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.
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.
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.
In the first part (Secs. I and II) of this paper, starting from the Pauli current, we obtain the decomposition of the nonrelativistic field velocity into two orthogonal parts: (i) the ``classical'' part, that is, the velocity w=p/m in the center of mass (c.m.), and (ii) the ``quantum'' part, that is, the velocity V of the motion of the c.m. frame (namely, the internal ``spin motion'' or Zitterbewegung). By inserting such a complete, composite expression of the velocity into the kinetic-energy term of the nonrelativistic classical (i.e., Newtonian) Lagrangian, we straightforwardly get the appearance of the so-called quantum potential associated, as it is known, with the Madelung fluid. This result provides further evidence of the possibility that the quantum behavior of microsystems is a direct consequence of the fundamental existence of spin. In the second part (Secs. III and IV), we fix our attention on the total velocity v=w+V, now necessarily considering relativistic (classical) physics. We show that the proper time entering the definition of the four-velocity ${v}^{\ensuremath{\mu}}$ for spinning particles has to be the proper time \ensuremath{\tau} of the c.m. frame. Inserting the correct Lorentz factor into the definition of ${v}^{\ensuremath{\mu}}$ leads to completely different kinematical properties for v2. The important constraint ${p}_{\ensuremath{\mu}}{v}^{\ensuremath{\mu}}=m,$ identically true for scalar particles but just assumed a priori in all previous spinning-particle theories, is herein derived in a self-consistent way.
In this note, we first obtain the decomposition of the non-relativistic field velocity into the classical part (i.e., the velocity w=p/m OF the center-of-mass (CM), and the so-called quantum part (i.e., the velocity V of the motion IN the CM frame (namely, the internal spin-motion or Zitterbewegung), these two parts being orthogonal. Our starting point is the Pauli current. Then, by inserting such a composite expression of the velocity into the kinetic energy term of the non-relativistic newtonian lagrangian, we get the appearance of the so-called "quantum potential" (which makes the difference between classical and quantum behaviour) as a pure consequence of the internal motion. Such a result carries further evidence about the possibility that the quantum behaviour of micro-systems be a direct consequence of the fundamental existence of spin.
The authors explore certain interconnections between density-functional theory and quantum fluid dynamics of many-electron systems, in the relativistic domain following the hydrodynamical approach adopted by Takabayashi for the one-particle Dirac equation. In order to build a 'classical' hydrodynamical interpretation, the spinor formulation is transformed into a tensor formulation by defining a number of density functions (local observables). These lead to six 'classical' fluid dynamic equations, together with two subsidiary conditions, for a complete specification of the system. The various density functions and the hydrodynamical equations are physically interpreted. The relativistic hydrodynamics discussed here correspond to a 'spinning' fluid. The net many-electron fluid consists of components each of which is characterised by fluid dynamic quantities corresponding to each spinor. The net hydrodynamical quantities are obtained by summing over the occupied spinors. Thus, the earlier nonrelativistic 'classical' picture of the many-electron fluid as a collection of individual fluid components is also valid in the relativistic domain.
One of the most satisfactory pictures for spinning particles is the Barut-Zanghi (BZ) classical theory for the relativistic extended-like electron, that relates spin to zitterbewegung (zbw). The BZ motion equations constituted the starting point for recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. This language results to be actually suited and fruitful for a hydrodynamical re-formulation of the BZ theory. Working out, in such a way, a ``probabilistic fluid'', we are allowed to re-interpret the original classical spinors as quantum wave-functions for the electron. Thus, we can pass to ``quantize" the BZ theory employing this time the tensorial language, more popular in first-quantization. ``Quantizing'' the BZ theory, however, does not lead to the Dirac equation, but rather to a non-linear, Dirac--like equation, which can be regarded as the actual ``quantum limit'' of the BZ classical theory. Moreover, an original variational approach to the the BZ probabilistic fluid shows that it is a typical ``Weyssenhoff fluid'', while the Hamilton-Jacobi equation (linking together mass, spin and zbw frequency) appears to be nothing but a special case of de Broglie's famous energy-frequency relation. Finally, after having discussed the remarkable correlation between the gauge transformation U(1) and a general rotation on the spin plan, we clarify and comment on the two-valuedness nature of the fermionic wave-function, and on the parity and charge conjugation transformations.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed partial derivatives on coordinates. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations is presented, which can serve as a model of the graviton. The results obtained may prompt changes in some aspects of the construction of Feynman diagrams. Among other things, it presents a purely mathematical derivation of Maxwell's inhomogeneous equations without reference to empirical data on the action of electric current, which is usually referred to when deriving equations. Section 4 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor having four indices rather than two.