Relativistically Extended Modification of the Schroedinger Equation

We propose a nonlinear modification of the Schr\"{o}dinger equation that possesses the main properties of this equation such as the Galilean invariance, the weak separability of composite systems, and the homogeneity in the wave function. The modification is derived from the relativistic relation between the energy and momentum of free particle and, as such, it is the best relativistic extension of the Schr\"{o}dinger equation that preserves the properties in question. The only change it effectively entails in the Schr\"{o}dinger equation involves the conserved probability current. It is pointed out that it partially retains the linear superposition principle and that it can be used to model the process of decoherence.Comment: Latex, 9 pages, extended, new references adde

Non-separability without Non-separability in Nonlinear Quantum Mechanics

We show an example of benign non-separability in an apparently separable system consisting of $n$ free non-correlated quantum particles, solitonic solutions to the nonlinear phase modification of the Schr\"{o}dinger equation proposed recently. The non-separability manifests itself in the wave function of a single particle being influenced by the very presence of other particles. In the simplest case of identical particles, it is the number of particles that affects the wave function of each particle and, in particular, the width of its Gaussian probability density. As a result, this width, a local property, is directly linked to the mass of the entire Universe in a very Machian manner. In the realistic limit of large $n$ if the width in question is to be microscopic, the coupling constant must be very small resulting in an ``almost linear'' theory. This provides a model explanation of why the linearity of quantum mechanics can be accepted with such a high degree of certainty even if the more fundamental underlying theory could be nonlinear. We also demonstrate that when such non-correlated solitons are coupled to harmonic oscilators they lead to a faster-than-light nonlocal telegraph since changing the frequency of one oscillator affects instantaneously the probability density of particles associated with other oscillators. This effect can be alleviated by fine-tuning the parameters of the solution. Exclusion rules of a novel kind that we term supersuperselection rules also emerge from these solutions. They are similar to the mass and the univalence superselection rules in linear quantum mechanics. The effects in question and the exclusion rules do not appear if a weakly separable extension to $n$-particles is employed.Comment: Latex, 13 page

On the Staruszkiewicz Modification of the Schroedinger Equation

We discuss Staruszkiewicz's nonlinear modification of the Schr\"{o}dinger equation. It is pointed out that the expression for the energy functional for this modification is not unique as the field-theoretical definition of energy does not coincide with the quantum-mechanical one. As a result, this modification can be formulated in three different ways depending on which physically relevant properties one aims to preserve. Some nonstationary one-dimensional solutions for suitably chosen potentials, including a KdV soliton, are presented, and the question of finding stationary solutions is also discussed. The analysis of physical and mathematical features of the modification leads to the conclusion that the Staruszkiewicz modification is a very subtle modification of the fundamental equation of quantum mechanics.Comment: Latex, 16 page

Is an Electromagnetic Extension of the Schroedinger Equation Possible?

The idea of equivalence of the free electromagnetic phase and quantum-mechanical one is investigated in an attempt to seek modifications of Schr\"{o}dinger's equation that could realize it. It is assumed that physically valid realizations are compatibile with the U(1)-gauge and Galilean invariance. It is shown that such extensions of the Schr\"{o}dinger equation do not exist, which also means that despite their apparent similarity the quantum-mechanical phase is essentially different from the electromagnetic one.Comment: Latex, 5 pages, the list of references update

Energy Ambiguity in Nonlinear Quantum Mechanics

We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the canonical energy-momentum tensor and the energy functional defined as the expectation value of the corresponding nonlinear Hamiltonian operator. Some examples of such ambiguity are presented for a particularly simple model and some known modifications. However, we point out that there exist a class of nonlinear modifications of the Schr\"{o}dinger equation where this difference does not occur, which makes them more consistent in a manner similar to that of the linear Schr\"{o}dinger equation. It is found that necessary but not sufficient a condition for such modifications is the homogeneity of the modified Schr\"{o}dinger equation or its underlying Lagrangian density which is assumed to be ``bilinear'' in the wave function in some rather general sense. Yet, it is only for a particular form of this density that the ambiguity in question does not arise. A salient feature of this form is the presence of phase functionals. The present paper thus introduces a new class of modifications characterized by this desirable and rare property.Comment: Slightly extended, new references added, Latex, 15 page

