Z Naturforsch 2017; 72(1)a: 51–58 Open Access Ingo Steinbach* Quantum-Phase-Field Concept of Matter: Emergent Gravity in the Dynamic Universe DOI 10.1515/zna-2016-0270 Received April 20, 2016; accepted October 27, 2016; previously published online December 23, 2016 Abstract: A monistic framework is set up where energy is the only fundamental substance Different states of energy are ordered by a set of scalar fields The dual elements of matter, mass and space, are described as volume- and gradient-energy contributions of the set of fields, respectively Time and space are formulated as background-indepen dent dynamic variables The evolution equations of the body of the universe are derived from the first principles of thermodynamics Gravitational interaction emerges from quantum fluctuations in finite space Application to a large number of fields predicts scale separation in space and repulsive action of masses distant beyond a marginal distance The predicted marginal distance is compared to the size of the voids in the observable universe Keywords: Fundamental Structure of Mass and Space; Modified Theories of Gravity; Phase-Field Theory PACS numbers: 04.20.Cv; 04.50.Kd; 05.70.Fh Introduction ‘Several recent results suggest that the field equations of gravity have the same conceptual status as the equations of, say, elasticity or fluid mechanics, making gravity an emergent phenomenon’, starts the review of P admanabhan and Padmanabhan on the cosmological constant problem [1] This point of view relates to the holographic principle [2–4], which treats gravity as an ‘entropic force’ derived from the laws of thermodynamics An even more radical approach is given by the ‘causal sets’ of Sorkin [5], which treats space–time as fundamentally discrete following the rules of partial order I will adopt from the latter that there is no fundamental multi-dimensional continuous *Corresponding author: Ingo Steinbach, Ruhr-University Bochum, ICAMS, Universitaetsstrasse 150, 44801 Bochum, Germany, E-mail: ingo.steinbach@rub.de space–time, but a discrete set of fields; from the first, that thermodynamics shall be the fundament of our understanding of the world The concept is based on a formalism that is well established in condensed matter physics, the so-called phase-field theory (for review, see [6, 7]) It is applied to investigate pattern formation in mesoscopic bodies where no length scale is given Mesoscopic in this context means ‘large compared to elementary particles or atoms’ and ‘small compared to the size of the body’ Then, the scale of a typical pattern is treated emergent from interactions between different elements of the body under investigation The general idea of the phase-field theory is to combine energetics of surfaces with volume thermo dynamics It is interesting to note that it thereby inherits the basic elements of the holographic principle, which relates the entropy of a volume in space–time to the entropy at the surface of this volume In the phase-field theory, the competition of the free energy of volume and surface drives the evolution of the system under consideration I will start out from the first principles of energy conservation and entropy production in the general form of [8] Energy is the only fundamental substance ‘Fundamental substance’ in this context means ‘a thingin-itself, regardless of its appearance’ [9] There will be positive and negative contributions to the total energy H They have to be balanced to zero, as there is no evidence, neither fundamental nor empirical, for a source where the energy could come from H = = 0 I will call this the ‘principle of neutrality’ Compare also the theory of Wheeler and DeWitt [10], which is, however, based on a fundamentally different framework in relativistic quantum mechanics The Hamiltonian Ĥ will be expanded as a function of the fields {φI}, I = 1, … N, and their gradients The wave function |w > will be treated explicitly in the limiting case of quasi-stationary elementary masses The time dependence of the Hamiltonian and the wave function is governed by relaxational dynamics of the fields according to the demand of entropy production Here, I will treat the interaction of neutral matter only Additional quantum numbers like charge and colour may be added to the concept later ©2016, Ingo Steinbach, published by De Gruyter This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License Unauthenticated Download Date | 1/18/17 10:09 PM 52 I Steinbach: Quantum-Phase-Field Concept of Matter Basic Considerations The new concept is based on the following statements: –– The first and the second laws of thermodynamics apply –– Energy is fundamental and the principle of neutrality applies, i.e the total energy of the universe is zero –– There is the possibility that energy separates into two or more different states –– Different states of energy can be ordered by a set of N dimensionless scalar fields {φI}, I = 1, …, N The fields have normalised bounds 0 ≤ φI ≤ 1 –– The system formed by the set of fields is closed in itself: N ∑φ I =1 I (1) = –– Two one-dimensional metrics evaluating distances between states of energy define space and time as dynamic variables –– Planck’s constant h, the velocity c, and the kinetic constant τ with the dimension of momentum are universal –– Energy and mass are proportional with the constant c2 One component of the set of fields {φI} is considered as an ‘order parameter’ in the sense of Landau and Lifshitz [11] The ‘0’ value of the field φI denotes that this state is not existing The value ‘1’ means that this state is the only one existing Intermediate values mean coexistence of several states There is obviously a trivial solution of (1): φI = For this solution, no ‘shape’ can be disN tinguished It is one possible homogeneous state of the body ‘There is, however, no reason to suppose that […] the body […] will be homogeneous It may be that […] the body […] separates into two (or more) homogeneous parts’ [11, p 251] In fact, we shall allow phase separation by the demand of entropy production Phase separation requires the introduction of a metric that allows distinguishing between objects (parts of the body): ‘space’ Now that we have already two fundamentally different states of the body, the homogeneous state and the phase-separated state, we need a second, topologically different metric to distinguish these states: ‘time’ Both coordinates, space and time, are dynamic, dependent only on the actual state of the body They are background independent having no ‘global’ meaning, in the sense that they would be independent of the observer For general considerations about a dynamical universe, see Barbour’s dynamical theory [12] For discussions about the ‘arrow of time’, see [13] Variational Framework The concept is based on the variational framework of field theory [14] The energy functional Ĥ is defined by the integral over the energy density ĥ as a function of the fields {φI} with a characteristic length η, to be determined: N Hˆ = η∑ ∫ dφI hˆ({ φI }) I =1 (2) The functional Ĥ has the dimension of energy and the density ĥ has the dimension of force The functional (2) shall be expanded in the distances sI N ∞ ∂φ Hˆ = η∑ ∫ dsI I hˆ({ φI }) −∞ ∂sI I =1 N = ∑ ∫ dsI hˆ({ φI }), I =1 (3) ∞ −∞ (4) where distances are renormalized according to sI = ηsI ∂φ I ∂sI (5) For readability, I will omit the field index I of the distances in the following The individual components of the field are functions in space and time φ = φ(s, t) They will be embedded into a higher-dimensional mathematical space in Section 4.2 The time evolution of one field is determined by relaxational dynamics: τ ∂ δ +∞ φI = − dt < w | Hˆ | w > ∂t δφI ∫0 (6) I use the standard form of the Ginzburg–Landau functional, or Hamiltonian Ĥ, in two-dimensional Minkowski n otation, the time derivative accounting for dissipation 4U η Hˆ = ∑ ∫ ds −∞ π I =1 N +∞ ∂ ∂ π2 φI − φI + | φI (1 − φI )| , η ∂s c ∂t (7) where U is a positive energy quantum to be associated with massive energy Note that the special analytical form of this expansion is selectable as long as isotropy in space–time is guaranteed, and the dual elements of gradient and volume contributions are normalised to observable physical quantities; see (16) and (17) below Unauthenticated Download Date | 1/18/17 10:09 PM I Steinbach: Quantum-Phase-Field Concept of Matter 53 Quasi-Static Solution τ Now, I will formally derive the individual contributions of the concept related to known physical entities in mechanics I will only treat the quasi-static limit where the dynamics of the wave function |w > and the dynamics of the fields φI decouple This means that the field is kept static for the quantum solution on the one hand The quantum solution on the other hand determines the energetics of the fields The expectation value of the energy functional (7) has three formally different contributions if the differen∂ ∂ tial operators and are applied to the wave function ∂s ∂t |w > or the field φI, respectively Applying the differential operators to the field components and using the normalisation of the wave function = 1 yields the force uI related to the gradient of the fields I: 2 4U η ∂φI ∂φ I π − 2 + | φI (1 − φI )| uI = π ∂s c ∂t η (8) The mixed contribution describes