Relativistic Hydrodynamics of the Dirac Matter

Despite various related attempts, it has not been attacked in any consequent manner to establish a complete tensor formulation for the Dirac electron field and also thereby to grasp this with a concrete model strictly. This paper is concerned with solving these problems. It is shown that the Dirac field can equivalently be represented in a closed form with the set of real field variables which directly mean density, velocity, momentum, and spin, and one more internal variable θ, respectively. Formally speaking, the Dirac spinor ψ and γ-matrices are here superseded completely by the set of two scalar (a scalar and a pseudoscalar) and three vector (two vector and a pseudovector) variables. The established formulation manifests a new kind of relativistic hydrodynamics. This offers the hydrodynamical model of the Dirac matter, to which one is led almost inevitably when one tries to depict it into some exact model. The structure of this formulation is clarified in detail. Equations of motion are brought to compact hydrodynamical forms [eqs. (d′) or (f) or (i)]. Besides them, an important subsidiary condition governs the behaviour of the matter to connect the vorticity of the momentum field to the velocity and spin fields [eq. (II)]. Those basic equations of the formulation are all deduced from its own Lagrangian. The basic equations are cast also in various different forms. Some characteristic features of the hydrodynamics are the fact that the distinctions between proper mass density and rest particle density and also between particle momentum and velocity are primarily specified by the θ, associated with the possibility of “ass” -like behaviours. The energy-momentum conservation law also manifests here a new structure. It may be interpreted as the Newtonian equation of motion for the particle momentum-energy vector under the usual Lorentz force plus certain quantum effects, and it has the distinct meaning from the Eulerian equation of flow. This quantum effect is interpreted as mechanical stress and flow of heat taking place inside the fluid. The theory also provides us with a new directly physical point of view concerning various transformation properties of the Dirac field, e.g., those for charge conjugation and time reversal. Furthermore it reveals a conspicuous quasi-symmetrical property of the Dirac field existing between velocity and spin. In particular this new symmetry becomes perfect for the case of zero rest mass (neutrino). The theory is formulated for cases of Dirac matter under external electromagnetic field and also of interacting Dirac and electromagnetic fields. It is manifestly gauge-independent in both cases. In this paper, however, the theory is worked out only for the case of c-number Dirac field, in other words, the quantum mechanics of a Dirac electron. The mathematical background of the method is supplied in the Appendix, where the mathematical apparatus is systematized to establish how one can manipulate the Dirac field solely with the set of tensor quantities which are related to the Dirac spinor as its bilinear covariants.