Introduction to Path Integrals

Transcription

1 Itroductio to Path Itegrals Path Itegrals i Quatum Mechaics Before explaiig how the path itegrals or rather, the fuctioal itegrals work i quatum field theory, let me review the path itegrals i the ordiary quatum mechaics of a sigle particle. I the coordiate basis, motio of a quatum particle is described by the propagatio amplitude Ut B, x B ; T A, x A x B e itb taĥ/ h x A 1 for movig from poit x A at time t A to poit x B at time t B ; this amplitude is also called the evolutio kerel. I the semi-classical regime, this kerel is give by the WKB approximatio where UB; A prefactor expis[x cl t]/ h 2 S[xt] t B t A Lxt, ẋt dt 3 is the actio itegral of the classical mechaics ad x cl t is the classical path from A to B that obeys the Euler Lagrage equatios of motio. I actio terms, this path miimizes the the fuctioal S[xt] uder coditios xt A x A ad xt B x B. If there are several classical paths from A to B, the S[x] has several local miima, they all cotribute to the evolutio kerel with appropriate phases, ad we get iterferece: UB; A classical paths i prefactor i expis[x i t]/ h. 4 I the exact quatum mechaics, a sum 4 over classical paths becomes a itegral over all possible path from A to B, xt Bx B UB; A D[xt] exp is[xt]/ h. 5 xt Ax A However, ulike the sum 4, the itegral here is ot limited to the classical paths that obey 1

2 the Euler Lagrage equatios of motio. Istead, we itegrate is over all differetiable paths xt from A to B, ad they do ot obey ay equatios of motio except by accidet. But i the semiclassical h 0 limit, the cotributios of most paths to the itegral is washed out by iterferece with similar paths whose actio differs by oly O h. The oly survivors of this wash-out are the statioary poits of the fuctioal S[xt], which are precisely the classical paths from A to B. This is how the WKB approximatio 4 ad evetually the classical mechaics emerge i the h 0 limit. The problem with the path itegral 5 is how to defie the itegratio measure D[xt] for paths. The basic method is to discretize the time: Slice the cotiuous time iterval t A t t B ito a large but fiite set of discrete times t 0, t 1, t 2,..., t N 1, t N, t t A + t, t t B t A N, t 0 t A, t N t B, 6 but evetually take the N limit. This gives us D[xt] def lim d3 x 1 d 3 x 2 d 3 x N 1 ormalizatio factor, where x xt. 7 Note that we do ot itegrate over the x 0 xt A ad x N xt B because they are fixed by the boudary coditios i eq. 5. The o-obvious part of eq. 7 is the ormalizatio factor. We shall see later i these otes that this factor depeds o N, o the et time T t B t A, ad eve o the particle s mass, ad the exact formula for this factor is ot easy to guess. Fortuately, there is a differet versio of path itegratio that does ot suffer from such ormalizatio factors. Let s cosider paths i the phase space x, p rather tha just the x-space. I other words, let s treat xt ad pt as idepedet variables ad write the actio itegral 3 i the Hamiltoia laguage S[xt, pt] B A [ pt dxt Hxt, pt dt ] 8 as a fuctioal of both xt ad pt. A classical path is a miimax of this fuctioal a local miimum with respect to variatios of the xt but a local maximum with respect to 2

3 variatios of the px. Also, the positio xt is subject to boudary coditios at the start A ad fiish B, but there are o boudary coditios for the mometum pt. I the quatum mechaics, UB; A xt Bx B xt Ax A D [xt] D[pt] exp is[xt, pt]/ h 9 where D [xt] D[pt] N 1 lim d 3 x N d 3 p 2π h This time, there are o fuy ormalizatio factors: all we have is the d 3 p/2π h 3 for each mometum variable, ad that s stadard covetio i quatum mechaics. Note that for a give N, we itegrate over N mometa but oly N 1 positios because of the boudary coditios o both eds; to make this differece explicit, I have marked the D [xt] with a prime. Derivig the Phase Space Path Itegral from the Hamiltoia QM Let s start with a mathematical lemma: N lim eâ/n eˆb/n e â+ˆb 11 eve if the operators â ad ˆb do ot commute with each other. Proof: eâ/n eˆb/n 1 + â + ˆb N + O1/N 2, 12 ad lim 1 + â + ˆb N N + O1/N 2 eâ+ˆb 13 regardless of the details of the O1/N 2 terms. 3

