A Method for Deriving the Dirac Equation from the Relativistic Newton’s Second Law

The derivation becomes possible when we find a new formalism which connects the relativistic mechanics with the quantum mechanics. In this paper, we explore the quantum wave nature from the Newtonian mechanics by using a concept: velocity field. At first, we rewrite the relativistic Newton’s second law as a field equation in terms of the velocity field, which directly reveals a new relationship connecting to the quantum mechanics. Next, we show that the Dirac equation can be derived from the field equation in a rigorous and consistent manner. PACS numbers: 03.65.Bz, 03.65.Pm, 11.10.Cd In the past century, many attempts were made to address to understand the quantum wave nature from the classical mechanics, however, much of the connection with the classical physics is rather indirect. In this paper, we propose a concept: velocity field, and show that the Dirac equation may be derived from the relativistic Newton’s second law in terms of the velocity field in a rigorous manner. According to the Newtonian mechanics, in a hydrogen atom, the single electron revolves in an orbit about the nucleus, its motion can be described with its position in an inertial Cartesian coordinate System S : (x1, x2, x3, x4 = ict). As the time elapses, the electron draws a spiral path (or orbit), as shown in Fig.1(a) in imagination. If the reference frame S rotates through an angle about the x2-axis in Fig.1(a), becomes a new reference frame S, there will be a Lorentz transformation linking the frames S and S. Then in the frame S, the spiral path of the electron tilts with respect to the x4-axis with the angle as shown in Fig.1(b). At one moment, for example, t4 = t0 moment, the spiral path pierces many points at E-mail: [email protected] the plane t4 = t0 , for example, the points labeled a, b and c in Fig.1(b), these points indicate that the electron can appear at many points at the time t0, in agreement with the concept of the probability in quantum mechanics. This situation gives us a hint for deriving quantum wave nature from the Newtonian mechanics. Because the electron pierces the plane t4 = t0 with 4vector velocity u, at every pierced point we can label a local 4-vector velocity . The pierced points may be numerous if the path winds up itself into a cell about the nucleus (due to a nonlinear effect in a sense), then the 4-vector velocities at the pierced points form a 4-vector velocity field. It is noted that the observation plane selected for the piercing can be taken at an arbitrary orientation, so the 4-vector velocity field may be expressed in general as u(x1, x2, x3, x4 = ict), i.e. the velocity u is of a function of position. At every point in the reference frame S the electron satisfies the relativistic Newton’s second law m duμ dτ = qFμνuν (1) the notations consist with the convention[1]. Since the Cartesian coordinate system is a frame of reference whose axes are orthogonal to one another, there is no distinction between covariant and contravariant components, only subscripts need be used. Here and below, summation over twice repeated indices is implied in all case, Greek indices will take on the values 1,2,3,4, and regarding the mass m as a constant. As mentioned above, the 4-vector velocity u can be regarded as a 4-vector velocity field, then duμ dτ = ∂uμ ∂xν dxν dτ = uν∂νuμ (2) qFμνuν = quν(∂μAν − ∂νAμ) (3) Substituting them back into Eq.(1), and re-arranging these terms, we obtain