A note on the Feynman Path integral for Electromagnetic External fields
aa r X i v : . [ phy s i c s . g e n - ph ] S e p A note on Feynman Path Integralfor Electromagnetic External FieldsLuiz C.L. Botelho
Departamento de Matem´atica AplicadaInstituto de Matem´atica, Universidade Federal FluminenseRua Mario Santos Braga, CEP 24220-140Niter´oi, Rio de Janeiro, Brasile-mail: [email protected]
Abstract:
We propose a Fresnel stochastic white noise framework to analyze the natureof the Feynman paths entering on the Feynman Path Integral expression for the FeynmanPropagator of a particle quantum mechanically moving under an external electromagnetictime-independent potential.
Key words:
Feynman Path Integral, Nelson Stochastic Mechanics, Stochastic Calcu-lus.
2. The Fresnel stochastic nature of Feynman path integral
Let us start our note by considering as a basic object associated to a quantum particleof mass m , the following white noise functional Fresnel-Feynman path integral definedon an ensemble of closed, quantum trajectories of a white noise process with correlationfunction depending solely on the particle classical mass particle I m [ j ] = Z ~n ( T )=0 ~n ( σ )=0 D F [ ~n ( σ )] exp (cid:18) im Z T
12 ( ~n ( σ ) dσ (cid:19) exp (cid:18) i Z T ~j ( σ ) ~n ( σ ) dσ (cid:19) (1)Where ~n ( σ ) are the quantum white-noise Feynman closed trajectories on R for thepropagation time interval σ ∈ [0 , T ] and satisfying the Dirichlet condition ~n (0) = ~n ( T ) =0. Here j ( σ ) denotes a fixed external real valued path source.1t is worth to call the reader attention that the normalized white-noise external sourceFeynman path integral eq(1) can be straightforwordly (heuristically) evaluated through arandom Fourier series expansion for the random white noise trajectory ~n ( σ ) 0 ≤ σ < T yelding 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 analyze the stochastic nature of the Feynman integral associated to thequantum system defined by a particle (of a fixed newtonian mass m ) (1) under the presenceof an external time-independent electromagnetic field we firstly consider the well-definedunique system classical trajectory connecting the spatial points ~x and ~x in time T .Namely: m d ~x ( σ ) dσ = e (cid:18) − ~ ∇ .φ + d~xdσ × ~ ∇ × ~A (cid:19) ( ~x ( σ )) ~x (0) = ~x ~x ( T ) = ~x (3)We now introduce what we call the effective classical potential, throught the Taylorexpansion below defined ~ φ eff (cid:0) x, [ x cL ( σ )] (cid:1) definition ≡ e Φ (cid:0) x cL ( σ )+ √ ~ x (cid:1) − e Φ( x cL ( σ )) −√ ~ (cid:0) [ ~ ∇ ℓ ( eφ )( x cL ( σ )] × ~x ℓ (4)Note that this effective classical potential also depends functionally on the system’sclassical trajectory { x cL ( σ ) } ≤ σ ≤ T and the Plank constant.We now introduce the Feynman quantum trajectories of our system defined mathemat-ically as those paths ~x q ( σ ), functionals of the Feynman white-noise path ~n ( σ ), through theHamilton-Jacobi