4 Now cosider a quatum particle livig i three space dimesios with a Hamiltoia operator of the form Ĥ Kˆp + V ˆx 14 where the kietic eergy ˆK Kˆp does ot deped o the positio ˆx ad the potetial eergy ˆV V ˆx does ot deped o the mometum ˆp. Usig the lemma 11, we may write the evolutio operator for the particle as Ût B t A e iĥtb ta/ h lim e i ˆV t/ h e i ˆK t/ h N 15 where t t B t A /N as i eq. 6. Cosequetly, i the coordiate basis x B Ût B t A x A lim d 3 x 1 d 3 x N 1 N x e i ˆV t/ h e i ˆK t/ h x 1 16 where we have idetified x 0 x A ad x N x B. Each Dirac bracket i the above product evaluates to x e i ˆV t/ h e i ˆK t/ h x 1 e iv x t/ h x e i ˆK t/ h x 1 d e iv x t/ h 3 p 2π h 3 x p e ikp t/ h p x 1 d 3 p 2π h 3 e iv x t/ h e ix p/ h e ikp t/ h e ix 1 p / h d 3 [ p i 2π h 3 exp p x x 1 V x t Kp t ]. h 17 Pluggig this formula back ito eq. 16 ad combiig all the expoetials, we arrive at UB; A d lim d 3 x 1 d 3 3 p 1 d 3 x N 1 2π h 3 p N 2π h 3 exp is/ h, 18 where S N N p x x 1 t V x + Kp 19 4

5 is the discretized actio for a discretized path. Ideed, i the large N limit N [ ] p x x 1 + V x + Kp t B A pt dxt Hxt, pt dt S[xt, pt]. Cosequetly, we should iterpret the product of coordiate ad mometum itegrals i eq. 18 as the discretized itegral over the paths i the mometum space, 20 d d 3 x 1 d 3 3 p 1 d 3 x N 1 2π h 3 p N 2π h 3 D [xt] D[pt] 21 i perfect agreemet with eq. 10. Ad eq. 18 itself is the proof of the path-itegral formula UB; A xt Bx B xt Ax A D [xt] D[pt] exp is[xt, pt]/ h. 9 A ote o discretizatio. Iterpretig the sum p x x 1 as the discretized itegral p dx calls for assigig the mometa p to mid-poit discrete times with respect to the coordiates x : x xt t A + t but p pt t A t. 22 As log as the Hamiltoia ca be split ito separate kietic ad potetial eergies accordig to eq. 14, such differet discrete times for the x ad p are OK because N N Hx, p dt V x dt + Kp dt t V x + t Kp 23 ad the details of the discretizatio do ot matter i the large N limit. However, whe the classical Hamiltoia is more complicated tha a sum of kietic ad potetial eergies, the path 5

6 itegral formalism suffers from the discretizatio ambiguity. For example, for Hx, p p 2 x 24 we could discretize the actio as S or or p x x 1 t p x x 1 t p x x 1 t or somethig else, p 2 x, p 2 x 1, p 2 Mx + Mx 1, 25 all these optios lead to differet evolutio kerels, ad there are o geeral rules how to resolve such ambiguities. Istead, the discretizatio ambiguities of the path-itegral formalism correspod to the operator-orderig ambiguities of the Hilbert-space formalism of quatum mechaics. For example, give the classical Hamiltoia of the form 24, we ca take the quatum Hamiltoia operators to be Ĥ Ĥ 1 ˆx ˆp2, or Ĥ 1 ˆp2 ˆx, or Ĥ ˆp 1 ˆx ˆp, or 1 ˆx ˆpMˆxˆp 1 26 Mˆx, or somethig else. The Lagragia Path Itegral I this sectio, I shall reduce the Hamiltoia path itegrals over both xt ad pt to the Lagragia path itegrals over the xt aloe by itegratig over the paths i mometum space. This works oly whe the kietic eergy is quadratic i the mometum, Hp, x p2 + V x Ĥ ˆp2 + V ˆx. 27 6

