A derivation of the Dirac equation in an external field based on the Poisson process

30 July 2001

Physics Letters A 286 (2001) 227–230 www.elsevier.com/locate/pla

A derivation of the Dirac equation in an external field based on the Poisson process T. Kudo a , I. Ohba a,b,c,∗ , H. Nitta d a Department of Physics, Waseda University, Tokyo 169-8555, Japan b Kagami Memorial Laboratory for Materials Science and Technology, Waseda University, Tokyo 169-0051, Japan c Advanced Research Center for Science and Technology, Waseda University, Tokyo 169-8555, Japan d Department of Physics, Tokyo Gakugei University, Tokyo 184-8501, Japan

Received 19 December 2000; received in revised form 14 June 2001; accepted 17 June 2001 Communicated by P.R. Holland

Abstract The Dirac equation has been derived from the master equation of Poisson process by analytic continuation. We extend it to the case where a particle moves in an external field. Furthermore, we show that the generalized master equation is intimately connected with three-dimensional Dirac equation in an external field.  2001 Elsevier Science B.V. All rights reserved. PACS: 02.50.Ey; 03.65.Ca; 03.65.Pm Keywords: Dirac equation; Klein–Gordon equation; Poisson process; Path integral; Telegrapher equation; External field

It is well known that the telegrapher’s equation is equivalent to a stochastic motion of Poisson processes. In his pioneering work, Kac showed its solution is described by the path-integral form [1]. Gaveau et al. derived one-dimensional Dirac equation from the Poisson process where the particle moves at the speed of light and flips its direction at a random time on the light cone. Here the particle is always moving forward in time [2]. There exist two components corresponding to chiral amplitude, because there is no spin in one dimension. A master equation related to Poisson process analytically continued produces the Dirac equation. By means of analytic continuation, a diffusion type of the master equation is changed to a

* Corresponding author.

E-mail address: [email protected] (I. Ohba).

hyperbolic type of equation [3]. A treatment of onedimensional Klein–Gordon equation for a free particle is discussed in [4]. McKeon and Ord extended it by assuming that a particle can move forward and backward in a real time, and no analytic continuation is needed [5]. In their framework, however, they assumed the detailed balance between forward and backward amplitudes and essentially there are two chiral components. They also discussed Dirac equation for a complex spinor moving in an external vector potential in one dimension [6]. It is worthwhile to note that Feynman derived onedimensional Dirac equation from a path-integral formulation of a Wiener process [7]. A “checkerboard” path-integral formulation of the Poisson process somewhat resembles to this Feynman path-integral formulation. The difference between these two formulations is that the master equation of the Poisson process is

0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 4 2 9 - 7

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T. Kudo et al. / Physics Letters A 286 (2001) 227–230

corresponding to the telegrapher’s equation, while that of the Wiener process is corresponding to the Fokker– Planck equation. In this Letter, we extend the path-integral formulation of the Poisson process and take into account the effect of the external field in the one-dimensional framework, and furthermore, in the three-dimensional version. First, we extend the Poisson process by Gaveau et al. to the case that a particle passes through a time dependent external potential V (t), where its one-dimensional sample path can be absorbed at most once in a time interval ∆t by the influence of the potential at a time t. A Poisson-distributed random variable Na (t) means the number of a reversal at time t. The Poisson formula is
 (at)k Prob Na (t) = k = e−at . k!

