Path integral formulation of quantum mechanics (and also other equivalent
formulations) depends on a Lagrangian and/or Hamiltonian function that is
chosen to describe the underlying classical system. The arbitrariness presented
in this choice leads to a phenomenon called Quantization ambiguity. For example
both $L_1=\dot{q}^2$ and $L_2=e^\dot{q}$ are suitable Lagrangians on a
classical level ($\delta L_1=0=\delta L_2$), but quantum mechanically they are
diverse. This paper presents a simple rearrangement of the path integral to a
surface functional integral. It is shown that the surface functional integral
formulation gives transition probability amplitude which is free of any
Lagrangian/Hamiltonian and requires just the underlying classical equations of
motion. A simple example examining the functionality of the proposed method is
considered.Comment: 4 pages, published version, references added, comments are welcom

Kochan, Denis

English

arXiv

Path integral formulation of quantum mechanics (and also other equivalent formulations) depends on a Lagrangian and/or Hamiltonian function that is chosen to describe the underlying classical system. The arbitrariness presented in this choice leads to a phenomenon called Quantization ambiguity. For example both L1 = ˙q2 and L2 = eq˙ are suitable Lagrangians on a classical level (δL1 = δL2), but quantum mechanically they are diverse. This paper presents a simple rearrangement of the path integral to a surface functional integral. It is shown that the surface functional integral formulation gives transition probability amplitude which is free of any Lagrangian/Hamiltonian and requires just the underlying classical equations of motion. A simple example examining the functionality of the proposed method is considered

Kochan, D.

