16 Path Integrals 16.1 Path Integrals and Classical Physics Since Richard Feynman invented them over 60 years ago, path integrals have been used with increasing frequency in high-energy and condensed-matter physics, in finance, and in biophysics (Kleinert, 2009). Feynman used them to express matrix elements of the time-evolution operator exp( itH/~)in − terms of the classical action. Others have used them to compute matrix elements of the Boltzmann operator exp( H/kT ) which in the limit of zero − temperature projects out the ground state E of the system | 0i 1 (H E0)/kT (En E0)/kT lim e− − =lim En e− − En = E0 E0 (16.1) T 0 T 0 | i h | | ih | ! ! n=0 X a trick used in lattice gauge theory. Path integrals magically express the quantum-mechanical probability am- plitude for a process as a sum of exponentials exp(iS/~) of the classical action S of the various ways that process might occur. 16.2 Gaussian Integrals The path integrals we can do are gaussian integrals of infinite order. So we begin by recalling the basic integral formula (5.167) b 2 ⇡ 1 exp ia x dx = (16.2) − − 2a ia Z1 " ✓ ◆ # r 16.2 Gaussian Integrals 671 which holds for real a and b, and also the one (5.168) c 2 ⇡ 1 exp r x dx = (16.3) − − 2r r Z1 ⇣ ⌘ r which is true for positive r and complex c. Equivalent formulas for real a and b, positive r, and complex c are ⇡ b2 1 exp iax2 + ibx dx = exp i (16.4) − ia 4a Z1 r ✓ ◆ ⇡ c2 1 exp rx2 + cx dx = exp . (16.5) − r 4r Z r ✓ ◆ 1 This last formula will be useful with x = p, r = ✏/(2m), and c = i✏q˙ p2 2⇡m 1 1 exp ✏ + i✏ qp˙ dp = exp ✏ mq˙2 (16.6) − 2m ✏ − 2 Z1 ✓ ◆ r ✓ ◆ as will (16.4)withx = p, a = ✏/(2m), and b = ✏q˙ p2 2⇡m 1 1 exp i✏ + i✏ qp˙ dp = exp i✏ mq˙2 . (16.7) − 2m i✏ 2 Z1 ✓ ◆ r ✓ ◆ Doable path integrals are multiple gaussian integrals. One may show (ex- ercise 16.1) that for positive r1,...,rN and complex c1,...,cN , the integral (16.5) leads to N N 2 1 2 ⇡ 1 ci exp rixi + cixi dxi = exp . (16.8) − ! ri ! 4 ri ! Z1 Xi Yi=1 Yi=1 r Xi If R is the N N diagonal matrix with positive entries r ,r ,...,r , and ⇥ { 1 2 N } X and C are N-vectors with real x and complex c entries, then this { i} { i} formula (16.8) in matrix notation is N N 1 T T ⇡ 1 T 1 exp X RX + C X dx = exp C R− C . (16.9) − i det(R) 4 Z i=1 s ✓ ◆ 1 Y Now every positive symmetric matrix S is of the form S = OROT for some positive diagonal matrix R.SoinsertingR = OTSO into the previous equa- tion (16.9) and using the invariance of determinants under orthogonal trans- formations, we find N N 1 T T T ⇡ 1 T T 1 exp X O SOX + C X dxi = exp C O S− OC . − sdet(S) 4 Z1 i=1 Y (16.10) 672 Path Integrals The jacobian of the orthogonal transformations Y = OX and D = OC is unity, and so N N 1 T T ⇡ 1 T 1 exp Y SY + D Y dy = exp D S− D (16.11) − i det(S) 4 Z i=1 s ✓ ◆ 1 Y in which S is a positive symmetric matrix, and D is a complex vector. The other basic gaussian integral (16.4) leads for real S and D to (exer- cise 16.2) N N 1 T T ⇡ i T 1 exp iY SY + iD Y dyi = exp D S− D . − sdet(iS) 4 Z1 i=1 ✓ ◆ Y (16.12) The vector Y that makes the argument iY TSY + iDTY of the expo- − nential of this multiple gaussian integral (16.12) stationary is (exercise 16.3) 1 1 Y = S− D. (16.13) 2 The exponential of that integral evaluated at its stationary point Y is T T i T 1 exp iY SY + iD Y =exp D S− D . (16.14) − 4 ✓ ◆ Thus, the multiple gaussian integral (16.12) is equal to its exponential eval- uated at its stationary point Y , apart from a prefactor involving the deter- minant det iS. Similarly, the vector Y that makes the argument Y TSY + DTY of − the exponential of the multiple gaussian integral (16.11) stationary is Y = 1 S− D/2, and that exponential evaluated at Y is T T 1 T 1 exp Y SY + D Y =exp D S− D . (16.15) − 4 ✓ ◆ Once again, a multiple gaussian integral is simply its exponential evaluated at its stationary point Y , apart from a prefactor involving