aa r X i v : . [ phy s i c s . g e n - ph ] S e p A note on the stochastic nature ofFeynman quantum paths
Luiz C.L. Botelho
Departamento de Matem´atica Aplicada,Instituto de Matem´atica, Universidade Federal Fluminense,Rua Mario Santos Braga24220-140, Niter´oi, Rio de Janeiro, Brazile-mail: [email protected]
Abstract
We propose a Fresnel stochastic white noise framework to analyze the stochastic natureof the Feynman paths entering on the Feynman Path Integral expression for the FeynmanPropagator of a particle quantum mechanically moving under a time-independent poten-tial.
Key words:
Feynman Path Integrals; Nelson Stochastic Mechanics; Stochasticx Calculus.
Let us start our note by considering as a basic object associated to a quantum particle of mass m the following white noise functional Fresnel-Feynman path integral defined on an ensemble ofclosed, quantum trajectories related to a white noise process with correlation function dependingon the particle classical mass mI m [ j ] = Z n ( T )=0 n (0)=0 D F [ η ( σ )] exp (cid:18) im Z T
12 [ η ( σ )] dσ (cid:19) exp (cid:18) i Z T j ( σ ) η ( σ ) dσ (cid:19) (1)ere η ( σ ) are the quantum white-noise Feynman closed trajectories on R N defined for thepropagation time interval σ ∈ [0 , T ] and satisfying, the vanishing and-point Dirichlet condition η (0) = η ( T ) = 0. j ( σ ) denotes a fixed external real valued path source on the Schwartz testfunction space D (0 , T ).It is worth to call the reader attention that the normalized white-noise external sourceFeynman path integral eq.(1) can be straightforwardly evaluated through a random Fourierseries expansion for the random white noise trajectory η ( σ ), 0 ≤ σ ≤ T with the exact result I m [ j ] I m [0] = exp (cid:26) i m Z T ( j ( σ )) dσ (cid:27) (2)In order to analyse the stochastic nature of the Feynman paths associated to the quantumsystem defined by a particle (of a fixed mentonion mass m ) under the presence of a potencial V ( x ) on R D , we firstly consider the well-defined unique system classical trajectory connectingthe spatial end-points x and x in the time interval T . Namely m d x CL dσ ( σ ) = ( −∇ V )( x CL ( σ )) x CL (0) = x x CL ( T ) = x (3)We now introduce what we call the effective potential, through the Taylor expansion belowdefined V eff ( x, [ x CL ( σ )]) definition ≡ V ( x CL ( σ ) + √ ~ x ) − V ( x CL ( σ )) − √ ~ [( ∇ V )( x CL ( σ ))] x. (4)Note that this effective potential also depends functionally on the system’s classical trajec-tory { x CL ( σ ) } .We now introduce the Feynman quantum trajectories of our system which are defined math-ematically as those paths x q ( σ ), formally functionals of the Feynman white-noise path n ( σ ),through the Hamilton-Jacobi equation for the quantum trajectory ([1]), where E denotes theclassical system total energy. Namelly12 m |∇ W eff ( x, [ x CL ]) | + V eff ( x, [ x CL ( σ )]) = E, (5)2ith the white-noise ODE’s stochastic equation. A formal Sturm-Liouville stochastic problem dx q ( σ ) dσ = (cid:18) m ( ∇ W eff )( x q ( σ ) , [ x CL ]) (cid:19) + η ( σ ) x q (0) = x q ( T ) = 0 η (0) = η ( T ) = 0 . (6)We claim that the full path defined below as quantum fluctuations around the classical pathwith “size” of order ( ~ ) / x ( σ ) = x CL ( σ ) + √ ~ ( x q ( σ )) (7)can be mathematically used to be the set of paths that enter in the Feynman path integral ex-pression for the quantum mechanical propagator, and leading straighforwardly to the expectedresult that on the asymptotic semi-classical limit ~ →