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Acta Polytechnica Vol. 50 No. 5/2010Does a Functional Integral Really Need a Lagrangian?D. KochanAbstractPath integral formulation of quantum mechanics (and also other equivalent formulations) depends on a Lagrangianand/or Hamiltonian function that is chosen to describe the underlying classical system. The arbitrariness presented inthis choice leads to a phenomenon called Quantization ambiguity. For example both L1 = q˙2 and L2 = eq˙ are suitableLagrangians on a classical level (δL1 = δL2), but quantum mechanically they are diverse.This paper presents a simple rearrangement of the path integral to a surface functional integral. It is shown that thesurface functional integral formulation gives transition probability amplitude which is free of any Lagrangian/Hamiltonianand requires just the underlying classical equations of motion. A simple example examining the functionality of theproposed method is considered.Dedicated to my friend and colleague Pavel Bo´na.Keywords: quantization of (non-)Lagrangian systems, path vs. surface functional integral.1 A standard path integralloreAccording to Feynman [1], the probability amplitudeof the transition of a system from the space-time con-ﬁguration (q0, t0) to (q1, t1) is given as follows:A(q1, t1 | q0, t0) ∝∫[Dγ˜] exp{ ih¯∫γ˜padqa −Hdt}. (1)Here the path-summation is taken over all trajecto-ries γ˜(t) = (q˜(t), p˜(t), t) in the extended phase spacewhich are constrained as follows:γ˜(t0) =(q˜(t0) = q0, p˜(t0) – arbitrary, t0),γ˜(t1) =(q˜(t1) = q1, p˜(t1) – arbitrary, t1).To obtain a proper normalization of the Feynmanpropagator, one requires:δ(q˜0 − q0) =∫Rn[q1]dq1A∗(q1, t1 | q˜0, t0)A(q1, t1 | q0, t0) ,δ(q1 − q0) = limt1→t0A(q1, t1 | q0, t0) .The ﬁrst equation asks for the conservation of thetotal probability and the second expresses the obvi-ous fact that no evolution takes place whenever t1approaches t0.It is a miraculous consequence (not a require-ment!) of the propagator deﬁnition (1) that it satis-ﬁes an evolutionary chain rule (Chapman-Kolmogo-rov equation)A(q1, t1 | q0, t0) =∫Rn[q]dq A(q1, t1 | q, t)A(q, t | q0, t0) ,whose inﬁnitesimal version is the celebrated Schro¨-dinger equation.It is a curious fact that Formula (1) was not originallydiscovered by Feynman. In his pioneering paper [2]he arrives at a functional integral in the conﬁgurationspace onlyA(q1, t1 | q0, t0) ∝∫[Dq] exp{ ih¯∫q(t)L(q, q˙, t)dt}. (2)Later, however, it was shown that this formula repre-sents a very special case of the most general prescrip-tion (1). Formula (1) is at the heart of our furtherdiscussion.2 DeHamiltonianizationA step beyond involves eliminating the Hamiltonianfunction H from Formula (1). The price to be paidfor this will be to replace the path summation thereinby a surface functional integration.Our aim is the transition probability amplitude be-tween (q0, t0) and (q1, t1). Let us suppose that thereexists a unique classical trajectory in the extendedphase space γcl(t) = (qcl(t), pcl(t), t) which connectsthese points (locally this assumption is always satis-ﬁed). Then for any curve γ˜(t) = (q˜(t), p˜(t), t) which57Acta Polytechnica Vol. 50 No. 5/2010enters the path integration in (1), we can assign twoauxiliary curves which we call λ0(s) and λ1(s). Theyare parameterized by s ∈ [0, 1] and speciﬁed as fol-lows:λ0(s) = (q0, π0(s), t0)whereπ0(0) = pcl(t0), π0(1) = p˜(t0),λ1(s) = (q1, π1(s), t1)whereπ1(0) = pcl(t1), π1(1) = p˜(t1).Let us emphasize that neither λ0(s) nor λ1(s) varieswith respect to the q and t coordinates in the ex-tended phase space. They are allowed to evolve onlywith respect to the momentum variables. There are,of course, inﬁnitely many of such curves, but as wewill see nothing in the theory will be dependent on aparticular choice of λ0(s) and λ1(s).Using these curves one can write:∫γ˜padqa −Hdt =∫γclpadqa −Hdt+∮∂Σpadqa −Hdt , (3)where ∂Σ = γ˜ − λ1 − γcl + λ0 is a contour spannedby four curves γ˜(t), γcl(t), λ0(s), λ1(s) counting theirorientations.The ﬁrst integral on the right is the classical actionScl(q1, t1 | q0, t0). While the contour integral in (3)can be rearranged to represent a surface integral:∮∂Σpadqa −Hdt =∫Σdpa ∧(dqa − ∂H∂padt)− ∂H∂qadqa ∧ dt =:∫ΣΩ . (4)Surface Σ spanning the contour ∂Σ is understoodhere as a map from the parametric space (t, s) ∈[t0, t1]× [0, 1] to the extended phase space, i.e.