We present a new interpretation of the Higgs field as a composite particle made up of a positive, with, a negative mass Planck particle. According to the Winterberg hypothesis, space, i.e., the vacuum, consists of both positive and negative physical massive particles, which he called planckions, interacting through strong superfluid forces. In our composite model for the Higgs boson, there is an intrinsic length scale associated with the vacuum, different from the one introduced by Winterberg, where, when the vacuum is in a perfectly balanced state, the number density of positive Planck particles equals the number density of negative Planck particles. Due to the mass compensating effect, the vacuum thus appears massless, charge-less, without pressure, energy density, or entropy. However, a situation can arise where there is an effective mass density imbalance due to the two species of Planck particle not matching in terms of populations, within their respective excited energy states. This does not require the physical addition or removal of either positive or negative Planck particles, within a given region of space, as originally thought. Ordinary matter, dark matter, and dark energy can thus be given a new interpretation as residual vacuum energies within the context of a greater vacuum, where the populations of the positive and negative energy states exactly balance. In the present epoch, it is estimated that the dark energy number density imbalance amounts to, , per cubic meter, when cosmic distance scales in excess of, , are considered. Compared to a strictly balanced vacuum, where we estimate that the positive, and the negative Planck number density, is of the order, particles per cubic meter, the above is a very small perturbation. This slight imbalance, we argue, would dramatically alleviate, if not altogether eliminate, the long standing cosmological constant problem.

We prove: (1) the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable, (2) the set of all Diophantine equations which have at most finitely many solutions in positive integers is not recursively enumerable, (3) the set of all Diophantine equations which have at most finitely many integer solutions is not recursively enumerable, (4) analogous theorems hold for Diophantine equations D(x₁, …, x_p) = 0, where p ∈ N\{0} and for every i ∈ {1, …, p} the polynomial D(x₁, …, x_p) involves a monomial M with a non-zero coefficient such that x_i divides M, (5) the set of all Diophantine equations which have at most k variables (where k ≥ 9) and at most finitely many solutions in non-negative integers is not recursively enumerable.

We work within a Winterberg framework where space, i.e., the vacuum, consists of a two component superfluid/super-solid made up of a vast assembly (sea) of positive and negative mass Planck particles, called planckions. These material particles interact indirectly, and have very strong restoring forces keeping them a finite distance apart from each other within their respective species. Because of their mass compensating effect, the vacuum appears massless, charge-less, without pressure, net energy density or entropy. In addition, we consider two variable models, where, , is Newton’s constant, and, , increases with an increase in cosmological time. We argue that there are at least two competing models for the quantum vacuum within such a framework. The first follows a strict extension of Winterberg’s model. This leads to nonsensible results, if increases, going back in cosmological time, as the length scale inherent in such a model will not scale properly. The second model introduces a different length scale, which does scale properly, but keeps the mass of the Planck particle as, the Planck mass. Moreover we establish a connection between ordinary matter, dark matter, and dark energy, where all three mass densities within the Friedman equation must be interpreted as residual vacuum energies, which only surface, once aggregate matter has formed, at relatively low temperatures. The symmetry of the vacuum will be shown to be broken, because of the different scaling laws, beginning with the formation of elementary particles. Much like waves on an ocean where positive and negative planckion mass densities effectively cancel each other out and form a zero vacuum energy density/ zero vacuum pressure surface, these positive mass densities are very small perturbations (anomalies) about the mean. This greatly alleviates, i.e., minimizes the cosmological constant problem, a long standing problem associated with the vacuum.

Backward electromagnetic waves are extraordinary waves with contra-directed phase velocity and energy flux. Unusual properties of the coherent nonlinear optical coupling of the phase-matched ordinary and backward electromagnetic waves with contra-directed energy fluxes are described which enable greatly-enhanced frequency and propagation direction conversion, parametrical amplification, as well as control of shape of the light pulses. Extraordinary transient processes that emerge in such metamaterials in pulsed regimes are described. The results of the numerical simulation of particular plasmonic metamaterials with hyperbolic dispersion are presented, which prove the possibility to match phases of such coupled guided ordinary and backward electromagnetic waves. Particular properties of the outlined processes in the proposed metamaterial are demonstrated through numerical simulations. Potential applications include ultra-miniature amplifiers, frequency changing reflectors, modulators, pulse shapers, and remotely actuated sensors.