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HADRONIC MATHEMATICS, MECHANICS AND CHEMISTRY 13
the system. The quantization of such a Hamiltonian then leads to a plethora of
illusions, such as the belief that the uncertainty principle for energy and time
is still valid while, for the example here considered, such a belief has no sense
because H does not represent the energy (see Refs. [9b] for more details).
Under the strict adoption of Assumption 1.2.1, all these ambiguities are absent
because H will always represent the energy, irrespective of whether conserved or
nonconserved, thus setting up solid foundations for correct physical interpretations.
1.2.3 General Inapplicability of Conventional
Mathematical and Physical Methods for Interior
Dynamical Systems
The impossibility of reducing interior dynamical systems to an exterior form
within the fixed reference frame of the observer causes the loss for interior dynamical
systems of all conventional mathematical and physical methods of the
20-th century.
To begin, the presence of irreducible nonselfadjoint external terms in the analytic
equations causes the loss of their derivability from a variational principle.
In turn, the lack of an action principle and related Hamilton-Jacobi equations
causes the lack of any possible quantization, thus illustrating the reasons why
the voluminous literature in quantum mechanics of the 20-th century carefully
avoids the treatment of analytic equations with external terms.
By contrast, one of the central objectives of this monograph is to review the
studies that have permitted the achievement of a reformulation of Eqs. (1.2.3)
fully derivable from a variational principle in conformity with Assumption 1.2.1,
thus permitting a consistent operator version of Eqs. (1.2.3) as a covering of
conventional quantum formulations.
Recall that Lie algebras are at the foundations of all classical and quantum
theories of the 20-th century. This is due to the fact that the brackets of the time
evolution as characterized by Hamilton’s equations,
dA
dt = ∂A
∂r k a
= ∂A
∂r k a
× drk a
dt + ∂A × dp ak
∂p ak dt
× ∂H
∂p ak
− ∂H
∂r k a
=
× ∂A
∂p ak
= [A, H], (1.2.6)
firstly, verify the conditions to characterize an algebra as currently understood
in mathematics, that is, the brackets [A, H] verify the right and left scalar and
distributive laws,
[n × A, H] = n × [A, H], (1.2.7a)
[A, n × H] = [A, H] × n,
(1.2.7b)