The expression E = h × f is misleading because it implies the distribution of a photon quantum over 299 792 458 m, while the expression E = (h × c)/λ enables us to explain the particle-wave duality

The two equations E = h × f and E = (h × c)/λ for the quantum of energy of electromagnetic radiation provide the same result but describe electromagnetic radiation very differently. E = (h × c)/λ describes the quantum of energy of electromagnetic radiation to be located already in one wavelength and therefore like a particle. E = h × f describes the quantum of energy distributed over 299 792 458 m and therefore like a wave. To obtain h × f for the quantum of energy, we have to refer the quantum of energy to 299 792 458 m. Only then we obtain from E = (h × c)/(299 792 458 m), as the distance of 299 792 458 m of the velocity c is cancelling out now, E = h × 1/s = h × Hz, which is the precondition to obtain the correct value for the quantum of energy by multiplying Planck’s constant h by the frequency f. This already indicates the necessity of today's physics to have to speak of a particle-wave duality. It turns out that electromagnetic radiation consists of the first wavelength that carries the quantum of energy and behaves like a particle, which today is called “photon,” and a few following wavelengths that do not carry a further quantum of energy and behave like a wave, which today is called “electromagnetic wave.” By this knowledge, the particle-wave duality vanishes, and we obtain one single physical phenomenon, which I call “photon-wave.” The strange behavior of quantum objects at a single slit, at double-slits, and at beam splitters can now be understood in a causal way. “God does not play dice!” Einstein was right.

Mass gap problem and Planck constant

The mass gap problem is about defining the constant that defines the minimal excitation of the vacuum. Planck’s constant is defining the minimal possible excitation of the vacuum from the point of quantum mechanics. The mass gap problem can be solved in quantum mechanics by the formulation of the photon’s mass according to the Planck‐Einstein relation.

Why space could be quantised on a different scale to matter

The scale of quantum mechanical effects in matter is set by Planck’s constant, \hbarℏ. This represents the quantisation scale for material objects. In this article, we present a simple argument why the quantisation scale for space, and hence for gravity, may not be equal to \hbarℏ. Indeed, assuming a single quantisation scale for both matter and geometry leads to the `worst prediction in physics’, namely, the huge difference between the observed and predicted vacuum energies. Conversely, assuming a different quantum of action for geometry, \beta \ll \hbarβ≪ℏ, allows us to recover the observed density of the Universe. Thus, by measuring its present-day expansion, we may in principle determine, empirically, the scale at which the geometric degrees of freedom should be quantised.

Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral

The constant ħ is elevated to a dynamical field, coupling to other fields, and itself, through the Lagrangian density derivative terms. The spatial and temporal dependence of ħ falls directly out of the field equations themselves. Three solutions are found: a free field with a tadpole term; a standing-wave non-propagating mode; a non-oscillating non-propagating mode. The first two could be quantized. The third corresponds to a zero-momentum classical field that naturally decays spatially to a constant with no ad-hoc terms added to the Lagrangian. An attempt is made to calibrate the constants in the third solution based on experimental data. The three fields are referred to as actons. It is tentatively concluded that the acton origin coincides with a massive body, or point of infinite density, though is not mass dependent. An expression for the positional dependence of Planck’s constant is derived from a field theory in this work that matches in functional form that of one derived from considerations of Local Position Invariance violation in GR in another paper by this author. Astrophysical and Cosmological interpretations are provided. A derivation is shown for how the integrand in the path integral exponent becomes Lc/ħ(r), where Lc is the classical action. The path that makes stationary the integral in the exponent is termed the “dominant” path, and deviates from the classical path systematically due to the position dependence of ħ. The meaning of variable ħ is seen to be related to the rate of time passage along each path increment. The changes resulting in the Euler-Lagrange equation, Newton’s first and second laws, Newtonian gravity, Friedmann equation with a Cosmological Constant, and the impact on gravitational radiation for the dominant path are shown and discussed.

Emergence of Planck’s Constant from Iterated Maps

Iterations of continuous maps are the simplest models of generic dynamical systems. In particular, circle maps display several key properties of complex dynamics, such as phase-locking and the quasi-periodicity route to chaos. Our work points out that Planck’s constant may be derived from the scaling behavior of circle maps in the asymptotic limit.