Path integral formulation of quantum mechanics

Quantum mechanics is an intrinsically probabilistic theory. Although the time evolution is deter-ministic, it is not possible to predict the outcome of an experiment. Even worse, the experiment has to be iterated many times to obtain meaningful information about the system under the same initial conditions. If an experimenter would be allowed to do an experiment only once, the result would be absolutely meaningless. This statistical nature will be made explicit in the functional integral formulation of quantum mechanics. In quantum mechanics the path integral defines expectation values of quantum operators. It encodes the basic principle of quantum mechanics, i.e. the superposition principle: we can view the path integral as a weighted summation over all possible paths of the system in phase space. Since the physical time is a true parameter in quantum mechanics (there is no time operatorˆt), we can understand this sum as the complete possible/accessible history of the physical system.² The derivation is quite simple and will be outlined with emphasis on the physical implications:

i

View the path integral as a limit of n-slits placed atN discrete points, which effectively describes free space in the limit n Ñ 8and N Ñ 8. Since we will construct the path integral in phase space (parametrized by the position q and the momentum p) we can apply thenÑ 8limit at the beginning.

ii

Take the unitary time evolution operator

U

pt¹, tq “ e^{´h¯iHpt1´tq}Θpt¹´tq, ˇˇΨpt¹qD

“

U

pt¹, tq |Ψptq y, (2.1) whereΘpt¹´tqdenotes the Heaviside distribution defined as

Θptq “

#1 tą0

0 tă0 (2.2)

The time evolution operator is applied N times with discrete time step ∆t “ ^pt1´tq{^N to describe the time evolution of the system from the fixed initial state|qiy att “0 to the final state|qfyatt¹“t. The transition/probability amplitude for this process is given by

B qf

ˇˇ ˇˇ

”e^´h¯iH∆tıNˇˇˇˇqi

F

“

ż

Nź´1

k“1

dq_kA qf

ˇˇ

ˇe^´¯hiH∆t ˇˇˇqN´1

E

ˆA q_N´1

ˇˇ ˇ ¨ ¨ ¨ˇˇˇq1

E A q1

ˇˇ

ˇe^´h¯iH∆t ˇˇ ˇq_iE

, (2.3) where we inserted the resolution of identity

ż

dq_k|q_ky xq_k| “1, (2.4)

at each intermediate time step in the q representation and omitted the limiting process N Ñ 8.

statement "point-like" should be taken mathematically,i.e. it is impossible to read off an absolutely exact value instantaneously.

2This idea goes back to Dirac and was later refined and truly appreciated by Feynman.

iii

Look at a single infinitesimal transition amplitude and note that the Hamiltonian generi-cally depends on both phase space variablesqandp, so we need to insert a complete set of

|pystates as well A

q_k`1ˇˇˇe^´h¯iH∆tˇˇˇq_kE

“ ż

dp_kA qk`1

ˇˇ ˇpk

E A pk

ˇˇ

ˇe^{´h¯iHpq,pq∆t}ˇˇˇq_kE

“ ż

dp_kA qk`1

ˇˇ ˇpk

E A pk

ˇˇ

ˇe^{´h¯iHpqk,pkq∆t`Op∆t2q} ˇˇˇqk

E

« ż

dp_kxq_k`1|p_ky xp_k|q_kye^{´h¯iHpqk,pkq∆t}, (2.5) where we used the Baker-Campell-Hausdorff formula to expand the product

e^{´h¯iHpq,pq∆t} “ e^{´h¯iTppq∆t}e^{´h¯iVpqq∆t}e¹²p^´¯hⁱq²r^Tppq,Vpqqs^∆t2 e^Op^∆t3q. (2.6) The factorization is exact only if“

