20. Path Integration¶
PDF pages 907–964
Chapter 20 Path Integration 20.1 Feynman Path Integral We will begin by developing the path-integral formulation of quantum mechanics,1 at least for the evolution of a single particle in a potential. Recall that we introduced the propagator [Eq. (15.17)] as the matrix representation of the evolution operator,
(20.1) (propagator) where the evolution operator corresponds to a time-independent Hamiltonian H(x, p):
(20.2) Here we will work more specifically with the position-representation propagator
(20.3) (propagator) In the last expression here, the evolution operator has been ‘‘hidden’’ inside the states according to the notation
(20.4) where the ‘‘wrong’’ time dependence comes not from |x, t\rangle denoting time-evolving states in the sense of |x(t)\rangle , but as eigenstates of the Heisenberg-picture operator x(t). Thus, the propagator here is roughly a ‘‘transition amplitude’’ from x0 at time t0 to x at time t. More precisely, if we recall that
(20.5) we can apply \langle x| and inserting an identity to give
Z
(20.6) or
Z dx0 K(x, t; x0, t0) \psi(x0, t0), (20.7) (evolution via propagator) so that K(x, t; x0, t0) is a convolution kernel corresponding to the evolution (i.e., K(x, t; x0, t0) is a repre- sentation of the Green function for the Schrödinger equation). 1R. P. Feynman, ‘‘Space-Time Approach to Non-Relativistic Quantum Mechanics,’’ Reviews of Modern Physics 20, 367 (1948) (doi: 10.1103/RevModPhys.20.367); Richard P. Feynman and A. R. Hibbs, ‘‘Quantum Mechanics and Path Integrals,’’ (McGraw–Hill, 1965).
20.1.2 Splitting the Evolution Operator¶
Chapter 20. Path Integration A key technique in the construction of the path integral is the splitting of the propagator at an intermediate time. This is easy to do, we simply insert the identity operator, expanded in the position basis at some intermediate time t′, into the propagator inner product,
= Z
(20.8) with the result
Z dx′K(x, t; x′, t′)K(x′, t′; x0, t0). (composition property of propagator) (20.9) This is the analogue of the Chapman–Kolmogorov equation (17.133) for conditional probability densities in diffusion problems. This relation basically says that we can regard the transition from x0 to x at the corresponding times to be as if the particle ‘‘passed through’’ the intermediate x′ point, so long as we sum the amplitude over all possible intermediate points x′. 20.1.1 Dividing the Evolution Now the idea is to do the same thing, but split the time interval (t −t0) into N equal subintervals of length
will assume N to be large (such that we will suppose \deltat to be infinitesimal). Sticking to the specific form (20.2) for the evolution operator, we find2
Z dxN−1|xN−1\rangle \langle xN−1|
Z
(20.10) Then we have an (N −1)-dimensional integral over a product of N matrix elements,
Z N−1 Y j=1 dxj !
(20.11) This is now an infinite-dimensional integral, since we have in mind the limit N −\rightarrow \infty. 20.1.2 Splitting the Evolution Operator To continue, we will assume a standard particle Hamiltonian of the form H = p2
(20.12)
dimensions is straightforward. The Baker–Campbell–Hausdorff expansion3 allows us to split the evolution operator with this Hamiltonian according to
(20.13) 2We are following here the phase-space construction of the path integral, as in Mark Srednicki, Quantum Field Theory (Cambridge, 2007), Chapter 6 (draft available at http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf). 3R. M. Wilcox, ‘‘Exponential Operators and Parameter Differentiation in Quantum Physics,’’ Journal of Mathematical Physics 8, 962 (1967) (doi: 10.1063/1.1705306).
20.1 Feynman Path Integral (For more on this operator splitting, see Section 26.1.) The correction term here is negligible in the following sense: there are N such correction terms that add together, which gives an error N\deltat2 = \deltat, which vanishes as \deltat −\rightarrow 0. Then considering any one of the matrix elements in the integrand of (20.11), we have
= Z
(20.14) where we have inserted a momentum-space identity operator between the operators. We should be careful to emphasize that the position and momentum appearing in the exponentials are operators. Since we have inner products of eigenstates of these operators, we can simply replace them by the appropriate eigenvalues,
Z
= 2\pi¯h Z dp1 eip1(x2−x1)/¯he−iT (p1)\deltat/¯he−iV (x1)\deltat/¯h, (20.15) where we have used the inner product
\sqrt 2\pi¯h , (20.16) which is just the momentum eigenstate expressed in the position basis. Note that in introducing the momen- tum integral, we have also introduced a Fourier transformation. In fact, by tacking on an extra |x2\rangle on the front of Eq. (20.14) and then integrating over x2, we have the recipe for propagating the initial state |x2\rangle over a small time \deltat: first propagate using the potential-energy exponential operator, which is diagonal in x, then Fourier transform to momentum space, where we may apply the kinetic operator, and finally transform back to position space to express the result in the x2 basis. This is in fact the basis for split-operator methods for numerically solving the Schrödinger equation (Chapter 26). Since T(p) is quadratic, the momentum integral in the mini-propagator (20.15) is a Gaussian integral, so we will go ahead and carry it out by completing the square. Letting \deltax1 := x2 −x1,
2\pi¯h Z dp1 eip1\deltax1/¯he−ip 2
=
Z
=
Z dp1 e−ip 2
(20.17) where we have shifted away the constant offset in the Gaussian integrand. The integral here is not strictly convergent, but we can help it by nudging it into the territory of convergent Gaussian integrals: Z \infty −\infty dx e−i\alphax2 = lim \beta\rightarrow 0 Z \infty −\infty
\beta\rightarrow 0 r \pi
r \pi i\alpha. (20.18) Thus,
r 2\pim¯h i\deltat = r m
= r m i2\pi¯h\deltat eim ˙x 2
= r m
(20.19)
20.1.4 Phase-Space Path Integral¶
Chapter 20. Path Integration where we have taken
\deltat (20.20) in view of the \deltat −\rightarrow 0 limit, and we have identified
2m ˙x2 −V (x) (20.21) as the classical Lagrangian. 20.1.3 Functional Integral Now collecting all N matrix elements (20.19) and putting them into the propagator (20.11), we have
m
N/2 Z N−1 Y j=1 dxj !
= m
N/2 Z N−1 Y j=1 dxj ! N−1 Y j=0 eiL(xj, ˙xj)\deltat/¯h ! . (20.22) Defining the path-integral differential Dx := m
N/2 N−1 Y j=1 dxj, (20.23) (functional-integral differential) we can simplify the notation in Eq. (20.22) and recombine the exponentials to write
Z Dx exp i ¯h Z t t0 dt L(x, ˙x) . (20.24) (Feynman propagator) This is a functional integral or path integral, as we are integrating the action functional
Z t t0 dt′ L[x(t′), ˙x(t′)], (20.25) (action functional) in terms of which the propagator has the even simpler form
Z Dx exp i ¯hS[x] , (20.26) (Feynman propagator) noting that the integration is with respect to the trajectory (or function) x(t). Recall that we discussed the related functional differentiation in Section 8.2.1, and we discussed the Lagrangian and classical variational principles in Section 8.2.2. Note that the normalization factor in (20.23) is highly divergent, but it is typically hidden inside the differential here since it typically drops out of calculations on the way to a physical result. 20.1.4 Phase-Space Path Integral An alternate form of the path integral, before carrying out the momentum integrals, comes about by rewriting
2\pi¯h Z dp1 eip1 ˙x1\deltat/¯he−iH(x1,p1)\deltat/¯h. (20.27)
20.1.5 Example: Free-Particle Propagator¶
20.1 Feynman Path Integral Then collecting all the matrix elements, we can now write the phase-space path integral4
Z Dx Dp exp i ¯h Z t t0 dt [p ˙x −H(x, p)] , (phase-space propagator) (20.28) where the integration measure in phase space is Dx Dp :=
N−1 Y j=1 dxj ! N−1 Y j=0 dpj ! . (functional-integration measure) (20.29) This looks similar to before, except that we now have the phase-space action—the integral of p ˙x−H—in the propagator instead of the Lagrangian action, and the integration is over both positions and momenta. (Note that like in the classical application of the phase-space action, as in Section 8.2.2, the endpoints of x(t) are pinned to x0 and x, while the momentum endpoints are not similarly restricted.) At this point, in the case of a Hamiltonian quadratic in p, the integral is Gaussian and can be carried out analytically as above. The stationary-phase condition on the p integral gives the classical Hamilton equation
(20.30) which can be used to eliminate p in favor of ˙x in p ˙x −H(x, p) to reproduce the Lagrangian, and thus the Lagrangian form of the propagator (20.24). Note that while the positions and momenta were introduced in similar ways, there is a fundamental asymmetry between them.5 In the infinitesimal propagator from xj to xj+1, we regard the position as moving smoothly between these two coordinates in a time \deltat. However, the momentum is a constant pj over the same time interval. So in this time-slicing construction (the construction with finite N), the path position follows a continuous, polygonal path, while the momentum is discontinuous and piecewise constant. 20.1.5 Example: Free-Particle Propagator
discrete form (20.22), which we can write explicitly as
m
N/2 Z N−1 Y j=1 dxj ! N−1 Y j=0 eim ˙x 2
! = m
N/2 Z N−1 Y j=1 dxj ! N−1 Y j=0
! . (20.31) Now let
(20.32) where ¯xj is the straight-line path from x0 to x = xN,
x −x0 t −t0
(20.33)
average. In particular, note that
(20.34) 4This general form of the path integral in phase space was given by H. Davies, ‘‘Hamiltonian approach to the method of summation over Feynman histories,’’ Mathematical Proceedings of the Cambridge Philosophical Society 59, 147 (1963) (doi: 10.1017/S0305004100002097). 5H. Davies, op. cit.; Claude Garrod, ‘‘Hamiltonian Path-Integral Methods,’’ Reviews of Modern Physics 38, 483 (1966) (doi: 10.1103/RevModPhys.38.483); M. S. Marinov, ‘‘Path integrals in quantum theory: An outlook of basic concepts,’’ Physics Reports 60, 1 (1980) (doi: 10.1016/0370-1573(80)90111-8).
Chapter 20. Path Integration For the increments in the path integral, we may write
(20.35) and thus the Gaussian factors in the path integral (20.31) involve squared differences of the form
(20.36) This means that the path integral (20.31) splits into a product of three components, each involving Gaussian functions of each of these terms. For example, the last term leads to the product N−1 Y j=0
(20.37) while the second term N−1 Y j=0
(20.38) in view of Eq. (20.34). Collecting these with the remaining product leads to the path integral
m
N/2 eim(x−x0)2/2¯h(t−t0) Z N−1 Y j=1 d\deltaj ! N−1 Y j=0
! , (20.39) where we should keep in mind that \delta0 = \deltaN = 0 in the j = 0 and j = N −1 factors. Also, note that since the coordinate transformation from xj to \deltaj was just a coordinate shift, there is no Jacobian factor to worry about. In making the coordinate transformation here, we have reduced the problem of computing K(x, t; x0, t0) to that of a periodic path integral of the form K(0, t; 0, t0). To finish up the path integral, we can begin by examining the \deltaj integral, which involves the terms
j−1
N−2 (20.40) If we add up these contributions, we have an integral of the form
m
N/2 eim(x−x0)2/2¯h(t−t0) Z N−1 Y j=1 d\deltaj ! exp im
, (20.41) where now M\alpha\beta is a matrix of dimension (N −1)\times(N −1), and has twos along the diagonal, and −1’s along the adjacent diagonals:
−1
−1 −1
−1
... ... ... ... ... ...
−1
−1 . (20.42) This matrix has determinant
(20.43)
20.2 Classical Limit as we will now show. Let detn denote the determinant of the n \times n version of the same matrix. Then expanding the determinant in terms of minors along the first row gives the recursion relation detn = 2detn−1 −detn−2. (20.44) With the initial values det1 = 2 and det2 = 3, we can see that the recursion and initial conditions are satisfied by
(20.45) This establishes the determinant (20.43). Now to finish up with the path integral, we can use the Gaussian integration formula Z
\piN/2 p
, (20.46) for an N \times N matrix S\alpha\beta [see, for example, the normalized Gaussian function in Eq. (4.165)], and adapt it for complex Gaussians as in Eq. (20.18). The result is
m
N/2 eim(x−x0)2/2¯h(t−t0)
m (N−1)/2 \sqrt N = r m i2\pi¯hN\deltat eim(x−x0)2/2¯h(t−t0) (20.47) or finally,
r m i2\pi¯h(t −t0) exp im(x −x0)2 2¯h(t −t0) (20.48) (free-particle propagator) Note that the removal of the mean ¯x(t) from the path x(t) to focus on the fluctuations \delta(t) here is the same
is subtracted away to produce a periodic path. 20.2 Classical Limit One nice feature of the path integral is that it gives a clear path to the classical limit, at least in a heuristic argument.6 Returning to the propagator (20.26),
Z Dx exp i ¯hS[x] , (20.49) note that as ¯h −\rightarrow 0, the exponential becomes highly oscillatory. Except, that is, where the rest of the expo- nent, S[x], vanishes. Thus, we might expect the oscillatory paths to roughly cancel, and the integral should be dominated by the ‘‘stationary’’ paths, for which slight variations don’t change the action. Dropping all but these stationary paths is known as the stationary-phase approximation, and amounts to restricting the path integral to classical paths satisfying
(20.50) (stationary-phase condition) which is just the variational principle that generates the Euler-Lagrange equation. That is, when ¯h is small compared to the action of the system, the path integral is dominated by the contribution from the classical path. Paths that deviate from the classical path by even a small amount on the classical scale are suppressed because they oscillate wildly and cancel other, slightly deviant paths. When ¯h is larger relative to the action, the paths have more leeway to ‘‘explore’’ around the classical path without suffering the fate of cancellation. 6Richard P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill, 1965), Section 2-3, p. 29.
20.2.1 Semiclassical Propagator¶
Chapter 20. Path Integration 20.2.1 Semiclassical Propagator With this picture in mind, we can go on and develop an approximate form for the propagator in the semiclassical regime.7 Since we expect the classical path xc to be important, we will consider small deviations \deltax from it:
(20.51) Then expanding the action in a functional Taylor series around the classical path, we find
(20.52) (series expansion of the action) up to second order. Note that by definition, \deltaS[xc] = 0, but we leave the first-order term here to illustrate the general pattern. Let’s take a moment to examine the notation here. Given the action functional (20.25),
Z t t0 dt′ L[x(t′), ˙x(t′)], (20.53) recall that we define the first variation \deltaS[x] of the functional as
ϵ\rightarrow 0
ϵ
ϵ=0 , (20.54) (first variation) in analogy to the usual derivative, but the perturbation to the argument is a function \deltax(t). Note that this variation is, roughly speaking, proportional to \deltax, and hence why we can use it in the series expansion. Then the second variation is the variation of the first variation:
ϵ\rightarrow 0
ϵ
ϵ=0
(second variation) (20.55) This pattern of definitions can obviously continue indefinitely for higher-order variations. The variations can then serve as the definition for functional derivatives. In terms of the first variation, the first functional derivative \deltaS/\deltax is defined such that the inner product with the perturbation \deltax gives the first variation: \deltaS
:= Z t t0 dt′ \deltaS
(20.56) (first functional derivative) The second functional derivative is similarly defined in terms of the second variation as
:= Z t t0 dt′ Z t t0 dt′′ \deltax(t′) \delta2S
(second functional derivative) (20.57) Note that these are all reasonably straightforward generalizations of the same derivatives for a scalar-valued function of a vector S(xj), where the first derivative is a vector of partial derivatives \partial S/\partial xj, and the second derivative is a matrix of derivatives \partial 2S/\partial xj\partial xk. 7L. S. Schulman, Techniques and Applications of Path Integration (Wiley, 1981), Chapter 13; Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed. (World Scientific, 2009), Chapter 4.