(1)

The velocity v(t) = v(−1)Na (t ) changes its sign by the reversal. Let a probability of the reversal in space be a∆t and a probability of the absorption at a time t be IV (t)∆t in a time interval ∆t. Using t s(t) =

(−1)Na (τ ) dτ,

(2)

0

conditional expectation (12) in [2] (1984) should be changed to f± (x, t) = (1 − IV ∆t)   × (1 − a∆t)E es(t )Lϕ(x) | (−1)Na (t ) = (−1)Na (t −∆t ) = ±1  + a∆tE es(t )Lϕ(x) |

 

−(−1)Na (t ) = (−1)Na (t −∆t ) = ∓1   + O ∆t 2 , (3) where ϕ(x) is a characteristic function of the interval between x and x + v∆t, and a space-shift generator becomes L = −v(∂/∂x). Eq. (3) satisfies ∂f± (4) = −a(f± − f∓ ) ± Lf± − IV f± , ∂t in the continuous limit of time. Gaveau et al. assumed that the effect of the external (scalar) potential should be taken into a space shift generator and

concluded that it is not possible. The potential can be included, however, if we assume that it should be taken into a time shift operator. In this case the evolution operator can be combined into a simple evolution as exp(s(t)L). Iteration of (4) means a telegrapher’s equation in an external field, 2    ∂ + a + IV f± = L2 + a 2 f± . (5) ∂t The case of vanishing IV in the generalized telegrapher’s equation (5) reduces the one-dimensional telegrapher’s equation for a free particle. It is worth while to mention that one can get a passage time of the particle in the potential using a solution of the telegrapher’s equation for the external field (5). Next we also extend the approach by McKeon and Ord as the same way above to the case where the absorption is occurred. A probability of the absorption is assumed to be IV (t)∆t. Since the particles can move not only forward, but also backward in time, Poisson distribution functions should be denoted by F± (x, t) and B± (x, t) which represent moving forward and backward in time, and + and − mean moving to right and left in space, respectively. The probabilities of turning right and left at right angles in space-time are aR ∆t and aL ∆t in a time interval ∆t. They assumed the equal probability aR ∆t at four types of turning from F− to F+ , from F+ to B+ , from B+ to B− and from B− to F− . For other four types they assumed the equal probability aL ∆t. The notation aR,L ∆tF∓ , for example, means that the upper part of the subscript ∓ is aR ∆tF− and the lower part is aL ∆t F+ . Following (3), Eq. (2) in [5] is extended to F± (x, t) = (1 − IV ∆t)

× (1 − aL ∆t − aR ∆t)F± (x ∓ ∆x, t − ∆t) + aL,R ∆tB± (x ∓ ∆x, t + ∆t)  + aR,L ∆tF∓ (x ± ∆x, t − ∆t) .

(6)

Furthermore, they assumed the condition of detailed balance F± (x, t) = B∓ (x ± ∆x, t + ∆t). This means, for example, that the forward right moving distribution function at a space-time point (x, t) is equal to the backward left moving one at the adjacent space-time point (x + ∆x, t + ∆t). By this condition Eq. (6) is rewritten as follows: B∓ (x ± ∆x, t + ∆t) = (1 − IV ∆t)

T. Kudo et al. / Physics Letters A 286 (2001) 227–230

× (1 − aL ∆t − aR ∆t)B∓ (x, t)

 + aL,R ∆tF∓ (x, t) + aR,L ∆tB± (x, t) .

(7)

In the limit ∆x, ∆t → 0 with ∆x = v∆t, (6) and (7) become ∂F± ∂F± + = aL,R (−F± + B± ) ∂x ∂t + aR,L (−F± + F∓ ) − IV F± , ∂B∓ ∂B∓ ±v + = aL,R (−B∓ + F∓ ) ∂x ∂t + aR,L (−B∓ + B± ) − IV B∓ .

±v

(8)

(9)

If one scale the two chiral components as g± = e(aL +aR )t (F± − B∓ ), the master equations (8) and (9) are transformed to ∂g± ∂g± ± = λg∓ ∓ IV g± , v ∂x ∂t