0, the leading contribution comes solelyfrom the classical path.To show these results, we firstly consider the formal object written in full below: G ( x , x , T ) = exp (cid:18) i ~ S [ x CL ( σ )] (cid:19) × n Z x q ( T )=0 x q (0)=0 D F [ x q ( σ )] h Z η ( T )=0 η (0)=0 D F [ η ( σ )] exp (cid:16) im Z T
12 ( η ( σ )) dσ (cid:17) × det F h d d ¯ σ − m ∇ x (cid:16) δδx q (¯ σ ) W eff ( x q ( σ ) , [ x CL ( σ )]) (cid:17)i × δ ( F ) h dx q ( σ ) dσ − m ( ∇ W eff )( x q ( σ ) , [ x CL ]) − η ( σ ) iio × exp( − iET ) exp( i [ W eff (0 , [ x CL ]) − W eff ( T, [ x CL ])]) (8)We will formally show that it satisfies the system’s quantum Feynman propagator equationfor the time interval T ∂∂T G ( x , x , T ) = (cid:16) − ~ m ∆ + V ) G ( x , x , T ) G ( x , x , T ) T → = δ ( N ) [ x − x ] . (9)In order to arrive at such result, we firstly compute exactly the white-noise path integtralwith the following result (written entirely on terms of the quantum path x q ( σ ))3 ( x , x , T ) = f ( E, T ) exp( i ~ S ( x CL ( σ ))) × n Z x q ( T )=0 x q (0)=0 D F [ x q ( σ )] exp (cid:16) im Z T h dx q dσ − m ( ∇ W eff ( x q ( σ ) , [ x CL ])) i dσ (cid:17) × def f h dd ¯ σ − m ∇ x (cid:16) δδx q (¯ σ ) W eff ( x q ( σ ) , [ x CL ]) (cid:17)io . (10)The action functional on the Feynman path integral thus posseses the following explicitfunctional form im Z T (cid:20) dx q ( σ ) dσ − m ∇ W eff ( x q ( σ ) , [ x CL ]) (cid:21) ! ( σ ) dσ = im Z T (cid:18) dx q ( σ ) dσ (cid:19) ! ( σ ) dσ + im m Z T ( ∇ W eff ( x q ( σ ) , [ x CL ])) dσ − i Z T (cid:18) ∇ W eff ( x q ( σ ) , [ x CL ]) · dx q ( σ ) dσ (cid:19) dσ. (11)By using the Itˆo’s rule for derivatives we can evaluate the stochastic integral on last line ofeq.(11) − i (cid:18)Z T (cid:18) ∇ W eff ( x q ( σ ) , [ x CL ]) dx q ( σ ) dσ (cid:19) dσ (cid:19) = − i (cid:16) W eff ( x q ( T ) , [ x CL ]) − W eff ( x q ( σ ) , [ x CL ])) (cid:17) + i (cid:18)Z T (∆ W eff )( x q ( σ ) , [ x CL ]) dσ (cid:19) . (12)Through eq.(5), we simpligy further eq.(11). Namely: im m Z T ( ∇ W eff ( x q ( σ ) , [ x CL ])) dσ = (cid:18) i m (cid:19) (2 m ) (cid:20)Z T ) (cid:0) E − V eff ( x q ( σ ) , [ x CL ] (cid:1) ( σ ) dσ (cid:21) = +( iET ) − i (cid:18)Z T V eff ( x q ( σ ) , [ x CL ])( σ ) dσ (cid:19) . (13)4y noting now that on the Itˆo’s stochastic calculus One must use the operational rule θ (0) = + , one has the following expression for the functional determinantdet f (cid:20) dd ¯ σ − m (∆ W eff )( x q ( σ ) , [ x CL ]) (cid:21) = exp (cid:18) − im +2 m Z T dσ (∆ W eff )( x q ( σ ) , [ x CL ]) (cid:19) . (14)It is worth call attention that this term canceals out exactly with similar term (howeverwith inversew signal) on eq.(12). Note that if one has used from the beginning the stochasticStratonovick calculus, with the prescription θ (0) = 0, eq.(14) would be one and the usual ruleof integration for parts would be applied to eq.(12). We think that this an important result:the Feynman path integral appears to be insensitive to the kind of Stochastic Calculus used todefine it. The point is just to be consistent with the stochastic prescription being used whichusually reflects it self on the choose of the “valve” of the distribution θ ( σ ) on the point σ = 0.By grouping togheter all the above results on eq.(10), one has the partial outcome: G ( x , x , T ) = exp (cid:18) i ~ S [ x CL ( σ )] (cid:19) × n Z x q ( T )=0 x q (0)=0 D F [ x q ( σ )] exp h i Z T (cid:16) m dx q dσ (cid:17) ( σ ) − i Z T V eff ( x q ( σ ) , [ x CL ])( σ ) dσ io . (15)We now observe that through the classical motion equation eq.