20.2 Classical Limit In terms of the functional derivatives, we can write the series expansion (20.52) for the action as
Z t t0 dt′ \deltaS \deltax(t′)
x=xc
Z t t0 dt′ Z t t0 dt′′ \delta2S
x=xc
(series expansion of the action) (20.58) as an alternative to Eq. (20.52). Again, \deltaS/\deltax vanishes when evaluated at xc, and again this just generalizes the discrete form of the Taylor series. Inserting this latter expansion into the path integral (20.49), we note again that the first-order term is zero, and we also note that the second-order term gives a Gaussian factor in the fluctuations \deltax(t):
Z Dx exp
i 2¯h Z dt′ Z dt′′ \deltax(t) \delta2S
x=xc \deltax(t′′) ! . (20.59) We will be careful and go back to the discrete (time-sliced) form for the path integral to evaluate it, as in Eq. (20.22):
m
N/2 eiS[xc]/¯h Z N−1 Y j=1 d\deltaxj ! exp i 2¯h N−1 X j=1 N−1 X j′=1 \deltaxj \partial 2S
x=xc \deltaxj′ . (20.60) Note that the sums in the exponential only run from 1 to N −1 because we regard the endpoints x0 and xN to be fixed. Then using the Gaussian-integral formula (20.46),
r m
m \deltat (N−1)/2 det−1/2 \partial 2S
exp i ¯hS[xc] , (20.61) where the determinant is over an (N −1) \times (N −1) matrix. In continuous language, we can write this result in terms of a functional determinant as
r m
\deltat m \delta2S \deltax2 exp i ¯hS[xc] , (semiclassical propagator) (20.62) where we may take the determinant to be defined by the discrete expression (20.61). Note that in deriving this propagator, we have considered lowest-order quantum fluctuations about the classical trajectory, in what amounts to making a Gaussian approximation for the fluctuations. While the exponential gives the quantum phase factor along the classical path, the determinant gives the correction to the amplitude as the size of the Gaussian fluctuations stretches or compresses in phase space. For the action (20.53), we can write out the first action variation
Z t t0 dt′ \partial L
= Z t t0 dt′ \partial L \partial x −d dt \partial L \partial ˙x \deltax (20.63) and the second variation
Z t t0 dt′ \partial 2L
, = Z t t0 dt′ Z t t0 dt′′ \delta(t′ −t′′) \times \partial 2L
\partial L
\partial 2L
, (20.64)
Chapter 20. Path Integration and thus the functional derivatives \deltaS[x]
\partial x −d dt \partial L \partial ˙x (20.65) and \delta2S[x]
\partial 2L
\partial L
d dt′ −d
\partial ˙x2(t) d dt′ . (20.66) In this last expression, note that all derivative operators d/dt operate on everything to the right. For a particle Lagrangian (20.21),
2m ˙x2 −V (x), (20.67) the first functional derivative is \deltaS[x]
(20.68) while the second derivative becomes \delta2S[x]
dt\delta(t −t′) d dt′ , (20.69) in which case the functional determinant in the propagator (20.62) becomes det−1/2 \deltat m \delta2S \deltax2
−d dt\delta(t −t′) d dt′ −\delta(t −t′) V ′′(xc) m
−\partial 2 t −V ′′(xc) m , (20.70) where we have used the discrete form of the delta function, which is the identity times \deltat−1. In fact, we have
second-order, finite-difference operator for the second derivative can be written ∆(2)
\deltat2 , (20.71) such that \partial 2
(20.72) the matrix M in Eq. (20.42) is the discrete representation for the operator −\deltat2\partial 2 t . Recalling from Eq. (20.43) that det M = N, then det −\partial 2 t
(20.73) and this means, for example, that we can write the propagator (20.62) in continuous notation in terms of a ‘‘renormalized’’ determinant:
r m i2\pi¯h(t −t0) s det(−\partial 2 t ) det[−\partial 2 t −V ′′(xc)/m] exp i ¯hS[xc] , (semiclassical propagator) (20.74) where we used t −t0 = N\deltat.
20.2 Classical Limit 20.2.1.1 Gel’fand–Yaglom Method To evaluate the functional determinant further here, we can extend the recursion method that we used in Section 20.1.5 for the free-particle determinant. In place of the matrix (20.42) for the operator −\deltat\partial 2 t , we can compute the determinant we need in terms of a similar (N −1) \times (N −1) matrix: det −\partial 2 t −V ′′(xc)/m det(−\partial 2 t ) = 1 N det
N−1 −1
−1
N−2 −1
−1
... ... ... ... ... ...
−1
−1
, (20.75) where \omega 2
m
m (20.76) defines an effective, time-dependent, harmonic-oscillator frequency based on the curvature of the potential along the classical path. As in the free-particle case, the n \times n version of the determinant satisfies the recursion relation detn =
n detn−1 −detn−2, (20.77) with initial values
1 ) −1. (20.78) To cast this recursion into continuous form, note that we can rewrite Eq. (20.77) as detn −2detn−1 + detn−2 \deltat2 + \omega 2 ndetn−1 = 0. (20.79) Defining Dn := \deltat detn, (20.80) the recursion is the same, Dn −2Dn−1 + Dn−2 \deltat2 −\omega 2 nDn−1 = 0. (20.81) but we can recast the initial conditions as
1 ) D2 −D1 \deltat
1 . (20.82) In the continuum limit \deltat −\rightarrow 0, D(t) then satisfies the differential equation \partial 2
(continuous representation of the determinant) (20.83) The determinant then is simply D(t)/\deltat, where t here is the final time of the propagator. This recasting of the functional determinant into the solution of an ordinary differential equation is specific to determinants of Schrödinger-type operators −\partial 2 x + V (x) in one dimension, and this is the Gel’fand–Yaglom method for computing functional determinants.8 8I. M. Gel’fand and A. M. Yaglom, ‘‘Integration in Functional Spaces and its Applications in Quantum Physics,’’ Journal Mathematical Physics 1, 48 (1960) (doi: 10.1063/1.1703636); Hagen Kleinert, op. cit., Section 2.4; M. Chaichian and A. Demichev, Path Integrals in Physics, Volume I: Stochastic Processes in Quantum Mechanics (Institute of Physics, 2001), p. 168.
Chapter 20. Path Integration 20.2.1.2 Van Vleck–Morette Determinant Now we will use the Gel’fand–Yaglom method to establish a simple form for the semiclassical functional
time leads to \partial 2 t + V ′′(xc) m ˙xc = 0. (20.84) This in turn implies that ˙xc(t) is a solution to the differential equation (20.83). To emphasize this, we will write the solution
(20.85) Since we have a second-order differential equation, there are two independent solutions D1 and D2, and the Wronskian determinant W := det D1 D2 ˙D1 ˙D2 = D1 ˙D2 −D2 ˙D1 (20.86) is constant, because the coefficient of \partial 2 t is constant and the coefficient of \partial t vanishes. Dividing through by D 2 1 , ˙D2 D1 −D2 D 2 ˙D1 = W D 2 , (20.87) and noting that the left-hand side is the derivative of D2/D1, we can integrate both sides to obtain D2(t) D1(t) −D2(t0)
Z t t0 dt′ W D 2 1 (t′), (20.88) or to simplify,
D1(t0)D1(t) + WD1(t) Z t t0 dt′ D 2 1 (t′). (20.89) The general solution D(t) can be written as a superposition c1D1 + c2D2, or
Z t t0 dt′ ˙x 2c (t′), (20.90) where we have absorbed some of the t0-dependent constants in the coefficients. The initial conditions from
Z t t0 dt′ ˙x 2c (t′) (20.91) to represent the functional derivative. We will continue by transforming this into a simpler form. First, recall that a classical particle in one dimension, by virtue of the conserved energy E = 1 2m ˙x 2
c 2m + V (xc), (20.92) can always be solved in the sense of obtaining an integral for t(x): t −t0 = Z x x0 dx′ m p 2m[E −V (x′)] = Z x x0 dx′ m p(x′). (20.93) Differentiating this gives \partial t
Z x x0 dx′ m2 2m[E −V (x′)]
Z x x0 dx′ m2 p3(x′). (20.94) 9Here we are mainly following M. Chaichian and A. Demichev, op. cit., Section 2.2.3; see also Hagen Kleinert, op. cit., Sections 2.4.5 and 4.3.
20.2 Classical Limit Then we can use this result to simplify the integral Z t t0 dt′
Z t t0 dt′ p 2c (t′) = m3 Z t t0 ˙xc dt′ p 3c (t′) = m3 Z x x0 dxc p 3c (xc)
\partial E . (20.95) Now to calculate \partial E\partial t. Starting with the phase-space action, we have S[xc] = Z t t0 dt′ pc(t′) ˙xc(t′) −H(xc, pc) i = Z x x0 dxc pc(xc) −(t −t0)E. (20.96) Differentiating with respect to the endpoint x, \partial S[xc] \partial x
Z x x0 dxc \partial pc
\partial x
Z x x0 dxc \partial pc \partial E −(t −t0) \partial E \partial x , (20.97) where we pulled \partial E/\partial x out of the integral, since we regard E as only depending on boundary conditions (e.g., on x0 and ˙x0, or x0 and x). Then \partial pc
\partial E \partial pc −1 = \partial pc m −1 = 1 ˙xc , (20.98) and so the second term in Eq. (20.97) vanishes, so that \partial S[xc] \partial x
(20.99) Differentiating with respect to the initial coordinate, \partial 2S[xc]
\partial x0 = \partial \partial x0 p
m pc(x) \partial E \partial x0 . (20.100) To find \partial E/\partial x0, we can differentiate Eq. (20.96) with respect to x0 to obtain \partial S[xc] \partial x0
\partial x0 , (20.101) and then differentiate with respect to t to obtain \partial E \partial x0
\partial t\partial x0 . (20.102) From Eq. (20.96), we can differentiate with respect to x0 \partial S[xc] \partial x0
Z x x0 dxc \partial pc \partial x0 −(t −t0) \partial E \partial x0
Z x x0 dxc \partial pc \partial E −(t −t0) \partial E \partial x0
(20.103)
Chapter 20. Path Integration in analogy to Eq. (20.99). Thus, Eq. (20.102) becomes \partial E \partial x0
\partial t
\partial E \partial E
m pc(x0) \partial E \partial t , (20.104) and putting this in Eq. (20.100) we have \partial 2S[xc]
m2 pc(x0)pc(x) \partial E \partial t . (20.105) Solving for \partial E/\partial t, \partial E
(20.106) Putting this into Eq. (20.95), Z t t0 dt′
˙xc(x0) ˙xc(x)\partial 2S[xc]
−1 , (20.107) so the determinant solution D(t) in Eq. (20.91) finally becomes
\partial 2S[xc]
−1 (20.108) The original determinant that we needed in Eq. (20.75) becomes det −\partial 2 t −V ′′(xc)/m det(−\partial 2 t ) =
m (t −t0) −\partial 2S[xc]
−1 (20.109) At long last, the semiclassical propagator (20.74) simplifies to
\sqrt 2\pii¯h s −\partial 2S[xc]
exp i ¯hS[xc] , (semiclassical propagator) (20.110) so that the determinant reduces to a derivative of the classical action at the propagator endpoints. More generally, in multiple dimensions, this result generalizes to
\sqrt 2\pii¯h det1/2 −\partial 2S[xc]
exp i ¯hS[xc] , (semiclassical propagator) (20.111) The determinant here, det −\partial 2S[xc]
= det −\partial p0 \partial x = det −\partial p \partial x0 , (Van Vleck–Morette determinant) (20.112) is called the Van Vleck–Morette determinant.10 This determinant has an interpretation as follows. In the variational spirit of the path integral, the classical trajectory xc is specified in terms of its endpoints x0 and x. Alternately, it can be specified in terms of the initial conditions (x0, p0) or the end conditions (x, p). The Van Vleck–Morette determinant is the determinant of the coordinate transformation between the variational and initial/end specifications. Since the phase-space area of a small bundle of paths around 10The determinant was introduced in the semiclassical context by J. H. Van Vleck, Proceedings of the National Academy of Sciences 14, 178 (1928) (doi: 10.1073/pnas.14.2.178); and introduced in the context of path integration by C. DeWitt-Morette, Physical Review 81, 848 (1951) (doi: 10.1103/PhysRev.81.848). See also, for example, Bryce DeWitt, The Global Approach to Quantum Field Theory, vol. 1 (Oxford, 2003), Chapter 14.
20.2.2 Example: Semiclassical Propagator for the Harmonic Oscillator¶
20.2 Classical Limit the classical path is invariant along the path (classically, at least), roughly speaking the determinant maps this area into position space, to represent the effect of local convergence or divergence of classical trajectories on the amplitude. (In optics terminology, this is akin to using Gaussian beams instead of rays for tracing optical systems, and the spreading or converging of the beam waist influences the intensity at the beam center.) We are also glossing over what happens if the functional determinant in Eq. (20.62) vanishes, such that the Van Vleck–Morette determinant diverges. This corresponds to the formation of caustics in the classical trajectories, and requires more careful treatment.11 20.2.2 Example: Semiclassical Propagator for the Harmonic Oscillator As an example of the semiclassical propagator, we will evaluate this explicitly for a harmonic oscillator of frequency \omega. Note that for this case, the semiclassical propagator (20.62) is exact, because it was based on a truncation at the second derivative, and higher derivatives will vanish anyway for the harmonic oscillator, which has a quadratic Lagrangian. 20.2.2.1 Classical Action Thus, starting with the classical action, S[xc] = Z t t0 dt′ m ˙x2 −m\omega2x2 , (20.113) we will integrate by parts on the first term to obtain S[xc] = mx ˙x
t t0 −m Z t t0 dt′ x
. (20.114) The second factor in the integrand vanishes under the classical equations of motion, so we only have the boundary term S[xc] = m 2 x ˙x t t0. (20.115) We will then use the classical trajectory
(20.116) to evaluate the action integral, with the result S[xc] = m x0 ˙x0 cos2 \omega(t −t0) −sin2 \omega(t −t0) + ˙x 2
cos \omega(t −t0) sin \omega(t −t0) −m 2 x0 ˙x0 = m ˙x 2
cos \omega(t −t0) sin \omega(t −t0) −2x0 ˙x0 sin2 \omega(t −t0) . (20.117) Now rearranging the general solution (20.113) to obtain ˙x0 at the final spacetime coordinates (x, t),
sin \omegat (20.118) we can eliminate the initial derivative. After a bit of algebra we find S[xc] = m\omega 2 sin \omega(t −t0) hx 2 0 + x2 cos \omega(t −t0) −2x0x i (harmonic-oscillator action) (20.119)
11M. V. Berry and K. E. Mount, ‘‘Semiclassical approximations in wave mechanics,’’ Reports on Progress in Physics 35, 315
Chapter 20. Path Integration 20.2.2.2 Determinant Then to obtain the Van Vleck–Morette determinant for the harmonic oscillator, we simply differentiate the action: \partial 2S[xc]
m\omega sin \omega(t −t0). (Van Vleck–Morette determinant, harmonic-oscillator) (20.120) Then assembling the parts of the semiclassical propagator (20.110), we have
r m\omega 2\pii¯h sin \omega(t −t0) exp im\omega 2¯h sin \omega(t −t0) hx 2 0 + x2 cos \omega(t −t0) −2x0x i . (propagator, harmonic oscillator) (20.121) Again, though we derived this expression via the semiclassical propagator, this is identical to the fully quantum propagator. Any higher-order corrections to the path integral go as higher powers of ¯h−1, and as functional derivatives of the action beyond second order. It is also easy to check that this expression reduces correctly to the free-particle propagator (20.48) in the limit \omega −\rightarrow 0. In deriving that result, we also used the classical (straight-line) path to simplify the calculation. 20.3 Monte-Carlo Methods in Quantum Statistical Mechanics One nice connection of the path-integral formulation of quantum mechanics is to statistical mechanics,12 as we will now review. This connection comes by recalling the definition of the partition function Z := Tr e−\betaH , (20.122) (partition function) where \beta := 1/kBT for temperature T and Boltzmann constant kB. We can then compare this to the propagator from combining Eqs. (20.1) and (20.2):
(20.123) In particular, choosing the initial and final states to be the same, and then choosing imaginary times t0 = 0 and t = −i¯h\beta, we have
(20.124) This imaginary-time replacement is called a Wick rotation of the propagator. Summing over all states |\alpha\rangle gives the trace we require to complete the partition function, so we identify Z = X \alpha
(imaginary-time propagator as partition function) (20.125) or in the position representation, Z = Z
Z
(imaginary-time propagator as partition function) (20.126) In particular, from the path integral (20.24), we can make the same replacement to obtain Z = Z dx Z Dx exp i ¯h Z t dt L(x, ˙x) t\rightarrow −i¯h\beta . (partition function as path integral) (20.127) 12Richard P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw–Hill, 1965), Chapter 10.
