Introduction to Path Integrals

Transcription

1 Introduction to Path Integrals Consider ordinary quantum mechanics of a single particle in one space dimension. Let s work in the coordinate space and study the evolution kernel Ut B, x B ; T A, x A ) = x B e itb ta)ĥ/ h x A ) the amplitude for moving from point x A at time t A to point x B at time t B. In the semi-classical regime, this kernel is given by the WKB approximation UB; A) prefactor expis[x cl t)]/ h) 2) where S[xt)] = t B t A Lxt), ẋt)) dt 3) is the action integral of the classical mechanics and x cl t) is the classical path from A to B i.e., x cl t) is the minimum of the functional S[xt)] under conditions xt A ) = x A and xt B ) = x B. If there are several classical paths from A to B because S[x] has several local minima, then they all contribute to the kernel with appropriate phases and we get interference, UB; A) classical paths i prefactor i expis[x i t)]/ h). 4) In the exact quantum mechanics, a sum 4) over classical path becomes an integral over all possible path from A to B, xt B)=x B ) UB; A) = D[xt)] exp is[xt)]/ h. 5) xt A)=x A Note that we integrate over all differentiable functions xt) that satisfy the boundary condition at t A and t B, and they do not obey any equations of motion except by accident. However, in the semiclassical h 0 limit, contribution of most paths is washed out by

2 interference with similar paths whose action differs by only O h). The only survivors of this wash-out stationary points of the functional S[xt)], which are precisely the classical paths from A to B. This is how the WKB approximation 4) and eventually the classical mechanics emerge in the h 0 limit. The problem with the path integral 5) is how to define the differential D[xt)] of a path. Obviously, we should discretize the time: Slice a continuous time interval t A t t B into a large but finite set of discrete times t 0, t, t 2,..., t N, t N ), t n = t A + n t, t = t B t A N, t 0 = t A, t N = t B, 6) but eventually take the N limit. This gives us D[xt)] def = lim dx dx 2 dx N normalization factor where x n xt n ). 7) Note that we do not integrate over the x 0 xt A ) and x N xt B ) because they are fixed by the boundary conditions in eq. 5). The non-obvious part of eq. 7) is the normalization factor. We shall see later in these notes that this factor depends on N, on the net time T = t B t A, and even on the particle s mass, and the exact formula for this factor is not easy to guess. Fortunately, there is a different version of path integration that does not suffer from such normalization factors. Let s consider paths in the phase space x, p) rather than just the x-space. In other words, let s treat xt) and pt) as independent variables and write the action integral 3) in the Hamiltonian language S[xt), pt)] = B A [ p dx Hxt), pt)) dt ] 8) as a functional of both xt) and pt). A classical path is a minimax of this functional a local) minimum with respect to variations of xt) but a local) maximum with respect to variations of px). Also, the coordinate xt) is subject to boundary conditions at the start A 2

3 and finish B, but there are no boundary conditions for the momentum pt). In the quantum mechanics, where UB; A) = xt B)=x B xt A)=x A D [xt)] D [xt)] D[pt)] = D[pt)] exp is[xt), pt)]/ h ) 9) N lim dx n N dp n 2π h. 0) This time, there are no funny normalization factors: all we have is /2π h for every dp n, and that s standard procedure in quantum mechanics. Note that for a given N, we integrate over N momenta but only N coordinates because of the boundary conditions on both ends; to make this difference explicit, I have marked the D [xt)] with a prime. Deriving the Phase Space Path Integral from the Hamiltonian QM Let s start with a mathematical lemma: ) N lim eâ/n eˆb/n = e â+ˆb ) even if the operators â and ˆb do not commute with each other. Proof: eâ/n eˆb/n = + â + ˆb N + O/N 2 ), 2) and lim + â + ˆb ) N N + O/N 2 ) = eâ+ˆb 3) regardless of the details of the O/N 2 ) terms. Now consider a quantum particle living in one space dimension with a Hamiltonian operator of the form Ĥ = Kˆp) + V ˆx) 4) where the kinetic energy ˆK Kˆp) does not depend on the position ˆx and the potential energy ˆV = V ˆx) does not depend on the momentum ˆp. Using the lemma ), we may 3

4 write the evolution 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 5) where t = t B t A )/N as in eq. 6). Consequently, in the coordinate basis x B Ût B t A ) x A = lim dx N dx N x n e i ˆV t/ h e i ˆK t/ h x n 6) where we have identified x 0 x A and x N x B. Each Dirac bracket in the above product evaluates to x n e i ˆV t/ h e i ˆK t/ h x n = = e iv xn) t/ h x n e i ˆK t/ h x n = e iv xn) t/ h dpn 2π h x n p n e ikpn) t/ h p n x n dpn = 2π h e iv xn) t/ h e ixnpn/ h e ikpn) t/ h e ixn p n/ h [ dpn i = 2π h exp p n x n x n ) V x n ) t Kp n ) t) ]. h 7) Plugging this formula back into eq. 6) and combining all the exponentials, we arrive at UB; A) = dp lim dx dx N 2π h dpn eis/ h, 8) 2π h where S = N N ) p n x n x n ) t V x n ) + Kp n ) 9) is the discretized action for a discretized path. Indeed, in the large N limit N [ ] p n x n x n ) + V x n ) + Kp n )) t B A [ p dx H dt ] S[xt), pt)]. Consequently, we should interpret the product of coordinate and momentum integrals in 20) 4

