BIRTH, DEVELOPMENT AND APPLICATIONS OF
ISO-MATHEMATICS, ISO-MECHANICS AND ISO-CHEMISTRY

By assuming that quantum mechanics is exactly valid for the electromagnetic interactions of point particles in vacuum, Sir Ruggero Maria Santilli ("Ruggero" for friends and colleagues) proposed in 1978 when he was at Harvard University the construction of the axiom-preserving isotopies of 20th century applied mathematics, today known as Iso-Mathematics, in the two 1978 monographs

which were written at the Lyman Laboratory of Physics of Harvard University under DOE support for the verification under strong interactions of the 1935 historical argument by Einstein-Podolsky-Rosen that "Quantum mechanics is not a complete theory" (EPR argument), whose study had been initiated during Santilli's Ph. D. studies in the 1960's at the University of Torino, Italy

Santilli notes in the above monographs that nuclear volumes are generally smaller than the sum of the volumes of their constituents, as a result of which, when members of a nuclear structure, protons and neutrons are generally in condition of mutual penetration of their hyperdense charge distributions, by therefore implying that strong interactions between extended nucleons are non-linear (in the wave functions), non-local (occurring in volumes) and not derivable from a potential, nowadays called non-Hamiltonian interactions (FTM-II) and technically identified as variationally non-selfadjoint interactions (FTM-I).

Since these interactions cannot be represented with a Hamiltonian by assumption, Santilli proposed in 1978 that they can be represented with a new operator T^* called the isotopic element, which is sandwiched in between the product of all quantum mechanical quantities (numbers, functions, operators, etc.), resulting in the new associativity-preserving product he called the iso-product < (FTM-II, Eq. (5), p. 71, on)

A* B = A T^* B,    T^* > 0,       (1)

where the isotopic operator T^* is solely restricted by the condition of being positive-definite, but otherwise possesses an arbitrary functional dependence on the coordinates r, momenta p, wave functions ψ and any other needed local variable T^*(r, p, ψ, ...) > 0, thus allowing indeed a representation of the indicated non-Hamiltonian interactions.

In the same 1978 proposal, Santilli constructed the isotopies of all branches of Lie's theory, today known as the Lie-Santilli iso-theory (see independent references below), with the iso-commutation rules (FTM-II, page 148 to 198)

[X_i, X_j]^* = X_i* X_j - X_j * X_i = C_ij^k X_k,       (2)

and initiated the isotopies of other aspects of 20th century applied mathematics under the full preservation of its axioms, merely realized in their most general possible form.

Thanks to the prior construction of the isotopies of Lie's theory (FTM-II), Santilli constructed in 1993 the axiom-preserving isotopies of the Lorentz symmetry, today known as the Lorentz-Santilli iso-symmetry, by achieving the invariance of arbitrary speeds of light propagating within inhomogeneous and anisotropic media. Subsequently, Santilli constructed the isotopies of all other space-time symmetries, including the rotational, SU(2), Poincare', and the spinorial covering of the Poincare' symmetry, see the review.

In the same FTM monographs, Santilli proposed the EPR completion of quantum mechanics into hadronic mechanics (FTM-II, p.112, 199, 254) with particular reference to its time reversal invariant isotopic branch today known as iso-mechanics which is based on the isotopic completion of the Schroedinger equation, today known as the Schroedinger-Santilli iso-equation on an iso-Hilbert iso-space H^* with iso-states |ψ(r)> (FTM-II, Eq. (14), p. 259)

H * |ψ(r)> = H T^*(r, p, ψ, ...) |ψ(r)> = E |*>,       (3)

and on the isotopy of Heisenberg's equation, today known as Heisenberg-Santilli iso-equation here expressed in the finite form (FTM-II, p.153, Eq. (18a)

i dA/dt = [A, H]^* = A* H - H*A, =

= A T^*(r, p, ψ, ...) H = H T^*(r, p, ψ, ...) A,       (4)

and in the finite form (FTM-II, Eq. (16), p. 260)

