Search

We show that the precision of an angular measurement or rotation (e.g., on the orientation of a qubit or spin state) is limited by fundamental constraints arising from quantum mechanics and general relativity (gravitational collapse). The limiting precision is $r^{-1}$ in Planck units, where $r$ is the physical extent of the (possibly macroscopic) device used to manipulate the spin state. This fundamental limitation means that spin states $S_1$ and $S_2$ cannot be experimentally distinguished from each other if they differ by a sufficiently small rotation. Experiments cannot exclude the possibility that the space of quantum state vectors (i.e., Hilbert space) is fundamentally discrete, rather than continuous. We discuss the implications for finitism: does physics require infinity or a continuum?, Comment: 10 pages. Version to appear in Physics Letters B

Recent cosmological measurements tend to confirm that the fine structure constant {\alpha} is not immutable and has undergone a tiny variation since the Big Bang. Choosing adequate units, this could also reflect a variation of Planck's constant h. The aim of this Letter is to explore some consequences of such a possible change of h for the pure and mixed states of quantum mechanics. Surprisingly enough it is found that not only is the purity of a state extremely sensitive to such changes, but that quantum states can evolve into classical states, and vice versa. A complete classification of such transitions is however not possible for the moment being because of yet unsolved mathematical difficulties related to the study of positivity properties of trace class operators., Comment: To appear in Phys. Lett. A

In geometrical universe models, orbital motion may derive from geometrical imperatives. In this article I outline a method to simulate gravitational orbits via rotating gravitational orbital pairs formed between particle masses in units of Planck mass. For each unit of Planck time, all particle masses will form orbital pairs with each other. These orbital pairs then rotate by a discrete step according to the radius of orbit, this incremental angle simulating the orbital momentum. All orbital co-ordinates are then summed and averaged before the next increment of time. Dimensioned constants $G, h, c$ are not required as gravitational orbits emerge as the sum of this orbital pair motion although for convenience the orbitals can be measured in Planck units. This is useful for simulation hypothesis modeling as the entire universe can be updated in real (Planck) time on a serial processor.

Assuming that a scalar field controls the inflationary era, we examine the combined effects of string and f ( R ) gravity corrections on the inflationary dynamics of canonical scalar field inflation, imposing the constraint that the speed of the primordial gravitational waves is equal to that of light’s. Particularly, we study the inflationary dynamics of an Einstein–Gauss–Bonnet gravity in the presence of α R 2 corrections, where α is a free coupling parameter. As it was the case in the pure Einstein–Gauss–Bonnet gravity, the realization that the gravitational waves propagate through spacetime with the velocity of light, imposes the constraint that the Gauss–Bonnet coupling function ξ ( ϕ ) obeys the differential equation ξ = H ξ , where H is the Hubble rate. Subsequently, a relation for the time derivative of the scalar field is extracted which implies that the scalar functions of the model, which are the Gauss–Bonnet coupling and the scalar potential, are interconnected and simply designating one of them specifies the other immediately. In this framework, it is useful to freely designate ξ ( ϕ ) and extract the corresponding scalar potential from the equations of motion but the opposite is still feasible. We demonstrate that the model can produce a viable inflationary phenomenology and for a wide range of the free parameters. Also, a mentionable issue is that when the coupling parameter α of the R 2 correction term is α 1 0 − 3 in Planck Units, the R 2 term is practically negligible and one obtains the same equations of motion as in the pure Einstein–Gauss–Bonnet theory, however the dynamics still change, since now the time derivative of ∂ f ∂ R is nonzero. We study in detail the dynamics of inflation assuming that the slow-roll conditions hold true and also we briefly address the constant-roll case dynamics, and by using several illustrative examples, we compare the dynamics of the pure and R 2 -corrected Einstein–Gauss–Bonnet gravity. Finally the ghosts issue is briefly addressed.

