Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006, страница 11

, . . .} called Poisson brackets:{A, B} ≡∂A ∂B ∂B ∂A−,∂pi ∂qi∂pi ∂qi(1.21)again with the Einstein summation convention for the repeated index i. The Poissonbrackets have the obvious properties{A, B} = − {B, A}{A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0antisymmetry,(1.22)Jacobi identity.(1.23)If two quantities have vanishing Poisson brackets, they are said to commute.The original Hamilton equations are a special case of (1.20):∂p ∂H∂Hd∂H ∂pi− i=−,pi = {H, pi} =dt∂pj ∂qj∂pj ∂qj∂qid∂H ∂qi∂q ∂H∂Hqi = {H, qi } =− i=.dt∂pj ∂qj∂pj ∂qj∂pi(1.24)By definition, the phase space variables pi , qi satisfy the Poisson bracketsnpi , qjnpi , pjnqi , qjooo= δij ,= 0,(1.25)= 0.A function O(pi, qi ) which has no explicit dependence on time and which, moreover, commutes with H (i.e., {O, H} = 0), is a constant of motion along the classicalpath, due to (1.20).

In particular, H itself is often time-independent, i.e., of the formH = H(pi , qi ).(1.26)H. Kleinert, PATH INTEGRALS51.1 Classical MechanicsThen, since H commutes with itself, the energy is a constant of motion.The Lagrangian formalism has the virtue of being independent of the particularchoice of the coordinates qi . Let Qi be any other set of coordinates describing thesystem which is connected with qi by what is called a local 2 or point transformationqi = fi (Qj , t).(1.27)Certainly, to be of use, this relation must be invertible, at least in some neighborhoodof the classical path,Qi = f −1 i (qj , t).(1.28)Otherwise Qi and qi could not both parametrize the same system.

Therefore, fimust have a nonvanishing Jacobi determinant:∂fi∂Qjdet!6= 0.(1.29)In terms of Qi , the initial Lagrangian takes the formL0 Qj , Q̇j , t ≡ L fi Qj , t , f˙i Qj , t , tand the action readsA =Z=Ztbtatbta(1.30)dt L0 Qj (t), Q̇j (t), t(1.31)dt L fi Qj (t), t , f˙i Qj (t), t , t .By varying the upper expression with respect to δQj (t), δ Q̇j (t) while keepingδQj (ta ) = δQj (tb ) = 0, we find the equations of motiond ∂L0∂L0−= 0.dt ∂ Q̇j∂Qj(1.32)The variation of the lower expression, on the other hand, givesδA =Ztb=Ztbtatadtdt∂L ˙∂Lδfi +δf∂qi∂ q̇i i!!∂Ld ∂L∂L tb−δfi +δf .∂qi dt ∂ q̇i∂ q̇i i ta(1.33)If δqi is arbitrary, then so is δfi . Moreover, with δqi (ta ) = δqi (tb ) = 0, also δfivanishes at the endpoints.

Hence the extremum of the action is determined equallywell by the Euler-Lagrange equations for Qj (t) [as it was by those for qi (t)].2The word local means here at a specific time. This terminology is of common use in fieldtheory where local means, more generally, at a specific spacetime point .61 FundamentalsNote that the locality property is quite restrictive for the transformation of thegeneralized velocities q̇i (t). They will necessarily be linear in Q̇j :∂fi∂fq̇i = f˙i (Qj , t) =Q̇j + i .∂Qj∂t(1.34)In phase space, there exists also the possibility of performing local changes ofthe canonical coordinates pi , qi to new ones Pj , Qj .

Let them be related bypi = pi (Pj , Qj , t),(1.35)qi = qi (Pj , Qj , t),with the inverse relationsPj = Pj (pi , qi , t),(1.36)Qj = Qj (pi , qi , t).However, while the Euler-Lagrange equations maintain their form under any localchange of coordinates, the Hamilton equations do not hold, in general, for any transformed coordinates Pj (t), Qj (t). The local transformations pi (t), qi (t) → Pj (t), Qj (t)for which they hold, are referred to as canonical .