5 eq. 8) as the discretized integral over the paths in the momentum space, dp dx dx N 2π h dpn 2π h D [xt)] D[pt)] 2) in perfect agreement with eq. 0). formula UB; A) = A note on discretization. xt B)=x B xt A)=x A D And eq. 8) itself is the proof of the path-integral [xt)] D[pt)] exp is[xt), pt)]/ h ). 9) Interpreting the sum n p nx n x n ) as the discretized integral pdx calls for assigning the momenta p n to mid-point discrete times with respect to the coordinates x n : x n xt = t A + n t) but p n pt = t A + n 2 ) t). 22) As long as the Hamiltonian can be split into separate kinetic and potential energies according to eq. 4), such different discrete times for the x n and p n are OK because N N Hx, p) dt = V x) dt + Kp) dt t V x n ) + t Kp n ) 23) and the details of the discretization do not matter in the large N limit. However, when the classical Hamiltonian is more complicated than a sum of kinetic and potential energies, the path integral formalism suffers from the discretization ambiguity. For example, for Hx, p) = p 2 x) 24) we could discretize the action as S or or n n n p n x n x n ) t n p n x n x n ) t n p n x n x n ) t n or something else, p 2 n x n ), p 2 n x n ), p 2 n Mx n ) + Mx n ), 25) all these options lead to different evolution kernels, and there are no general rules how to 5

6 resolve such ambiguities. In fact, the discretization ambiguities of the path-integral formalism correspond to the operator-ordering ambiguities of the Hilbert-space formalism of quantum mechanics. For example, given the classical Hamiltonian of the form 24), we can take the quantum Hamiltonian operators to be Ĥ = Ĥ = ˆx) ˆp2, or Ĥ = ˆp2 ˆx), or Ĥ = ˆp ˆx) ˆp, or 26) ˆx) ˆpMˆx)ˆp Mˆx), or something else? Lagrangian Path Integral In this section, I shall reduce the Hamiltonian path integrals over both xt) and pt) to the Lagrangian path integrals over the xt) alone by integrating over the paths in momentum space. This works only when the kinetic energy is quadratic in the momentum, Hp, x) = p2 + V x) = Ĥ = ˆp2 + V ˆx). 27) For such Hamiltonians, pẋ Hp, x) = pẋ p2 V x) = p Mẋ)2 + Mẋ2 2 V x) = Lẋ, x) p Mẋ)2 28) and consequently S Ham [xt), pt)] = S Lagr [xt)] dt p Mẋ) 2. 29) Therefore, in the path integral 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 On the second line here, we integrate over the coordinate-space paths xt) after integration over the momentum-space paths pt), so as far as D[pt)] is concerned, we can treat the 30) 6

7 coordinate-space path xt) as a constant. Also, the path-integral measure is linear so we may shift the integration variable by a constant, 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 = const. Plugging this formula back into eq. 30) gives us a Lagrangian path integral UB; A) = const xt B)=x B xt A)=x A D 3) ) i [xt)] exp h SLagr [xt)]. 32) In this formalism there is no independent momentum-space path pt), we integrate only over the coordinate-space path xt), and the action is given by the Lagrangian formula 3). However, the price of this simplification is the un-known overall constant multiplying the path integral 32). To calculate this constant we should first discretize time and only then integrate out the discrete momenta p n. For finite N, the discretized Hamiltonian-formalism action 9) can be written as S Ham discr x 0,..., x N ; p,..., p N ) = n where = t t n p n M x n x n t p n x n x n ) n + M 2 t = t n p 2 n ) 2 x n x n ) 2 t n n p n M x n x n t t n V x n ) V x n ) ) 2 + S Lagr discr x 0,..., x N ) S Lagr discr x 0,..., x N ) = t [ ) 2 M xn x n V x n )] 2 t n [ ) 2 M dx dt V x)] = S Lagr [xt)] 2 dt 33) 34) 7

8 is the discretized action for of the Lagrangian formalism. In light of eq. 33) we may write the discretized path integral 8) as dp dx dx N 2π h dpn i 2π h exp i = dx dx N exp ) discr x 0,..., x N ; p,..., p N ) ) h SHam discr x 0,..., x N ) h SLagr N dpn 2π h exp i t p n M x ) ) 2 n x n h t where we integrate over all momenta p n before we integrate over the coordinates. Consequently, in each integral on the last line of eq. 35) we may shift the integration variable from p n to p n = p n M x n / t, thus dpn 2π h exp i t p n M x ) ) 2 n x n h t = dp ) n i t 2π h exp h p 2 n Plugging this formula back into eq. 35), we arrive at the Lagrangian path integral UB; A) = ) N/2 MN lim 2πi ht B t A ) xt B)=x B xt A)=x A D ) i [xt)] exp h SLagr [xt)]. = = 35) M 2πi h t. 36) ) i dx dx N exp h SLagr discr x 0,..., x N ) Note however that in the Lagrangian formalism, the D [xt)] is not just the limit of d N x dx dx N but also includes the normalisation factor 37) CN, M, t B t A ) = ) N/2 MN. 38) 2πi ht B t A ) This normalization factor depends on N, on the net time T = t B t A, and on the particle s mass M, but it does not depend on the potential V x) or the initial and final points x A and x B. Consequently, without discretizing time, a Lagrangian path integral calculation yields the amplitude UB; A) up to an unknown overall factor F M, T ). However, we may obtain 8