20.3.1 Path Integral as Ensemble Average¶
20.3 Monte-Carlo Methods in Quantum Statistical Mechanics We can absorb the dx integral into the path integral as follows. Recall that in the integration path measure (20.23), we had path coordinates x1 . . . xN−1 for the integration, with initial and final coordinates x0 and x ≡xN. For the purposes of the diffusion integral (20.127), recall that we should identify the endpoints
(20.128) which is the subject of the last integration dx. Alternately, for reasons we will come to, we can enforce this by keeping the integral over x, calling it an integral over x0, and then introducing an integral over xN, so
Z = Z ˜Dx \delta[x(¯h\beta) −x0] exp i ¯h Z t dt L(x, ˙x) t\rightarrow −i¯h\beta , (partition function as path integral) (20.129) where we have modified the integration measure to read ˜Dx := m
N/2 N Y j=0 dxj, (functional-integral differential for periodic paths) (20.130) and in the continuum limit, we have identified the endpoint xN ≡x(t) −\rightarrow x(¯h\beta). To see what all this means, note that the Hamiltonian (20.12) implies the Schrödinger equation
2m\partial 2
(20.131) which under the same imaginary-time replacement becomes
2m\partial 2
(20.132) (diffusion equation in imaginary time) which is a diffusion equation in the imaginary time \beta, with a damping term (corresponding to a space- dependent cooling in a heat equation) due to V (x). Thus, we expect the paths in Eq. (20.127) to correspond to diffusive paths that as a whole satisfy the diffusion equation (20.132). We will develop this notion further in the next section, but this is also the content of the Feynman–Kac formula, as we developed in the form of Eqs. (17.447) and (17.448) in the context of stochastic calculus. 20.3.1 Path Integral as Ensemble Average Now we will explore in more detail the paths in the diffusive path integral (20.127). Writing out the Lagrangian explicitly in the path integral, we have Z = Z ˜Dx \delta[x(¯h\beta) −x0] exp i ¯h Z t dt 1 2m ˙x2 −V (x) t\rightarrow −i¯h\beta = Z ˜Dx \delta[x(¯h\beta) −x0] exp " −1 ¯h Z ¯h\beta d˜\beta m
# , (20.133) where ˜\beta := ¯h\beta. Now the idea is to split the two terms in the exponential, giving different interpretations to each one. Suppose that f(x) is a unit-normalized, nonnegative distribution: Z
(20.134)
Chapter 20. Path Integration Then we may write any integral involving the product of f(x) with another function as an average, Z
(20.135) where the expectation value is an ensemble average chosen from the ‘‘probability’’ distribution f(x). We can apply this to the path integral (20.133) Z = Z ˜Dx \delta[x(¯h\beta) −x0] exp " −m 2¯h Z ¯h\beta
exp " −1 ¯h Z ¯h\beta d˜\beta V (x) # , (20.136) where, after appropriate normalization, the first exponential factor (the kinetic factor) will play the role of the probability measure f(x). This probability measure will be normalized by introducing a normalization constant \eta, such that the resulting path integral is normalized. Then the generalization of Eq. (20.134) is Z¶
\eta Z ˜D′x exp " −m 2¯h Z ¯h\beta
= 1, (20.137) where ˜D′x is the same as ˜Dx, but omits the integration over the endpoint x(¯h\beta) (or xN), because this integration is already taken care of by the \delta function. In discrete form, this reads¶
Z ˜D′x exp " −m 2¯h Z ¯h\beta
= m¶
N/2 Z N−1 Y j=0 dxj exp − m
, (20.138) where \delta\beta = \beta/N. Note that this has the form of N −1 Gaussian convolutions with one final integral over the result. We can proceed by carrying out first the x0 integral, which is a Gaussian integral of the form Z dx0 exp − m
= s
m , (20.139) where we have simply recentered the x0 integration by x0 −\rightarrow x0 −x1 and performed the Gaussian integral. Then Eq. (20.138) becomes
m
N/2 s
m Z N−1 Y j=1 dxj exp − m
. (20.140) Iterating this procedure N −1 more times, we have
m
m N/2 = 1, (20.141) and we see that we already have a normalized integral from keeping the momentum-integration factors (had we dropped these factors, we could easily use this normalization procedure to restore them). Then applying Eq. (20.135) to the path integral (20.136), Z = Z dx0 **
" −1 ¯h Z ¯h\beta d˜\beta V (x)
, (20.142) where we changed the remaining xN integration over to an integration over x0 via the \delta function, and the double brackets here refer to an ensemble average over x(˜\beta) chosen from the probability functional f[x(t)] defined in Eq. (20.137).
20.3 Monte-Carlo Methods in Quantum Statistical Mechanics Now before proceeding, we can interpret things a bit. The probability functional f[x(t)] that defines the ensemble average is, in its finite representation in Eq. (20.138), a product of Gaussian factors of the form exp − m
. (20.143) This factor is a Gaussian probability density for the separation of neighboring points xj+1 and xj, and says that a pair of such points will only be chosen with significant probability if |xj+1 −xj| is small, of the order of the standard deviation q ¯h2\delta\beta/m or less. Thus, the points xj define a continuous path in the N −\rightarrow \infty limit, of random, Gaussian steps of variance ¯h2\delta\beta/m. This is just a Wiener process (or Brownian motion). However, a standard Wiener path W(t) takes a step of variance dt in a time interval dt, so we can identify x(˜\beta) as a scaled, displaced Wiener process:
r ¯h m W(˜\beta), (20.144) since dW(˜\beta) has an rms average of \sqrt¯h\delta\beta. Furthermore, the \delta function in the expectation value ties the endpoint x(¯h\beta) of the path to the initial point x0. This is just a Brownian bridge tied down at ‘‘time’’ ¯h\beta to its initial position. We have in fact already worked out the conversion of the \delta function to a bridge in Eq. (17.561), with result
\delta[W(T)] F[W(t)]
= \sqrt 2\piT
F[BT (t)]
, (20.145)
a factor ¯h/m longer in view of Eq. (20.144). Thus, the path integral (20.142) becomes Z = r m
Z dx0 ** exp " −1 ¯h Z ¯h\beta d˜\beta V (x)
p
, (partition function as ensemble average over Brownian bridges) (20.146) where now, the ensemble average is over Wiener loops or Brownian bridges that start and end at x0, where we sum over all possible x0. The exponential factor involves an integral of the potential V evaluated along the stochastic loop. Note that the overal \beta-dependent normalization factor can be dropped, as overall factors in partition functions are ignorable. Also, note that since we have defined the Brownian bridges or stochastic loops to be periodic in ˜\beta, with period ¯h\beta, we may regard the ‘‘imaginary time’’ ˜\beta itself to be periodic (i.e., representing a compact time dimension), with the same period. 20.3.1.1 Alternate Normalization of the Periodic Path It is constructive to consider alternate ways to compute the normalization of the path integral (20.136). We already covered one method in Section 20.1.5, where we evaluated a determinant as part of a direct matrix- Gaussian integration to integrate a periodic path integral. Another way is to simply normalize against the
Z0 = Z ˜Dx \delta[x(¯h\beta) −x0] exp " −m 2¯h Z ¯h\beta
= Z ˜D′x exp " −m 2¯h Z ¯h\beta¶
. (20.147) This second form looks superficially the same as the integral (20.138) that we computed before, but because the xN integral changed the \delta function into a pinned endpoint on the path, this integral also includes the contribution that we obtained from the \delta function in Eq. (20.145). Rather than deal with the path integral,¶
Chapter 20. Path Integration we will go back to the original partition function (20.122), writing out the free-space Hamiltonian and the momentum-space trace: Z0 = Z dx \langle x| exp −\betap2 2m |x\rangle . (20.148) To evaluate this, we insert a momentum-space identity, to reduce the momentum operators in the exponential to eigenvalues: Z0 = Z dx Z
−\betap2 2m |x\rangle = 2\pi¯h Z dx Z dp exp −\betap2 2m = r m
Z dx. (20.149) The remaining position integral is a divergent volume factor, which is telling us that we shouldn’t normalize with respect to Z0, but rather to the free-partition-function density Z0(x), which is just Z0 without the extra integration factor:
r m
(20.150) This gives the proper normalization of the kinetic part of the partition function (20.136), including the \delta function, but without the integral over x0. Thus, in converting Eq. (20.136) to an ensemble average, we associate a factor of 1/Z0(x) (i.e., 1/\eta in our previous notation) with the probability functional f[x(t)], leaving a factor of Z0(x) in the remaining ensemble average. The result is Z = r m
Z dx0 ** exp " −1 ¯h Z ¯h\beta d˜\beta V (x)
, (ensemble average via alternate normalization) (20.151) which agrees exactly with Eq. (20.146), implementing the normalization due to the change to an ensemble average and the loop-closure \delta function all in one go. However, we don’t get the clear distinction of the factor here as being due to the pinning of the loop end. 20.3.1.2 YANPP (Yet Another Normalization of the Periodic Path) Finally, we will compute the normalization yet another way, illustrating how to perform the explicit, iterated convolution with the \delta function up front (this method is more along the lines of Section 20.1.5, but with direct evaluation of the individual integrals). We need to compute the free-partition-function density, which we can infer from Eqs. (20.147):
Z ˜D′x \delta[x(¯h\beta) −x0] exp " −m 2¯h Z ¯h\beta
. (20.152) In discrete form, this expression becomes¶
m
N/2 Z N−1 Y j=0 dxj \delta(xN −x0) exp − m
, (20.153) Note that this is the same as \eta in Eqs. (20.138), but now with the loop-closure \delta-function included. Carrying out the x0 integral implements the \delta function, leaving a somewhat more complicated integrand than before:
m
N/2 Z N−1 Y j=1 dxj exp − m
exp − m
. (20.154)
20.3.2 Feynman-Kac Formula¶
20.3 Monte-Carlo Methods in Quantum Statistical Mechanics We can then carry out the x1 integral Z dx1 exp − m
exp − m
= Z dx1 exp − m
exp − m
= Z dx1 exp −m
exp − m
= s
m exp − m
, (20.155) where we complete the square via
2 + x 2 N
2 + x 2 N
(20.156) For the x2 integral, we then complete a square (x3 −x2)2 + (x2 −xN)2/2, so let’s guess that the general case will be a square of the form
N/j
N/j
(20.157) Therefore, the jth integral will contribute a factor of q j2\pi¯h2\delta\beta/(j + 1)m, and change the ‘‘outside expo- nential’’ from an argument of (xj −xN)2/j to (xj+1 −xN)2/(j + 1). After the (N −1)th integral, all the exponential factors vanish, leaving
m
m (N−1)/2 N−1 Y j=1 s j j + 1 = r m
= r m
(20.158) This is, of course, the same factor we obtained in Eq. (20.150) in the previous section, but with distinctly more work. In fact, we have just reproduced the Brownian-bridge construction via square-completion of Section 17.7.1, suitably scaled for the present stochastic paths.The normalization of the partition function then proceeds as in the previous section. 20.3.2 Feynman–Kac Formula The Feynman–Kac formula that we treated before in Section 17.11 is an equivalent formalism for diffusive path integrals, though we derived it before with a quite different approach using It¯o calculus. In particular, the form we will consider here is the solution (17.448)
** f0[x + W(t)] exp − Z t dt′ V [x + W(t′), t −t′]
(20.159)
20.3.3 Thermal Density Matrix¶
Chapter 20. Path Integration to the diffusion equation (17.447)
2\partial 2 x f −V (x, t)f. (20.160) To demonstrate the equivalence of this formula with the present formalism, we will need to transform this equation into precisely the same diffusion equation as Eq. (20.132), which we will write as
2\partial 2 ˜x \psi − V ˜x p ¯h/m ¯h \psi, (20.161) where ˜\beta = ¯h\beta as usual, and ˜x = p m/¯h x. Then the solution of this equation according to Eq. (20.159) is at
**
−1 ¯h Z ¯h\beta d˜\beta V hp
p
. (20.162) Note that what we have here is already the diffusive propagator, giving the probability density (amplitude) at (˜x, ¯h\beta), given that the ensemble started at (˜x0, 0). What remains from Eq. (20.125) is to set the final point x equal to the initial point x0, and then integrate over the initial point to obtain the partition function: Z = Z
Z d˜x0 **
−1 ¯h Z ¯h\beta d˜\beta V hp
p
. (20.163) Then applying Eq. (20.145), we have Z =
Z d˜x0 ** exp
−1 ¯h Z ¯h\beta d˜\beta V hp
p
, (20.164) and putting ˜x0 = p m/¯h x0, we find Z = r m
Z dx0 ** exp
−1 ¯h Z ¯h\beta d˜\beta V h x0 + p
, (20.165) which is equivalent to the partition-function expression (20.146) that we already derived. Thus, the Feynman– Kac formula reproduces what we derived quite differently from the imaginary-time propagator, and in fact it indicates somewhat more generality in allowing for time dependence of the potential, which is more useful in the Schrödinger propagator rather than the (equilibrium) partition function. 20.3.3 Thermal Density Matrix To review the utility of the partition function, recall that the canonical partition function is given in terms of the energies Ej of a system as Z := X n e−\betaEn, (20.166) where the exponential terms in this sum represent the relative probabilities (according to the Boltzmann distribution) of an element of the canonical ensemble to occupy state n. The partition function neatly gives the normalization factor for the Boltzmann probabilities, such that the normalized probability of occupying state n is
Z , (20.167) and useful quantities such as the thermal energy may be computed as derivatives of Z:
Z = X n En e−\betaEn Z = X n
(20.168)
20.3 Monte-Carlo Methods in Quantum Statistical Mechanics For a quantum system, the definition (20.122) follows by identifying state energies as expectation values En = \langle n|H|n\rangle with respect to energy eigenstates |n\rangle , such that Z = X n
X n
e−\betaH . (20.169) Then note also that in thermal equilibrium, the quantum state, or thermal density operator is closely related, as
X n
Z X n
Z X n
(20.170) This is an expression of \rho in the energy basis; the basis-independent expression is therefore
Z . (20.171) (thermal density operator) In the position representation, the density matrix is then simply
Z . (20.172) (thermal density operator) Examining the steps leading up to Eq. (20.125), we see that the exponential matrix element is precisely the Wick-rotated propagator:
(imaginary-time propagator as thermal density matrix) (20.173) and thus the propagator gives the thermal state of a system, up to a factor of Z, or simply the unnormalized thermal density matrix. The next step in computing the partition function in terms of the propagator was to carry out the spatial trace, which follows here as the simple normalization of the density operator:
Z
Z
(20.174) Thus, all of the above path-integration results apply also to Z\rho(x, x′), if we drop the overall spatial integral and interpret the paths as traveling from x′ to x in imaginary time from 0 to ¯h\beta. Thus, for example, Eq. (20.127) becomes
Z Z Dx exp i ¯h Z t dt L(x, ˙x) t\rightarrow −i¯h\beta , (partition function as path integral) (20.175) where as in the original propagator, the paths run from x0 to x. Similarly, the ensemble-averaged expression (20.146) translates to
Z r m
** exp " −1 ¯h Z ¯h\beta d˜\beta V (x)
x( ˜\beta) , (partition function as ensemble average over Brownian bridges) (20.176) where x(˜\beta) travels from x0 to x as ˜\beta goes from 0 to ¯h\beta with steps of variance ¯h\delta ˜\beta/m in each time step \delta ˜\beta. The extra exponential factor here bears a bit more explanation. Because we want to pin the endpoint of the path at x, and not x0, we must still carefully handle this pinning. To do this, we can take Eq. (20.175) and introduce an integration over the endpoint xN as in Eq. (20.129), inserting a \delta function to fix the endpoint,
Z Z dxN Dx \delta[xN −x¯h\beta] exp " −m 2¯h Z ¯h\beta
exp " −1 ¯h Z ¯h\beta d˜\beta V (x) # , (20.177)¶
Chapter 20. Path Integration in mixed continuous-discrete notation, and where we have changed the notation for the endpoint from x to x¯h\beta to avoid confusion with the paths x(˜\beta) in the path integral. There are N integrals implied by dxN Dx, and the normalization factor in Dx is again the appropriate normalization factor for the kinetic term to act as the probability density, so that the analogue of Eq. (20.142) becomes
Z **
" −1 ¯h Z ¯h\beta d˜\beta V (x)
. (20.178) Now we can evaluate the contribution of the \delta function here by modifying Eq. (20.145) to handle the offset \delta function here. Examining the derivation of this relation, we can see that the factor of 1/ \sqrt 2\pit there came
point Wt, then we have [cf. Eq. (17.592)]
\delta[W(t) −Wt] F[W(t)]
= e−W 2 t /2t \sqrt 2\pit
F[Bt(t)]
, (20.179) which is what we require here. As in Eq. (20.146), to adapt this to the (unpinned) paths x(t), we set t −\rightarrow ¯h2\beta/m, and then count W(t) as the (scaled) separation of x¯h\beta from the initial point x0, which yields Eq. (20.176). 20.3.3.1 Example: Free-Particle Thermal State The simplest application of the formalism here is to work out the thermal density operator for a free particle.