(10)

where λ = −aL + aR . Setting v = c, λ = −mc2/h¯ , IV = V /h¯ and ψ T = (g+ , g− ), (10) becomes ∂ψ ∂ψ = mc2 σy ψ − ich¯ σz − iV (x, t)ψ. (11) ∂t ∂x This is the one-dimensional Dirac equation in an optical potential in the Weyl representation. Since we have derived (11) without analytic continuation, we can say that the Poisson process with relaxation of the particles is fundamental description for the Dirac equation in an optical potential. We can make an alternative choice of chiral components, such as z± = F± + B∓ in (8) and (9), which satisfy

i h¯

∂z± ∂z± + = −a(z± − z∓ ) − IV z± , (12) ∂x ∂t with a = aL + aR . Iteration of (12) becomes the telegrapher’s equation in the external field,  2 ∂ 2 z± ∂ + a + IV z± = v 2 (13) + a 2 z± , ∂t ∂x 2 ±v

identifying (5). In this case we have to connect the probabilities a and IV to imaginary values by analytic continuation. As “checkerboard” path-integral formulation, if we chose v = c, a = −imc2/h¯ , IV = iV /h¯ and ψ T = eat (z+ , z− ), the master equation (12) becomes i h¯

∂ψ ∂ψ = mc2 σx ψ − ich¯ σz + V (t)ψ. ∂t ∂x

(14)

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This is the one-dimensional Dirac equation in the real potential. Later we shall show the same choice is useful for the three-dimensional scheme. Let us now extend formulation (3) to the threedimensional case. For a spin-1/2 particle we have to prepare two spinor functions φ± corresponding to two chiral components f± . Therefore, the onedimensional space-shift generator L = −v(∂/∂x) has to be generalized to the three-dimensional space-shift generator L = −vσ · ∇, where σ is the Pauli matrix. In this case f+ φ+ and f− φ− mean the four-component Dirac spinor,  f+ φ+ F= (15) , f− φ− where φ+ and φ− are arbitrary two-spinors. A lefthanded component switches a right-handed component at some random time and vice versa. Threedimensional version of (4) becomes a master equation in an external potential,   ∂F σ 0 0 1 = −v · ∇F + a F 0 −σ 1 0 ∂t − (a + IV )F. (16) Iteration of (16) becomes the generalized telegrapher’s equation  2 ∂ (17) + a + IV F = v 2 ∆F + a 2 F. ∂t If we set v = c, a = −imc2/h¯ , IV = iV /h¯ and ψ = eat F , the telegrapher’s equation (17) becomes threedimensional Klein–Gordon equation in an external potential,  2  1 ∂ mc V (t) 2 ∆ψ − 2 (18) ψ= ψ. +i h¯ h¯ c ∂t A generalization to the three-dimensional case for a particle moving in an electromagnetic field is considered in the same way. We extend the three-dimensional generator to the case where a particle moves in a static external potential A(x). In this case we use a minimal coupling generator L = −vσ · (∇ − ieA(x)/h¯ ) as a space-shift generator. It should be noted that the evolution operator is described by a simple form, such as exp(s(t)L) [2]. Three-dimensional version of (4) becomes   e ∂F σ 0 = −v · ∇−i A F 0 −σ h¯ ∂t

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T. Kudo et al. / Physics Letters A 286 (2001) 227–230

 +a

0 1

1 F − (a + IV )F. 0

(19)

If we set v = c, a = −imc2/h¯ , IV = iV /h¯ and ψ = eat F , the generalized master equation (19) becomes the three-dimensional Dirac equation in an external field,  ∂ψ 2 0 1 = mc i h¯ ψ 1 0 ∂t   e σ 0 − ich¯ · ∇ − i A(x) ψ 0 −σ h¯ + V (t)ψ.

Acknowledgements T.K. thanks K. Hara for calling our attention to [3], K. Imafuku, K. Yuasa and G. Kimura for helpful discussions. I.O. is supported partially by the Grant-inAid for COE Research, MEXT, and by the Waseda University Grant for Special Research Projects.

References

(20)

The “checkerboard” path-integral formulation of the Poisson process for the particle moving in an external field produces the three-dimensional Dirac equation in an external field. It is possible to get the passage time for the particle passing through an external field using the solution of the generalized telegrapher’s equation.

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