9 this factor by comparing with a similar path integral for a free particle: the overall F M, T ) factor is the same in both cases, and the free amplitude is known to be U free B; A) = M 2πi ht exp imxb x A ) 2 ). 39) 2 ht Alternatively, all kind of quantities can be obtained from the ratios of path integrals, and such ratios do not depend on the overall normalization of the D[xt)]; this is the method most commonly used in the quantum field theory. Partition Function The partition function of a quantum system with a Hamiltonian Ĥ is the trace Zt) def = Tr Ût; 0) Tr exp itĥ/ h) = eigenvalues E n exp ite n / h). 40) This time-dependent partition function is related to the temperature-dependent partition function of Statistical Mechanics Zβ) = Tr exp βĥ) 4) via analytical continuation of time t to imaginary values t i hβ = i h k B Temperature. 42) In the path integral formalism, the partition function is given by ZT ) = dx Ut, x; 0, x) = D[xt)] e is[xt)]/ h. 43) xt )=x0) Note no prime over D because the paths xt) are subject to only one boundary condition periodicity in time, xt ) = x0). Without discretizing time, the path integral 43) can be calculated up to an overall normalization constant. Consequently, when we extract the Hamiltonian s spectrum {E n } from the partition function ZT ), the multiplicity of all the eigenvalues can be determined only up to some unknown overall factor. 9

10 For example, consider a harmonic oscillator with action S[xt)] = M 2 dt ẋ 2 t) ω 2 x 2 t) ). 44) This action is a quadratic functional of the xt), and it can be diagonalized via Fourier transform, xt) = S[xt)] = C n = C n + n= + n= = MT 2 y n e 2πint/T, y n = y n, 45) C n y ny n, 46) 2πn ) ) 2 ω 2. 47) T Note that the discrete frequencies 2πn/T of the Fourier transform 45) are completely determined by the boundary conditions xt ) = x0) and have nothing to do with the oscillator s frequency ω. By linearity of the transform 45), D[xt)] = periodic + n= = J dy 0 dy n a constantjacobian d Re y n d Im y n. 48) To be precise, the Jacobian J here depends on T and on the mass M via the normalization of the Lagrangian path integral, but it does not depend on any of the y n variables, and it does not depend on the oscillator s frequency ω. In terms of the Fourier variables y n, the path integral 43) becomes Z = J dy 0 = J n=0 πi h C 0 i d Re y n d Im y n exp h S = ic 0 h y2 0 + πi h. 2C n ) 2iC n h y n 2 49) 0

11 The coefficients C n are spelled out in eq. 47), but it s convenient to rewrite them as C 0 = M 2T ωt )2, C n>0 = 2π2 Mn 2 T Consequently, the partition function 49) becomes ) ) 2 ωt. 50) 2πn ZT ) = F ωt ) ) ωt 2 ) 5) 2πn where F = J 2π ht im i ht 4π 2 Mn 2 52) combines all the factors that do not depend on the oscillator s frequency ω. A priori, F could be a function of M or T, but by the non-relativistic dimensional analysis, a dimensionless function F M, T, h) which does not depend on anything else must be a constant. It is not clear whether this constant is finite or infinite: it contains an infinite product over n that is badly divergent, and the Jacobian J is also badly divergent. To resolve this issue, we need to discretize time and then go through a calculation similar to the above but more complicated; I ll presented in a separate write up you should read as a part of your next homework. For now, just take it without proof that all the divergences cancel out and F is finite. The remaining infinite product in the denominator of eq. 5) is absolutely convergent, and it may be evaluated just by looking at its poles and zeros. The analytic function sx) = x x/n) 2 = n n x n ) n + x has no zeroes, it has simple poles at all integers positive, negative, and zero), it does not have any worse-than-pole singularities in the complex x plane, and it does not grow when x ±i. These facts completely determine this function to be 53) x x/n) 2 = π sinπx) 54) where the normalization comes from the residue of the pole at x = 0. In eq. 5) we have a

12 similar product for x = ωt/2π, hence ZT ) = 2 F sinωt/2). 55) To extract the oscillator s eigenvalues from this partition function, we expand it as ZT ) = 2 F sinωt/2) = if e iωt/2 e iωt/2 = if e iωt n+ ) 2. 56) n=0 Comparing to eq. 40) we immediately see that the eigenvalues are E n = hωn+ 2 ) and they all have the same multiplicity if. Of course, we all new those facts back in the undergraduate school if not earlier), but now we know how to derive them in the path-integral formalism. 2