the correlation between the field and the wave function, and shall be set to in the quasi-static limit: (1 − φI ) 4U η ∂φI ∂ ∂s π ∂s ∂φ ∂ − I < w w > = ∂t c ∂t (9) The force eI related to the volume of field I is defined as eI |φ I =1 = 4U η ∂2 ∂2 φI < w − 2 w > ∂s c ∂t π ∂ φ v π2 ∂ ∂ 1 φ = τv φ = U η − + φ − ∂t ∂s η ∂ s c +mφ ∆e (11) I have transformed the time derivative of the field ∂ ∂ ∂ φ into the moving frame with velocity v, =v and ∂t ∂t ∂s used the Euler–Lagrange relation +∞ δ +∞ ∂ ∂ ∂ ∂ ∂ ds∫ dt → − − ∫ −∞ δφ ∂φ ∂t ∂φt ∂s ∂φs (12) The contributions of (11) proportional to U dictate from their divergence in the limit η → 0 the special solution for the field, which is the well-known ‘solution of a traveling wave’, or ‘traveling wave solution’ (see Appendix of [15]) We find, besides the trivial solution φ(s, t) ≡ 0, the primitive solution (s1 , ∂t ∂s j = ∂s ∑ ∂s Ωi , repel each other Ωi separates interactions from attractive to repulsive Thereby, a Discussion and interpretation In the previous section, a rigorous derivation has been presented from which generalised Newton’s equations, invariance of speed of light, and repulsive gravitational action on ultra-long distances are derived The latter is, Unauthenticated Download Date | 1/18/17 10:09 PM I Steinbach: Quantum-Phase-Field Concept of Matter 57 of course, consistent with Einstein’s equation with a finite cosmological constant, though the approach is fundamentally different The question is how to ‘adjust’ such a cosmological constant; see [20] In the present concept, there is no ‘global’ constant The marginal length is formulated from a quasi-local energy balance Let me explain this in more detail As stated in the beginning, there is no fundamental, absolute space, neither one-dimensional nor multi-dimensional Space is defined by the (negative) energy content of the volume of the fields φI ≡ 1 on the one hand Within one particle φI is decomposed in single component wave functions |wI > in the limiting case of quasi-stationary fields and constructed explicitly Space is attributed with negative energy and massive particles are attributed with positive energy The physical space is a one-dimensional box between two elementary particles forming the end points of space Quantum fluctuations in finite space with discrete spectrum define the negative energy of space The junctions between individual components of the field define elementary particles with positive energy The energy of mass is the condensation of those fluctuations that not fit into finite space Comparison of the energy of mass to the energy of space defines the coupling coefficient Gi between an individual elementary particle i and the spaces it is embedded in It depends on the position of one elementary mass i in space and time relative to all other masses By varying the energy of space with respect to distance, the action on the state of masses is derived This leads to a generalised law of gravitation that shows attractive action for close masses and repulsive action for masses more distant than a marginal distance ΩE This distance is correlated to the size of the largest structures in the universe observed in the reference frame of our solar system The predicted marginal length ΩE correlates well with the observed size of the voids in the universe It must be stated clearly that the new ‘generalised law of gravitation’ (30) is not a priori in conflict with general relativity, as it has no restriction concerning the topology of a global multi-dimensional space of cognition, except the quasi-local limit of flat Euclidean space The new contribution of the present concept is the quasi-local mechanism of balancing in- and outgoing quantum fluctuations on the field at the position of the observer The concept sticks strictly to the demand of energy conservation It makes a prediction for gravitational action on ultra-long distances This prediction can be verified experimentally by investigating trajectories of large structures in the universe The presented concept might open a door towards a new perception of physics where thermodynamics, quantum mechanics, and cosmology combine naturally Acknowledgements: The author would like to thank Claus Kiefer, Cologne, for helpful suggestions and discussions; Dmitri Medvedev, Bochum/Novosibirsk, for providing the velocity dependent traveling wave solution; Friedrich Hehl, Cologne, for revealing some inconsistencies in the original manuscript and grounding him to reality; Fathollah Varnik, Bochum, for critical reading of the manuscript References [1] T Padmanabhan and H 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