T

p

p

q,

V

p

q

q‰

“ 0. Alternatively and more general, we can write all

p

operators to the left of all

q

operators in the Hamiltonian up to linear order in∆t. Interestingly, this implies that for very short timescales the kinetic/momentum con-tributions disentangle from the potential/position concon-tributions. Since we are considering infinitesimal time steps in the limitN Ñ 8we can discard allp∆t²qterms. Additionally, the momentum space representation is the Fourier transform of the position space repre-sentation

|qy “ ż

dp|py xp|qy “ ż dp

2π¯he^¯hiqp|py, xp|qy “ 1

2π¯he^h¯iqp, (2.7) so we finally get

A q_k`1

ˇˇ

ˇe^´h¯iH∆t ˇˇ ˇq_kE

« ż

dp_kxq_k`1|p_ky xp_k|q_kye^{´h¯iHpqk,pkq∆t}

“ ż dp_k

2π¯he^´h¯iqk`1pk e^¯hiqkpk e^{´h¯iHpqk,pkq}

“ ż dp_k

2π¯he^´

i h¯

”_qk`1´qk

∆t pk`Hpqk,pkqı

∆t. (2.8)

Insert (2.8) into (2.3) to obtain B

qf

ˇˇ ˇˇ

”e^´¯hiH∆tıNˇˇˇˇqi

F

«

ż

^˜Nź´1

k“1

dq_k

¸ ˜_N´1 ź

k“0

dp_k 2π¯h

¸

ˆexp

«

´i

¯ h∆t

Nÿ´1 k“0

q_k`1´q_k

∆t p_k`Hpq_k, p_kq ff

. (2.9) Note that we have an additional integration overp₀coming fromxq₁|. . .|q_i“q₀yas well as an additional summandk“0in the exponential.

iv

Finally, take the N Ñ 8 and ∆t Ñ 0 limit (keepingt “ ^N{^∆t fixed) which yields the continuous path integral. The measure will be denoted by

Drq, ps “ lim

NÑ8 N´ź1

k“1

dq_k

Nź´1 k“0

dp_k

2π¯h. (2.10)

Furthermore, the sum over the discrete set turns into an integral over the dense time intervalr0, tsand the difference quotient^∆q{^∆tconverges to the derivative

∆t

Nÿ´1 k“0

∆tÑ0

ÝÝÝÝÝÑ

NÑ8

żt 0

dt¹, q_k`1´q_k

∆t

∆tÑ0

ÝÝÝÝÝÑ

NÑ8

Bqpt¹q

Bt¹ ”q9ptq, (2.11) so the Hamiltonian form of the path integral is given by

A

q_fˇˇˇe^´¯hiHtˇˇˇq_iE

“

ż

qf“qptq qi“qp0q

Drq, psexp

„i

¯ h

ż0 t

dt¹´

ppt¹qq9pt¹q ´Hpqpt¹q, ppt¹qq¯

. (2.12)

The path integral can be viewed as a collection of infinitely many integrals summed up for each point in time and weighted by the classical action in Hamiltonian form for all paths/configura-tions starting inq_i and ending inq_f. Quantum mechanically it describes the propagation of the quantum system from the state|q_iyto the state|q_fywith respect to the Hamiltonian

H

There are mathematical issues which may impact physical calculations as well, so it is advisable to carry out the steps

iii

_and

iv

_{on page15}explicitly for the problem at hand. Even for cal-culating the free quantum propagator the path integral should be evaluated in discretized form in order to regularize divergences. For quantum mechanical calculations we will ignore (at least for all practical purposes) the following mathematical problems:

• The path integral measure is in general ill-defined for arbitrary path integrals. However, for Gaussian integrals³ we can define a Wiener measure which is mathematically sound (see Section2.1.3)

• The weight function is a complex function and so weighting is strictly speaking not possible because the complex numbers defy any ordering. So in principle we are not able to dis-tinguish important contributions from insignificant contribution. It is also not clear if the integral does converge at all. We can argue that physically the contributions with large clas-sical action will be highly fluctuating and should average to zero. This argument is based on the stationary phase approximation (see Infobox Stationary Phase Approximationon page18)

After introducing functional integrals we will see that there are remedies to the problem when considering stochastic functional integrals (see Section2.1.4and Table2.1). Apart from the fact that the path integral is (maybe the most) “natural” representation of the quantum superposition principle mentioned above, there are further advantages:

• The classical limit can be easily obtained. In fact we can view the path integral as being composed of the classical solution (obeying the Hamilton equations of classical mechanics) and the quantum corrections due to quantum fluctuations.