Σ : (t, s) → (qa(t, s), pa(t, s), t(t, s) = t) .Partial derivatives of the initial Hamiltonian functioncan be substituted using the velocity-momentum re-lations and classical equations of motion:∂H∂pa= T abpb(= q˙a)and∂H∂qa= −Fa(= −p˙a).Here we consider the physically mostly relevant sit-uation only. In this case the velocities and mo-menta become related linearly by the metric tensorTab(q) (and its inverse) deﬁned by the kinetic energyT =12Tabq˙aq˙b =12T abpapb of the system, thenΩ = dpa ∧ dqa −(T abpadpb − Fadqa)∧ dt . (5)This object represents a canonical two-form in theextended phase space. It is a straightforward gen-eralization of the standard closed two-form dθ =dp ∧ dq − dH ∧ dt to the case when the forces arenot potential-generated.It is clear that for a given pair of trajectories(γ˜, γcl) there exists inﬁnitely many Σ surfaces. Theyform a set which we call Uγ˜ . Since the surface inte-gral∫ΣΩ is only boundary dependent and Formulas(3) and (4) are satisﬁed, we can write:exp{ ih¯∫γ˜padqa −Hdt}=eih¯Scl∞γ˜∫Uγ˜[DΣ] exp{ ih¯∫ΣΩ}.Here ∞γ˜ stands for the number of elements pertain-ing to the corresponding stringy set Uγ˜ . Assumingno topological obstructions from the side of the ex-tended phase space,∞γ˜ becomes an inﬁnite constantindependent of γ˜. Taking all of this into account wecan rewrite (1) as follows:A(q1, t1 | q0, t0) ∝ e ih¯Scl∫U[DΣ] exp{ ih¯∫ΣΩ}. (6)In this formula the undetermined normalization con-stant ∞ was included into the integration measure[DΣ] and the path integral over γ˜’s was converted tothe surface functional integral as was promised:∫[Dγ˜]∫Uγ˜[DΣ] . . . =∫⋃γ˜Uγ˜[DΣ] . . . =:∫U[DΣ] . . .The set U =⋃γ˜Uγ˜ over which the functional integra-tion is carried out contains all extended phase spacestrings which are anchored to the given classical tra-jectory γcl.To eliminate Hamiltonian H completely we needto express Scl(q1, t1 | q0, t0) in terms of the force ﬁeld.Such a quantity may not exist in general, however wewill see that in special cases one can recover an appro-priate analog of Scl(q1, t1 | q0, t0) requiring a certainbehavior ofA(q1, t1 | q0, t0) ∝eih¯Scl∫U[DΣ] exp{ ih¯∫ΣΩ}.in the limit h¯→ 0.58Acta Polytechnica Vol. 50 No. 5/20103 FunctionalityA major advantage of the surface functional inte-gral formulation rests in its explicit independence onHamiltonian H . From the point of view of classi-cal physics, dynamical equations and the force ﬁeldsentering them seem to be more fundamental thanthe Hamiltonian and/or Lagrangian function, whichprovide these equations in a relatively compact butambiguous way, see [3]. Therefore from the concep-tual point of view, Formula (6) gives us transitionprobability amplitude from a diﬀerent and hopefullynew perspective. It is clear that for the potentialgenerated forces the surface functional integral for-mula (6) gives nothing new compared to (1), sincein this case Ω is closed and can be represented asΩ = d(padqa−Hdt). There are, of course, some hid-den subtleties which we pass over either quickly or insilence, however, all of them are discussed in [4].To show functionality we need to analyze either astrongly non-Lagrangian system [5] or a weakly non-Lagrangian one. For the sake of simplicity let usfocus on the second case. To this end, let us considera system consisting of a free particle with unit massaﬀected by friction:q¨ = −κq˙ ⇔ Ω = dp ∧ (dq − p dt)− κp dq ∧ dt .In the considered example, the surface functionalintegral can be carried out explicitly (for detailssee [4]). At the end one arrives at the path integralin the conﬁguration space with a surprisingly trivialresult: ∫U[DΣ] exp{ ih¯∫ΣΩ}∝exp{− ih¯t1∫t0(12q˙2cl − κpclqcl)dt}×∫[Dq] exp{ ih¯t1∫t0(12q˙2 − κpclq)dt}.If we deﬁne Scl to beScl(q1, t1 | q0, t0) =t1∫t0(12q˙2cl − κpclqcl)dt , (7)thenA(q1, t1 | q0, t0) ∝∫[Dq] exp{ ih¯t1∫t0(12q˙2 − κpclq)dt}and in the classical limit h¯→ 0 we arrive at the sad-dle point equation which is speciﬁed by the functionalterm in the exponent above:q¨ = −κq˙cl .This diﬀerential equation is diﬀerent from the equa-tion q¨ = −κq˙ that we started with initially, butboth of them coincide when a solution q(t) satisfy-ing q(t0) = q0 and q(t1) = q1 is looking for. In thepresent situation we gain:Scl =κ4(q1 − q0) (q0 + 3q1)e−κt1 − (q1 + 3q0)e−κt0e−κt0 − e−κt1andA(q1, t1|q0, t0) =√κ4πih¯ tanh κ2 (t1 − t0)eih¯Scl . (8)Here we have already employed the normaliza-tion conditions speciﬁed in the ﬁrst paragraph.One can immediately verify that in the frictionlesslimit (κ→ 0) the transition probability amplitudeA(q1, t1 | q0, t0) matches the Schro¨dinger propaga-tor for a single free particle.4 ConclusionQuantization of dissipative systems has been very at-tractive problem from the early days of quantum me-chanics. It has been revived again and again acrossthe decades. Many phenomenological techniques