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NATO ASI series. Series B : Physics

Field Theory of the Spinning Electron: II — The New Non-Linear Field Equations

Erasmo Recami, Giovanni Salesi

Published Jan 1, 1997• 3 citations
Publisher
DOI
10.1007/978-1-4899-0081-4_20

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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.
We describe a new microscopic approach for analyzing interacting electron systems with local moments or, in principle, any local order parameter. We specialize attention to the doped Mott insulator phase of the Hubbard model, where standard weak-coupling perturbation methods fail. A rotationally invariant Stratonovich-Hubbard field is introduced to decouple the static spin components of the interaction at each site. Charge degrees of freedom are then treated by ``slave'' Hartree-Fock in the presence of a spatially varying random spin field. The effective action reduces to a classical Heisenberg model at half filling, insuring that the system has (i) finite-range order at T>0 with an exponentially diverging correlation length and (ii) a one-electron Mott-Hubbard gap in the presence of disorder. Away from half-filling properties are determined by strongly non-Gaussian fluctuations in the amplitude and orientation of the local spin fields. The saddle point equations of the theory at zero temperature reduce to inhomogeneous Hartree-Fock, so that disordering of domain walls at finite temperature is in principle included. We present preliminary small system results for the intermediate and large interaction regimes obtained by Monte Carlo simulation of random spin field configurations.
We formulate the Migdal-Eliashberg theory of electron-phonon interactions in terms of classical spins by mapping the free energy to a Heisenberg spin chain in a Zeeman magnetic field. Spin components are energy-integrated normal and anomalous Green's functions and sites of the chain are fermionic Matsubara frequencies. The Zeeman field grows linearly with the spin coordinate and competes with ferromagnetic spin-spin interaction that falls off as the square of the inverse distance. The spin-chain representation makes a range of previously unknown properties plain to see. In particular, infinitely many new solutions of the Eliashberg equations both in the normal and superconducting states emerge at strong coupling. These saddle points of the free-energy functional correspond to spin flips. We argue that they are also fixed points of kinetic equations and play an essential role in far from equilibrium dynamics of strongly coupled superconductors. Up to an overall phase, the frequency-dependent gap function that minimizes the free energy must be non-negative. There are strong parallels between our Eliashberg spins and Anderson pseudospins, though the two sets of spins never coincide.