A(t) = e ^{H T*(r, p, ψ, ...) t i}A(0) e^{- i T*(r, p, ψ, ...) H} =

= U(t) A(0) U^†(t),

U U^† ≠ I.       (5)

As one can see, Santilli achieved in 1978 the foundations of first known isotopic completion of 20th century applied mathematics as well as of the first known quantitative representation of non-Hamiltonian strong interactions represented by the isotopic element T^*(r, p, ψ, ...) which is directly contained in the basic dynamical equations (3) (4) (5). Additionally, these discoveries were achieved by preserving all mathematical and quantum mechanical axioms to such an extent, that at the abstract realization-free level, Eqs. (3) (4) (5) coincide with the conventional quantum equations.

Iso-Chemistry is the axiom-preserving EPR completion of quantum chemistry with basic dynamical equations (1)-(5) elaborated by iso-mathematics to achieve a deeper treatment of molecular structures via the representation of the non-Hamiltonian interactions due to the overlapping of wave packets of valence electrons.

It should be noted that iso-mechanics and iso-chemistry are invariant over time because they are invariant under anti-Hermiticity, e.g., [A, B]^* = - [A, B]^{* †}. Consequently, iso-mechanics and iso-chemistry can only represent stable systems with non- Hamiltonian internal interactions, e.g., stable nuclei or molecules. For this reason, Santilli proposed the Lie-isotopic formulations in FTM-II as a particular case of the broader irreversible Lie-admissible/geno-formulations for the representation of nuclear fusions or combustion with product (A, B) = A < B - B > A = A R B - B S A, forward H^> > |> = E^>|> and backward H^< < |> = E^<|> geno- equations whose irreversibility is assured by different values of the forward and backward genotopic elements, R ≠ S. Subsequently, Santilli constructed the hyperstructural and isodual banches of hadronic mechanics indicated below (see the review

DEVELOPMENT OF ISO-MATHEMATICS, ISO-MECHANICS AND ISO-CHEMISTRY

The development of iso-mathematics and iso-mechanics up to maturity for applications has been reported in Santilli's 1995 monographs

EHM-I: R. M. Santilli, Elements of Hadronic Mechanics, Vol. I,

EHM-II: R. M. Santilli, Elements of Hadronic Mechanics, Vol. II,

the development of iso-chemistry has been reported in the 2001 monograph

FHC: R. M. Santilli, Foundation of Hadronic Chemistry,

and applications to biology have been reported in the. monograph

FHB: R. M. Santilli, Foundation of Hadronic Biology,.

These books remain to this day the most comprehensive mathematical-physical or mathematical-chemical presentations in their field.

Following the birth of isotopic methods in 1978, Santilli realized that Eqs. (3) (4) (5) do not maintain a crucial property of quantum mechanics, namely, the preservation of the eigenvalues under the same conditions but at different times, which property is necessary for the credibility of the experimental verifications. Additionally, Eqs. (3) (4) (5) have a non-unitary structure which is known to violate causality, e.g., admitting solutions in which effects precede their causes.

I^* = 1/T^* > 0,       (6)

I^* * A = A * I^* = A,       (7)

n^* = n I^*,       (8)

where, depending on the application, I^* can be: an ordinary positive number as used in iso-cryograms (EHM-I, Appendix 2C, p. 84); a matrix representing the extended dimension, shape and density of hadrons (EHM-II, Chapter 6, p. 209 on); or a non-linear combination of wave functions to overcome the huge Coulomb repulsion between the identical electrons in molecular bonds (FHC, Chapter 4 on).

The axiomatically important property is that the set of iso-numbers (8) for all possible numbers "n" and a given positive-definite iso-unit I^* verifies all axioms of a numeric field, thus being fully usable for experiments.

