The method of Kramers to represent a two-component spinor is revised and developed into a consequent formulation of the quantum mechanical theory of a non-relativistic spinning particle (§4). Furthermore the method is unified with the hydrodynamical representation of the Pauli electron theory established in the previous article, giving the picture of the Pauli field as an assembly of very small rotating bodies (distinct from usual rigid bodies) continuously distributed in space (§3). The formalistic and physical features of these new representations are considered. Also some remarks are given on the characteristics of the former hydrodynamical formulation (§2). Particularly it is pointed out that Planck's constant enters this scheme only as the constant signifying the magnitude of the spin.
The method of Kramers to represent a two-component spinor is revised and developed into a consequent formulation of the quantum mechanical theory of a non-relativistic spinning particle (§4). Furthermore the method is unified with the hydrodynamical representation of the Pauli electron theory established in the previous article, giving the picture of the Pauli field as an assembly of very small rotating bodies (distinct from usual rigid bodies) continuously distributed in space (§3). The formalistic and physical features of these new representations are considered. Also some remarks are given on the characteristics of the former hydrodynamical formulation (§2). Particularly it is pointed out that Planck's constant enters this scheme only as the constant signifying the magnitude of the spin.
A quantum-mechanical non-relativistic spinnig particle is represented equivalently with a non-linear vector field which defines a new kind of hydrodynamics of a spinning fluid. In this hydrodynamics, quantum effects are separated as non-linear terms which mean the occurrence of “internal potential” and “internal magnetic field”. Mathematically, the method is based upon the replacement of the original calculus in terms of a spinor ψ by the calculus in terms of tensors formed as the bilinear expressions in ψ*. This article (hereafter called I) and also the major part of the subsequent one (hereafter called II) which will appear in succession, are the translation of the paper which appeared in Soryusiron-Kenkyu 7, pp. 600–640, (January 1955) in Japanese, while a brief account of these works was reported in Prog. Theor. Phys. 12 (1954), 810. Very recently articles by Bohm, Schiller, and Tiomno appeared (Nuovo Cimento 1, Suppl. (1955), 48, 67), in which they investigated, independently of us, the similar hydrodynamical method for the purpose of a “ causal interpretation ” of quantum theory in the case of a non-relativistic spinning particle. However, their method stands upon Euler angle representation of the spinor (cf II), not directly concerned with the self-contained tensor formulation of the spinor equation and ψ. This replacement is ensured by setting up the identities which should hold among those tensor quantities.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and c. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.
The Pauli theory of electrons is formulated in the language of multivector calculus. The advantages of this approach are demonstrated in an analysis of local observables. Planck's constant is shown to enter the theory only through the magnitude of the spin. Further, it is shown that, when obtained as a limiting case of the Pauli theory, the Schrödinger theory describes a particle with constant local spin. An important consequence of this result is the realization that the usual interpretations of the Dirac and Schrödinger theories are mutually inconsistent in certain details.
Introduction The claim that it is impossible to construct a realistic physical model of quantum matter is often presented as definitively and finally proved in the case of systems with spin ½, or internal angular momentum of magnitude ħ/2, which exhibit a characteristic ‘nonclassical two-valuedness’. Nevertheless, the extension of the causal interpretation to embrace nonrelativistic spin ½ systems was successfully carried through in the 1950s (Bohm, Schiller & Tiomno, 1955; Bohm & Schiller, 1955; Takabayasi, 1955; Takabayasi & Vigier, 1957) and developed more recently (Dewdney, Holland & Kyprianidis, 1986, 1987a, b; Dewdney, Holland, Kyprianidis & Vigier, 1988). The idea is to assume that the spinor wave represents a new type of physical field propagating in spacetime that exerts an influence on a corpuscle moving within it. The additional information conveyed by the spinor beyond that contained in the spinless Schrödinger wave implies that the state of the particle is not exhausted by specifying its position and momentum, but that it is endowed with further properties such as an internal angular momentum. The equation governing the dynamics of the spinor wavefunction is the Pauli equation: where the Hamiltonian operator is given by Here e, m and μ are respectively the charge, mass and magnetic moment of the particle, A 0 and A are the electromagnetic potentials, B = ∇ × A is the magnetic field, V is any other external scalar potential, and σ is a vector whose components are the 2 × 2 Pauli matrices. The Hamiltonian (9.1.2) istherefore a 2 × 2 matrix operator and the wavefunction ψ( x , t ) has two complex components.