Relativistic Electron Motion and Spin Precession

It is widely believed that the effect of the spin on electron motion cannot be accurately described within the frame of validity of geometrical charged-particle optics. However, there is no convincing reason that prevents one from incorporating the spin into the formalism of relativistic mechanics if an appropriate interaction Hamiltonian is found. To achieve a proper calculation procedure, it is advantageous to describe the relativistic motion and the spin precession of the electron in Minkowski space. By using x₄ = ict as the fourth spatial coordinate of the four-dimensional Euclidian space, we avoid difficulties in constructing relativistic covariant Lagrangians and Hamiltonians. We describe the motion of the electron by considering its four coordinates xμ(τ) as functions of the independent Lorentz-invariant variable τ, which we conceive as the world time or universal time. This time increases monotonically, whereas the timelike position coordinate x₄ = ict needs not, contrary to classical mechanics. The four-dimensional Minkowski space is composed of the three-dimensional space with coordinates x₁ = x, x₂ = y, x₃ = z and the imaginary timelike coordinate x,₄. The imaginary character of this coordinate is ultimately connected with certain properties of the time as experienced by men.

The extension of the space from three to four dimensions is accompanied by a change of the properties of physical quantities. For example, an axial vector in three-dimensional space becomes an antisymmetric tensor in four-dimensional space because the four-dimensional cube is enclosed by 12 two-dimensional plane surfaces. The transition of different physical quantities is shown in Table 14.1.