hadronic mathematics, mechanics and chemistry - Institute for Basic ...

840 RUGGERO MARIA SANTILLI= U × [A, H] × U † = [ˆ,Ĥ] = Â × ˆT r × Ĥ − Ĥ × ˆT r × Â, (6.1.24)where one should note isounits of time and space denoted with the subindecest, r, respectively (generally ignored whenever there is no ambiguity).Similarly, we have the lifting of canonical commutation rules into isocanonicalisocommutation rules also introduced for the first time in memoir [14][r i , p j ] = i × δ i j → [ˆr iˆ,ˆp j ] = îˆδ i j = i × Î × δi j, (6.1.25)Similarly, we have the lifting of the Schrödinger equations into the Schrödinger-Santilli isoequations first formulated in an invariant form in memoir [15]i × × ∂ |ψ >= H × |ψ >→∂t→ î ˆ× ˆ∂ˆ∂ˆt | ˆψ(ˆt, ˆr) >= i × Ît × ∂ ∂ˆt == Ĥ ˆ×| ˆψ >= Ĥ(ˆr, ˆp) × ˆT r (ˆt, ˆr, ˆp, Ê, ˆψ, ...) × | ˆψ > . (6.1.26)and the lifting of the linear momentum into isolinear isomomentum (reachedfor the first time in memoir [15] following decades of search due to the precedingabsence of the isodifferential calculusp k × |ψ >= −i × × ∂ k |ψ >→ U × (p k × |ψ >) == U × p k × (U × I † ) −1 × U × |ψ >= ˆp k ˆ×| ˆψ >= −U × (i × × ∂ k |ψ >) == −î ˆ× ˆ∂ k | ˆψ >= −i × Îi k × ∂ i| ˆψ >, (6.1.27)We should also recall the new invariance of the conventional inner productunder isotopic transforms here expressed for a non-null constant z ∈ R< ψ| × |ψ > ×I ≡< ψ| × z 2 × | psi > ×(z − 2 × I) ≡< ψ| ˆ×| psi > ×Î, (6.1.28)with extension to an arbitrary positive-definite nonunitary transform and isounitU × U † = Î > 0 via the techniques of Volume I.Note the abstract identity of hadronic and quantum mechanics as illustrated bythe property that all relative equations and physical laws are merely differentiatedby a ”hat” denoting the existence of a broader realization of the same axioms.The above occurrences forcefully establishes the validity of nonrelativistic andrelativistic hadronic mechanics in the conditions of their applicability, evidentlybecause of the preservation of the conventional axioms of quantum mechanics.In turn, this forcefully establishes the validity of the Minkowski-Santilli isospacesfor interior particle conditions as verified below.Alternatively, the preservation of the abstract axioms in the transition fromquantum to hadronic mechanics renders nonscientific the aprioristic selection of