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, we trace the origin of this principle in quantum mechanics. We find that it is related to the existence of a radical lengthR and the impossibility of measuring particle positions with unlimited accuracy. We formulate «R-inherent quantum mechanics» which incorporates these ideas. We derive a newR-inherent uncertainty relation which, in addition to the usual restriction, also implies that the ultimate position accuracy is Δq ult≈R rather than Δq ult=0. It indicates a further departure from the classical physics. Thep andq representations are connected by a new transform which reduces to the Fourier transform asR→0. The usual Hilbert space with symmetry betweenp andq 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» we have the usual momentum−iJ∂/∂q and co-ordinateq. However, in thep-representation, we have the momentump and the co-ordinateq -iJ∂/∂q-iR p/2(p 2+m 2)1/2. This indicates a violation of theq-q symmetry. The theory indicates that a particle can never be confined in a volume much smaller than 4πR 3/3 and that matter cannot be squeezed into a singularity in space with infinite mass density by any force.