1 method of Planck scale Simulation Hypothesis modeling has the programmer God using geometrical objects for the base units such that particle to particle interactions result from geometrical imperatives. Described in this article is a method for programming the cosmic microwave background parameters at the Planck scale. A Planck size micro black-hole is defined as the minimum discrete entity and comprises the Planck units. The simulation clock-rate is measured in units of Planck time whereby for each unit of time a 4-axis hypersphere (the universe space) expands by 1 Planck step which constitutes the addition to the sum universe of a micro black-hole. This constant expansion is the origin of the speed of light. The mass-space parameters increment linearly, the electric parameters in a sqrt-progression, thus for electric parameters the early universe transforms most rapidly. The velocity of expansion is constant and is the origin of the speed of light, the Hubble constant becomes a measure of the universe radius and the CMB radiation energy density correlates to the Casimir force. A peak frequency of 160.2GHz correlates to a 14.624 billion year old universe (sans-particles). The cosmological constant, being the age when the simulation reaches the limit, approximates $t$ = 10$^{123} t_p$.

By combining a renormalization group argument relating the charge e and mass m of the proton by e^2 ln m ~ 0.1 pi (in Planck units) with the Carter-Carr-Rees anthropic argument that gives an independent approximate relation m ~ e^20 between these two constants, both can be crudely estimated. These equations have the factor of 0.1 pi and the exponent of 20 which depend upon known discrete parameters (e.g., the number of generations of quarks and leptons, and the number of spatial dimensions), but they contain NO continuous observed parameters. Their solution gives the charge of the proton correct to within about 8%, though the mass estimate is off by a factor of about 1000 (16% error on a logarithmic scale). When one adds a fudge factor of 10 previously given by Carr and Rees, the agreement for the charge is within about 2%, and the mass is off by a factor of about 3 (2.4% error on a logarithmic scale). If this 10 were replaced by 15, the charge agrees within 1.1% and the mass itself agrees within 0.7%., Comment: 12 pages, LaTeX

Large extra dimensions lower the Planck scale to values soon accessible. Not only is the Planck scale the energy scale at which effects of modified gravity become important. The Planck length also acts as a minimal length in nature, providing a natural ultraviolet cutoff and a limit to the possible resolution of spacetime. In this paper we examine the influence of the minimal length on the Casimir energy between two plates., references and comment on the figure in section III added

We derive an expression for the mean square displacement (MSD) of a particle whose motion is governed by a uniform, periodic, quantum multi-baker map. The expression is a function of both time, t , and Planck’s constant, h , and allows a study of both the long time, t →∞, and semi-classical, h →0, limits taken in either order. We evaluate the expression using random matrix theory as well as numerically, and observe good agreement between both sets of results. The long time limit shows that particle transport is generically ballistic for any fixed value of Planck’s constant. However, for fixed times, the semi-classical limit leads to diffusion. The mean square displacement for non-zero Planck’s constant, and finite time, exhibits a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck’s constant. We argue that these results are generic for a large class of 1D quantum random walks, similar to the quantum multi-baker, and that a sufficient condition for diffusion in the semi-classical limit is classically chaotic dynamics in each cell. Some connections between our work and the other literature on quantum random walks are discussed. These walks are of some interest in the theory of quantum computation.

Recent developments in quantum computing motivate investigations of the ultimate limits to computation. By combining results in cosmic dynamics with entropy bounds it is possible to present limits to the amount of information and its processing rate in the universe up to the present. We find that both the total number of bits and number of operations for the universe does not exceed O (10 123 ). Possible extensions of the information bounds are discussed, as is the nature of the holographic entropy bound.

The Simulation Hypothesis proposes that all of reality is an artificial simulation, analogous to a computer simulation. This essay summaries a project to determine the feasibility of assigning geometrical objects to the Planck units such that the universe can be simulated using geometrical imperatives at the Planck scale instead of the dimensioned physical constants. This approach is suitable for coding a mathematical universe that can grow autonomously. If these geometrical objects cannot be distinguished from our physical Planck units then they can be used in Planck scale simulation hypothesis models. These objects are derived from an electron formula (a mathematical electron) that, although itself dimension-less, embeds geometrical objects for mass, length, time and charge. Consequently although the attributes of the electron (mass, charge, frequency, wavelength) derive from these embedded objects, this mathematical electron itself does not exist in any dimensioned sense. The sum particle universe (the simulation) is therefore dimension-less, but internally it has measurable parameters that could give the appearance of a physical reality to self-aware structures from within the simulation. The essay is divided into 4 sections; the Planck units and a magnetic monopole electron, a discretely expanding Planck unit black-hole scaffolding as the origin of the Planck units, relativity as the mathematics of perspective and gravitational orbits as the sum of particle-particle gravitational orbitals. There are only 2 dimension-less physical constants used; the fine structure constant alpha and Omega, however Omega can be further defined using mathematical constants, pi and ln(e) and so the information required to construct this Planck universe model could be embedded in the electron formula and the magnitude of alpha.