3Unfortunately, these are the only integrals we are able to solve, so we will use good approximation and/or transfor-mations to obtain a Gaussian integral from a more complex path integral.

• Non-perturbative effects such as the instanton solution to the quantum double well/tun-neling problem can be easily constructed.

• Path integrals can be easily extended to functional integrals describing physics ind`1 dimensional spacetime. In this case the time slicing in the construction of the path integral (see step

ii

_{on page} 14) naturally gives rise to a time ordered correlations function of operators.

Lagrangian formulation of the path integral

The path integral in Hamiltonian form (2.12) can be transformed into a Lagrangian version if the momentum dependence of the Hamiltonian is purely quadratic

Hpq, pq “ p²

2m `Vpqq, (2.13)

or can be brought into quadratic form by completing the square and shifting the integration variable such that we have a factorization of the position integral and a true Gaussian integral over the momentump

A q_f

ˇˇ ˇe^´h¯iHt

ˇˇ ˇq_iE

“ ż

qf“qptq qi“qp0q

Drqse^{´h¯iş0tdt1Vpqpt1qq} ż

Drpse

i

¯h

ş0 tdt¹

´

ppt¹qq9pt¹q´^ppt1q2m²

¯

. (2.14) The momentum integral evaluates to (usingpÝÑa

¯h{ⁱpand Gaussian integrals defined in Ap-pendixB.1)

Irps “ lim

NÑ8

ż

^˜_N´ź1

k“0

dp_k 2π?

i¯h

¸ exp

#_N´1 ÿ

k“0

«ci

¯

h∆qkp_k´1 2p_k

ˆ t N m

˙ p_k

+ff

“ lim

NÑ8

$’

&

’%

p2πq^N{2

´ 2π?

i¯h¯N det

„ˆ t N m

˙ 1_N

´¹{²

¨exp

«1 2

t N

Nÿ´1 k“0

˜ci

¯ h∆qk

¸ ˆ t N m

˙_´1˜c i

¯ h∆qk

¸ff+

“ lim

NÑ8

# 1

p2πi¯hq^N{2 ˆ t

N m

˙´^N{²

exp

«i

¯ h

Nÿ´1 k“0

1

2mp∆qkq² ff+

. (2.15)

Here we strictly evaluate the discrete time sliced integral and take the continuum limitN Ñ 8. This is the reason why the timetsneaked into the denominator of the prefactor, which could have been easily missed by just applying the Gaussian integration formula. Executing the continuum limit on both sides of (2.15) and inserting it into (2.14) yields the path integral formulation with the classical action in Lagrangian from

A qf

ˇˇ

ˇe^´¯hiHtˇˇˇqi

E“ ż

qf“qptq qi“qp0q

Drqse^{¯hiş0tdt112mq9pt1q2´Vpqpt1qq}

“ ż

qf“qptq qi“qp0q

Drqse^{¯hişdt1Lpqpt1q,q9pt1qq} “ ż

qf“qptq qi“qp0q

Drqse^i{¯hSrq,q9s. (2.16)

The path integral measureDrqshas been rescaled such that the prefactor in (2.16) is absorbed Drqs “ lim

NÑ8

ˆN m t2πi¯h

˙^N2 Nź´1 k“1

dq_k. (2.17)

For non-quadratic momentum integrals we need to employ the so-called stationary phase ap-proximation:

Stationary Phase / Saddle Point Approximation

For a contour integral over a complex analytic functionfpzqof the form I“

ż

C

dz hpzqe^tfpzq «

tÑ8hpz0qe^tfpz0q, where d dzfpzq

ˇˇ ˇˇ

z0

“0,

in the limittÑ 8the value ofIis best approximated at points wherefpzqis extremized or equivalently, the first derivative offpzqvanishes. The contourCmust be chosen such that

• the real part offpzqis maximized,

• the imaginary part is stationaryÝÑsmall fluctuations,

in order to obtain the correct approximation/asymptotic behavior ofI.