Z r m
(20.180) (free particle thermal state) Note that the partition function here involves a divergent volume integral over a constant, so we won’t bother to write it down. Note also that the density operator here agrees with the free-space propagator (20.48) under the replacement (t −t0) −\rightarrow −i¯h\beta (if we take away the factor of Z). 20.3.3.2 Example: Partition Function of an Infinite Square Well Consider an infinite square well, open from x = 0 to L,
lim V0\rightarrow \inftyV0 [\Theta(x −L) + \Theta(−x)] , (20.181) and let’s use world lines to compute the partition function (20.146). In that expression, we have an ensemble average over a particular statistic: the exponentiated line integral of −V (x) over each path. If the path
touch either wall of the well. In the language of stochastic processes, this can be phrased in terms of the escape probability of a Brownian bridge from the interval (−a, L −a) [see Eq. (17.415)]: Pescape = 1 + \infty X j=−\infty h e−2(a+jL)2 −e−2(jL)2i . (20.182) Then the partition function is Z = r m
Z dx0 n 1 −Pescape[x(˜\beta)] o . (20.183)
20.3 Monte-Carlo Methods in Quantum Statistical Mechanics To adapt the escape probability here, we should let a −\rightarrow x0, but since we are dealing with a bridge that is effectively pinned at time ¯h2\beta/m, we should compensate by setting a −\rightarrow x0/¯h p
p
Z = r m
Z L dx0 \infty X j=−\infty h
. (20.184) The integral over the second term vanishes, because it is of the form Z 1
r \pi 4a erf
−erf j\sqrta , (20.185) which vanishes under an infinite sum over j. (Note that the individual terms technically give rise to divergent sums, which vanish independently if defined in the sense of a Cauchy principal value.) Then the partition function is Z = r m
\infty X j=−\infty
(20.186) which is technically correct, but we will take this a bit further. Starting with the Poisson sum rule in the form13 \infty X n=−\infty
\infty X n=−\infty
\infty X n=−\infty cos(2\pint), (20.187) (Poisson sum rule) we multiply by exp(−at2) and integrate over t, \infty X n=−\infty e−an2 = \infty X n=−\infty Z dt e−at2 ei2\pint, (20.188) so that \infty X n=−\infty e−an2 = r\pi a \infty X n=−\infty
(20.189) (Poisson sum rule for Gaussians) Using this formula, Eq. (20.186) becomes Z = 1 \infty X j=−\infty
(20.190) or finally after simplifying the sum, Z = \infty X n=1
\infty X n=1 e−\betaEn, (partition function of infinite square well) (20.191) which is what we expect for a partition function for a set of states with energies En = n2\pi2¯h2/2mL2. This is the just another way to derive the eigenenergies for the infinite square well! Note that we just needed the escape probability for a Brownian bridge here; we can also derive the density matrix with this technique, using the more general formula (17.415) for a nonperiodic Brownian bridge. 13Daniel A. Steck, Classical and Modern Optics (2006), available online at http://steck.us/teaching.
20.3.4 Propagator Paths¶
Chapter 20. Path Integration 20.3.4 Propagator Paths In developing path integrals in statistical mechanics, due to the imaginary-time identification t = −i¯h\beta leading to Eq. (20.125), each step in the path integral has a factor of the form exp[−m(\deltax)2/2¯h|\deltat|], where |\deltat| = ¯h\beta. In Section 20.3.1, this allowed us to develop the path integral as a Monte-Carlo average over Wiener paths, where adjacent steps in the path were close, with \deltax ∼ \sqrt \deltat. However, the original quantum propagator has oscillatory factors of the form exp[im(\deltax)2/2¯h(\deltat)] in Eq. (20.17), and it is less obvious that a similar interpretation in terms of diffusive paths is possible there. The two amplitude distributions are contrasted in the plot below (showing the real Gaussian in the imaginary-time case, and the real part of the imaginary Gaussian in the real-time case). real imaginary
-5 amplitude -1 Even though the factor exp[im(\deltax)2/2¯h(\deltat)] oscillates without decaying, note that it is possible to integrate it in the sense of normalizing it, because the tail oscillations increase in frequency [in the same sense that the Fresnel integrals C(x) and S(x) converge to well-defined values as x −\rightarrow \infty]. Nevertheless, the undamped, oscillatory character of the step amplitudes cause problems for rigorous handling of the propagator path integral, and there has been a great deal of work in establishing rigorous foundations for the complex path integral.14 In a heuristic approach, we can still say that for a Gaussian distribution,
(20.192) the convolution of multiple Gaussians for successive steps in the path integral
\sqrt
(20.193) still works whether the phase of \alpha is 0 or −\pi/4, so the complex Gaussian still works out as a stable ‘‘step’’ distribution. In the imaginary-time (real Gaussian) case, the Wiener-path interpretation came from the decay of the Gaussian tails for large steps. In the real-time (imaginary Gaussian) case, the ‘‘unlikely’’ paths from the diffusion picture still have large amplitude, but being in the oscillatory tails, they have a rapidly oscillating phase, and tend not to contribute via cancellation with other unlikely paths. This is an argument similar to the one used in Section 20.2.2 for the semiclassical limit of the path integral. In this heuristic way we can still think of the important paths for the propagator as being the Wiener paths. A reasonably safe and less heuristic approach to the propagator path integral is to realize that from the treatment of path integrals for statistical mechanics in Section 20.3, propagators in imaginary time, K(x2, −i\tau2; x1, −i\tau1), are mathematically well-defined objects with an interpretation in Wiener paths. The 14Sergio Albeverio and Sonia Mazzucchi, Scholarpedia 6, 8832 (2011) (doi: 10.4249/scholarpedia.8832).
20.4.1 Model Problem¶
20.4 Ordering Issues quantum-mechanical propagators in real time, K(x2, t2; x1, t1), can then be defined via analytic continuation of the imaginary-time function. (Similar arguments are possible for analytic continuation of imaginary mass or ¯h.) In this sense, even the quantum propagator is calculated in terms of Wiener paths. This gives another way to think about how the important quantum paths are diffusive in nature despite the oscillatory amplitude factors. 20.4 Ordering Issues Going back to the derivation of the path-integral propagator (20.24), we made the assumption of a particle Hamiltonian (20.12), where the Gaussian integration changed the momentum kinetic term into the velocity kinetic term—but with the opposite sign—in the exponential. The question is, what happens if we have a more general Hamiltonian H(x, p) that doesn’t separate out into position and momentum terms so easily? That is, what if there is an ordering ambiguity, as in pxp, where different orderings are connected by correction terms given in terms of commutators? 20.4.1 Model Problem As an example, let us work out path integrals for a one-degree-of-freedom Hamiltonian of the form
2mg(x)p + V (x), (20.194) (variable-mass Hamiltonian) for some ‘‘space-dependent mass’’ or ‘‘metric’’ function g(x), and consider the case of various orderings of the kinetic-energy term. The various orderings are equivalent up to commutator terms, which we will indicate. Then, we will continue with a particular ordering (Weyl ordering) and work out the path integral in more detail. We will keep the particular symmetric ordering here on the kinetic-energy term in mind as a reference case. Actually, what turns out to be a more natural ordering is what we will refer to as ‘‘Laplace–Beltrami’’ or ‘‘covariant’’ ordering,
2m 1 g1/4 p 1 \sqrtg p 1 g1/4 + V (x), (Laplace–Beltrami Hamiltonian) (20.195) as we will see below [Eq. (20.233)]; despite the more complicated form, the path-integral representations in this ordering are more natural and simple, so this will be our more important reference case. 20.4.1.1 Basic Structure The reason we need to be somewhat specific about the form of the Hamiltonian is that it will affect some of the basic rules in quantum mechanics (normalization integrals, form of the plane wave, etc.). Before continuing with the path integral, then, we will take a moment to work out the consequences of having a Hamiltonian of the form (20.194), or one of its reorderings. Starting with the classical Lagrangian
2 g(x) ˙x2 −V (x), (20.196) (classical Lagrangian) we can define the canonical momentum
(20.197) and the classical Hamiltonian
p2 2mg(x) + V (x). (20.198) In quantizing this Hamiltonian, there is an ambiguity in the ordering of the kinetic-energy term. We can regard the ordering in Eq. (20.194) as an arbitrary choice for the sake of an example, with other orderings obtainable by commuting operators.
Chapter 20. Path Integration The classical Poisson bracket then has the form
\partial x \partial g
\partial p \partial g \partial x, (20.199)
which means that in quantum mechanics we still have the commutator [x, p] = i¯h. (20.200) This will define the form of the quantum-mechanical momentum operator. 20.4.1.2 Coordinate Transformation and Curvature We have to be careful about interpreting the classical system in quantum mechanics, because of the difference between orderings of x and p that does not happen classically, as we noted above.15 Note in particular that the form of the Lagrangian and Hamiltonian that give rise to the ordering problem are analogous to a curved structure in the phase and state space.16 The Lagrangian in particular implies the line element
(20.201)
equivalent to a flat-space line element ds2 = dq2, (20.202) given the nonlinear coordinate transformation dq dx = p g(x). (20.203) In these coordinates, the Lagrangian (20.196) becomes
2 g(x) ˙q2 dx dq 2
2 ˙q2 −V (q), (20.204)
2m + V (q) (20.205) with canonical momentum
(20.206) In these coordinates, we know how to quantize the Hamiltonian: the canonical variables become operators with commutator [q, pq] = i¯h. The Schrödinger equation in flat-space coordinates is then
−¯h2 2m\partial 2 q + V (q) \psi. (20.207) 15Our discussion here is a simplified version of the more general multidimensional case. This point of view of using point- coordinate transformations to understand quantum systems with ordering problems and quantum mechanics on curved manifolds was pioneered by Bryce Seligman DeWitt, ‘‘Point Transformations in Quantum Mechanics,’’ Physical Review 85, 653 (1952) (doi: 10.1103/PhysRev.85.653); Bryce S. DeWitt, ‘‘Dynamical Theory in Curved Spaces. I. A Review of the Classical and Quantum Action Principles,’’ Reviews of Modern Physics 29, 377 (1957) (doi: 10.1103/RevModPhys.29.377). See also Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed. (World Scientific, 2009), Section 1.13; Christian Grosche, ‘‘An Introduction into the Feynman Path Integral,’’ arXiv.org preprint (arXiv: hep- th/9302097v1). 16at least in 2 or more dimensions; in one dimension the manifold cannot be regarded as ‘‘intrinsically’’ curved. The other distinction is that in a truly curved manifold, the transformation that we will do can only be performed locally not globally; if there is a global transformation to flat space, then the space is just a flat space in complicated coordinates.