To study nuclear quantum effects on electron transfer reactions, the non-adiabatic rate constant is expanded in terms of Planck's constant. The expansion to the second order reproduces well the quantum correction in the activation free energy for a quantum Monte Carlo simulation of water. In addition, the spin-boson model is discussed using the expansion. The validity of the rate constant given by the spin-boson model is investigated in non-linear systems. Conditions for obtaining exact values of the rate constant are established in the second order of Planck's constant. In liquids, the conditions are satisfied if the one-body distribution function does not change during the electron transfer reaction.

An analysis is presented of Planck scale effects through non-renormalizable dimension-five operators on the coupling constant unification in the standard and supersymmetric standard model. These effects turn out to be small.

In this letter we introduce a model of interacting strings in which the usual ideal gas approximations are not made. The model is constructed in analogy with nucleation models, the formation of droplets in a supersaturated gas. We consider the strings to be interacting and their number not fixed. The equilibrium configuration is the one for which the time derivative of the number of strings in the various energies vanishes. We evaluate numerically the equilibrium configurations for various values of the energy density. We find that at a density of order one in Planck units there is a sharp transition, from a “gas” phase in which there are many strings, all in the massless or first few excited states, to a “liquid” phase in which all strings have coalesced into one (or few) highly excited string.

We combine the assumption of environmental decoherence, as the mechanism generating the classical (i.e. no quantum interferences) nature of spacetime, with the limit on its other classical feature, point-like continuity, namely Planck length. As a result, quantum extended objects with masses larger than the Planck mass have to derive their quantum behaviour from long-range correlations; objects with masses smaller than the Planck mass cannot display classical behaviour.

The significance of Planck length in a quantum gravity model is investigated by concentrating on the conformal degree of freedom. It is shown that Planck length is a lower bound to physical proper length in any space-time. It is impossible to construct an apparatus which will measure length scales smaller than Planck length. These effects exist even in flat space-time because of vacuum fluctuations of gravity. It is shown that these fluctuations lead to a high energy cut-off in flat space field theories, thereby removing the divergence problem. The one-loop corrections to a self interacting scalar field is computed and shown to be finite.

The ethereal nature and origin of all the TWs of a particle is emphasized. The notion of superluminal pure momentum carrier radiation is introduced. Planck's constant as fundamental factor of correlation and interconnection of the TWs is presented and discussed.

We present a model of gravity in which the Planck mass is generated spontaneously by a similar mechanism that is responsible for the tension in a recent membrane model of strings. The action describes very floppy fluctuations of the physical space in some large flat embedding space. The coupling constant is dimensionless but the fluctuations are so violet that they produce spontaneously a mass which plays the role of the dimensionally transmuted coupling constant and can be identified with the Planck mass.

The relationship between the various Planck mean coefficients in the optically thin limit is discussed in terms of isothermal curves of growth. The specification of a general thin limit is shown to require two restrictive statements—one specifies the optical depth to be small and the other states the degree of radiative nonequilibrium. In general the limit is scaled by both the ordinary and modified Planck means. For linearized problems near radiative equilibrium, however, these two means reduce to the linear Planck mean that alone acts as the inverse scaling length.

Starting from the hypothesis of a material carrier of the EMF, the Planck formula E = hv is generalized, statistical sense is ascribed to the Planck action quantum, and a general scheme is proposed for removing the divergencies in QFT.

The scattering process of two pointlike particles at CM energies in the order of Planck units or beyond, is very well calculable using known laws of physics, because graviton exchange dominates over all other interaction processes. At energies much higher than the Planck mass black hole production sets in, accompanied by coherent emission of real gravitons.

A lower bound for the numerical value of Planck's constant is estimated on theoretical grounds alone, if elementary particles are gravitating entities.