a determinant, det S. 16.3 Path Integrals in Imaginary Time At the imaginary time t = iβ~, the time-evolution operator exp( itH/~) − − is exp( βH) in which the inverse temperature β =1/kT is the reciprocal of − Boltzmann’s constant k =8.617 10 5 eV/K times the absolute temperature ⇥ − T . In the low-temperature limit, exp( βH) is a projection operator (16.1) − 16.3 Path Integrals in Imaginary Time 673 on the ground state of the system. These path integrals in imaginary time are called euclidian path integrals. Let us consider a quantum-mechanical system with hamiltonian p2 H = + V (q) (16.16) 2m in which the commutator of the position q and momentum p operators is [q, p]=i in units in which ~ = 1. For tiny ✏, the corrections to the approxi- mation p2 p2 exp ✏ + V (q) exp ✏ exp ( ✏V(q)) + (✏2) (16.17) − 2m ⇡ − 2m − O ✓ ◆ ✓ ◆ are of second order in ✏. To evaluate the matrix element q exp( ✏H) q ,weinserttheidentity h 1| − | 0i operator I in the form of an integral over the momentum eigenstates 1 I = p0 p0 dp0 (16.18) | ih | Z1 and use the inner product q p =exp(iq p )/p2⇡ so as to get as ✏ 0 h 1| 0i 1 0 ! 2 1 p q exp( ✏H) q = q exp ✏ p0 p0 exp ( ✏V(q)) q dp0 h 1| − | 0i h 1| − 2m | ih | − | 0i Z1 ✓ ◆ 2 ✏V(q0) 1 p0 dp0 = e− exp ✏ + ip0 (q q ) . (16.19) − 2m 1 − 0 2⇡ Z1 If we adopt the suggestive notation q q 1 − 0 =˙q (16.20) ✏ 0 and use the integral formula (16.6), then we find 2 1 ✏V(q0) 1 p0 q exp( ✏H) q = e− exp ✏ + i✏p0 q˙ dp0 h 1| − | 0i 2⇡ − 2m 0 Z1 ✓ ◆ m 1/2 = exp ✏ 1 m q˙2 + V (q ) (16.21) 2⇡✏ − 2 0 0 ⇣ ⌘ ⇥ ⇤ in which q1 enters through the notation (16.20). The next step is to link two of these matrix elements together 2✏H 1 ✏H ✏H q e− q = q e− q q e− q dq (16.22) h 2| | 0i h 2| | 1ih 1| | 0i 1 Z1 m = 1exp ✏ 1 m q˙2 + V (q )+ 1 m q˙2 + V (q ) dq . 2⇡✏ − 2 1 1 2 0 0 1 Z 1 ⇥ ⇤ 674 Path Integrals Linking three of these matrix elements together and using subscripts instead of primes, we have 3✏H 1 ✏H ✏H ✏H q e− q = q e− q q e− q q e− q dq dq (16.23) h 3| | 0i h 3| | 2ih 2| | 1ih 1| | 0i 1 2 ZZ1 2 m 3/2 = 1exp ✏ 1 m q˙2 + V (q ) dq dq . 2⇡✏ 8 − 2 j j 9 2 1 ZZ j=0 ⇣ ⌘ 1 < X ⇥ ⇤= Boldly passing from 3 to n and suppressing: some integral signs,; we get n✏H 1 ✏H ✏H qn e− q0 = qn e− qn 1 ... q1 e− q0 dqn 1 ...dq1 (16.24) h | | i h | | − i h | | i − ZZZ 1 n 1 n/2 m 1 − 1 2 = exp ✏ m q˙j + V (qj) dqn 1 ...dq1. 2⇡✏ 8 − 2 9 − ZZZ j=0 ⇣ ⌘ 1 < X ⇥ ⇤= Writing dβ for ✏ and taking the: limits ✏ 0 and n β/;✏ ,wefind ! ⌘ !1 that the matrix element q e βH q is a path integral of the exponential h β| − | 0i of the average energy multiplied by β − β βH 1 2 q e− q = exp mq˙ (β0)+V (q(β0)) dβ0 Dq (16.25) h β| | 0i − 2 Z Z0 2 n/2 in which Dq (n m/2⇡~ ) dqn 1 ...dq2 dq1 andq ˙ is the β-derivative of ⌘ − the coordinate q(β). We sum over all paths q(t) that go from q(0) = q0 at inverse temperature β =0toq(β)=qβ at inverse temperature β. In the limit β , the operator exp( βH) becomes proportional to a !1 − projection operator (16.1) on the ground state of the theory. In three dimensions with q˙(β)=dq(β)/dβ, and ~ = 1, the analog of 6 equation (16.25) is (exercise 16.28) β βH m 2 q e− q = exp q˙ (β0)+V (q(β0)) dβ0 Dq (16.26) h β| | 0i − 2 2 Z Z0 ~ 2 3n/2 where Dq (n m/2⇡~ ) dqn 1 ...dq2 dq1. ⌘ − Path integrals in imaginary time are called euclidian mainly to distinguish them from Minkowski path integrals, which represent matrix elements of the time-evolution operator exp( itH) in real time. − 16.4 Path Integrals in Real Time 675 16.4 Path Integrals in Real Time Path integrals in real time represent the time-evolution operator exp( itH). − Using the integral formula (16.7), we find in the limit ✏ 0 ! 2 i✏H 1 p q e− q = q exp i✏ p0 p0 exp [ i✏V(q)] q dp0 h 1| | 0i h 1| − 2m | ih | − | 0i Z1 2 1 i✏V(q0) 1 p0 = e− exp i✏ + ip0 (q q ) dp0 2⇡ − 2m 1 − 0 Z1 2 1 i✏V(q0) 1 p0 = e− exp i✏ + i✏p0q˙ dp0 2⇡ − 2m 0 Z1 m 1/2 m q˙2 = exp i✏ 0 V (q ) .