Title Participants Abstract
"A Note on Fractional Curl Operator" "T Stefański, Marek Czachor" "In this letter, we demonstrate that the fractional curl operator, widely used in electromagnetics since 1998, is essentially a rotation operation of components of the complex Riemann-Silberstein vector representing the electromagnetic field. It occurs that after the wave decomposition into circular polarisations, the standard duality rotation with the angle depending on the fractional order is applied to the left-handed basis vector while the right-handed basis vector stems from the complex conjugation of the left-handed counterpart. Therefore, the fractional curl operator describes another representation of rotations of the electromagnetic field decomposed into circular polarisations. Finally, we demonstrate that this operator can describe a single-qubit phase-shift gate in quantum computing."
"Cosmic-Time Quantum Mechanics and the Passage-of-Time Problem" "Marek Czachor" "Abstract A new dynamical paradigm merging quantum dynamics with cosmology is discussed. We distinguish between a universe and its background space-time. The universe here is the subset of space-time defined by Ψτ(𝑥)≠0, where Ψτ(𝑥) is a solution of a Schrödinger equation, x is a point in n-dimensional Minkowski space, and τ≥0 is a dimensionless ‘cosmic-time’ evolution parameter. We derive the form of the Schrödinger equation and show that an empty universe is described by a Ψτ(𝑥) that propagates towards the future inside some future-cone 𝑉+. The resulting dynamical semigroup is unitary, i.e., ∫𝑉+𝑑4𝑥|Ψτ(𝑥)|2=1 for τ≥0. The initial condition Ψ0(𝑥) is not localized at 𝑥=0. Rather, it satisfies the boundary condition Ψ0(𝑥)=0 for 𝑥∉𝑉+. For 𝑛=1+3 the support of Ψτ(𝑥) is bounded from the past by the ‘gap hyperboloid’ ℓ2τ−−√=𝑐2𝑡2−𝑥2, where ℓ is a fundamental length. Consequently, the points located between the hyperboloid and the light cone 𝑐2𝑡2−𝑥2=0 satisfy Ψτ(𝑥)=0, and thus do not belong to the universe. As τ grows, the gap between the support of Ψτ(𝑥) and the light cone increases. The past thus literally disappears. Unitarity of the dynamical semigroup implies that the universe becomes localized in a finite-thickness future-neighbourhood of ℓ2τ−−√=𝑐2𝑡2−𝑥2, simultaneously spreading along the hyperboloid. Effectively, for large τ the subset occupied by the universe resembles a part of the gap hyperboloid itself, but its thickness Δτ is non-zero for finite τ. Finite Δτ implies that the three-dimensional volume of the universe is finite as well. An approximate radius of the universe, 𝑟τ, grows with τ due to Δτ𝑟3τ=Δ0𝑟30 and Δτ→0. The propagation of Ψτ(𝑥) through space-time matches an intuitive picture of the passage of time. What we regard as the Minkowski-space classical time can be identified with 𝑐𝑡τ=∫𝑑4𝑥𝑥0|Ψτ(𝑥)|2, so 𝑡τ grows with τ as a consequence of the Ehrenfest theorem, and its present uncertainty can be identified with the Planck time. Assuming that at present values of τ (corresponding to 13–14 billion years) Δτ and 𝑟τ are of the order of the Planck length and the Hubble radius, we estimate that the analogous thickness Δ0 of the support of Ψ0(𝑥) is of the order of 1 AU, and 𝑟30∼(𝑐𝑡𝐻)3×10−44. The estimates imply that the initial volume of the universe was finite and its uncertainty in time was several minutes. Next, we generalize the formalism in a way that incorporates interactions with matter. We are guided by the correspondence principle with quantum mechanics, which should be asymptotically reconstructed for the present values of τ. We argue that Hamiltonians corresponding to the present values of τ approximately describe quantum mechanics in a conformally Minkowskian space-time. The conformal factor is directly related to |Ψτ(𝑥)|2. As a by-product of the construction, we arrive at a new formulation of conformal invariance of 𝑚≠0 fields."
"Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics" "Marek Czachor" "Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ (MOND) change the dynamics, but do not alter the calculus.However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition β 1⊕ β 2= tanh (tanh - 1(β 1) + tanh - 1(β 2)) , although multiplication β 1⊙ β 2= tanh (tanh - 1(β 1) · tanh - 1(β 2)) , and division β 1⊘ β 2= tanh (tanh - 1(β 1) / tanh - 1(β 2)) do not seem to appear in the literature. The map f X(β) = tanh - 1(β) defines an isomorphism of the arithmetic in X= (- 1 , 1) with the standard one in R. The new arithmetic is projective and non-Diophantine in the sense of Burgin (Uspekhi Matematicheskich Nauk 32:209–210 (in Russian), 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov (Technika Molodezhi 10:16–19 (11:30–33 in Russian), 1961). Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Treating the above example as a paradigm, we ask what can be said about the set X of ‘real numbers’, and the isomorphism f X: X→ R, if we assume the standard form of Newtonian mechanics and general relativity (formulated by means of the new calculus) but demand agreement with astrophysical observations. It turns out that the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if fX(t/tH)≈0.8sinh(t-t1)/(0.8tH). The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with Ω Λ= 0.7 , Ω M= 0.3. Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element 0 X of addition, is nonzero from the point of view of the standard arithmetic of R. This implies fX-1(0)=0X>0. The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters. Physics of the dark universe seems to be both geometry and arithmetic."
"Arithmetic loophole in Bell's Theorem: Overlooked threat to entangled-state quantum cryptography" "Marek Czachor" "Bell’s theorem is supposed to exclude all local hidden-variable models of quantum correlations. However,an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical oftwo-electron singlet states. Observable probabilities are consistent with the usual arithmetic employedby macroscopic observers but counterfactual aspects of Bell’s theorem are sensitive to the choice ofhidden-variable arithmetic and calculus. The model is classical in the sense of Einstein, Podolsky,Rosen and Bell: elements of reality exist and probabilities are modeled by integrals of hidden-variableprobability densities. Probability densities have a Clauser–Horne product form typical of local realistictheories. However, neither the product nor the integral nor the representation of rotations are the usualones. The integral has all the standard properties but only with respect to the arithmetic that definesthe product. Certain formal transformations of integral expressions found in the usual proofs à la Belldo not work, so standard Bell-type inequalities cannot be proved. The system we deal with is deterministic, local-realistic, rotationally invariant, observers have free will, detectors are perfect, hencethe system is free of all the canonical loopholes discussed in the literature."
"A Loophole of All ‘Loophole-Free’ Bell-Type Theorems" "Marek Czachor" "Bell’s theorem cannot be proved if complementary measurements have to be represented by random variables which cannot be added or multiplied. One such case occurs if their domains are not identical. The case more directly related to the Einstein–Rosen–Podolsky argument occurs if there exists an ‘element of reality’ but nevertheless addition of complementary results is impossible because they are represented by elements from different arithmetics. A naive mixing of arithmetics leads to contradictions at a much more elementary level than the Clauser–Horne–Shimony–Holt inequality."
"Unifying Aspects of Generalized Calculus" "Marek Czachor" "Non-Newtonian calculus naturally unifies various ideas that have occurred over the years in the field of generalized thermostatistics, or in the borderland between classical and quantum information theory. The formalism, being very general, is as simple as the calculus we know from undergraduate courses of mathematics. Its theoretical potential is huge, and yet it remains unknown or unappreciated."
