Fermi Quantization of Tensor Systems

Abstract

In this paper we characterize classical tensor systems which admit Fermi quantization as those having unitary Lie Poisson brackets. Examples include Euler’s tensor equation for a rigid body and Dirac’s equation in tensor form, for which we give a new derivation which is simple and direct.

Geometric Origin of Physical Constants in a Kaluza-Klein Tetrad Model

“…For this case, as well as its teleparallel equivalent, the elimination of Newton's constant along with all coupling constants is possible only when a special relation holds between various terms in the Lagrangian (see when the tetrad is expressed in terms of the spinor." [14] These orthogonality and completeness conditions give rise to the oscillator modes that lead directly to fermion creation and annihilation operators [17]. Indeed, the oscillator modes of a bispinor field are precisely the oscillator modes of a tetrad field.…”

“…On such space-times, spinor structures and homotopy classes of global tetrad fields are synonymous [20]. The second assumption, required for a consistent interpretation within quantum mechanics, is that we restrict to solutions of the tensor Dirac equation having "unique continuation" [17], [24], [25]. That is, for any observer, the history of a tetrad field in the past must uniquely determine its evolution into the future [24] − [29].…”

“…Non-unique continuation of solutions occurs in other areas of physics, such as in fluid dynamics, but is not considered to be appropriate for quantum mechanics. Note that there can be a bispinor field having unique continuation, whose tetrad field cannot be uniquely predicted into the future [17], [26], [27]. However, in a Minkowski spacetime, continuation of the tetrad field is unique if we restrict to physically realizable bispinor solutions of the Dirac equation whose energy spectrum is bounded from below [17], [24], [25], [28], [29].…”

Hestenes' Tetrad and Spin Connections

“…For a deformable body, the elastic modes are self-adjoint and the rigid modes are isometric with respect to the Euclidean metric on R 3 . This analogy extends into the quantum realm since rigid modes satisfying Euler's equation can be Fermi quantized [13], [14].…”

Measuring a Kaluza-Klein radius smaller than the Planck length

“…This analogy extends into the quantum realm since rigid modes satisfying Euler's equation can be Fermi quantized [4]. As with Euler's equation for a rigid body, the tetrad formulation of Dirac's partial differential bispinor equation is a classical Hamiltonian system, with (noncanonical) unitary Lie-Poisson brackets [4]. Fermi quantization of such classical systems is possible whenever the Lie algebra can be represented by fermion creation and annihilation operators.…”