They are characterized by the forminvariance of the action, up to an arbitrary surface term,tbZtaZdt [pi q̇i − H(pi , qi , t)] =tbtahdt Pj Q̇j − H 0 (Pj , Qj , t)tb+ F (Pj , Qj , t)tai(1.37),where H 0 (Pj , Qj , t) is some new Hamiltonian. Its relation with H(pi , qi , t) must bechosen in such a way that the equality of the action holds for any path pi (t), qi (t)connecting the same endpoints (at least any in some neighborhood of the classicalorbits). If such an invariance exists then a variation of this action yields for Pj (t)and Qj (t) the Hamilton equations of motion governed by H 0 :Ṗi = −Q̇i∂H 0,∂Qi(1.38)∂H 0.=∂PiThe invariance (1.37) can be expressed differently by rewriting the integral on theleft-hand side in terms of the new variables Pj (t), Qj (t),Ztbtadt(pi∂qi∂qi∂qṖj +Q̇j + i∂Pj∂Qj∂t!)− H(pi (Pj , Qj , t), qi (Pj , Qj , t), t) ,(1.39)and subtracting it from the right-hand side, leading toZtbta(∂qP j − pi i∂Qj!dQj − pi∂qidP∂P!j )j∂q− H + pi i − H dt∂t0=tb−F (Pj , Qj , t) .(1.40)taH.

Kleinert, PATH INTEGRALS71.1 Classical MechanicsThe integral is now a line integral along a curve in the (2N + 1)-dimensional space,consisting of the 2N-dimensional phase space variables pi , qi and of the time t.The right-hand side depends only on the endpoints. Thus we conclude that theintegrand on the left-hand side must be a total differential. As such it has to satisfythe standard Schwarz integrability conditions [2], according to which all secondderivatives have to be independent of the sequence of differentiation. Explicitly,these conditions are∂pi ∂qi∂q ∂pi− i= δkl ,∂Pk ∂Ql ∂Pk ∂Ql∂q ∂pi∂pi ∂qi− i= 0,∂Pk ∂Pl ∂Pk ∂Pl(1.41)∂pi ∂qi∂qi ∂pi−= 0,∂Qk ∂Ql ∂Qk ∂Qland∂pi ∂qi∂q ∂p− i i∂t ∂Pl∂t ∂Pl=∂(H 0 − H),∂Pl(1.42)∂qi ∂pi∂(H 0 − H)∂pi ∂qi−=.∂t ∂Ql∂t ∂Ql∂QlThe first three equations define the so-called Lagrange brackets in terms of whichthey are written as(Pk , Ql ) = δkl ,(Pk , Pl ) = 0,(Qk , Ql ) = 0.(1.43)Time-dependent coordinate transformations satisfying these equations are calledsymplectic.

After a little algebra involving the matrix of derivativesits inverseJ =J −1 = ∂Pi /∂pj∂Pi /∂qj∂Qi /∂pj∂Qi /∂qj∂pi /∂Pj∂pi /∂Qj∂qi /∂Pj∂qi /∂Qjand the symplectic unit matrixE=0−δijδij0!,(1.44)(1.45),,(1.46)we find that the Lagrange brackets (1.43) are equivalent to the Poisson brackets{Pk , Ql } = δkl ,{Pk , Pl } = 0,{Qk , Ql } = 0.(1.47)81 FundamentalsThis follows from the fact that the 2N × 2N matrix formed from the Lagrangebrackets−(Qi , Pj )−(Qi , Qj )L≡(1.48)(Pi , Pj )(Pi , Qj )can be written as (E −1 J −1 E)T J −1 , while an analogous matrix formed from thePoisson bracketsono  nP,Q−P,Pijij(1.49)P ≡ nono Qi , Qj− Qi , Pjis equal to J(E −1 JE)T . Hence L = P −1 , so that (1.43) and (1.47) are equivalent toeach other.

Note that the Lagrange brackets (1.43) [and thus the Poisson brackets(1.47)] ensure pi q̇i − Pj Q̇j to be a total differential of some function of Pj and Qj inthe 2N-dimensional phase space:pi q̇i − Pj Q̇j =dG(Pj , Qj , t).dt(1.50)The Poisson brackets (1.47) for Pi , Qi have the same form as those in Eqs.

