ap-5-10.dvi Acta Polytechnica Vol. 50 No. 5/2010 Does a Functional Integral Really Need a Lagrangian? D. Kochan Abstract 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 = q̇ 2 and L2 = e q̇ 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 transitionprobability amplitudewhich is free of anyLagrangian/Hamiltonian and requires just the underlying classical equations of motion. A simple example examining the functionality of the proposed method is considered. Dedicated to my friend and colleague Pavel Bóna. Keywords: quantization of (non-)Lagrangian systems, path vs. surface functional integral. 1 A standard path integral lore According to Feynman [1], the probability amplitude of the transition of a system fromthe space-time con- figuration (q0, t0) to (q1, t1) is given as follows: A(q1, t1 | q0, t0) ∝∫ [Dγ̃]exp { i h̄ ∫ γ̃ padq a − Hdt } . (1) Here the path-summation is taken over all trajecto- ries γ̃(t) = (q̃(t), p̃(t), t) in the extended phase space which are constrained as follows: γ̃(t0) = ( q̃(t0)= q0, p̃(t0)–arbitrary, t0 ) , γ̃(t1) = ( q̃(t1)= q1, p̃(t1)–arbitrary, t1 ) . To obtain a proper normalization of the Feynman propagator, one requires: δ(q̃0 − q0)=∫ Rn[q1] dq1A ∗(q1, t1 | q̃0, t0)A(q1, t1 | q0, t0) , δ(q1 − q0)= lim t1→t0 A(q1, t1 | q0, t0) . The first equation asks for the conservation of the total probability and the second expresses the obvi- ous fact that no evolution takes place whenever t1 approaches 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 Schrö- dinger equation. It is a curious fact thatFormula (1)wasnot originally discovered by Feynman. In his pioneering paper [2] he arrives at a functional integral in the configuration space only A(q1, t1 | q0, t0) ∝∫ [Dq]exp { i h̄ ∫ q(t) L(q, q̇, t)dt } . (2) Later, however, it was shown that this formula repre- sents a very special case of themost general prescrip- tion (1). Formula (1) is at the heart of our further discussion. 2 DeHamiltonianization A step beyond involves eliminating the Hamiltonian function H from Formula (1). The price to be paid for thiswill be to replace the path summation therein by a surface functional integration. Our aim is the transition probability amplitude be- tween (q0, t0) and (q1, t1). Let us suppose that there exists a unique classical trajectory in the extended phase space γcl(t) = (qcl(t), pcl(t), t) which connects these points (locally this assumption is always satis- fied). Then for any curve γ̃(t) = (q̃(t), p̃(t), t) which 57 Acta Polytechnica Vol. 50 No. 5/2010 enters the path integration in (1), we can assign two auxiliary curveswhichwe call λ0(s) and λ1(s). They are parameterized by s ∈ [0,1] and specified as fol- lows: λ0(s) = (q0, π0(s), t0)where π0(0) = pcl(t0), π0(1)= p̃(t0), λ1(s) = (q1, π1(s), t1)where π1(0) = pcl(t1), π1(1)= p̃(t1). Let us emphasize that neither λ0(s) nor λ1(s) varies with respect to the q and t coordinates in the ex- tended phase space. They are allowed to evolve only with respect to the momentum variables. There are, of course, infinitely many of such curves, but as we will see nothing in the theory will be dependent on a particular choice of λ0(s) and λ1(s). Using these curves one can write:∫ γ̃ padq a − Hdt = ∫ γcl padq a − Hdt + ∮ ∂Σ padq a − Hdt , (3) where ∂Σ = γ̃ − λ1 − γcl + λ0 is a contour spanned by four curves γ̃(t), γcl(t), λ0(s), λ1(s) counting their orientations. The first integral on the right is the classical action Scl(q1, t1 | q0, t0). While the contour integral in (3) can be rearranged to represent a surface integral:∮ ∂Σ padq a − Hdt = ∫ Σ dpa ∧ ( dqa − ∂H ∂pa dt ) − ∂H ∂qa dqa ∧ dt =: ∫ Σ Ω . (4) Surface Σ spanning the contour ∂Σ is understood here 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 initialHamiltonian function can be substituted using the velocity-momentum re- lations and classical equations of motion: ∂H ∂pa = T abpb ( = q̇a ) and ∂H ∂qa = −Fa ( = −ṗa ) . Here we consider the physically mostly relevant sit- uation only. In this case the velocities and mo- menta become related linearly by the metric tensor Tab(q) (and its inverse) defined by the kinetic energy T = 1 2 Tabq̇ aq̇b = 1 2 T abpapb of the system, then Ω=dpa ∧ dqa − ( T abpadpb − Fadqa ) ∧ dt . (5) This object represents a canonical two-form in the extended 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 are not potential-generated. It is clear that for a given pair of trajectories (γ̃, γcl) there exists infinitely many Σ surfaces. They form a set which we call Uγ̃. Since the surface inte- gral ∫ Σ Ω is only boundary dependent and Formulas (3) and (4) are satisfied, we can write: exp { i h̄ ∫ γ̃ padq a − Hdt } = e i h̄ Scl ∞γ̃ ∫ Uγ̃ [DΣ]exp { i h̄ ∫ Σ Ω } . Here ∞γ̃ stands for the number of elements pertain- ing to the corresponding stringy set Uγ̃. Assuming no topological obstructions from the side of the ex- tended phase space, ∞γ̃ becomes an infinite constant independent of γ̃. Taking all of this into account we can rewrite (1) as follows: A(q1, t1 | q0, t0) ∝ e i h̄ Scl ∫ U [DΣ]exp { i h̄ ∫ Σ Ω } . (6) In this formula the undetermined normalization con- stant ∞ was included into the integration measure [DΣ] and the path integral over γ̃’s was converted to the surface functional integral as was promised: ∫ [Dγ̃] ∫ Uγ̃ [DΣ] . . . = ∫ ⋃ γ̃ Uγ̃ [DΣ] . . . =: ∫ U [DΣ] . . . The set U = ⋃ γ̃ Uγ̃ overwhich the functional integra- tion is carried out contains all extended phase space strings which are anchored to the given classical tra- jectory γcl. To eliminate Hamiltonian H completely we need to expressScl(q1, t1 | q0, t0) in termsof the forcefield. Such a quantitymaynot exist in general, howeverwe will see that in special casesone canrecoveranappro- priate analog of Scl(q1, t1 | q0, t0) requiring a certain behavior of A(q1, t1 | q0, t0) ∝ e i h̄ Scl ∫ U [DΣ]exp { i h̄ ∫ Σ Ω } . in the limit h̄ → 0. 58 Acta Polytechnica Vol. 50 No. 5/2010 3 Functionality A major advantage of the surface functional inte- gral formulation rests in its explicit independence on Hamiltonian H. From the point of view of classi- cal physics, dynamical equations and the force fields entering them seem to be more fundamental than the Hamiltonian and/or Lagrangian function, which provide these equations in a relatively compact but ambiguous way, see [3]. Therefore from the concep- tual point of view, Formula (6) gives us transition probability amplitude from a different and hopefully new perspective. It is clear that for the potential generated forces the surface functional integral for- mula (6) gives nothing new compared to (1), since in this case Ω is closed and can be represented as Ω= d(padq a − Hdt). There are, of course, some hid- den subtleties whichwe pass over either quickly or in silence, however, all of them are discussed in [4]. To show functionalitywe need to analyze either a strongly non-Lagrangian system [5] or a weakly non- Lagrangian one. For the sake of simplicity let us focus on the second case. To this end, let us consider a system consisting of a free particle with unit mass affected by friction: q̈ = −κq̇ ⇔ Ω=dp ∧ (dq − pdt) − κpdq ∧ dt . In the considered example, the surface functional integral can be carried out explicitly (for details see [4]). At the end one arrives at the path integral in the configuration space with a surprisingly trivial result: ∫ U [DΣ]exp { i h̄ ∫ Σ Ω } ∝ exp { − i h̄ t1∫ t0 ( 1 2 q̇2cl − κpclqcl ) dt } × ∫ [Dq]exp { i h̄ t1∫ t0 ( 1 2 q̇2 − κpclq ) dt } . If we define Scl to be Scl(q1, t1 | q0, t0)= t1∫ t0 ( 1 2 q̇2cl − κpclqcl ) dt , (7) then A(q1, t1 | q0, t0) ∝∫ [Dq]exp { i h̄ t1∫ t0 ( 1 2 q̇2 − κpclq ) dt } and in the classical limit h̄ → 0 we arrive at the sad- dlepoint equationwhich is specifiedby the functional term in the exponent above: q̈ = −κq̇cl . This differential equation is different from the equa- tion q̈ = −κq̇ that we started with initially, but both of them coincide when a solution q(t) satisfy- ing q(t0) = q0 and q(t1) = q1 is looking for. In the present situation we gain: Scl = κ 4 (q1 − q0) (q0 +3q1)e −κt1 − (q1 +3q0)e−κt0 e−κt0 − e−κt1 and A(q1, t1|q0, t0)= √ κ 4πih̄tanh κ2(t1 − t0) e i h̄ Scl . (8) Here we have already employed the normaliza- tion conditions specified in the first paragraph. One can immediately verify that in the frictionless limit (κ → 0) the transition probability amplitude A(q1, t1 | q0, t0) matches the Schrödinger propaga- tor for a single free particle. 4 Conclusion Quantization of dissipative systems has been very at- tractive problem from the early days of quantumme- chanics. It has been revived again and again across the decades. Many phenomenological techniques and effectivemethodshavebeen suggested. References [6] and [7] provideaverybasic list of papersdealingwith this point. We have developed here a new quantization method that generalizes the conventional path inte- gral approach. We have focused only on the nonrela- tivistic quantummechanicsof 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 possesses several 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 “dissipative quantumevolution”will not remainMarkovian. This fact is again confirmed, since the probability ampli- tude under consideration does not satisfy the memo- ryless Chapman-Kolmogorov equation mentioned in the 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 dictum quantization is not a science, quantization is an art! 59 Acta Polytechnica Vol. 50 No. 5/2010 Acknowledgement This researchwas supported inpart byMŠMTGrant LC06002,VEGAGrant1/1008/09andbytheMŠSR program for CERN and international mobility. A. M. D. G. 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Das, U., Ghosh, S., Sarkar, P., Talukdar, B.: QuantizationofDissipativeSystemswithFriction Linear in Velocity, Physica Scripta 71 (2005), 235–237. Mgr. Denis Kochan, Ph.D. E-mail: kochan@fmph.uniba.sk Department of Theoretical Physics FMFI, Comenius University 842 48 Bratislava, Slovakia 61