20.4 Ordering Issues The one-dimensional Laplacian transforms as \partial 2
dq \partial x dx dq \partial x =
1
, (20.208) and so the Schrödinger equation in the original variables, corresponding to the Lagrangian (20.196), is
−¯h2 2m△+ V (x) \psi, (20.209) (Schrödinger equation) where the curved, one-dimensional Laplacian △:=
1
(20.210) (Laplace–Beltrami operator, 1D) is the one-dimensional form of the Laplace–Beltrami operator. Note that this implies a certain natural ordering of the kinetic-energy term in the Hamiltonian, to which we will return below. The coordinate transformation also induces some other changes in the structure of the Hilbert space. In the flat coordinates, the inner product has the usual form
Z dq \psi∗ 1(q) \psi2(q), (20.211) which in the context of the curved Hamiltonian transforms to
Z dx p g(x) \psi∗ 1(x) \psi2(x), (20.212) (inner product)
Z dx p
(20.213) (identity in position representation) By inserting this identity in the inner product
Z dx′ p
Z dx′ p
(20.214) and comparing to
Z
(20.215) we can conclude
p g(x) \delta(x −x′) (20.216) (inner product of position states) for the orthogonality relation of the position states. In terms of the propagator, which we can rederive by using Eq. (20.212) to insert the identity in
Z dx0 p
Z dx0 p
(20.217) so that we have generalized Eq. (20.7) to
Z dx0 p g(x0) K(x, t; x0, t0) \psi(x0, t0), (20.218) (evolution via propagator)
Chapter 20. Path Integration where the propagator is
(20.219) (propagator) as it was before in the case where g is independent of x. By extension, in our application of the propagator to statistical mechanics, by transforming Eq. (20.126) we can write Z = Z dx p
Z dx0 Z dx p g(x) \delta(x −x0) K(x, −i¯h\beta; x0, 0), (partition function) (20.220) due to the change in integration measure. Turning now to momentum, the momentum operator is defined in part by the commutator rule (20.200) which we note is satisfied for the usual momentum operator, modified by inserting factors of g, p = ¯h
(20.221) as we can quickly check: xp = ¯h
(20.222) However, the momentum operator must be Hermitian with respect to the inner product (20.212),
(20.223) which as an inner product reads
(20.224) and in explicit integral form, we have Z
1(x) ¯h
= Z dx ¯h
1(x)
(20.225) This is satisfied only when \alpha = 1/2 −\alpha, or \alpha = 1/4, p = ¯h i
= ¯h i \partial x −¯hg′(x) 4g(x) = ¯h i \partial x −¯h 4 [\partial x log g(x)], (20.226) (momentum operator) and so we have the usual momentum operator with a ‘‘curvature correction’’ given in terms of a derivative of g(x). From this form of the momentum operator, we can infer the momentum eigenstates in the position representation
eipx/¯h \sqrt
(20.227) (momentum eigenstate) with a factor of g−1/4 to ‘‘adapt’’ to the corresponding factors in the momentum operator. Then to infer the momentum-representation identity operator, we can consider
eip(x−x′)/¯h \sqrt
, (20.228)
20.4 Ordering Issues whence integration over p yields Z¶
\delta(x −x′)
p g(x) . (20.229) comparing to Eq. (20.216) then gives Z
(20.230) (momentum completeness) for completeness in momentum states, and similarly we have
(20.231) (momentum orthonormality) for the momentum orthnormality relation. 20.4.1.3 “Natural” Ordering Recall again that the classical Lagrangian (20.196) led to the Laplace–Beltrami operator (20.210) as the natural generalization of the Laplacian, in the sense of connecting to the usual Laplacian under a coordinate change into the standard form of the Lagrangian. Writing this operator in terms of the corresponding momentum operator (20.226), we find −¯h2△= g1/4 p 1 \sqrtg p 1 g1/4 . (20.232) (Laplace–Beltrami operator, 1D) Thus, the Hamiltonian induced by the Laplace–Beltrami operator is
2m 1 g1/4 p 1 \sqrtg p 1 g1/4 + V (x), (Laplace–Beltrami Hamiltonian) (20.233) which in turn yields the Schrödinger equation (20.209). The kinetic energy differs from that of our original (product-ordered) Hamiltonian (20.194) by ordering terms, as we can see by employing the commutation
g1/4 p 1 \sqrtg p 1
g1/4 p 1
g1/4 p g′ 4g7/4 = p1 g p −i¯h g′ 4g2 p + i¯h 1 g1/4 p g′ 4g7/4 = p1 g p −i¯h g′ 4g2 p + i¯hp g′ 4g2 −(i¯h)2 g′ 4g5/4 g′ 4g7/4 = p1 g p + i¯h p, g′ g2 + ¯h2g′2 16g3 = p1 g p + ¯h2 4g3 gg′′ −2g′2 + ¯h2g′2 16g3 , (20.234) so that
(20.235) (ordering relation) where
8mg3 −¯h2g′2 32mg3 (20.236) (effective ordering potential) is an effective potential due to reordering our original Hamiltonian into the Laplace–Beltrami ordering. Any ordering is valid as a problem, given that it is imposed by the physics. The Laplace–Beltrami Hamiltonian
20.4.2 Point Transformations of the Path Integral¶
Chapter 20. Path Integration is ‘‘special’’ in that it is the correct choice for the Schrödinger equation corresponding to the classical Lagrangian, and thus generated the modified structure of the inner product, momentum operators, and so on. However, any other ordering can be regarded as valid in the sense of being the same as H△plus ¯h-dependent ‘‘quantum-correction’’ potentials. 20.4.2 Point Transformations of the Path Integral Now we would like to understand the path integral for a quantum system corresponding to the Lagrangian (20.196), and thus the Hamiltonian H△(x, p) in Eq. (20.195) or (20.233). At first we will do this beginning from the ‘‘flat-space’’ propagator for the Lagrangian
2 ˙q2 −V (q), (20.237) and thus the action functional
Z t t0 dt Lq(q, ˙q). (20.238) We have already worked out this problem; to review, in terms of the action functional, the propagator in continuous-time notation is [Eq. (20.26)]
Z Dq exp i ¯hS[q(t)] , (20.239) or more precisely, in discrete form,
m
N/2 Z N−1 Y j=1 dqj ! N−1 Y j=0
! , (20.240) where the short-time action is
Z tj+1 tj dt Lq(q, ˙q)
= m
j −V (qj) \deltat = m
(20.241) Now we wish to consider the coordinate transformation dq dx = p g(x) (20.242) that maps us to the quantum system defined by H△(x, p), and use this transformation to work out the path integral. In terms of the new coordinate x, we can write the short-time action as
2\deltat [q(xj+1) −q(xj)]2 −V [q(xj)] \deltat, (20.243)
transformed kinetic energy in the action, we will have to write it in terms of the coordinate x at some time in the range [tj, tj+1], expanding the path integral about these points. Here we will carry out this expansion about both tj and tj+1/2, which we will refer to as the pre-point and midpoint choices.17 In terms of stochastic calculus, these choices correspond to It¯o and Stratonovich calculi, respectively, and we will also see how the different choices and calculi here correspond to different possible orderings in the Hamiltonian. 17We are working out a simplified version of the calculation presented by H. O. Girotti and T. J. M. Simões, ‘‘A Generalized Treatment of Point Canonical Transformations in the Path Integral,’’ Il Nuovo Cimento 74, 59 (1983) (doi: 10.1007/BF02721685), for the transformation in multiple dimensions about an arbitrary time point.
20.4 Ordering Issues 20.4.2.1 Midpoint (Stratonovich) Expansion of the Path Integral We will start by expanding the short-time action (20.243) around ¯xj to order \deltat, regarding \deltaxj as O( \sqrt \deltat). Starting with the expansions
\deltax 2 j −q′′′(¯xj) \deltax 3 j
\deltax 2 j + q′′′(¯xj) \deltax 3 j , (20.244) the expansion for the short-time action becomes
2\deltat [q(xj+1) −q(xj)]2 −V [q(xj)] \deltat = m 2\deltat q′(¯xj) \deltaxj + q′′′(¯xj) \deltax 3 j 2 −V [q(xj)] \deltat = m
j + mq′(¯xj) q′′′(¯xj) 24\deltat \deltax 4 j −V [q(xj)] \deltat = m
j + m(2gg′′ −g′2) 96g \deltat \deltax 4 j −V [q(xj)] \deltat, (20.245) where we have used the derivatives
q′′ = g′ 2\sqrtg , q′′′ = g′′ 2\sqrtg − g′2
4g3/2 . (20.246) Then we can also transform and then expand the integration measure about the midpoints, using
\deltax 2 j
1 −q′′(¯xj) 2q′(¯xj) \deltaxj + q′′′(¯xj) 8q′(¯xj) \deltax 2 j = q g(¯xj) 1 −g′(¯xj) 4g(¯xj) \deltaxj + 2gg′′ −g′2 32g2 \deltax 2 j , (20.247) so that the measure becomes N−1 Y j=1 dqj = N−1 Y j=1 dxj q′(xj) = q′(x0) N−1 Y j=1 dxj ! N−1 Y j=0 q′(xj) = p g(x0) N−1 Y j=1 dxj ! N−1 Y j=0 q g(¯xj) 1 −g′(¯xj) 4g(¯xj) \deltaxj + 2gg′′ −g′2 32g2 \deltax 2 j . (20.248) Then collecting the various terms from Eqs. (20.245) and (20.248), and inserting them into the propagator (20.240), we find
p g(x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i ¯h mg(¯xj) 2\deltat \deltax 2 j −V [q(¯xj)] \deltat \times 1 −g′(¯xj)
2gg′′ −g′2 32g2 \deltax 2 j + im(2gg′′ −g′2) 96¯hg \deltat \deltax 4 j ) , (20.249)
Chapter 20. Path Integration after expanding out everything in the exponential except for the explicit kinetic and potential energies, so that the other expansion terms appear as moment-type components in quasi-Gaussian integrals. We can deal with the ‘‘mean’’ term of the form \deltaxj by converting it into a prefactor as follows: N−1 Y j=0 1 −g′(¯xj)
32g2 \deltax 2 j = N−1 Y j=0 1 −g′(xj)
32g2 \deltax 2 j −gg′′ −g′2 8g2 \deltax 2 j = N−1 Y j=0 exp −g′(xj) 4g(xj)\deltaxj −gg′′ −g′2 8g2 \deltax 2 j = N−1 Y j=0 exp h
i = N−1 Y j=0 g(xj) g(xj+1) 1/4 = g(x0) g(x) 1/4 . (20.250) Here we used the conversion g′(¯xj)
\deltaxj
g(xj) \deltaxj + gg′′ −g′2 2g2 \deltax 2 j , (20.251) and note that we only track the xj vs. ¯xj dependence of the coefficients of \deltaxj, not \deltax 2 j , since the distinction does not matter in the latter case to O(\deltat). Then the propagator (20.249) simplifies to
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i ¯h mg(¯xj) 2\deltat \deltax 2 j −V [q(¯xj)] \deltat \times 1 + gg′′ −g′2 16g2 \deltax 2 j + im(2gg′′ −g′2) 96¯hg \deltat \deltax 4 j ) (20.252) upon transforming the \deltax piece. Using the replacements \deltax2 −\rightarrow i¯h mg(x) \deltat, \deltax4 −\rightarrow − 3¯h2 m2g2(x) \deltat2, (20.253)
20.4 Ordering Issues to be justified later [Eq. (20.273), Section 20.4.2.3], the propagator becomes
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i ¯h mg(¯xj) 2\deltat \deltax 2 j −V [q(¯xj)] \deltat \times 1 + i¯h(gg′′ −g′2) 16mg3 \deltat −i¯h(2gg′′ −g′2) 32mg3 \deltat )
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i ¯h mg(¯xj) 2\deltat \deltax 2 j −V [q(¯xj)] \deltat 1 −i¯hg′2 32mg3 \deltat ) , (20.254) where the replacements are valid at any point in the time interval, because the difference is beyond order \deltat. Then if we finish the coordinate transformation of the potential, V [q(¯xj)] ≡V (¯xj), (20.255) and define the usual velocity
\deltat , (20.256) (velocity) then the path integral finally becomes18
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j −V (¯xj) −¯Veff(¯xj) ) , (midpoint path integral) (20.257) where we exponentiated the last term in the path integral, and we have defined the effective potential
32mg3 . (20.258) (midpoint effective potential) As in the q variables, we can write this path integral in short-hand continuous notation as
Z D h x p g(¯x) i exp i ¯h ¯S[x(t)]
Z t t0 dt L(¯x, ˙x) −¯Veff(¯x)
2 g(¯x) ˙x2 −V (¯x), (midpoint effective potential, continuous shorthand) (20.259) but it is particularly important to keep the discrete-form (20.257) as the fundamental object, since certain things such as the factors of g must be counted carefully for the prefactor to make sense. Note also that the midpoint path-integral action ¯S(¯x, ˙x) does not agree with the classical action with only the Lagrangian (20.196), being different by an effective potential that depends on the time-slicing expansion (and ordering assumptions regarding the Hamiltonian). 18Note that this expression agrees with the more general one given by Christian Grosche, ‘‘An Introduction into the Feynman Path Integral,’’ arXiv.org preprint (arXiv: hep-th/9302097v1), Eq. (2.24); and in M. Chaichian and A. Demichev, Path Integrals in Physics, Volume I: Stochastic Processes in Quantum Mechanics (Institute of Physics, 2001), p. 249, Eq. (2.5.19).