equation for the quantum trajectory ([1]), where E denotes the classicalsystem total energy (2) m h(cid:16) ~ ∇ .W eff − ec ~A eff (cid:17) ( ~x, ~x cL ( σ ) i + ¯ V eff ( x, [ ~x cL ( σ ]) = E (5) (1) The same mass parameter of the (vectorial) white noise eq(1). (2) ~ (cid:0) ~A effi ( ~x, [ ~x cL ( σ )]) (cid:1) = ~A i (cid:0) ~x cL ( σ ) + √ ~ ~x (cid:1) − ~A i ( ~x cL ( σ ) − √ ~ (cid:20) P k =1 (cid:18) ∂∂x k ~A i (cid:19) ( ~x cL ( σ )) x k (cid:21) d~x q ( σ ) dσ = (cid:18) m (cid:16) ~ ∇ W eff − ec ~A eff (cid:17) ( ~x q ( σ ) , [ x cL ]) + ~n ( σ ) (cid:19) ~x q (0) = ~x q ( T ) = 0 ~n (0) = ~n ( T ) = 0 (6)We claim that the full Feynman path defined below as quantum fluctuations aroundthe classical path with “size” of order ( ~ ) / , ~x ( σ ) = ~x cL ( σ ) + √ ~ ( ~x q ( σ )) , (7)can be mathematical used to be the set of paths that enters in the Feynman path integralexpression for the quantum mechanical propagator, and leading straighforwardly to theexpected result that on the asymptotic semi-classical limit ~ →
0, the leading contributioncomes solely from the classical path.To show these results, we firstly consider the formal object written fully below: G ( x , x , T ) = exp (cid:18) ~ S [ x cL ( σ )] (cid:19) × exp (cid:26) − i ~ Z T dx i,cL ( σ ) dσ E ijk ∂∂x j A k ( x cL ( σ )) dσ (cid:27) × (cid:26) Z x q (0)=0 x q (0)=0 D F [ ~x q ( σ )] (cid:20) Z ~n ( T )=0 ~n (0)=0 D F ( ~n ( σ ) exp (cid:18) im Z T
12 ( ~n ( σ )) dσ (cid:19) (cid:21) × det F (cid:20) dd ¯ σ − m ~ ∇ x (cid:18) δδ~x q (¯ σ ) W eff (cid:19) + δδ~x q ( σ ) (cid:16) ecm ~A eff (cid:17)(cid:21) × δ ( F ) (cid:20) d~x q ( σ ) dσ − m (cid:16) ~ ∇ W eff − ec ~A eff (cid:17) ( ~x q ( σ ) , [ ~x cL ( σ )]) − ~n ( σ ) (cid:21) (cid:27) × (exp( − iET )) (8)We will heuristically show that it satisfies the system’s quantum Feynman propagatorequation: + i ~ ∂∂T G ( ~x , ~x , T ) = ( − i ~ ~ ∇ − ec ~A ) m + e φ ! G ( ~x , ~x , T ) (9-a) G ( ~x , ~x , T ) = δ (3) ( ~x − ~x ) (9-b)(div ~A )( ~x ) ≡ ~x q ( σ )) G ( ~x , ~x , T ) = exp( iET ) × exp (cid:18) i ~ S ( x cL ( σ ) (cid:19) × (Z ~x q ( T ) ~x q (0)=0 D F [ ~x q ( σ )] × exp im Z T (cid:20) d~x q dσ − m (cid:16) ~ ∇ W eff − ec ~A eff (cid:17) ( ~x q , ~x cL ) (cid:21) dσ !) × det F (cid:20) dd ¯ σ − m ∇ x (cid:18) δδx q (¯ σ ) W eff (cid:19) + δδ~x q ( σ ) (cid:16) ecm ~A eff (cid:17)(cid:21) (10)The action functional on the Feynman path integral thus posseses the following explicitfunctional form im Z T (cid:20) (cid:18) d~x q dσ (cid:19) − m d~x q dσ (cid:16) ~ ∇ W eff − ec ~A eff (cid:17) + 1 m (cid:16)(cid:16) ~ ∇ W eff − ec ~A eff (cid:17)(cid:17) (cid:21) ( σ ) dσ = im Z T (cid:18) d~x q dσ (cid:19) ( σ ) dσ ! + (cid:18) im Z T (cid:18) − m d~xdσ · ~ ∇ W eff − emc d~x q dσ · ~A eff (cid:19) ( σ ) dσ + (cid:18) im Z T m (cid:18) ( ~ ∇ W eff ) + e c ( ~A eff ) − ec ~A eff · ~ ∇ W eff (cid:19) ( σ ) dσ (cid:19) (11)By noting the result (cid:18)Z T (cid:18) d~x