This follows from the analyticity condition of complex functions, where both the real and imagi-nary part have to solve the Laplace equations. Thus, neither the imagiimagi-nary part nor the real part can have an absolute extremum on the complex plane (except at the origin). In fact, Laplace equations for the real and imaginary part yields a saddle point such that the maximum along a certain contour is a minimum along another. The stationary curves forImfpzq “ const. are tangent to the gradient ofRefpzq, so the optimal contour follows the curve where the absolute value of the function is maximally decreasing when moving through the saddle point. Therefore, this approximation is sometimes called the “steepest descent method”.

Applying the stationary phase approximation to the Hamiltonian version of the path integral (2.12) we see that the Hamiltonian equations of motion

B Bpptq

´

pptqq9ptq ´Hpqptq, pptqq¯

“q9ptq ´BHpqptq, pptqq Bpptq

“! 0, (2.18)

satisfies the stationarity condition. Solving the Hamilton equations forq9ptqand inserting this into the integrand, removes thepptqdependence since it corresponds to the Legendre transfor-mation from the Hamiltonian formulations to the Lagrangian formulation. Thus, the momentum integration overpis dropped and the remaining factors can be absorbed into the definition of Drqsfollowing (2.17). In general this can always be done since the Hamiltonian equation (2.18) is local inq9 andpand so it must hold for arbitraryk or at each time step, respectively. In the case of quadratic integrals (2.13) the stationary phase approximation becomes exact (up to the previous calculated normalization factor of the Gaussian integral).

Semi-classical approximation

The stationary phase approximation can be used to define the validity of the semi-classical limit and allows the “splitting” into the classical path and quantum fluctuations about the classical

solution which arise as corrections in ¹{^¯h. The solution to the saddle point equation are the classical equations of motion (2.18) that minimizes the classical action. Thus, we can expand about the saddle point solution up to quadratic order in the fluctuations

Srqs «Srqcls `1 2 ż

dtdt¹pq´qclqptq δ²Srqs δqptqδqpt¹q

ˇˇ ˇˇ

q“qcl

pq´qclqpt¹q. (2.19) Note that Srqsis strictly a functional and so we need to apply second order functional deriva-tives. The first order functional derivative ofSrqsvanishes when evaluated on solutions of the equations of motion q_cl. Note also that the second order functional derivative is positive defi-nite since the classical solution minimizes the action and hence the convergence of the Gaussian integral is ensured. The stationary path integral reads then

ż

qf“qptq qi“qp0q

Drqse^i{¯hSrqs “ÿ

ℓ

e^{i{¯hSrqpℓqcl s} det

˜ i 2π¯h

δ²Srq^pℓqs δq^pℓq_f δq_i^pℓq

¸´¹{²

, (2.20)

where the sum over ℓ runs over the contribution from different saddle point solutions. For

¯

h Ñ0 this is the dominant/leading term in the asymptotic expansion. To make the argument mathematically more rigorous one should start from the analytic continuation and expand to higher orders in the fluctuations. Then, the method of steepest descend ensures the convergence of the Gaussian integrals and enables us to choose the constant phase such that the derivatives of the function are real. For an even complex functionfpzqwe find

I“ ż

C

dz e^tfpzq « d 2π

tf²pzqexp

„

tfpz0q `3f^pivqpz0q tf²pz0q



, (2.21)

where we see that higher order terms are suppressed with higher powers of t. This power counting carries over to non-Gaussian integrals arising from higher order fluctuation terms in the actionSrqs.⁴ In general we would find contributions to the path integral (2.20) scaling with

ż

qf“qptq qi“qp0q

Drqse^i{¯hSrqs „exp

«

¯ h^´1

˜

Srq_cls ` ÿ

ną2 neven

¯

h^n{2 δⁿSrqs δqⁿ

Nδ^n/2Srqs δq^n/2

ˇˇ ˇˇ

q“qcl

¸ff

, (2.22)

wherendenotes the (even!) expansion order in the fluctuation. Again for¯h Ñ0the classical action is the leading term in the asymptotic expansion. Since this is an asymptotic expansion, we can only truncate the series for systems where quantum fluctuations are small. With this final remark we conclude our discussion on path integrals. For a more in-depth discussion and detailed insight see the excellent textbooks [44,45].

Im Dokument Gauge/Gravity duality (Seite 34-39)