Regularized QFT that emerges as a consequence of a quantum weak law of large numbers

Department of Physics, Politechnika Gdanska, Narutowicza 11/12, 90-952 Gdansk, Poland

Harmonic oscillators whose frequencies are OBSERVABLES and not PARAMETERS naturally lead to reducible representations of harmonic-oscillator Lie algebras, whose central element I(k) at the rhs of

[a(k),a(k')*]=I(k) delta(k,k')

is not an identity but the operator of frequency of successes known from quantum weak laws of large numbers. I(k) has eigenvalues 0/N,1/N,2/N... (N-1)/N, N/N where N is a (finite and Poincare invariant) natural number that characterizes the representation, and which is NOT related to the (infinite) number of k's. In the weak limit N->infinity I(k) gets replaced by Prob(k), the probability of finding k in a given vacuum state. The limiting formalism is thus equivalent to the one that would start from very beginning with

[a(k),a(k')*]=Prob(k) delta(k,k')

and Prob(k) would play here a role of a cutoff function.

So, can we employ this observation in construction of a finite, but not necessarily Wightmanian QFT, where the weak law of large numbers N->infinity would be the correspondence principle between a more fundamental finite-N QFT and the more standard but automatically REGULARIZED QFT? A relatively complete answer will be given in my talk, but partial results can be found also in these lecture notes: http://www.mif.pg.gda.pl/kft/Regularyzacja/rbq.pdf