q dσ · ~ ∇ W eff (cid:19) ( σ ) dσ (cid:19) = 0 (12)togheter with the delta function distribution prescription at coincident points below tomake the functional determinant equals to 1 O (0) = 0 . (13)4e have thus the result G ( x , x , T ) = exp (cid:18) i ~ S [ ~x cL ( σ )] (cid:19) × (cid:26) Z ~x q ( T )=0 ~x q (0)=0 D F [ ~x q ( σ )] exp i Z T (cid:18) m dx q dσ (cid:19) ( σ ) dσ ! × exp (cid:18) − iec (cid:18)Z T ~A eff ( ~x q , [ ~x cL ( σ )]) d~x q dσ ( σ ) dσ (cid:19)(cid:19) × exp (cid:18) − ie Z T φ eff ( x q ( σ )) dσ (cid:19) (cid:27) (14)We now observe that through the classical motion equation eq(3) and the definitionof the effective potential eq(4) and the formal invariance of translation under classicaltrajectories of the Feynman path measure (see ref[2]) one can re-write eq(14) in the usualFeynman form G ( ~x , ~x , T ) = Z ~x ( T )= ~x ~x (0)= ~x D F [ ~x ( σ )] exp (cid:26) i ~ S [ ~x ( σ )] (cid:27) , (15)with the classical system action(3) (3) To arrive at the final result eq(16) we suppose the “orthogonality” conditions among the classicaland quantum (fluctuation) Feynman path when in presence of an external time-independent classicalexternal electromagnetic field: iec √ ~ Z T d~x cLi dσ · ∂A i ∂x ℓ ( ~x cL ) ~x q,ℓ ( σ ) dσ ! = 0 iec Z T d~x cLi ( σ ) dσ A i,eff ( ~x q , ~x cL ( σ )) dσ ! = 0 iec √ ~ Z T d~x qi dσ A i ( ~x cL ( σ ) dσ ! = 0 iec Z T d~x q,i dσ · ∂A i ∂x ℓ ( ~x cL ( σ )) .~x q,ℓ ( σ ) dσ ! = 0Otherwise we have all of ours results above written holding true for the quantum closed trajectories( ~x cL ( σ ) ≡ [ ~x ( σ )] = m Z T (cid:18) d~xdσ (cid:19) ( σ ) dσ + ec Z T (cid:18) ~A ( ~x ( σ )) · d~x ( σ ) dσ (cid:19) ( σ ) dσ − e Z T φ ( ~x ( σ )) dσ (16)The above heuristic (from a rigorous mathematical point of view [3]) manipulationshows our claims.
3. Conclusions
As a general conclusion of our note on the nature of the Feynman path integral in thepresence of on external electromagnetic potential we stress that we have substituted thewhole machinery of sliced steps for defining Feynman path integrals for the somewhat clas-sical stochastic equation eq(6) driven by (still mathematically formal) a Fresnel quantumwhite noise ~n ( σ ). This result may be of practical use for Monte-Carlo sampling evalu-ations of observables since the numerical approximated solution of the quantum Fresnelstochastic eq(6) appears a less formidable task than evaluating the Feynman path integraleq(16) directly. These claims are result that eq(5) is a first order system of usual non-linear partial differential equations and eq(6) reduces to non-linear algebraic equationsfor the Fourier coefficients of the expansion of the closed quantum trajectory in Fourierseries (note that ~n ( σ ) = ∞ P n =1 ~e n sin (cid:18) nπT σ (cid:19) ) · Acknowledgements:
We are thankfull to CNPq for a fellowship and thankfull alsoto Professor Jos´e Helayel - CBPF and Professor W. Rodrigues - IMEC-UNICAMP.