One more note on the properties of spin-1 matrices

Samuel M. Harris

Vishnu K. Agrawala

ELENA M. GEORGIEVA, Y. S. Kim

It is noted that 2x2 "S" matrices in multilayer optics can be represented by the Sp(2) group whose algebraic property is the same as the group of Lorentz transformations applicable to two spacelike and one timelike dimensions. It is also noted that Wigner's little groups have a slide-rule-like property that allows us to perform multiplications by additions. It is shown that these two mathematical properties lead to a cyclic representation of the S matrix for multilayer optics, as in the case of ABCD matrices for laser cavities. It is therefore possible to write the N-layer S matrix as a multiplication of the N single-layer S matrices resulting in the same mathematical expression with one of the parameters multiplied by N. In addition, it is noted, as in the case of lens optics, that multilayer optics can serve as an analog computer for the contraction of Wigner's little groups for internal space-time symmetries of relativistic particles.

Richard J. Blume

Francesco R. Ruggeri

In Part I of this note, we argued that the exp(ipx) portion of a free quantum particle’s wavefunction follows from d/dx A(x,t) = p extended to an eigenvalue equation: -id/dx exp(iA) = p exp(iAc) “p” appears as a scalar because it is the momentum magnitude, but one may consider flow in 3-space i.e. px,py,pz. In such a case, one might think of using basis vectors ei such that ei dot ei =1 (i=1,2,3). Given that one has an operator equation, one uses basis operators or matrices instead leading to the Pauli matrices or 4x4 matrices as found by Dirac if one imposes the restriction that upon “squaring” the operators disappear because one only has pp. As a result, we argued that intrinsic spin is linked to 3-space recognition for a free particle. The resulting spin, however, is spin ½ so what about other spins say 0 or 1? This seems to imply that free recognition of 3-space does not apply to other spin cases as it does to spin ½. The Stern-Gerlach experiment already shows that spin ½ is interesting in that one may have spin up or down project along any 3-space axis. In Part I, we showed that a wave-equation developed for the photon from d/dt (partial) b (ElEl + BB) + grad dot ElxB = 0 contains an eijk (-i d/dxi) term with eijk being the Levi-Civita symbol as well as the spin matrix. This is not the Pauli matrix (or matrix containing it) and suggests a different geometry or basis scheme. Thus, there is physical angular momentum (spin) in a plane perpendicular to the momentum and the (1,0) spin component is “hidden”, but associated with angular momentum in another spatial direction. As a result, 3-space is not treated in an independent way as in the spin ½ case. In this note, we investigate this idea further by considering the deuteron, He4 and He3 and try to argue that intrinsic spin still seems to have a connection to 3-space recognition, but that the fully independent 3-space recognition corresponds to spin ½. We only consider spin 0, ½ and 1 in this note.