andeﬀective methods have been suggested. References [6]and [7] provide a very basic list of papers dealing withthis point.We have developed here a new quantizationmethod that generalizes the conventional path inte-gral approach. We have focused only on the nonrela-tivistic quantum mechanics of spinless systems. How-ever, the generalization to the ﬁeld theory is straight-forward.Let us stress that the proposed method repre-sents an alternative approach to [7] and possessesseveral qualitative advantages. For example, prop-agator (8) is invariant with respect to time trans-lations, the same symmetry property which is pos-sessed by the underlying equation of motion. More-over, it is reasonable to expect that the “dissipativequantum evolution” will not remain Markovian. Thisfact is again conﬁrmed, since the probability ampli-tude under consideration does not satisfy the memo-ryless Chapman-Kolmogorov equation mentioned inthe ﬁrst paragraph.Finally, let us believe that the simple geometri-cal idea behind the surface functional integral quan-tization will ﬁt within the Ludwig Faddeev dictumquantization is not a science, quantization is an art!59Acta Polytechnica Vol. 50 No. 5/2010AcknowledgementThis research was supported in part by MSˇMT GrantLC06002, VEGA Grant 1/1008/09 and by the MSˇ SRprogram for CERN and international mobility.A. M. D. G.References[1] Feynman, R. P., Hibbs, A. R.: Quantum Mechan-ics and Path Integrals, McGraw-Hill Inc., NewYork, 1965.[2] Feynman, R. P.: Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. of Mod.Phys. 20 (1948), 367–387.[3] Wigner, E. P.: Do the Equations of Motion De-termine the Quantum Mechanical CommutationRelations?, Phys. Rev. 77 (1950), 711–712.Okubo, S.: Does the Equation of Motion deter-mine Commutation Relations?, Phys. Rev. D 22(1980), 919–923.Henneaux, M.: Equations of motion, Commuta-tion Relations and Ambiguities in the LagrangianFormalism, Annals Phys. 140 (1982), 45–64.[4] Kochan, D.: Direct quantization of equations ofmotion, Acta Polytechnica 47, No. 2–3 (2007),60–67, arXiv:hep-th/0703073.Kochan, D.: How to Quantize Forces(?): An Aca-demic Essay on How the Strings Could EnterClassical Mechanics, J. Geom. Phys. 60 (2010),219–229, arXiv:hep-th/0612115.Kochan, D.: Quantization of Non-Lagrangiansystems: some irresponsible speculations AIPConf. Proc. 956 (2007), 3–8.Kochan, D.: Quantization of Non-LagrangianSystems, Int. J. Mod. Phys. A 24 Nos. 28 & 29(2009), 5 319–5 340.Kochan, D.: Functional integral for non-Lagrangian systems, Phys. Rev. A 81, 022112(2010), arXiv:1001.1863.[5] Douglas, J.: Solution of the inverse problem inthe calculus of variations, Trans. Am. Math. 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J.: Path integralapproach to quantum Brownian motion, Physica121A (1983), 587–616.Geicke, J.: Semi-classical quantisation of dissipa-tive equations, J. Phys. A: Math. Gen. 22 (1989)1 017–1025.Weiss, U.: Quantum Dissipative Systems (2nded.), Series in Modern Condensed Matter Physics,Vol. 10, World Scientiﬁc, Singapore, 1999.Tarasov, V. E.: Quantization of non-Hamiltonianand Dissipative Systems, Phys. Lett. A 288(2001), 173–182, quant-ph/0311159.Breuer, H. P., Petruccione, F.: The Theoryof Open Quantum Systems. Oxford UniversityPress, Oxford, 2002.Razavy, M.: Classical and Quantum DissipativeSystems. Imperial Colleges Press, London, 2005.Chrus´cin´ski, D., Jurkowski, J.: Quantum dampedoscillator I: dissipation and resonances, Ann.Phys. 321 (2006), 854–874, quant-ph/0506007.Gitman, D. M., Kupriyanov, V. G.: Canoni-cal quantization of so-called non-Lagrangian sys-tems, Eur. Phys. J. C 50 (2007), 691–700, arXiv:hep-th/0605025.Tarasov, V. E.: Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. ElsevierScience, 2008.[7] Caldirola, P.: Forze non conservative nella mecca-nica quantistica, Nuovo Cim. 18 (1941), 393–400.Kanai, E.: On the Quantization of the DissipativeSystems, Prog. Theor. Phys. 3 (1948), 440–442.Moreira, I. C.: Propagators for the Caldirola-Kanai-Schro¨dinger Equation, Lett. Nuovo Cim.23 (1978), 294–298.60Acta Polytechnica Vol. 50 No. 5/2010Jannussis, A. D., Brodimas, G. N., Strectlas, A.:Propagator with friction in Quantum Mechanics,Phys. Lett. 74A (1979), 6–10.Das, U., Ghosh, S., Sarkar, P., Talukdar, B.:Quantization of Dissipative Systems with FrictionLinear in Velocity, Physica Scripta 71 (2005),235–237.Mgr. Denis Kochan, Ph.D.E-mail: kochan@fmph.uniba.skDepartment of Theoretical PhysicsFMFI, Comenius University842 48 Bratislava, Slovakia61

Does a functional integral really need a Lagrangian?

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