"Non-Diophantine Arithmetics in Mathematics, Physics and Psychology" "Marek Czachor, Mark Burgin" "For a long time, all thought there was only one geometry — Euclidean geometry. Nevertheless, in the 19th century, many non-Euclidean geometries were discovered. It took almost two millennia to do this. This was the major mathematical discovery and advancement of the 19th century, which changed understanding of mathematics and the work of mathematicians providing innovative insights and tools for mathematical research and applications of mathematics.A similar event happened in arithmetic in the 20th century. Even longer than with geometry, all thought there was only one conventional arithmetic of natural numbers — the Diophantine arithmetic, in which 2+2=4 and 1+1=2. It is natural to call the conventional arithmetic by the name Diophantine arithmetic due to the important contributions to arithmetic by Diophantus. Nevertheless, in the 20th century, many non-Diophantine arithmetics were discovered, in some of which 2+2=5 or 1+1=3. It took more than two millennia to do this. This discovery has even more implications than the discovery of new geometries because all people use arithmetic.This book provides a detailed exposition of the theory of non-Diophantine arithmetics and its various applications. Reading this book, the reader will see that on the one hand, non-Diophantine arithmetics continue the ancient tradition of operating with numbers while on the other hand, they introduce extremely original and innovative ideas."
"Time travel without paradoxes: Ring resonator as a universal paradigm for looped quantum evolutions" "Marek Czachor" "A ring resonator involves a scattering process where a part of the output is fed again into the input. The same formal structure is encountered in the problem of time travel in a neighborhood of a closed timelike curve (CTC). We know how to describe quantum optics of ring resonators, and the resulting description agrees with experiment. We can apply the same formal strategy to any looped quantum evolution, in particular to the time travel. The argument is in its essence a topological one and thus does not refer to any concrete geometry. It is shown that the resulting paradigm automatically removes logical inconsistencies associated with chronology protection, provided all input-output relations are given by unitary maps. Examples of elementary loops and a two-loop time machine illustrate the construction. In order to apply the formalism to quantum computation one has to describe multi-qubit systems interacting via CTC-based quantum gates. This is achieved by second quantization of loops. An example of a multiparticle system, with oscillators interacting via a time machine, is explicitly calculated. However, the resulting treatment of CTCs is not equivalent to the one proposed by Deutsch in his classic paper"
"WAVES ALONG FRACTAL COASTLINES: FROM FRACTAL ARITHMETIC TO WAVE EQUATIONS" "Marek Czachor" "Beginning with addition and multiplication intrinsic to a Koch-typecurve, we formulate and solve wave equation describing wave propagationalong a fractal coastline. As opposed to examples known from the literature, we do not replace the fractal by the continuum in which it is embedded. This seems to be the first example of a truly intrinsic descriptionof wave propagation along a fractal curve. The theory is relativisticallycovariant under an appropriately defined Lorentz group."
"Swapping Space for Time: An Alternative to Time-Domain Interferometry" "Marek Czachor" "Young's double-slit experiment [1] requires two waves produced simultaneously at two different points in space. In quantum mechanics the waves correspond to a single quantum object, even as complex as a big molecule. An interference is present as long as one cannot tell for sure which slit is chosen by the object. The more we know about the path, the worse the interference. In the paper we show that quantum mechanics allows for a dual version of the phenomenon: self-interference of waves propagating through a single slit but at different moments of time. The effect occurs for time-independent Hamiltonians and thus should not be confused with Moshinsky-type time-domain interference [2], a consequence of active modulation of parameters of the system (oscillating mirrors, chopped beams, time-dependent apertures, moving gratings, etc.). The discussed phenomenon is counterintuitive even for those who are trained in entangled-state quantum interferometry. For example, the more we know about the trajectory in space, the better the interference. Exactly solvable models lead to formulas deceptively similar to those from a Youngian analysis. The models are finite dimensional, with interaction terms based on two-qubit CNOT quantum gates. The effect is generic and should be observable in a large variety of experimental configurations. Moreover, there are reasons to believe that this new type of quantum interference was in fact already observed in atomic interferometry almost three decades ago, but was misinterpreted and thus rejected as an artifact of unknown origin."