Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

P in the above paper refers to the Poincare group, and Wigner
established a correspondence between unitary irreducible
representations of P and relativistic (stable) particles via their mass
and spin.

The authors point out that it is generally believed that relativistic
resonances and unstable particles are associated with simple poles of
an S-matrix which is analytic, the location of the pole being:

1) s_R = (M - i GAMMA/2)^2

containing information about mass and width of the quasistable state
and the spin j being defined by the particular partial wave in which
the quasistable state occurs.

The set P is defined by:

2) P+ = {(L, a): (L, a) is an element of P, ao > = 0, a^2 > = 0}

and P+ is a subset and semigroup of P that remains invariant under:

3) (L1, a1)(L2, a2) = (L1L3, a1+ L1a2) for P

The operators defined by the (U(L, a)f)(u, j3) equation earlier provide
a continuous representation of P+ by by contractions in Lj^2(R^3) and
there aren't nonzero probabilities for t < 0 since the time evolution
is now given by a semigroup with nonnegative t

Osher Doctorow

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