7 For such Hamiltoias, p ẋ Hp, x p ẋ p2 ad cosequetly Mẋ2 V x p + Mẋ2 V x 2 p Mẋ2 Lẋ, x 28 S Ham [xt, pt] S Lagr [xt] 1 dt p Mẋ Therefore, i the path itegral formalism, UB; A B A B A D [xt] i D[pt] exp h SHam [xt, pt] i D [xt] exp h SLagr [xt] i D[pt] exp dt p Mẋ 2. h 30 O the secod lie here, we itegrate over the coordiate-space paths xt after itegratig over the mometum-space paths pt, so as far as D[pt] is cocered, we ca treat the coordiate-space path xt as a costat. Also, the path-itegral measure is liear so we may shift the itegratio variable by a costat, thus i i D[pt] exp dt p Mẋ 2 D[pt Mẋt] exp dt p Mẋ 2 h h i D[p t] exp dt p 2 t h cost. Pluggig this formula back ito eq. 30 gives us the Lagragia path itegral UB; A cost xt Bx B xt Ax A D 31 i [xt] exp h SLagr [xt]. 32 I this formalism there is o idepedet mometum-space path pt, we itegrate oly over the coordiate-space path xt, ad the actio is give by the Lagragia formula 3. However, the price of this simplificatio is the u-kow overall costat multiplyig the path itegral 32. 7

8 To calculate this costat we should first discretize the time ad oly the itegrate out the discrete mometa p. For fiite N, the discretized Hamiltoia-formalism actio 19 ca be writte as S Ham discr x 0,..., x N ; p 1,..., p N where t t p M x x 1 t p x x 1 + M 2 t t p 2 2 x x 1 2 t p M x x 1 t t V x V x 2 + S Lagr discr x 0,..., x N S Lagr discr x 0,..., x N t [ 2 M x x 1 V x ] 2 t [ 2 M dx dt V x] S Lagr [xt] 2 dt is the discretized actio for of the Lagragia formalism. I light of eq. 33 we may write the discretized path itegral 18 as d d 3 x 1 d 3 3 p 1 d 3 x N 1 2π h 3 p N i 2π h 3 exp d 3 x 1 d 3 x N 1 exp N discr x 0,..., x N ; p 1,..., p N h SHam i h SLagr d 3 p 2π h 3 exp discr x 0,..., x N i t p M x 2 x 1 h t where we itegrate over all the mometa p before we itegrate over the coordiates. Cosequetly, i each itegral o the last lie of eq. 35 we may shift the itegratio variable from p to p p M x / t, thus d 3 p 2π h 3 exp i t p M x 2 x 1 h t d 3 p 2π h 3 exp M 2πi h t 3/2. i t h p

9 Pluggig this formula back ito eq. 35, we arrive at the Lagragia path itegral UB; A 3N/2 MN lim 2πi ht B t A xt Bx B xt Ax A D i [xt] exp h SLagr [xt]. i d 3 x 1 d 3 x N 1 exp h SLagr discr x 0,..., x N Note however that i the Lagragia formalism, the D [xt] is ot just the limit of d 3N 1 x d 3 x 1 d 3 x N 1 but also icludes the ormalisatio factor 37 CN, M, t B t A 3N/2 MN. 38 2πi ht B t A This ormalizatio factor depeds o N, o the et time T t B t A, ad o the particle s mass M, but it does ot deped o the potetial V x or the iitial ad fial poits x A ad x B. Cosequetly, without discretizig time, a Lagragia path itegral calculatio yields the amplitude UB; A up to a ukow overall factor F M, T. However, we may obtai this factor by comparig with a similar path itegral for a free particle: the overall F M, T factor is the same i both cases, ad the free amplitude is kow to be U free B; A M 2πi ht 3/2 exp imxb x A 2 2 ht. 39 Alteratively, all kid of quatities ca be obtaied from the ratios of path itegrals, ad such ratios do ot deped o the overall ormalizatio of the D[xt]; this is the method most commoly used i the quatum field theory. The Partitio Fuctio The partitio fuctio of a quatum system with a Hamiltoia Ĥ is the trace Zt def Tr Ût; 0 Tr exp itĥ/ h eigevalues E exp ite / h. 40 This time-depedet partitio fuctio is related to the temperature-depedet partitio fuc- 9

