanequalittosee figure 1.8): D[x a ]D[x b ]D[x c ]dx 1 dx e i S[xa]+S[x b ]+S[xc] ħ O x )O 1 x 1 ) 1.11) t 1 t i a b t t f Figure 1.8: The path from x i,t i )tox f,t f )isseparatedintothreepathsa, b and c. Wehavetodistinguisht 1 <t from t <t 1. c This can be rewritten: x f,t f x,t O x ) x,t x 1,t 1 O 1 x 1 ) x 1,t 1 x i,t i dx dx 1 1.113) Using the equation 1.109), it can be rewritten: x f e iht f t ) ħ Ô ˆx)e iht t 1 ) ħ Ô 1 ˆx)e iht 1 t i ) ħ x i = x f,t f Ôt )Ô1t 1 ) x i,t i 1.114) Recall that this result is only valid for t >t 1. Exercise: Check that for t 1 >t,onegetsinstead: x f,t f Ô 1 t 1 )Ô t ) x i,t i 1.115) Thus, the final result is: D[x]O 1 xt 1 ))O xt ))e i S[x] ħ = x f,t f T [Ô1t 1 )Ôt )] x i,t i 1.116) where the time-ordered product T [Ô1t 1 )Ôt )] is defined as: T [Ô1t 1 )Ôt )] = θt t 1 )Ôt )Ô1t 1 )+θt 1 t )Ô1t 1 )Ôt ) 1.117) where θt) istheheavisidestepfunction.
1.4. OPERATOR MATRIX ELEMENTS 9 Exercise: Consider a system with classical Lagrangian L = m ẋ V x). Treating V as a perturbation and using the path integral, derive the well-known formula for the time evolution operator in the interaction picture Ut) =e ih0t/ħ ˆT [e i ħ where H 0 = ˆp m and ˆV t, ˆx) =e ih0t/ħ V ˆx)e ih0t/ħ. R t 0 ˆV t,ˆx)dt ] 1.118)
30 CHAPTER 1. THE PATH INTEGRAL FORMALISM
Chapter Functional and Euclidean methods In this chapter, we will introduce two methods: The functional method, which allows a representations of perturbation theory in terms of Feynman diagrams. The Euclidean path integral, which is useful for, e.g., statistical mechanics and semiclassical tunneling. We will show a common application of both methods by finding theperturbative expansion of the free energy of the anharmonic oscillator..1 Functional method Experimentally, to find out the properties of a physical system for example an atom), we let it interact with an external perturbation a source, e.g. an electromagnetic wave), and study its response. This basic empirical fact has its formal counterpart in the theoretical study of classical and) quantum systems. To give an illustration, let us consider a system with Lagrangian Lx, ẋ) and let us make it interact with an arbitrary external forcing source: Lx, ẋ) Lx, ẋ) +Jt)xt) Kx f,t f ; x i,t i J) = D[x]e i ħ R Lx,ẋ)+Jt)xt))dt.1) K has now become a functional of Jt). This functional contains important physical information about the system all information indeed) as exemplified by the fact that the functional derivatives of K give the expectation values of ˆx and thus of all operators that are a function of x): ħδ iδjt 1 ) ħδ iδjt n ) Kx f,t f ; x i,t i J) = x f,t f T [ˆxt 1 ),...,ˆxt n )] x i,t i J=0 31.)
3 CHAPTER. FUNCTIONAL AND EUCLIDEAN METHODS As an example, let us see how the effects of a perturbation can be treated with functional methods. Let us look at the harmonic oscillator, with an anharmonic self-interaction which is assumed to be small enough to be treated as a perturbation: L 0 x, ẋ) = m ẋ mω x, L pert x, ẋ) = λ 4! x4.3) The propagator K 0 [J] fortheharmonicoscillatorwithasourcejt) is: K 0 [J] = D[x]e i R ħ L0+Jt)xt))dt.4) Consider now the addition of the perturbation. By repeated use of the functional derivatives with respect to J see eq..)) the propagator can be written as: K[J] = D[x]e i R ħ L0 λ 4! x4 +Jt)xt))dt = D[x] ) n 1 iλ x 4 t 1 ) x 4 R t n )e i L0+Jt)xt))dt ħ dt 1 dt n n! ħ4! n = ) n 1 iλ ħ 4 δ 4 n! ħ4! δj 4 t n 1 ) ħ 4 δ 4 δj 4 t n ) K 0[J]dt 1 dt n exp dt iλħ3 δ 4 ) 4! δj 4 K 0 [J].5) t) Thus, when we take the J 0limit,wegetthepropagatoroftheanharmonic hamiltonian in terms of functional derivatives of the harmonic propagator in presence of an arbitrary source J.. Euclidean Path Integral Instead of computing the transition amplitude x f e iht ħ computed the imaginary time evolution: x i,wecouldhave K E x f e βh ħ xi, β R +.6) which corresponds to an imaginary time interval t = iβ. 0 ϵ ϵ N 1)ϵ β = Nϵ Figure.1: The interval between 0 and β is divided into N steps. It is straightforward to repeat the same time slicing procedure we employed in section 1.1.4 see figure.1) and write the imaginary time amplitude as a path integral. The result is: K E x f,x i ; β) = D[x] E e 1 ħ SE[x].7)