(3) and the definition of theeffective potential eq.(4) and the formal invariance of translation under classical trajectories ofthe Feynman path measure (see ref. [2]) one can re-write eq.(14) in the usual Feynman form G ( x , x , T ) = Z x ( T )= x x (0)= x × exp (cid:26) i ~ S [ x ( σ )] (cid:27) , (16)with the classical system action S [ x ( σ )] = 12 m Z T (cid:18) dxdσ (cid:19) ( σ ) dσ − Z T V ( x ( σ )) dσ. (17)5he above heuristic-mathematical methods (from a rigorous mathematical point of view[3]) manipulations shows our claims. As a general conclusion of our short note on the stochastic nature of the Feynman path integral,we stress that we have substituted the whole machinery of Feynman path integrals for thesomewhat classical stochastic equation eq.(6) driven by (still mathematically formal) a Fresnelquantum white noise n ( σ ). This result may be of practical use for Monte-Carlo samplingevaluations of observable since the numerical approximated solution of eq. the quantum Fresnelstochastic eq.(6) appears a less formadable task than evaluating the Feynman Path integraleq.(16) directly. These claims are result that eq.(5) is a first order system of usual non-linearpartial differential equations and eq.(6) reduces to non-linear algebraic equations for the FourierCoeficients of the expansion of the closed quantum trajectory in Fourier series. Note that η ( σ ) = ∞ X n =1 e n sin (cid:18) nπT σ (cid:19) . As a last remark we call the reader attention that in the case of a time-dependent potential(a non-conservative classical system), the own quantum nature is not a straightforward concept(it is an open quantum system). If in this case we adopt the natural generalization of ourHamilton-Jacobi equation for time-dependent potentials12 m (cid:12)(cid:12) ∇ W eff ( x, t, [ x CL ]) (cid:12)(cid:12) + V eff ( x, t, [ x CL ]) = ∂∂t W eff ( x, t, [ x CL ]) (18)6nd the time-dependent Itˆo’s rule for the integration by parts below − i (cid:18)Z T ( ∇ W eff ( x q ( σ ) , [ x CL ])) · dx q ( σ ) dσ (cid:19) dσ = − i (cid:0) W eff ( x q ( T ) , T, [ x CL ]) − W eff ( x q (0) , , [ x CL ]) (cid:1) + i (cid:18)Z T (∆ W eff )( x q ( σ ) , σ, [ x CL ]) dσ (cid:19) + i (cid:18)Z T (cid:18) ∂W eff dt (cid:19) ( x q ( σ ) , σ, [ x CL ]) dσ (cid:19) (19)one has an additional anomaly factor I anomaly on the path-integral result and fully dependingon the function W eff ( x, t, [ x CL ]); namely I anomaly = exp (cid:20) i Z T (cid:18) ∂W eff ∂t (cid:19) ( x q ( σ ) , σ ) dσ (cid:21) . (20)That could signals that one should modify eq.(5) or point at that quantization of time-dependent mechanical point particle systems on the real m of path integrals (Feynman propa-gators) need additional care or worse, time-dependent one particle quantum system are some-what problematic in their quantum mechanical interpetration ([4]). Additional studies on thiscase will appears elsewhere. 7 EFERENCES [1] Luiz C.L. Botelho, Modern Physics Letter B, vol. , n o
3, 73–78, (2000).[2] L.S. Schulman, Techniques and Applications of Path Integration, John Erley & Sons(1981).[3] Luiz C.L. Botelho, A note on Feynman-Kac path integral representations for scalarwave motions-Random Operators and Stochastic Equations, vol. , 271–292, (2013),DOI:10.1515/rose-2013-0012.[4] Luiz C.L. Botelho, Il Nuovo Cimento, vol. B, 37–59, n o
1, Gennaio 2002.– Luiz C.L. Botelho, Modern Physics Letters B, vol. , n oo