10 tio of Statistical Mechaics Zβ Tr exp βĥ 41 via aalytical cotiuatio of time t to imagiary values t i hβ i h k B Temperature. 42 I the path itegral formalism, the partitio fuctio is give by ZT dx Ut, x; 0, x D[xt] e is[xt]/ h. 43 xt x0 Note o prime over D because the paths xt are subject to oly oe boudary coditio periodicity i time, xt x0. Without discretizig time, the path itegral 43 ca be calculated up to a overall ormalizatio costat. Cosequetly, whe we extract the Hamiltoia s spectrum {E } from the partitio fuctio ZT, the multiplicity of all the eigevalues ca be determied oly up to some ukow overall factor. For example, cosider a harmoic oscillator with actio S[xt] M 2 dt ẋ 2 t ω 2 x 2 t. 44 This actio is a quadratic fuctioal of the xt, ad it ca be diagoalized via Fourier trasform, xt S[xt] C C + + MT 2 y e 2πit/T, y y, 45 C y y, 46 2π 2 ω T Note that the discrete frequecies 2π/T of the Fourier trasform 45 are completely determied by the boudary coditios xt x0 ad have othig to do with the oscillator s 10

11 frequecy ω. By liearity of the trasform 45, D[xt] periodic + J dy 0 dy a costatjacobia d Re y d Im y. 48 To be precise, the Jacobia J here depeds o T ad o the mass M via the ormalizatio of the Lagragia path itegral, but it does ot deped o ay of the y variables, ad it does ot deped o the oscillator s frequecy ω. I terms of the Fourier variables y, the path itegral 43 becomes Z J dy 0 J 0 πi h C 0 i d Re y d Im y exp h S ic 0 h y2 0 + πi h. 2C 2iC h y 2 49 The coefficiets C are spelled out i eq. 47, but it s coveiet to rewrite them as C 0 M 2T ωt 2, C >0 2π2 M 2 T 1 2 ωt. 50 2π Cosequetly, the partitio fuctio 49 becomes ZT J 2πi ht/m ωt i ht /4π 2 M 1 ωt 2 2π if ωt 1 ωt π where F J 2πi ht M i ht 4πM 2 52 combies all the factors that do ot deped o the oscillator s frequecy ω. A priori, F could be a fuctio of M or T, but by the o-relativistic dimesioal aalysis, a dimesioless fuctio F M, T, h which does ot deped o aythig else must be a costat. It is ot clear whether 11

12 this costat is fiite or ifiite: it cotais a ifiite product over that is badly diverget, ad the Jacobia J is also badly diverget. To resolve this issue, we eed to discretize time ad the go through a calculatio similar to the above but more complicated; I have writte it dow i a separate supplemetary ote, ad you should read as a part of your ext homework. For ow, just take it without proof that all the divergeces cacel out ad F is fiite. The remaiig ifiite product i the deomiator of eq. 51 is absolutely coverget, ad it may be evaluated just by lookig at its poles ad zeros. The aalytic fuctio sx 1 x 1 1 x/ 2 x + x 53 has o zeroes, it has simple poles at all itegers positive, egative, ad zero, it does ot have ay worse-tha-pole sigularities i the complex x plae, ad it does ot grow whe Im x ±. These facts completely determie this fuctio to be 1 x 1 1 x/ 2 π siπx 54 where the ormalizatio comes from the residue of the pole at x 0. I eq. 51 we have a similar product for x ωt/2π, hece ZT if/2 siωt/2. 55 To extract the oscillator s eigevalues from this partitio fuctio, we expad it as ZT F 2i siωt/2 F e iωt/2 e iωt/2 F e iωt Comparig this series to eq. 40, we immediately see that the eigevalues are E hω ad they all have the same multiplicity F. Of course, we all ew those facts back i the udergraduate school if ot earlier, but ow we kow how to derive them i the path-itegral formalism. 12