Chapter 20. Path Integration 20.4.2.2 Pre-Point (It¯o) Expansion of the Path Integral We can again expand the path integral (20.240), but now around the pre-point xj to obtain a different (but equivalent) expression for the path integral. The recipe is the same as in the midpoint case: 1. Expand the kinetic energy in the action, keeping terms to O(\deltat), regarding \deltaxj as O( \sqrt \deltat). 2. Expand the integration measure, to the extent necessary and also keeping terms to O(\deltat). 3. Write all extra terms as a product outside the exponential. 4. Replace powers of \deltaxj outside the Gaussian path measure with powers of \deltat. 5. Exponentiate the converted terms. 6. Change velocity-like terms into prefactors if possible/desired. Starting with the first item, we can expand the post-point coordinate,
\deltax 2 j + q′′′(xj) \deltax 3 j , (20.260) so that the short-time action (20.243) becomes (to order \deltat)
2\deltat [q(xj+1) −q(xj)]2 −V [q(xj)] \deltat = m 2\deltat q′(xj) \deltaxj + q′′(xj) \deltax 2 j + q′′′(xj) \deltax 3 j 2 −V [q(xj)] \deltat = m
j + m 2\deltatq′(xj)q′′(xj) \deltax 3 j + m \deltat q′′2(xj) + q′(xj)q′′′(xj) \deltax 4 j −V [q(xj)] \deltat = m
j + mg′(xj) 4\deltat \deltax 3 j + mg′2
j + m(2gg′′ −g′2) 24g \deltat \deltax 4 j −V [q(xj)] \deltat = m
j + mg′(xj) 4\deltat \deltax 3 j + m(8gg′′ −g′2) 96g \deltat \deltax 4 j −V [q(xj)] \deltat, (20.261) where we have again used the derivatives (20.246). Now we proceed on to the measure, which doesn’t need much except for a careful counting of factors: N−1 Y j=1 dqj = N−1 Y j=1 dxj q′(xj) = q′(x0) N−1 Y j=1 dxj ! N−1 Y j=0 q′(xj) = p g(x0) N−1 Y j=1 dxj ! N−1 Y j=0 q g(xj). (20.262) Collecting all terms from the action (20.261) and measure (20.262), and expanding out the exponential in the propagator (20.240), we have
p g(x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i ¯h mg(xj) 2\deltat \deltax 2 j −V (xj) \deltat \times 1 + img′(xj) 4¯h\deltat \deltax 3 j + im(8gg′′ −g′2) 96¯hg \deltat \deltax 4 j −m2g′2
j ) , (20.263)
20.4 Ordering Issues where we have carried out the expansion of the \deltax3/\deltat part to second order (generating a \deltax6/\deltat2 term) to keep the expansion consistent, and we have dropped the dependence on q in the potential. Then using the replacements (20.273) and (20.282), the propagator becomes
p g(x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i ¯h mg(xj) 2\deltat \deltax 2 j −V (xj) \deltat \times 1 −3g′(xj) 4g(xj) \deltaxj −3i¯h(8gg′′ −g′2) 96mg3
32mg3 \deltat ) . (20.264) In preparation to exponentiate the last factor, we can add and subtract a \deltax2 term,
p g(x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i ¯h mg(xj) 2\deltat \deltax 2 j −V (xj) \deltat \times 1 −3g′(xj)
32g2 \deltax 2 j −9i¯hg′2 32mg3 \deltat −3i¯h(8gg′′ −g′2) 96mg3
32mg3 \deltat ) = p g(x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i ¯h mg(xj) 2\deltat \deltax 2 j −V (xj) \deltat \times 1 −3g′(xj)
32g2 \deltax 2 j −i¯h(gg′′ −g′2) 4mg3 \deltat −i¯hg′2 32mg3 \deltat ) , (20.265) where we have used the remaining replacement in Eqs. (20.273): \deltax2 −\rightarrow i¯h mg(x) \deltat. (20.266) Along the same lines as Eq. (20.250), we can change the \deltaxj term into a prefactor via N−1 Y j=0 1 −3g′(xj)
32g2 \deltax 2 j = N−1 Y j=0 exp −g′(xj) 4g(xj)\deltaxj = N−1 Y j=0 exp
8g2 \deltax 2 j = N−1 Y j=0 g(xj) g(xj+1) 3/4 exp 3(gg′′ −g′2) 8g2 \deltax 2 j = g(x0) g(x) 3/4 N−1 Y j=0 exp i\deltat ¯h 3¯h2(gg′′ −g′2) 8mg3 . (20.267)
Chapter 20. Path Integration Carrying this out in the propagator and exponentiating the remaining parts of the polynomial factor, we have the desired form19
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h m 2 g(xj) ˙x 2 j −V (xj) −V<(xj) ) , (pre-point propagator) (20.268) where the pre-point-ordering effective potential from the expansion is
32mg3 −¯h2(gg′′ −g′2) 8mg3 , (20.269) (pre-point effective potential) and the velocity is as defined in Eq. (20.256). In continuous-notation shorthand, we can write this as
Z D h x p g(x) i exp i ¯hS<[x(t)]
Z t t0 dt [L(x, ˙x) −V<(x)]
2 g(x) ˙x2 −V (x), (pre-point effective potential, continuous shorthand) (20.270) where x is to be read as the pre-point, and again the more precise discrete form (20.268) is important to keep in mind for proper counting of factors. Note also that the action must be modified with an extra effective potential, which is different from the potential that came out of the midpoint discretization above. 20.4.2.3 “Moment” Relations Now to justify the ‘‘moment’’ formulae of the quasi-Gaussian integrals that we have used. To handle even powers, we will quickly derive a form of the standard Gaussian moment theorem by integrating the product of a Gaussian and an arbitrary power once by parts:
\sqrt
Z
= \sqrt
Z
= \sqrt
Z
µ \sqrt
Z
\sqrt
Z
xn−1
xn−2 + µ
xn−1 . (20.271) In the path integral, we can think of the coordinate as \deltax, with \deltax2 appearing in the Gaussian weight, so we can set µ = 0 in the above recursion, obtaining
xn−2 . (20.272) 19Note that this form generalized to multiple dimensions is quoted by Christian Grosche, ‘‘An Introduction into the Feynman Path Integral,’’ arXiv.org preprint (arXiv: hep-th/9302097v1), Eq. (1.7). Note, however, that the form quoted there appears
is no prefactor. That same form is quoted by M. Chaichian and A. Demichev, Path Integrals in Physics, Volume I: Stochastic Processes in Quantum Mechanics (Institute of Physics, 2001), p. 246, Eq. (2.5.7).
20.4 Ordering Issues This means that we can replace \deltaxn by (n −1)[i¯h\deltat/mg(x)] \deltaxn−2, which implies the replacements \deltax2 −\rightarrow i¯h mg(x) \deltat, \deltax4 −\rightarrow − 3¯h2 m2g2(x) \deltat2, \deltax6 −\rightarrow −15i¯h3 m3g3(x) \deltat3. (even-order rules) (20.273) Technically, the variance-weight g is a function of the position, but here the width of the Gaussian is small [O(\deltax)], and the rest of the integrand (beyond the normalized Gaussian weight) varies slowly over this range—any corrections to the moments due to this slow variation is ignorable at order \deltat. However, this is not true for the odd ‘‘moments,’’ which would otherwise vanish, because the leading contribution comes from this variation. We can handle this more carefully as follows. First, we will define the ‘‘effective expectation’’
Z dx xn \sqrt
(20.274) which differs from a Gaussian expectation because the integral is not a normalized Gaussian distribution
expansions
1 −\sigma′(0)
x2
x2 \sigma2(0) 1 −2\sigma′(0)
, (20.275) we can write the nth effective moment (for n > 2) as
\sqrt 2\pi\sigma(0) Z dx xn 1 −\sigma′(0) \sigma(0) x
\sigma3(0)x3
\approx \sqrt 2\pi\sigma(0) Z dx xn 1 −\sigma′(0)
\sigma3(0)x3
(20.276) where we are working to order x in the expansions, keeping in mind that the contribution of the integral comes from x of the order \sigma, so we are counting \sigma and its derivative as being of the same order as x. Then writing the effective moment in terms of moments with respect to a Gaussian probability density of mean zero and variance \sigma2(0),
\sigma(0)
xn+1
\sigma3(0)
xn+3
(n −1)
xn−2 −n\sigma′(0) \sigma(0)
xn−1
\sigma3(0)
xn+1 , (20.277) where we have effectively integrated by parts by using the moment formula (20.272). Then the strategy is to match the coefficients of the three terms. Separating the n and the 2 in the last term and again using Eq. (20.272), we find that we can merge this last piece with the middle term:
\sigma2(0) xn−2 (n −1) −n\sigma′(0)
\sigma3(0)x3 =
\sigma2(0) xn−2 (n −1) −n\sigma′(0)
\sigma3(0)x3
=
\sigma2(0) xn−2 (n −1) −n\sigma′(0)
\sigma3(0)x3
\sigma(0) xn−1
=
\sigma2(0) xn−2
\sigma3(0)x3 = n
\sigma2(0) xn−2
\sigma3(0)x3 −\sigma2(0)
xn−2 . (20.278)
Chapter 20. Path Integration Now using the expansion
, (20.279) we can restore the x-dependence of the \sigma2 coefficient:
\sigma2(0)
\sigma(0) x xn−2
\sigma3(0)x3 −\sigma2(0)
xn−2 \approx n
\sigma2(x) xn−2
\sigma3(0)x3 −\sigma2(0)
xn−2
\sigma2(x) xn−2 −\sigma2(0)
xn−2 . (20.280) In the last step, we used Eq. (20.276) to restore the effecive expectation, and we have consistently kept our expansions to order x. Thus, we have the rule for effective moments (n > 2)
\sigma2(x) xn−2 −\sigma2(0)
xn−2 , (20.281) where the equality holds asymptotically for small \sigma(x). In the even-n case, this moment rule reduces to Eq. (20.272), since the difference between the effective and normal expectations is negligible at order \deltat. In the odd-n case, the last term vanishes, and this justifies the replacement rule \deltax3 −\rightarrow 3i¯h
(20.282) (odd-order rule) under the quasi-Gaussian integral, which is the only odd ‘‘moment’’ that we will need. 20.4.2.4 Midpoint–Pre-point Conversion Now that we have derived two equivalent forms—Eqs. (20.257)–(20.259) for the midpoint form and Eqs. (20.268)–(20.270) for the pre-point form—we should check their equivalence explicitly. Starting with the mid-point path inte- gral (20.257),
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j −V (¯xj) −¯Veff(¯xj) ) , (20.283) we will then need to expand out the g(¯xj) in the exponential as
\deltax 2 j , (20.284) and the measure factor as q
q g(xj) 1 + g′(xj) 4g(xj)\deltaxj + g′′(xj) 16g(xj)\deltax 2 j −g′2(xj) 32g2(xj)\deltax 2 j , (20.285)
20.4 Ordering Issues Writing out the expanded terms outside the exponential, we have
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h m 2 g(xj) ˙x 2 j −V (xj) −¯Veff(xj) \times 1 + img′(xj) 4¯h\deltat \deltax 3 j + img′′
j −m2g′2
j + g′(xj)
16g \deltax 2 j −g′2 32g2 \deltax 2 j + img′2
j ) , (20.286) where we have consistently multiplied out and expanded to order \deltat as usual, and all quantities are evaluated at the pre-point xj except as noted. Then using the replacements (20.273) and (20.282), we find
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h m 2 g(xj) ˙x 2 j −V (xj) −¯Veff(xj) \times 1 −3g′(xj) 4g(xj) \deltaxj −3i¯hg′′
32mg3 \deltat + g′(xj) 4g(xj)\deltaxj + i¯hg′′ 16mg2 \deltat −i¯hg′2 32mg3 \deltat −3i¯hg′2 16mg3 \deltat )
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h m 2 g(xj) ˙x 2 j −V (xj) −¯Veff(xj) \times 1 −g′(xj) 2g(xj)\deltaxj −i¯hg′′
4mg3 \deltat ) , (20.287) Then preparing to re-exponentiate, we add and subtract the \deltax2 term,
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h m 2 g(xj) ˙x 2 j −V (xj) −¯Veff(xj) \times 1 −g′(xj)
8g2 \deltax 2 j −i¯hg′2 8mg3 \deltat −i¯hg′′
4mg3 \deltat )
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h m 2 g(xj) ˙x 2 j −V (xj) −¯Veff(xj) \times 1 −g′(xj)
8g2 \deltax 2 j −i¯h(gg′′ −g′2) 8mg3 \deltat ) . (20.288)
Chapter 20. Path Integration As in Eq. (20.267), we can handle the \deltaxj term as N−1 Y j=0 1 −g′(xj)
8g2 \deltax 2 j = N−1 Y j=0 exp −g′(xj) 2g(xj)\deltaxj = N−1 Y j=0 exp −\delta log q g(xj) + gg′′ −g′2 4g2 \deltax 2 j = N−1 Y j=0 g(xj) g(xj+1) 1/2 exp gg′′ −g′2 4g2 \deltax 2 j = g(x0) g(x) 1/2 N−1 Y j=0 exp i\deltat ¯h ¯h2(gg′′ −g′2) 4mg3 . (20.289) Implementing this and exponentiating the remaining polynomial terms, we have
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h m 2 g(xj) ˙x 2 j −V (xj) −¯V<(xj) ) , (20.290) where the effective pre-point potential we derived from the mid-point propagator is
8mg3 −¯h2(gg′′ −g′2) 4mg3
8mg3 . (20.291) Putting in the midpoint effective potential (20.258), we find
32mg3 −¯h2(gg′′ −g′2) 8mg3 , (20.292) which is exactly the effective potential V<(x) from Eq. (20.269) that we derived from the direct pre-point expansion. 20.4.2.5 Discussion: Integration Under Different Path Measures To summarize the results of this section so far: beginning with the flat-space propagator (20.240), we have derived two different propagators—Eqs. (20.257) and (20.268)—by expanding about different points in the interval between the jth and (j + 1)th time slices. We have also explicitly shown that they are equivalent, in that one can be derived from the other by changing the expansion point. In the language of stochastic differential equations, these two path integrals correspond to Stratonovich (midpoint) and It¯o (prepoint) calculi, at least in the imaginary-time forms of the propagators. In a broad sense, a different choice of calculus will correspond to a different family of paths. More explicitly, the midpoint propagator (20.257) generates imaginary-time paths according to the Stratonovich SDE dx = s ¯h mg(¯x) dW ≡ s ¯h mg(x) ◦dW, (20.293) while the prepoint propagator generates paths via the It¯o SDE dx = s ¯h mg(x) dt. (20.294) Note that we have not written a drift term, as in both propagators we have explicitly eliminated velocities and recast them in terms of prefactors and potentials. We know from the theory of SDEs in Section 17.4.2
20.4 Ordering Issues that these two SDEs do not produce equivalent solutions. For example, casting the SDE (20.293) into It¯o form would produce a drift term:
4mg2(x) dt + s ¯h mg(x) dW. (20.295) Again, this should be compared to the no-drift SDE (20.294) to understand why the paths are different. Then the reason that we obtain the quantum effective potentials in the propagators—and in particular different effective potentials for the same system but different choice of expansion point—is to compensate for these differences in paths. Since the paths sample space differently, the potential weights them differently, essentially reweighting the path integral for the new set of paths. That is, paths under one path measure that less likely under the other measure must be ‘‘penalized’’ with a reduced amplitude to compensate for the difference in measure. We can also interpret the correction potentials at a more ‘‘microscopic’’ level. In arriving at the path integrals, we made the coordinate transformation defined by g(x) in Eq. (20.203). When this is a nonlinear transformation, it maps the flat-space Gaussian probability densities for the steps into distorted Gaussian distributions. But the kinetic-energy factor (Gaussian in ˙x) is derived by expanding about the zero-slope point of the distorted Gaussian, which does not necessarily coincide with the mean of the distribution. This introduces an effective drift in the paths, which must be compensated for by the extra potential terms. 20.4.2.6 Cameron–Martin–Girsanov Transformation and Drift Potentials Another common example of reweighting path integrals for different measures in the mathematical theory of SDEs is based on a change in the drift term of the paths. This is called a Girsanov transformation, or a Cameron–Martin transformation, or a Cameron–Martin–Girsanov transformation of measure.20 Roughly speaking, this transformation applies when comparing driftless paths
(20.296) to paths with a drift:
Z t dt′ a[˜x(t′)]. (20.297) Note that this last expression can be written in differential form as
(20.298) A path integral corresponding to the drifting trajectories in Eqs. (20.297) and (20.298) is then
Z Dx exp − Z t dt′ 1 ˙x(t′) −a[x(t′)] 2 , (20.299) (model path integral) where we have dropped the twiddle on ˜x. We can interpret the part of the measure inside the exponential here in discrete form as ˙x(t′) −a[x(t′)] 2 dt = 1 \deltax \deltat −a(x) 2
2\deltat , (20.300) so that each step in the walk still has variance \deltat, but now also a mean of a(x) \deltat. Then the Cameron–Martin–Girsanov transformation boils down to expanding the quadratic in the exponential as follows:
Z Dx exp − Z t dt′ ˙x2(t′) exp Z t dx(t′) a(x) −1 Z t dt′ a2(x) (transformed path integral) (20.301) 20Andrei N. Borodin and Paavo Salminen, Handbook of Brownian Motion—Facts and Formulae, 2nd ed. (Birkhäuser, 2002), pp. 49-50.
