3 Dirac [mathematical expression not reproducible]-matrices as a four-vector
If [??] is a vector, then the matrices [[gamma].sup.l] must be components of a 3-vector, so must the matrix [[gamma].sup.0] be the component of the time-vector in the usual four-vector formalism, hence [[gamma].sup.[mu]] must be a four-vector.
To represent this condition in a covariant form in an arbitrary frame, we assume that in our model there exists a four-vector [p.sub.[mu]] which for the particle at rest has the components ([p.sub.0], 0).
Frenkel spin-tensor obeys [S.sup.[mu]v][p.sub.v] = 0, that is, [[PHI].sup.[mu]] = 0, and can be used to construct four-vector of spin (below we also call it Pauli-Lubanski vector)
Explicit breaking of the Lorentz symmetry may induce faster-than-light dynamics for neutrino wave packets, with a time-like four-vector product [p.sup.[mu]][p.sub.[mu]] > 0 (see [27, 28]).
Upon promotion to a four-vector, one has [[??].sup.[mu]] = (0, [??]).
For example, if we take five generators Sa, where four are components of some four-vector
S [mu] and the fifth is scalar [S.sub.[mu]] the matrices [A.sub.[mu]] are connected with the s = 0 Kemmer-Duffin matrices .
Some applications have reached over 100Mflops for 16 processors, which is about the speed a four-vector
processor system of comparable speed achieves.
However, according to Padmanabhan , there exists a suitable definition for what we will call an energy current four-vector G of the gravitational field, such that the total energy current P := G + P is conserved:
This four-vector P should not be confused with the three-vector P in [(22).sub.1].
where [A.sup.[micro]] is the electromagnetic potential four-vector
([phi],[??]), [phi] is the scalar potential and [??] the vector potential.
By analogy with the current density four-vector
[j.sup.v] = (c[??], j), where g is the charge density, and j is the current density vector, which obeys a similar conservation relation, we define the Poynting four-vector