According to our current understanding of quantum mechanics, a `measurement' violates unitarity. In other words as the act of measurement `resets' the probabilities described by the Schrödinger equation, in the very `moment' of the collapse of the wavefunction, conservation of probability does not hold. To make progress in our understanding of the measurement problem it is thus expected that one must encompass in a single equation both otherwise mutually-exclusive processes of measurement and unitary evolution. In this light, turning to the continuity equation, we realize the possibility that if we allow for existence of sources and sinks in the `fluid of probability' we get closer to such a theory by arriving at nonlinear generalizations of Schrödinger and Klein-Gordon equations. The generalized equations derived are then shown to satisfy all conditions that are expected from a proper generalization: simplification to their linear counterparts by a well-defined dynamical condition, and Galilean and Lorentz invariance.

There is no formal difference between particles and black holes. This formal similarity lies in the intersection of gravity and quantum theory; quantum gravity. Motivated by this similarity, `wave-black hole duality' is proposed, which requires having a proper energy-momentum tensor of spacetime itself. Such a tensor is then found as a consequence of `principle of minimum gravitational potential'; a principle that corrects the Schwarzschild metric and predicts extra periods in orbits of the planets. In search of the equation that governs changes of observables of spacetime, a novel Hamiltonian dynamics of a Pseudo-Riemannian manifold based on a vector Hamiltonian is adumbrated. The new Hamiltonian dynamics is then seen to be characterized by a new `tensor bracket' which enables one to finally find the analogue of Heisenberg equation for a `tensor observable' of spacetime.

After proposing a natural metric for the space in which particles spin which implements the principle of maximum frequency, E=hf is generalised and its inverse Fourier transform is calculated. As a necessary result, a metric is found for the space in which quantum particles spin, hence the possibility of explanation of correlation of spacelike-separated particles is opened up.

After proposing the Principle of Minimum Gravitational Potential, in a pursuit to find the explanation behind the correction to Newton's gravitational potential that accounts for Mercury's orbit, by finding all the higher-order corrections it is shown that the consequences of the existence of speed of light for gravity are not yet fully explored.

An axiomatic theory is proposed that reconciles the existence of an absolute scale for time (Planck time) and special relativity. According to this theory speed of light c becomes a variable which is proposed to be taken as the fifth dimension.

Motivated by the well-known contradiction of special relativity and the heat equation, a wave equation for temperature scalar field is presented that also resolves the old controversy of (Lorentz) transformation of temperature and entropy. After showing that the current dogma of temperature and entropy being emergent concepts is based on but a logical fallacy, it is proposed that single particles posses entropy. This principle of fundamentality of entropy is then shown to be compatible with the equipartition theorem by yielding corrections in the quantum gravity regime.

In this short note we present a technique using which one attributes frequency and wavevector to (almost) arbitrary scalar fields. Our proposed definition is then applied to the classical wave equation to yield a novel nonlinear PDE.