20.4.3 Operator Orderings¶
Chapter 20. Path Integration Now the first exponential factor is the appropriate measure for trajectories without drift, as in Eq. (20.296). The second exponential factor acts as a ‘‘potential’’ that corrects for switching to the different path measure, and is called a Radon–Nikodym derivative. [The ‘‘derivative’’ here refers to being a ratio of the two path measures, in analogy to transforming between probability distributions fy(y) and fx(x) by setting
that don’t (randomly) move in the same way as the drifting paths, but rewards the ones that (randomly) do move in the right way. However, the correction factor in Eq. (20.301) does not correspond directly to a potential as we have discussed it, in view of the stochastic integral [the first integral in this factor, with respect to dx(t′)]. However, we can conveniently change this to a conventional integral as follows.21 Let A(x) be an antiderivative of a(x); that is,
(20.302) Then
2A′′(x) (dx)2
2a′(x) dt, (20.303) where we used the It¯o rule dx2 = dW 2 = dt. Solving for a dx and integrating gives Z t
Z t a′[x(t′)] dt′. (20.304) Thus, Eq. (20.301) becomes
Z Dx exp − Z t dt′ ˙x2(t′) exp −1 Z t dt′ a′(x) + a2(x) exp A[x(t)] −A[x(0)] . (transformed integral, potential form) (20.305) Now the second exponential factor has the form of an effective potential
a′(x) + a2(x) , (20.306) (effective drift potential) and the last exponential factor represents an endpoint correction [which vanishes, for example, in the case
To wrap up this discussion of drift potentials, we should reexamine subtleties in some of the steps in the above treatment that we glossed over, which will connect with the discussion on operator orderings to follow below. In particular, note that the first integral in the last exponential factor of Eq. (20.301) is a stochastic integral, and thus we must specify a calculus to evaluate the integral. Given the expansion of Eq. (20.303), we are clearly interpreting this as an It¯o integral [i.e., a(x) in the path integral (20.299) is evaluated at the prepoint x(t′)]. More precisely, given the path integral (20.305), it is equivalent to the path integral (20.299) provided we interpret the latter integral in the prepoint sense. In the path-integral language, the difference between, for example, a prepoint and midpoint interpretation comes about by introducing a correction of order dx to a(x) in Eq. (20.301). The stochastic integral has a differential of dx, so the first-order correction must be kept in this term (while these corrections are negligible in the last integral). In the language of operator orderings below, such a term arises in the path integral from a term of the form p a(x) in the corresponding quantum Hamiltonian, which can be ordered in different ways. 20.4.3 Operator Orderings We have so far constructed the path integrals for the system corresponding to the Laplace–Beltrami Hamilto- nian (20.195) by starting with a simpler Hamiltonian with no ordering issues, and then effecting a coordinate change that transforms both the Hamiltonian and the path integral into the desired form. However, we should 21Barry Simon, Functional Integration and Quantum Physics (Academic Press, 1979), pp. 172-3 (ISBN: 0126442509).
20.4 Ordering Issues also be able to follow a modified version of the basic path-integral construction of Section 20.1, using the modified rules from Section 20.4.1.1 corresponding to the Laplace–Beltrami Hamiltonian (20.195). This requires some care due to the operator-ordering issues, and we will see that different operator orderings nat- urally give rise to different choices of stochastic calculus, or time-expansion points (midpoint vs. prepoint) from the coordinate-transformation derivations of the path integrals in the previous section. 20.4.3.1 It¯o Calculus and p-x Ordering As our first ordering example, consider the ‘‘p-x’’-ordered version of (20.194),
2mg(x) + V (x), (variable-mass Hamiltonian, It¯o ordering) (20.307) will all momenta to the left and all positions to the right, which is called anti-standard ordering, or which we will call It¯o ordering or pre-point ordering, for reasons we will soon see. By commuting one momentum, we have
2mg(x)p + i¯hp g′
g′ 2mg2 , (20.308) so that the product-ordered Hamiltonian is related to the It¯o-ordered Hamiltonian via a commutator term linear in p. We can then relate this Hamiltonian to the Laplace–Beltrami Hamiltonian via Eq. (20.235), so that
g′ 2mg2 , (20.309) where the ordering potential V△is defined in Eq. (20.236). Now let’s revisit the construction of the path integral beginning with Eqs. (20.10) and (20.11):
= Z N−1 Y j=1 q g(xj) dxj !
(20.310) The change here—compared to the original path-integral construction of Section 20.1—is due to the modified position identity (20.213), where we have extra factors of \sqrtg to keep track of. Then in analogy with Eq. (20.14), we can insert the momentum identity (20.230) into each mini-propagator. Because of the ordering choice, the natural place to insert the identity is to the left of the exponential, as the exponential itself is p-x-ordered, at least to order \deltat. So we have
Z
= Z
=
Z dp1 eip1(x2−x1)/¯he−iHpx(x1,p1)\deltat/¯h =
Z dp1 ei[p1 ˙x1−Hpx(x1,p1)]\deltat/¯h, (20.311) where as before
˙x1 ≡\deltax1 \deltat , (20.312) and we have used the inner product (20.227). This is really the crucial step in the derivation: because of the ordering of the Hamiltonian, there was only one natural place to insert the momentum identity into the mini- propagator. That, as a result, led to the position operators in the Hamiltonian resolving to the eigenvalues
Chapter 20. Path Integration x1, rather than x2, and so we expect the result here to be consistent with the pre-point expansion of Section 20.4.2.2. Thus, in constructing path integrals, p-x ordering in the Hamiltonian leads naturally to It¯o calculus, at least when we think about the equivalent path integral in the diffusion picture. Notice that this matches the intuition from casting SDEs directly into the Fokker–Planck (Kolmogorov forward) equation [see Eqs. (17.129) and (17.139)], which is a diffusion equation (Schrödinger equation in imaginary time). There the derivative operators in the diffusion term appear to the left of the state-dependent diffusion coefficient, just as the momentum operators appear to the left of the ‘‘diffusion coefficient’’ [which appears in the SDE (20.295)] in the Hamiltonian (20.308). Proceeding to collect all the matrix elements as we did before to derive the phase-space path integral (20.28), we then have
Z N−1 Y j=1 q g(xj) dxj !( N−1 Y j=0 dpj [g(xj+1)g(xj)]1/4 ei[pj ˙xj−Hpx(xj,pj)]\deltat/¯h ) = p
Z N−1 Y j=1 dxj !( N−1 Y j=0 dpj g(xj) g(xj+1) 1/4 ei[pj ˙xj−Hpx(xj,pj)]\deltat/¯h ) =
Z N−1 Y j=1 dxj !( N−1 Y j=0 dpj ei[pj ˙xj−Hpx(xj,pj)]\deltat/¯h ) . (20.313) Thus, the phase-space path integral becomes
Z N−1 Y j=1 dxj !( N−1 Y j=0 dpj exp i\deltat ¯h h pj ˙xj −Hpx(xj, pj) i )
Z Dx Dp exp i ¯h Z t t0 dt h p ˙x −Hpx(x, p) i (phase-space propagator for Hpx) (20.314) in discrete and continuous notations. We can proceed by working out the momentum integrals in Eq. (20.314), recalling that we are working with eigenvalues and thus there is no longer any concern about ordering. The result of each momentum integral is the same as in Eq. (20.19), if we replace m with mg(xj), Z dpj exp i\deltat ¯h pj ˙xj −Hpx(xj, pj) = r mg(xj) i\deltat eiL(xj, ˙xj)\deltat/¯h (20.315) in terms of the Lagrangian (20.196). Thus, we obtain the propagator
m
N/2 Z N−1 Y j=1 dxj !( N−1 Y j=0 q g(xj) exp i\deltat ¯h L(xj, ˙xj) ) (propagator for Hpx) (20.316) for the p-x-ordered Hamiltonian. On the other hand, in view of Eq. (20.309), we can instead compute the propagator for the Hamiltonian
Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 dpj exp i\deltat ¯h pj ˙xj −Hpx(xj, pj) + V△(xj) + i¯hpj g′ 2mg2 ) . (20.317)
20.4 Ordering Issues The result is the same as for the Kpx propagator, then, but with a modified velocity and potential:
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h L xj, ˙xj + i¯h g′(xj) 2mg2(xj)
¯h V△(xj) ) . (20.318) Expanding out the Lagrangian, we have L xj, ˙xj + i¯h g′(xj) 2mg2(xj) = m 2 g(xj) ˙xj + i¯h g′(xj) 2mg2(xj) 2 −V (xj) = m 2 g(xj) ˙x 2 j −V (xj) + i¯h g′(xj) 2g2(xj) ˙xj −¯h2g′2 8mg3
2g2(xj) ˙xj −¯h2g′2 8mg3 , (20.319) and so the propagator becomes
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h L(xj, ˙xj) + V△(xj) −¯h2g′2 8mg3 −g′(xj) 2g2(xj)\deltaxj ) . (20.320) From Eq. (20.289) we have N−1 Y j=0 exp −g′(xj) 2g(xj)\deltaxj = g(x0) g(x) 1/2 N−1 Y j=0 exp i\deltat ¯h ¯h2(gg′′ −g′2) 4mg3 , (20.321) and using this to replace the \deltaxj term in the exponential, we find
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h L(xj, ˙xj) + V△(xj) + ¯h2(gg′′ −g′2) 4mg3 −¯h2g′2 8mg3 ) . (20.322) From Eq. (20.236) we had
8mg3 −¯h2g′2 32mg3 , (20.323) so the propagator finally becomes
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj) exp i\deltat ¯h L(xj, ˙xj) + ¯h2(gg′′ −g′2) 8mg3 −¯h2g′2 32mg3 ) . (pre-point propagator for H△) (20.324) This exactly matches the pre-point (corresponding to integration in the It¯o sense) propagator (20.268) that we derived by the point transformation, with the same effective potential (20.269). Again, this corresponds to the Hamiltonian H△(x, p), and follows in the operator formalism by first reordering this Hamiltonian in anti-standard (p-x) form, and then introducing momentum identities to derive the path integral.
Chapter 20. Path Integration 20.4.3.2 Anticipating Calculus and x-p Ordering By contrast, if we have a Hamiltonian Hxp(x, p) that is ‘‘x-p’’-ordered or in standard ordering,
2mg(x)p2 + V (x), (variable-mass Hamiltonian, anticipating ordering) (20.325) with all x’s to the left, all p’s to the right, then we can modify Eqs. (20.311) to become
Z
= Z
=
Z dp1 eip1(x2−x1)/¯he−iHxp(x2,p1)\deltat/¯h =
Z dp1 ei[p1 ˙x1−Hxp(x2,p1)]\deltat/¯h, (20.326) with the difference being that H(x, p) is now evaluated at the final point x2 in the interval [x1, x2]. The path integral has the same form (20.28) as for p-x ordering, but now in the integration, the x in the Hamiltonian is evaluated at the final time in each time interval [t, t+dt]. In the imaginary-time diffusion picture, this again corresponds to integrating with respect to diffusive (Wiener) paths, where the integrand is always evaluated at the final point of each time interval, which is called anticipating calculus. Thus, for diffusive path integrals, x-p ordering leads naturally to anticipating calculus. The rest of the derivation carries through in the same way, and for example we obtain the post-point propagator
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(xj+1) exp i\deltat ¯h L(xj+1, ˙xj) + ¯h2(gg′′ −g′2) 8mg3 −¯h2g′2 32mg3 ) . (post-point propagator for H△) (20.327) in place of the pre-point form (20.324), essentially by interchanging xj \leftarrow \rightarrow xj+1 in the integrand and prefactor. 20.4.3.3 Stratonovich Calculus and Weyl Ordering Perhaps the most useful ordering of the Hamiltonian to consider is Weyl (symmetrized) ordering (see Sec- tion 4.3.5). We can write the Weyl-ordered version of the Hamiltonian (20.194) or (20.195) as
p 2mg(x)p W + V (x), (variable-mass Hamiltonian, Weyl ordering) (20.328) where the subscript denotes Weyl ordering. To relate this form to other orderings, we can take advantage of the product rule for Weyl correspondence (4.121)
x −¯h
2i\partial x
x + ¯h 2i\partial p, p −¯h 2i\partial x B(x, p) (20.329)
20.4 Ordering Issues for an operator product ˆA = ˆB ˆC. Applying this rule twice, ˆp g(ˆx) ˆp \leftarrow \rightarrow p + ¯h 2i\partial x p −¯h 2i\partial x g(x) = p g(x)p + ¯h2 1 g ′′ = p g(x)p −¯h2(gg′′ −2g′2) 4g3 \leftarrow \rightarrow ˆp g(ˆx) ˆp W −¯h2(gg′′ −2g′2) 4g3 (20.330) where arrows denote Weyl correpondence between classical functions and operators, and we are being careful for the moment to mark operators with hats. That is, we can write
(20.331) where the effective ordering potential to switch between product and Weyl orderings is
8mg3 . (20.332) Then using Eqs. (20.235) and (20.236), we can also relate the Weyl and Laplace–Beltrami Hamiltonians via
32mg3 , (20.333) so that the ordering potential here is simpler than either V△or VW. Now suppose we construct the propagator for the Weyl-ordered Hamiltonian by HW(x, p). In adapting Eqs. (20.311) to this case, it is not obvious where to insert the momentum identity, but there is a trick we can use to make this simple.22 Recall that in terms of the characteristic operator (4.92), ˆ
(20.334) an arbitrary Weyl-ordered operator, for example the Hamiltonian, is obtained from the classical function from the Weyl correspondence (4.97)
(2\pi¯h)2 Z \infty −\infty d\pix Z \infty −\infty d\pip Z \infty −\infty dx Z \infty −\infty dp HW(x, p) ˆ
(20.335) (Note that we are again being careful for the moment about distinguishing classical variables from oper- ators.) In this representation, the only operator-dependence in the Hamiltonian comes in the form of the characteristic operator. Consider the analogous matrix element of the characteristic operator: \langle x2| ˆ
= Z
= Z
=
Z dp1 eip1(x2−x1)/¯hei\pixx2/2¯hei\pipp1/¯hei\pixx1/2¯h =
Z
(20.336) 22see, e.g., Christian Grosche, ‘‘An Introduction into the Feynman Path Integral,’’ arXiv.org preprint (arXiv: hep- th/9302097v1).
Chapter 20. Path Integration where we have used a symmetric splitting, similar to Eq. (4.101), which is exact here, because the argument of the exponential is linear in x and p. Then the similar matrix element of the Hamiltonian becomes
(2\pi¯h)2 Z d\pix Z d\pip Z dx Z dp HW(x, p) \langle x2| ˆ
=
Z dp1 eip1 ˙x1\deltat/¯h \times Z d\pix Z d\pip Z dx Z
=
Z dp2 eip1 ˙x1\deltat/¯h Z dx Z dp HW(x, p) \delta x −x1 + x2 \delta(p −p1) =
Z dp2 eip1 ˙x1\deltat/¯hHW(¯x1, p1), (20.337) where as before we have defined the midpoint
. (20.338) Now since
(20.339) with the second-order contributions negligible, the same argument applies to matrix elements of the mini- evolution operators, and thus Eqs. (20.337) become
Z dp1 ei[p1 ˙x1−HW(¯x1,p1)]\deltat/¯h. (20.340) Again proceeding to collect all the matrix elements as we did before to derive the phase-space path integral (20.313), we have
Z N−1 Y j=1 q g(xj) dxj !( N−1 Y j=0 dpj [g(xj+1)g(xj)]1/4 ei[pj ˙xj−HW(¯xj,pj)]\deltat/¯h ) = p
Z N−1 Y j=1 dxj !( N−1 Y j=0 dpj g(xj) g(xj+1) 1/4 ei[pj ˙xj−HW(¯xj,pj)]\deltat/¯h ) =
Z N−1 Y j=1 dxj !( N−1 Y j=0 dpj ei[pj ˙xj−HW(¯xj,pj)]\deltat/¯h ) . (20.341) Thus, the phase-space path integral becomes
Z N−1 Y j=1 dxj !( N−1 Y j=0 dpj exp i\deltat ¯h h pj ˙xj −HW(¯xj, pj) i )
Z Dx Dp exp i ¯h Z t t0 dt h p ˙x −HW(¯x, p) i (phase-space propagator for Weyl-ordered Hamiltonian) (20.342) in discrete and continuous notations. Now the path integral appears with the position in H(x, p) evaluated at the middle time t + dt/2 in the integration interval [t, t + dt]. Thus, Weyl ordering leads naturally to Stratonovich calculus in the sense of a diffusive path integral. We have indicated the spatial dependence in the Hamiltonian by ¯x to emphasize this, and again, HW(x, p) here is the classical Hamiltonian obtained from the Weyl-ordered Hamiltonian by replacing position and momentum operators by eigenvalues. This ordering is convenient, as Stratonovich integration proceeds according to the rules of ‘‘normal’’ calculus, and the integral is explicitly symmetric under time-reversal.
