Erratum: Quantum-Phase-Field Concept of Matter: Emergent Gravity in the Dynamic Universe (Zeitschrift für Naturforschung A. (2020) 72:1 (51-58) DOI: 10.1515/zna-2016-0270)

Steinbach, I.

Volume: 75 Pages: 89-91
DOI: 10.1515/zna-2019-0326
Published: 2020

The work "Quantum-phase-field concept of matter: Emergent gravity in the dynamic universe", published in [1], outlines a framework to describe physical matter from the solution of a one-dimensional non-linear wave equation. Unfortunately, a central result, the presented analytic form of this solution, (13) in [1] is incorrect. The corrected form can be found on arXiv [2]. In the present Erratum, besides the correct solution, some background information about these types of non-linear wave equations and their solutions is added. We start from a functional fDW, the famous "double-well potential", according to Landau's theory of phase transitions [3] with positive constants r and u, expanded in temperature T around the critical temperature Tc. φ. is the "order parameter" in the original Ginzburg-Landau theory. In this theory the potential function is expanded to the fourth order, keeping only even contributions of the order parameter φ: (Equation Presented) The potential has a minimum at φ = 0 for T > Tc and two minima at φm = ±√r(Tc-T)/u for T < Tc. These minima describe the disordered and ordered state of the system, respectively. Here only the ordered state T < Tc is discussed and to be consistent with conventions in the phase-field literature the order parameter is normalized to (Equation Presented), see Figure 1. (Equation Presented) Also, we will understand φ = φ(s, t) as a field variable in space s and time t and define the Hamiltonian H as an integral over the potential density and the Ginsburg gradient operators accounting for fluctuations: (Equation Presented) U is a constant with dimension of energy, ϵ is a constant with dimension of length and γ an inverse length. The well-known minimum solution δ/δφ G = 0 is for the boundary conditions φ(-∞) = 0 and φ(+∞) = 1: (Equation Presented) with η = √72 ϵ/γ for a traveling wave with speed ν. Note that for ν ≠ 0 a symmetry breaking contribution has to be added to the potential (2), which is omitted here to keep the focus on the type of potential. Details can be found in appendix of [4]. For the boundary conditions φ(-∞) = φ(+∞) = 0 we have, besides the trivial solution φ ≡ 0: (Equation Presented) where the waves, peaked at s1 and s2 at t = 0, respectively, s1 < s2, travel with opposite speed, see Figure 2. It is also well established that the special form of the potential (1) or (2) is not fixed from basic principles, besides that, between the two minima there should be a potential barrier to separate the minima with a given activation energy ∝ U. Only close to the critical point, i.e. where the activation energy U → 0, a rigorous renormalization group treatment may be applied to show that higher order contributions will become irrelevant [5]. Aside from the critical point, no argument exists to truncate the Landau expansion of the potential to the fourth order or to select the given form. In the "multiphase-field theory" [6] there arises, however, an additional constraint for the potential. In this theory a set of fields φI, I = 1 ⋯ N is defined which form junctions by the condition ΣN I=1 φI = 1. The functional is replaced by H = ΣN I=1 HI where each term HI has the special form (3). In the center of the junction, where φI = φJ = 1/N for all I, J, the maximum of the potential fDW m can be evaluated: (Equation Presented) i.e. for N > 3 the energy of the junction decreases with the order N and approaches 0 for large N. This must [Figure Presented] be termed "unphysical", as junctions between objects loose their penalty and the system would return to the disordered state. To remedy this problem, the so-called "double-obstacle potential" is introduced [6]: (Equation Presented) It has the same topology as (2) (see Fig. 1) but a maximum power of 2. Further on it has the advantage that it defies a linear wave aside from the breakpoints. We calculate the maximum potential of the junction fDO m: (Equation Presented) i.e. the energy of the junction increases with the order N and approaches a constant ∝ γ for large N, as it should. The main drawback of this potential is the non-analytical form with the absolute signs. The "non-linearity" of (2) is hidden in the breakpoints at φ = 0 and φ = 1. Only a piece-wise solution is possible for the boundary conditions φ(-∞) = 0 and φ(+∞) = 1, η = π √ϵ.γ, (Equation Presented) For φ(-∞) = φ(+∞) = 0, one finds (see Fig. 2): (Equation Presented) The last (10) replaces (13) in [1]. © 2019 De Gruyter. All rights reserved.

« back