(1.25)for the original phase space variables pi , qi .The other two equations (1.42) relate the new Hamiltonian to the old one. Theycan always be used to construct H 0 (Pj , Qj , t) from H(pi , qi , t). The Lagrange brackets (1.43) or Poisson brackets (1.47) are therefore both necessary and sufficient forthe transformation pi , qi → Pj , Qj to be canonical.A canonical transformation preserves the volume in phase space.

This followsfrom the fact that the matrix product J(E −1 JE)T is equal to the 2N × 2N unitmatrix (1.49). Hence det (J) = ±1 andYZ[dpi dqi ] =iYZ hidPj dQj .j(1.51)It is obvious that the process of canonical transformations is reflexive. It maybe viewed just as well from the opposite side, with the roles of pi , qi and Pj , Qjexchanged [we could just as well have considered the integrand (1.40) as a completedifferential in Pj , Qj , t space].Once a system is described in terms of new canonical coordinates Pj , Qj , weintroduce the new Poisson brackets{A, B}0 ≡∂A ∂B∂B ∂A−,∂Pj ∂Qj∂Pj ∂Qj(1.52)and the equation of motion for an arbitrary observable quantity O Pj (t), Qj (t), tbecomes with (1.38)dO n 0 o0 ∂O= H ,O +,(1.53)dt∂tH.

Kleinert, PATH INTEGRALS91.1 Classical Mechanicsby complete analogy with (1.20). The new Poisson brackets automatically guaranteethe canonical commutation rulesnPi , QjnPi , Pjno0= δij ,o0= 0.o0Qi , Qj(1.54)= 0,A standard class of canonical transformations can be constructed by introducinga generating function F satisfying a relation of the type (1.37), but dependingexplicitly on half an old and half a new set of canonical coordinates, for instanceF = F (qi , Qj , t).(1.55)One now considers the equationZtbtadt [pi q̇i − H(pi , qi , t)] =replaces Pj Q̇j by −Ṗj Qj +Ztbta"#ddt Pj Q̇j − H (Pj , Qj , t) + F (qi , Qj , t) , (1.56)dtdPQ,dt j j0definesF (qi , Pj , t) ≡ F (qi , Qj , t) + Pj Qj ,and works out the derivatives.

This yieldsZtbta=nodt pi q̇i + Ṗj Qj − [H(pi , qi , t) − H 0 (Pj , Qj , t)]Ztbta()∂F∂F∂F(qi , Pj , t)q̇i +(qi , Pj , t)Ṗj +(q , P , t) .dt∂qi∂Pj∂t i j(1.57)A comparison between the two sides renders for the canonical transformation theequations∂F (qi , Pj , t),pi =∂qi(1.58)∂Qj =F (qi , Pj , t).∂PjThe second equation shows that the above relation between F (qi , Pj , t) andF (qi , Qj , t) amounts to a Legendre transformation.The new Hamiltonian isH 0 (Pj , Qj , t) = H(pi, qi , t) +∂F (qi , Pj , t).∂t(1.59)Instead of (1.55) we could, of course, also have chosen functions with other mixturesof arguments such as F (qi , Pj , t), F (pi , Qj , t), F (pi, Pj , t) to generate simple canonicaltransformations.101 FundamentalsA particularly important canonical transformation arises by choosing a generating function F (qi , Pj ) in such a way that it leads to time-independent momentaPj ≡ αj .

Coordinates Qj with this property are called cyclic. To find cyclic coordinates we must search for a generating function F (qj , Pj , t) which makes thetransformed H 0 in (1.59) vanish identically. Then all derivatives with respect to thecoordinates vanish and the new momenta Pj are trivially constant. Thus we seekfor a solution of the equation∂F (qi , Pj , t) = −H(pi , qi , t),∂t(1.60)where the momentum variables in the Hamiltonian obey the first equation of (1.58).This leads to the following partial differential equation for F (qi , Pj , t):∂t F (qi , Pj , t) = −H(∂qi F (qi , Pj , t), qi , t),(1.61)called the Hamilton-Jacobi equation.A generating function which achieves this goal is supplied by the action functional(1.14).