Model of the gravitational dipole

A model of the gravitational dipole is proposed in a close analogy to that of the global monopole. The physical properties and the range of validity of the model are examined as is the motion of test particles in the dipole background. It is found that the metric of the gravitational dipole describes a curved space-time, so one would expect it to have a more pronounced effect on the motion of the test particles than the spinning cosmic string. It is indeed so and in the generic case the impact of repulsive centrifugal force results in a motion whose orbits when projected on the equatorial plane represent unfolding spirals or hyperbolas. Only in one special case these projections are straight lines, pretty much in a manner observed in the field of the spinning cosmic string. Even if open, the orbits are nevertheless bounded in the angular coordinate $\theta$.Comment: Latex, 10 pages, minor typesetting correction

Extension of the Staruszkiewicz Modification of the Schroedinger Equation

We present an extension of Staruszkiewicz's modification of the Schr\"{o}dinger equation which preserves its main and unique feature: in the natural system of units the modification terms do not contain any dimensional constants. The extension, similarly as the original, is formulated in a three-dimensional space and derives from a Galilean invariant Lagrangian. It is pointed out that this model of nonlinearity violates the separability of compound systems in the fundamentalist approach to this issue. In its general form, this modification does not admit stationary states for all potentials for which such states exist in linear quantum mechanics. This is, however, possible for a suitable choice of its free parameters. It is only in the original Staruszkiewicz modification that the energy of these states remains unchanged, which marks the uniqueness of this variant of the modification.Comment: Latex, 12 pages, extended, new references adde

Statistical Bias in the Distribution of Prime Pairs and Isolated Primes

Computer experiments reveal that twin primes tend to center on nonsquarefree multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of nonsquarefree multiples of 6 to the number of squarefree multiples of 6 equal $\pi^2/3-1$, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427, a relative difference of ca $6.0\%$ measured against the expected value. A deviation from the expected value of this ratio, ca $1.9\%$, exists also for isolated primes. This shows that the distribution of primes is biased towards nonsquarefree numbers, a phenomenon most likely previously unknown. For twins, this leads to nonsquarefree numbers gaining an excess of $1.2\%$ of the total number of twins. In the case of isolated primes, this excess for nonsquarefree numbers amounts to $0.4\%$ of the total number of such primes. The above numbers are for the first $10^{10}$ primes, with the bias showing a tendency to grow, at least for isolated primes.Comment: 7 page

Higher Order Modification of the Schroedinger Equation

We modify the Schr\"{o}dinger equation in a way that preserves its main properties but makes use of higher order derivative terms. Although the modification represents an analogy to the Doebner-Goldin modification, it can differ from it quite distinctively. A particular model of this modification including derivatives up to the fourth order is examined in greater detail. We observe that a special variant of this model partially retains the linear superposition principle for the wave packets of standard quantum mechanics remain solutions to it. It is a peculiarity of this variant that a periodic structure emerges naturally from its equations. As a result, a free particle, in addition to a plane wave solution, can possess band solutions. It is argued that this can give rise to well-focused particle trajectories. Owing to this peculiarity, when interpreted outside quantum theory, the equations of this modification could also be used to model pattern formation phenomena.Comment: Latex, 11 pages, extended, new references adde

Nonlinear Phase Modification of the Schroedinger Equation

A nonlinear modification of the Schr\"{o}dinger equation is proposed in which the Lagrangian density for the Schr\"{o}dinger equation is extended by terms polynomial in $\Delta^{m}\ln (\Psi^{*}/{\Psi})$ multiplied by $\Psi^{*}{\Psi}$. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase $S$ rather than the amplitude $R$ of the wave function $\Psi =R\exp (iS)$. From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schr\"{o}dinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to $n$ particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schr\"{o}dinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.Comment: Latex, 21 pages, extended and slightly modified, new references adde