_{1}

^{*}

Quantum
geometrodynamics (QGD) has established the following fundamental facts: First, every
elementary particle is the physical realization of a certain irreducible
4-quantum operator of spin (rank) 0, 1/2 or 1.
A photon (boson) is the physical realization of an irreducible 4-quantum operator
of spin zero. A fermion is the physical realization of an irreducible 4-quantum
operator of spin 1/2. A graviton (boson) is the physical realization
of an irreducible 3-quantum operator of spin zero, and the Ws and mesons
(bosons) are the physical realizations of irreducible 3-quantum operator of
rank one. Second, the particles of every composite fermion system (nuclei,
atoms, and molecules) reside in a certain 4-quantum space which is partitioned
into an infinite set of subspaces of dimension 4^{n }(*n* = 1, 2, 3, L, ∞; *n* is the index of the subspace and *n* is
called principal quantum number by physicists, and period by chemists) each of
which is reducible to a set of 2-level cells [1]. With these two fundamental facts, the
complexities associated with atomic, nuclear, and molecular many-body problems
have evaporated. As an application of the reducibility scenario we discuss in
this paper the explicit construction of the periodic table of the chemical
elements. In particular we show that each chemical element is characterized by
a state ket ｜E_{n};
l, m_{1}; s, m_{s}〉where *l* is
orbital angular momentum, *s *= 1/2, E_{n
= }E_{1 }+ khv (k = 1, 2, 3, L, ∞, *E*_{1} is the
Schr?dinger first energy level, and *v* is the
Lamb-Retherford frequency).

Democritus, the father of Greek Chemistry, was the originator of the atomic hypothesis. In 4C B.C. he came up with his atomic hypothesis that atoms are elementary indivisible and indestructible particles of which all matter was composed. The ideas of Democritus were expressed in the poem “De Rerum Natura” by Lucretius in the IC B.C.

There was, however, a rival hypothesis due to Aristotle which asserted that all matter was composed of varying proportion of the four Aristotelian elements, namely, earth, water, air, and fire. Because of the authority and overwhelming influence of Aristotle, the Democritian hypothesis was rejected, and the Aristotelian hypothesis was in force for about 2000 years.

Through the pioneering works of restless chemists of the 17, 18, and 19 centuries AD, notably Robert Boyle, Antoine Lavoisier, and John Dalton, the Aristotelian hypothesis was roundly condemned and eventually rejected. Their efforts, backed up by experimental results, helped to reinstate the Democritian hypothesis. Between 1860 and 1913 AD the empirical works of Chemists J.A.R. Newlands, D.I, Mendeleeve, H.G.J. Mosley and many others were summarized in a table called the Periodic

The periodic law implied that the atom, unlike the Democritian atom, had to be composite. The desire to determine what basic constituents of the atom could give rise to such a fundamental law brought physicists into the atomic fray. The fundamental experiments done between 1887 and 1932 by H. Hertz, A.H. Becquerel, Marie and Pierre Curie, E. Rutherford, J. Chadwick and many others brought about a new understanding of the structure and the nature of the forces in the atom. The atom consists of a central core called the nucleus surrounded by electrons, and the forces therein are non-Newtonian. By the 1920s classical field theory had made a remarkable transition to quantum field theory through the pioneering works of theorists M. Planck, L. de Broglie, W. Heisenberg, M. Born, P. Jordan, W. Pauli, E. Schrodinger, O. Klein, W. Gordon, P.A.M. Dirac, and others.

In 1926 Erwing Schrodinger took a giant step in the development of quantum mechanics with the construction of the Schrodinger equation which he applied to the Hydrogen atom. Schrodinger’s theory of the hydrogen atom recorded some spectacular results, but was flawed by the fact that it was a non-relativistic theory, amongst many other defects. In 1928 P.A.M. Dirac created his relativistic theory of the hydrogen atom called the Dirac equation [

As far as we know, no formal approach exists today for the treatment of many-electron atoms. The problem is so complex that only Schrodinger-based approximation methods have been developed: Variation calculations are used to obtain the ground states of the lightest atoms: for heavier atoms, the central-field approximation, Thomas-Fermi model, Hartree and Hartree-Fock approximations; and for nuclei, various many-body techniques are employed depending on the structure of the system. One major reason why these methods cannot yield satisfactory results is that, as palpably demonstrated by the Dirac’s theory, atoms are embedded in a 4-dimensional pseudoeuclidean background. Further, atoms, nuclei, and molecules are discrete systems, hence differential equations are not applicable. The Schrodinger theory is therefore not expected to apply to many-electron atoms! It is for this reason that we decide to consider other methods for the treatment of many-particle atoms and nuclear systems.

The situation that confronts us in many-fermion systems is similar to the problem that gave birth to statistical mechanics at the beginning of the 20th century. Like that case a completely new approach is needed here.

Our solution is based on the reducibility theorem according to which a 4-quantum subspace of index

The “2” outside the summation sign in (1) comes from the subspace of index

The Formula (1) gives the structural design for all fermion systems in the universe. The structure can be interpreted by analogy with gravitation: Each term (period) gives the “floor” of a “building” in an electrostatic environment established by the nucleus and electrons. The first period (first term) is the “ground floor”, with antifermions (positrons and antineutrinos) forming the “foundation”; the second period (second term), “^{th} period. The other periods are interpreted similarly. We shall now construct the periodic table of the chemical elements using the foregoing prescription and the dimensionality theorem. Here (X(a)) gives element “X” of atomic number

First period

Geometrical structure:

This is a “foundation” structure, the “ground floor” of the atomic world built with fermions and antifermions. For the electronic world the structure is (2), the s-shell having two levels, with the electronic configuration.

State Kets:

Chemical elements:

Second period

This is the first floor which has two levels (s-shell) as the deck and six levels (p-shell) as the first floor, with electronic configuration

State Kets:

Chemical elements:

Third period

The admissible structure here is

with the 3d-shell outstanding, as explained before.

State Kets:

Chemical elements:

4th period n = 4:

Taking cognizance of the 3d-shell, the admissible structure is

with 4d and 4f shells outstanding because there is no provision for “decks” for them.

State Kets:

Chemical elements:

5th period

The procedure now has a familiar ring: The admissible structure is

State Kets:

Chemical elements:

6th period

Given the deck “2” here “structural” stability requires that the 4f, 5d, and 6p shells constitute part of a floor on the 6s deck. Thus, the admissible structure is

with the

State Kets:

Chemical elements:

7th period

The 7th period is similar to the 6th period.

The structure (electronic configuration) that improves stability is

State Kets:

Chemical elements:

This procedure can in principle be continued indefinitely so that one can construct any atom one desires; but the atomic structures become progressively unstable. Actual laboratory “construction” of these unstable atoms is possible but they have only a fleeting existence.

We infer from the foregoing that fermions reside in definite quantum states of definite angular momentum:

The fermion ground state, a pure quantum state, is described geometrically by

The so called first excited state

The electronic states of molecules are more complex than the atomic and nuclear states. These states can, however, be deduced from the electronic states of atoms. As an example we consider H_{z}-molecule. The state ket of H_{2} is just the tensor product of state kets of two hydrogen atoms

Our theoretically derived periodic table (PT) confirms and generalizes the conventional PT of the chemical elements. The horizontal rows of elements are called periods, numbered from

The vertical columns define families of elements: The first 2 columns correspond to elements of angular momentum (spin)

Lastly, we call the world constructed with leptons and nucleons matter world. A world analogous with the matter world, called antimatter world, can be constructed in a similar way with antileptons and antinucleons. Thus, there exists matter―antimatter symmetry in the fermion world.