.. EUCLIDEAN PATH INTEGRAL 33 where D[x] E is a path integration measure we shall derive below and where S E [x] = β 0 m ) dx + V x)) dτ βe.8) dτ Notice that S E is normally minimized on stationary points. From a functional viewpoint K E is the continuation of K to the imaginary time axis K E x f,x i ; β) =Kx f,x i ; iβ)..9) Notice that more sloppily we could have also gotten K E by analytic continuation to imaginary time directly in the path integral: Kt) = D[x]e i ħ R t 0 Lx,ẋ)dt.10) Then, with the following replacements t = iτ dt = idτ ẋ = i dx dτ t = iβ we obviously get the same result. Notice that the proper time interval in the relativistic description ds = dt + dx i,correspondingtoaminkovskymetric with signature, +, +, +, gets mapped by going to imaginary time into a metric with Euclidean signature ds E = dτ + dx i.hencethe Euclidean suffix. Let us now compute explicitly by decomposing the imaginary time interval [0,β]intoN steps of equal length ϵ see figure.1). As in section 1.1.4 we can write: K E x f,x i ; β) = x N e Hϵ ħ where x f = x N and x i = x 0. We can compute, just as before: x e Hϵ ħ x = x p p e Hϵ ħ 1 πħ = 1 e ϵ πħ m = = πϵħ e 1 ħ xn 1 x 1 e Hϵ ħ x0 dx 1 dx N 1.11) e ip ħ x x) ϵ ħ x dp p m +V x) dp mħp i m ϵ x x)) m ϵħ x x) ϵ ħ V x) dp» m x x +V ϵ x) ϵ m πϵħ e 1 ħ LEx,x )ϵ 1 A E e S E x,x) ħ.1)
34 CHAPTER. FUNCTIONAL AND EUCLIDEAN METHODS and then proceed to obtain the Euclidean analogue of eq. 1.16) K E x f,x i ; β) =lim ϵ 0 N 1 =lim 1 ϵ 0 A E k=1 1 dx k A N e P N 1 ħ l=0 SEx l+1,x l ) E N 1 k=1 dx k A E }{{} R D[x]E e S E x f,x i ) ħ.13) Synthetizing: to pass from the real time path integral formulation to the Euclidean time formulation, one simply makes the change iϵ ϵ in the factor A defining the measure of integration, and modifies the integrand according to is = i t 0 ) 1 β ) 1 mẋ V x) dt S E = 0 mẋ + V x) dτ e is/ħ e SE/ħ.14)..1 Statistical mechanics Recalling the result of the previous section we have K E x f,x i ; β) = n Ψ nx f )Ψ n x i )e βen/ħ.15) where {Ψ n } is an othonormal basis of eigenvectors of the Hamiltonian. We can then define the partition function: Z[β] = dxk E x, x; β) = e βen/ħ.16) n We recognize here the statistical mechanical definition of the thermal partition function for a system at temperature k B T = ħ/β, wherek B is the Boltzmann constant. We can observe the equivalence: Z[β] = dxk E x, x; β) = {x0)=xβ)} D[x]e SE.17) Thus, the thermal partition function is equivalent to a functional integral over a compact Euclidean time: τ [0,β]withboundariesidentified. The integration variables xτ) arethereforeperiodicinτ with period β. Let us now check that K iβ) leadstotherightpartitionfunctionforthe harmonic oscillator. Using equation 1.10), we get: Z[β] = Kx, x; iβ)dx = e βωn+ ) 1.18) n which is indeed the expected result.
.3. PERTURBATION THEORY 35.3 Perturbation theory The purpose of this section is to illustrate the use of functional methods in the euclidean path integral. We shall do so by focussing on the anharmonic oscillator and by computing in perturbation theory the following physically relevant quantities the time ordered correlators on the ground state: 0 T [ˆxτ 1 )...ˆxτ n )] 0 the free energy and the ground state energy..3.1 Euclidean n-point correlators Consider the euclidean path integral defined between initial timeτ i = β and final time τ f =+ β.aswedidfortherealtimepathintegral,wecanconsider the euclidean n-point correlator x β )=x f x β )=xi D[xτ)] E xτ 1 )...xτ n )e S E [xτ)] ħ.19) By working out precisely the same algebra of section 1.4, the euclidean correlator is also shown to be equal to the operatorial expression x f e βh/ħ T [ˆx E τ 1 )...ˆx E τ n )]e βh/ħ x i.0) where ˆx E τ) =e Hτ/ħˆxe Hτ/ħ.1) represents the continuation to imaginary time of the Heisenberg picture position operator. Indeed, comparing to eq. 1.105) we have, ˆx E τ) =ˆx iτ). It is interesting to consider the limit of the euclidean path integral and correlators when β.thisismostdirectlydonebyworkingintheoperator formulation. By using the complete set of energy eigenstatets the euclidean amplitude can be written as K E x f, β ; x i, β ) = x f e βh ħ xi =.) = ψ n x f )ψ n x i ) e βen/ħ n [ ] β = ψ0 x f )ψ 0 x i ) e βe0/ħ 1+Oe βe1 E0)ħ ) where E 0 and E 1 are the energies of respectively the ground state and the first excited state. From the above we conclude that for β ħ/e 1 E 0 )the operator e βh/ħ acts like a projector on the ground state. We arrive at the same conclusion also for the correlators lim x f e βh ħ T [ˆxE τ 1 )...ˆx E τ n )]e βh ħ xi.3) β = 0 T [ˆx E τ 1 )...ˆx E τ n )] 0 ψ 0 x f )ψ 0 x i ) e βe 0 ħ [ ] 1+Oe βe1 E0)/ħ )