20.4 Ordering Issues Proceeding to the Lagrangian form of the path integral by carrying out the momentum integrals, we can again note that the result of each momentum integral is the same as in Eq. (20.19), if we replace m with mg(¯xj), Z dpj exp i\deltat ¯h pj ˙xj −HW(¯xj, pj) = r mg(¯xj) i\deltat eiL(¯xj, ˙xj)\deltat/¯h (20.343) in terms of the Lagrangian (20.196). Thus, we obtain the propagator
m
N/2 Z N−1 Y j=1 dxj !( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h L(¯xj, ˙xj) ) (propagator for Weyl-ordered Hamiltonian) (20.344) for the Weyl-ordered Hamiltonian HW(x, p). If we instead want the propagator corresponding to H△(x, p), we can use relation (20.333) to make
is much easier here than in the It¯o-ordered case, and we directly obtain the propagator
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h L(¯xj, ˙xj) −¯h2g′2 32mg3 ) . (propagator for H△(x, p)) (20.345) This propagator is the same as the propagator (20.257) that we obtained by expanding the short-time action about the midpoint in the coordinate-transformation derivation. 20.4.3.4 Symmetrized Ordering Note that the Weyl ordering is different from the other obvious symmetric choice, that of symmetric order- ing: H{xp} := Hpx + Hxp = 4m p2, g(x) + + V (x). (Hamiltonian for symmetrized ordering) (20.346) Here we used the It¯o and anticipating orderings from Eqs. (20.325) and (20.307), respectively. In this case, the Hamiltonian corresponding to the matrix element (20.311) is evaluated at each endpoint and then averaged, as we will see shortly. This is closer to Stratonovich calculus than either It¯o or anticipating, in that it focuses on the middle of the interval (by averaging the endpoints). To derive the path integral for this ordering, we will follow the procedure in Section 20.4.3.1. The propagator still has the form of Eq. (20.310), but upon inserting momentum identities, we have matrix elements
= Z
= Z
=
Z dp1 ei{p1 ˙x1−[H(x1,p1)+H(x2,p1)]/2}\deltat/¯h, (20.347) where the equalities here are only up to O(\deltat). The derivation then carries through as in the It¯o case through the Hamiltonian path integral (20.314). Then carrying out the momentum integrals, the idea is the same as
Chapter 20. Path Integration in Eq. (20.315), but g(xj) is replaced by 2g(xj) + 2g(xj+1) −1
(20.348) so that (20.315) becomes in this case Z dpj exp i\deltat ¯h pj ˙xj −Hpx(xj, pj) = r m[Mg−1(xj)]−1 i\deltat ei{m[Mg−1(xj)]−1 ˙x 2 j −MV (xj)}\deltat/¯h, (20.349) where we have introduced the averaging operator on the time lattice:
. (20.350) Thus, we obtain the propagator
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q [Mg−1(xj)]−1 exp i\deltat ¯h m 2 [Mg−1(xj)]−1 ˙x 2 j −MV (xj) ) (propagator for symmetrized Hamiltonian) (20.351) for the symmetrized Hamiltonian (20.346). Note that we can also rewrite this path integral in terms of the midpoint
, (20.352) using the expansions
\deltax 2 j
\deltax 2 j . (20.353) Thus, we will need
gg′′ −2g′2 8g \deltax 2 j + g′2g′′ 32g2 \deltax 4 j , (20.354) up to fourth order, where all instances of g on the right-hand side are evaluated at the midpoint. Expanding the parts of the path integral that need it, q [Mg−1(xj)]−1 exp i\deltat ¯h m 2 [Mg−1(xj)]−1 ˙x 2 j = q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j \times 1 + gg′′ −2g′2 16g2
16¯hg \deltax4 \deltat = q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j \times 1 + i¯hgg′′ −2g′2 16mg3 \deltat −3i¯hgg′′ −2g′2 16mg3 \deltat = q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j 1 −i¯hgg′′ −2g′2 8mg3 \deltat , (20.355)
20.4 Ordering Issues where we used the replacements (20.273) and consistently expanded to order \deltat. Thus, in midpoint form, the propagator (20.351) becomes
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j −V (¯xj) −¯h2(gg′′ −2g′2) 8mg3 ) , (propagator for symmetrized Hamiltonian, midpoint form) (20.356) where we have also noted that MV (xj) can be replaced by V (¯xj) at order \deltat. To compare with other Hamiltonians, we can consider reordering the original Hamiltonian (20.362) in symmetrized form, using p
p2 1 g + 1 g p2 + i¯h p, −g′ g2 = 1 p2 1 g + 1 g p2 −¯h2 gg′′ −2g′2 g3 , (20.357) so that
4mg3 . (20.358) Then with Eqs. (20.235) and (20.236), we can compare the Laplace–Beltrami ordering to the symmetrized form as
8mg3 + ¯h2g′2
8mg3 + ¯h2g′2 32mg3 . (20.359) Then putting in the extra terms as correction potentials in the propagator (20.316), we can obtain the propagator for H△(x, p) as
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h L(¯xj, ˙xj) −¯h2g′2 32mg3 ) . (symmetrized propagator for H△(x, p)) (20.360) Note that this is identical to the propagator (20.345) that we derived in Weyl ordering. Thus, in terms of stochastic differential equations, both orderings generate trajectories that correspond to Stratonovich calculus. 20.4.3.5 Product Ordering Yet another symmetric-type ordering for the Hamiltonian (20.194) is called product ordering, and the product-ordered analogue of the model Hamiltonian is Hgpg := p g(x) p2 2m p g(x) + V (x). (Hamiltonian for symmetrized ordering) (20.361) We will leave it as an exercise to derive the propagator for this Hamiltonian and for the Laplace–Beltrami Hamiltonian H△(x, p) (see Problem 20.1). However, the upshot is that, once written in midpoint form, the latter propagator is identical to the version derived in Weyl and symmetrized orderings. This conclusion also holds for more general path integrals in curved spacetime.23 23C. Grosche and F. Steiner, Handbook of Feynman Path Integrals (Springer, 1998), p. 170.
20.4.4 Normalizing the Weyl-Ordered Path Integral¶
Chapter 20. Path Integration 20.4.3.6 Equivalence Classes of Orderings An important conclusion from the above path integrals is that although we can define several path integrals that treat the beginning and end time slices (xj and xj+1) symmetrically, they end up generating identical path integrals when written in terms of the midpoint ¯xj, and thus all correspond to Stratonovich calculus with the same effective potential. Because of the need to Taylor-expand the path integral to fourth order in \deltax, it seems as if there are many more possible path integrals than choices for stochastic calculus (where we only expand to second order). However, we see that different path integrals that correspond to the same calculus turn out to be equivalent. That is, there is really only one path integral for each calculus, but many possible ways to write it down. 20.4.4 Normalizing the Weyl-Ordered Path Integral Now we want to return to our original model Hamiltonian (20.194) for a variable mass,
2mg(x)p + V (x), (20.362) (variable-mass Hamiltonian) and consider how to develop a normalized path integral for Monte–Carlo calculations in the sense of Sec- tion 20.3.1. In previous sections, we considered propagators for the Hamiltonian in Laplace–Beltrami form H△(x, p), but we need the propagators for H(x, p). From Eq. (20.235), we only need to add the ordering correction V△(x, p) from Eq. (20.236) to the potential V (x) in the propagator. Hence, from Eqs. (20.257), (20.258), and (20.259), or from Eq. (20.345), we have the propagator
m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j −V (¯xj) + ¯h2(gg′′ −2g′2) 8mg3 )
Z D h x p g(¯x) i exp i ¯h Z t t0 dt m 2 g(¯x) ˙x2 −V (¯x) + ¯h2(gg′′ −2g′2) 8mg3 (midpoint path integral) (20.363) for the Hamiltonian (20.362) in midpoint form, in both discrete and continuous notations. To have a better- defined problem to normalize in Monte-Carlo form, we will calculated the partition function for the same Hamiltonian. Using Eq. (20.220), we can write the partition function as Zpgp = Z dx0 Z dx p g(x) \delta(x −x0) Kpgp(x, −i¯h\beta; x0, 0) = Z dx0 Z dx \delta(x −x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j −V (¯xj) + ¯h2(gg′′ −2g′2) 8mg3 ) , (20.364) where we are still writing imaginary time in terms of t, rather than \beta, for the moment. Unfortunately, due to the presence of g(¯xj), the normalization procedures of Sections 20.3.1–20.3.1.2 do not directly apply here (i.e., the integrals are not Gaussian). However, we can make things simpler by rescaling time on a path-dependent basis. That is, we change to time t′, where
(20.365) (temporal rescaling for path integral)
20.4 Ordering Issues The path integral transformed thusly is Zpgp = Z dx0 Z dx \delta(x −x0) m i2\pi¯h N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q \deltat′ j exp i\deltat′ j ¯h m 2 ˙x 2 j −g(¯xj) V (¯xj) + ¯h2(gg′′ −2g′2) 8mg2 ) , (20.366) where the increments \deltat′ j in the new variable are no longer uniform. This is still a source of inconvenience, so rather than choose the time slices to be uniformly spaced in t, we will choose them (again, in a path-dependent way) to be uniformly spaced in t′, so that Zpgp = Z dx0 Z dx \delta(x −x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 exp i\deltat′ ¯h m 2 ˙x 2 j −g(xj) V (xj) + ¯h2(gg′′ −2g′2) 8mg2 ) . (20.367) This has the form of a path integral in flat space, with a potential and effective potential modified by a factor of g(xj). Note that in the potentials, we have switched from the midpoint to the prepoint, since for a flat-space path integral (additive noise), we need not make such distinctions. Now switching from t′ to −i¯h\beta, we have Zpgp = Z dx0 Z dx \delta(x −x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 exp "
¯h m 2 (\partial ˜\betaxj)2 + g(xj) V (xj) −¯h2(gg′′ −2g′2) 8mg2 # ) , (20.368)
Eq. (20.146), we can write down the normalized form Zpgp = r m
Z dx0 ** exp " −1 ¯h Z ¯h\beta d˜\beta g(x) V (x) −¯h2(gg′′ −2g′2) 8mg2
p
. (normalized partition function) (20.369) Note that due to the time rescaling, the paths here are (scaled and translated) Brownian bridges, just as in a flat-space path integral. In reviewing this derivation, an obvious question arises: why choose Weyl ordering? We could, of course, have picked any other ordering. Choosing symmetrized or product orderings would lead to the same result, as we have seen. However, It¯o or anticipating orderings would have led to prepoint or postpoint path integrals, with different effective potentials. The time rescaling would proceed [but with, e.g., g(xj) instead of g(¯xj)], and once in flat space, the distinction between prepoint, midpoint, and postpoint is lost, and yet we are stuck with a different potential. The resolution to this apparent problem is to note that it is only in a Stratonovich-type path integral that the normalization turns out to be simple. The reason is that, in the language of SDEs, it is only in the Stratonovich case with no drift that a temporal rescaling preserves the closure of loops (Brownian bridges). It¯o SDEs, for example, do not in general preserve loop closure, even when equivalent to a driftless Stratonovich SDE (see Section 17.7.4.3). In terms of the path integral, this would mean that in any other ordering or expansion point, the temporal rescaling would upset the path closure, and thus there would be an extra normalization contribution due to the delta function that enforces the path closure. In any Stratonovich path integral, we don’t have to worry about computing any such contribution.
Chapter 20. Path Integration 20.4.4.1 A Similar Path Integral As an additional example of a normalized path integral, consider the alternate ‘‘Hamiltonian’’
(20.370) (variable-mass ‘‘Hamiltonian’’) Superficially, this appears to be the same as the Hamiltonian (20.362) for which we just developed the normalized path integral. However, it is not, due to the space-dependent mass. Applying the momentum operator from Eq. (20.226), we readily see that
2mg(x)p
(variable-mass ‘‘Hamiltonian’’) (20.371) which is somewhat odd, as this Hamiltonian is not even Hermitian. Then rearranging the kinetic term, g1/4(x) p g(x)p
g3/4 p
g7/4 p 4g1/4 = p1 g p + i¯hp g′ 4g2 + i¯h g′ 4g7/4 p g1/4 = p1 g p + i¯hp g′ 4g2 + i¯h g′ 4g2 p −¯h g′2 16g3 = p1 g p + i¯hp g′ 2g2 −¯h2(gg′′ −2g′2) 4g3 −¯h2g′2 16g3 . (20.372)
this Hamiltonian to the original model Hamiltonian (20.362), we have
g′ 4mg2 −¯h2(gg′′ −2g′2) 8mg3 −¯h2g′2 32mg3 . (20.373) Relative to the propagator (20.363) for H(x, p), the last two terms here become extra potential terms. The second term here leads to an extra velocity term. To handle this, we can revisit the development of the path integral in Section 20.4.3.1, where a similar potential term −i¯hpg′/2mg2 arose. From Eq. (20.320), we can see that this potential will contribute an extra potential ¯h2g′2/32mg3. From Eq. (20.321), we see that there is an additional potential −¯h2(gg′′ −g′2)/16mg3, as well as an overall factor [g(x)/g(x0)]1/4. Collecting these parts and adapting Eq. (20.363), we find the propagator
p g(x0) m
N/2 Z N−1 Y j=1 dxj ! \times ( N−1 Y j=0 q g(¯xj) exp i\deltat ¯h m 2 g(¯xj) ˙x 2 j −V (¯xj) + ¯h2(2gg′′ −3g′2) 8mg3 ) . (midpoint path integral) (20.374) We can then adapt the derivation of the partition function (20.369), with the result
r m
Z dx0 ** exp " −1 ¯h Z ¯h\beta d˜\beta g(x) V (x) −¯h2(2gg′′ −3g′2) 8mg2
p
. (normalized partition function) (20.375) Note that the effective potential is different than in the partition function (20.369). Note also that the path integral and effective potential are perfectly well-defined, despite the odd nature of the Hamiltonian (20.371).
20.5 Exercises 20.5 Exercises Problem 20.1 (a) Derive the propagator for the Hamiltonian (20.361) Hgpg := p g(x) p2 2m p g(x) + V (x). (20.376) (b) Convert the path integral from (a) to midpoint form. (c) Use the path integral from (b) to derive a product-form path integral for the Laplace–Beltrami Hamiltonian H△(x, p), and show that it is equivalent to the analogous path integral derived from Weyl and symmetrized orderings.