The 1993 discovery of the iso-numbers triggered the reformulation of all preceding mathematical (EHM-I), physical (EHM-II) and chemical (FHC) advanced on iso-spaces S^* over an iso-field F^*. These advances did achieve the needed numerical invariance of the iso-eigenvalues E^* = E I^* under the same conditions but at different times. In fact, when correctly formulated on the iso-Hilbert iso-space H^* over an iso-field F^*, the Schroedinger-Santilli isoequation (3) becomes

H * |ψ^*(r^*)> =

= [ (1/2m) p^* T^* p^* + V^* (r^* ) ] T^* (r, p, ψ^*, ...) |ψ^*(r^*)> =

= E^* * |ψ^*(r^*)> = E |ψ^*(r^*)>,       (9)

where the iso-eigenvalues E^* is now iso-invariant under the correct reformulation of time evolution (5) given by,

A(t) = e ^{H T*(ψ*, ...) t i}A(0) e^{- i t T*(ψ*, ...) H} =

= I^* (e^{*,H t i}) * A (0) * (e^*,-itH) I ^*= U^*(t) A(0) U^*†,       (10)

(where e^* is the iso-exponentiation, see EHM-I, p. 136), because the above time evolution verifies the conditions of iso-unitarity (EHM-I, p. 335)

U^* * U^{* †} = U^{* †} * U^* = I^*.       (11)

In fact, as expected from the need of applying the isotopies to all aspects of conventional theories, the class of equivalence of isotopic methods is given by class of all infinitely possible iso-unitary transformations. Causality is then regained on iso-spaces over iso-fields due to the evident topological equivalence between unitarity and iso-unitarity axioms.

It should be recalled that, given a quantum mechanical or chemical model, its EPR completion into an iso-mechanical or iso-chemical model is fully achieved via a simple non-unitary transform which maps the unit into the iso-unit, the product into the iso-product, ordinary numbers into iso-numbers, functions into iso-functions, etc. (EHM-I)

U U^† = I^* ≠ I,

I → U I U^† = I^*, AB → U(A B)U^† = A^* T^* B^* = A^* * B^*,

n → U n U = n I^*, r → U r U^* = r I^* = r^*, f(r) → f^*(r^*), etc.       (12)

Once an iso-mathematical iso-mechanical or iso-chemical formulation has been achieved in this way, any additional transform has to be iso-unitary resulting in the numerical invariance of the iso-unit and of the isotopic element (EHM-II)

U^* * I^* * U^{* †} = U^{* †} * I^* * U^* = I^*,

U^* * (A * B) * U^{* †} = A T^* B' = A' * B'.    (13)

Despite all the above advances, Santilli communicated to colleagues in the early 1990's that isotopic methods could not yet be applied to specific physical or chemical problems. This was due to the fact that the Schroedinger-Santilli iso-equation (9) generally represents interactions on volumes, rather than at isolated points. Consequently, the use of the conventional Newton-Leibnitz differential calculus for the characterization of the linear momentum

p |ψ(r)> = - i ∂_{r |ψ(r)>,    (14)}

would lead to catastrophic inconsistencies. Hence, Santilli had no other option than that of generalizing the Newton-Leibnitz differential calculus from its historical definition at isolated points r, to a covering formulation defined on volumes r^* = r I^*, which he achieved in the 1996 memoir published at the Italian Rendiconti Circolo Matematico Palermo Vol. 42, 7-82 (1996),

IDC: R. M. Santilli, "Nonlocal-Integral Isotopies of Differential Calculus, Mechanics and Geometries."

In this way, Santilli introduced the novel iso-differential calculus which is characterized by all the infinitely possible iso-differentials of iso-coordinates d^* r^* which are such to recover the conventional differential dr when I^* is independent from r (IDC, Eq. 1.27), p.20)

Lim(_{I^*→ 1}) d^*r^* = dr,    (15)

with solution used in physical and chemical applications

d^*r^* = T^* d r^* = dr + r T^*dI^*,    (16)

and consequential iso-partial iso-derivative

∂^* _r^* f^*(r^*) = I^* ∂_r^* f^*(r^*),    (17)

that finally allowed the consistent definition of the iso-linear iso-momentum

p^* * | ψ(r^*)> = - i ∂^*_r^* = - i I^* ∂ _r^* | ψ(r^*)>.      (18)