Four-dimensional symmetry from a broad viewpoint: Pt. 3

Four-dimensional symmetry (from a broad viewpoint) actually allows for a new principle of universal probability, which dictates the distribution of quantized oscillators of all free fields, so that one can overcome divergence difficulties without upsetting unitarity. In order to gain a new perspective, it is traced the origin of this principle in quantum mechanics. It is found that it is related to the existence of a radical length R and the impossibility of measuring particle positions with unlimited accuracy. ''R-inherent quantum mechanics'' which incorporates these ideas is formulated. A new R-inherent uncertainty relation which, in addition to the usual restriction, also implies that the ultimate position accuracy is indicates a further departure from the classical physics. The p and q representations are connected by a new transform which reduces to the Fourier transform as R → 0. The usual Hilbert space with symmetry between p and q representations is no longer adequate to be the mathematical foundation. It has to be the generalized to a ''R-inherent Hilbert space'' whose co-ordinates, in terms of the momentum wave functions, appear to be ''curved'' with the universal probability function as the ''metric''. In the ''q-representation'' there are the usual momentum -iJdelta/deltaq and co-ordinate q. However, in the p-representation, there are the momentum p and the co-ordinate q = iJdelta/deltap - iRp/2(p2 + m2)sup(1/2). This indicates a violation of the p-q symmetry. The theory indicates that a particle can never be confined in a volume much smaller than 4πR3/3 and that matter cannot be squeezed into a singularity in space with infinite mass density by any force