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Zeitschrift für Naturforschung A

A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board: Fetecau, Corina / Kiefer, Claus

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Volume 72, Issue 1


Quantum-Phase-Field Concept of Matter: Emergent Gravity in the Dynamic Universe

Ingo Steinbach
Published Online: 2016-12-23 | DOI: https://doi.org/10.1515/zna-2016-0270


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-independent 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: 04.20.Cv; 04.50.Kd; 05.70.Fh

1 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 Padmanabhan and Padmanabhan on the cosmological constant problem [1]. This point of view relates to the holographic principle [2], [3], [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 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 thermodynamics. 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 thing-in-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=<w|Ĥ|w>=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.

2 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:


  • 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=1N. For this solution, no ‘shape’ can be distinguished. 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].

3 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:


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 s˜I



where distances are renormalized according to


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:


I use the standard form of the Ginzburg–Landau functional, or Hamiltonian Ĥ, in two-dimensional Minkowski notation, the time derivative accounting for dissipation.


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.

4 Quasi-Static Solution

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 differential operators s and t are applied to the wave function |w> or the field ϕI, respectively.

Applying the differential operators to the field components and using the normalisation of the wave function <w|w>=1 yields the force uI related to the gradient of the fields I:


The mixed contribution describes the correlation between the field and the wave function, and shall be set to 0 in the quasi-static limit:


The force eI related to the volume of field I is defined as


Next, we need to elaborate the structure of the fields. I do this for the special case of N=2 in a linear setting with periodic boundary conditions, for simplicity. The general case is a straightforward extension that cannot be solved analytically, however. For the analytic solvability, it is also convenient to replace the coupling function ϕI2 in (10) by the function m(ϕ)=14{(2ϕ1)ϕ(1ϕ)+12arcsin2ϕ1}, which is monotonous between the states 0 and 1 and has the normalisation to 0 and 1 for these states. Differences in both coupling functions become irrelevant in the sharp interface limit η→0 to be investigated here. For N=2 and ϕ1=1−ϕ2=ϕ, the equation of motion (6) read, with Δe=e1e2, mϕ=mϕ, and τ=4π2τ˜:


I have transformed the time derivative of the field tϕ into the moving frame with velocity v, t=vs and used the Euler–Lagrange relation


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<s2, s1<s<s2)


where ηv=η1v2c2 is the effective size of the transition region, or junction, between the fields, which I will call ‘particle’ in the following. It is a function of velocity. s1 and s2 are the spatial coordinates of the particles in the quasi-static picture related to the distance Ω=|s1s2|. Figure 1 depicts the solution for two fields where the particles travel with velocity v and have finite extension ηv. It will be treated in the ‘thin interface limit’ η→0, where its extension is negligible compared to distances between objects, but finite as discussed in Section 6.

Travelling wave solution for two fields in linear arrangement and with periodic boundary conditions.
Figure 1:

Travelling wave solution for two fields in linear arrangement and with periodic boundary conditions.

Continuing the analysis of (13), one easily proves



and we find, as a check for consistency, the energy of two particles from the integral


4.1 Volume Energy of the Fields

From the solution (13), we see that the field in the sharp interface limit forms a one-dimensional box with fixed walls and size ΩI for field I. According to Casimir [16], we have to compare quantum fluctuations in the box with discrete spectrum p and frequency ωp=πcp2ΩI to a continuous spectrum. This yields the negative energy EI of the field I:


where α is a positive, dimensionless coupling coefficient to be determined. I have used the Euler–MacLaurin formula in the limit ϵ→0 after renormalisation ppeϵp .

4.2 Multidimensional Interpretation

As stated at the beginning, the present concept has no fundamental space. The distance ΩI is intrinsic to one individual field I, and there is a small transition region of order η where different fields are connected. These regions are interpreted as elementary particles. The position of one particle related to an individual component of the field is determined by the steep gradient ϕIs. The parity of the particle is related to the parity of the field components. The individual components of the field, therefore, must be seen as spinors, and the particles must be attributed by a half-integral spin. From the isomorphism to the three-dimensional SU(2) symmetry group, we may argue that all components can be ordered in a three-dimensional Euclidean space. This ordering shall only be postulated in a small quasi-local environment around one particle. I will call this mathematical space the ‘space of cognition’, as our cognition orders all physical objects in this space. No assumption about a global space, its topology, or dimension has to be made. Figure 2 sketches this picture. Individual fields form a network of fields. Each field is expanded along a one-dimensional line coordinate and bound by two end points described by gradients of the field. Due to the constraint (1), the coordinates of different fields have to be synchronised within the particles of small but finite size η along the renormalisation condition (5). The constraint (1) also dictates that there is no ‘loose end’. The body is closed in itself, forming a ‘universe’.

Scheme of a number of seven fields connected by three particles. The particles have an uncertainty ηv depending on the velocity v in the orientation of the fields in the space of cognition. The junctions and fields can be pictured as knots and ropes respectively, forming a multi-dimensional network.
Figure 2:

Scheme of a number of seven fields connected by three particles. The particles have an uncertainty ηv depending on the velocity v in the orientation of the fields in the space of cognition. The junctions and fields can be pictured as knots and ropes respectively, forming a multi-dimensional network.

5 Generalised Newton Laws

In Section 3, the basic relations of the field theoretical framework of the dynamical universe have been set up, based on general consideration and the laws of thermodynamics. They shall now be applied to derive generalised Newton’s equation of acceleration and gravitation. Finally, a prediction of the structure of the observable universe on ultra-long distances will be given.

5.1 Generalised Newton’s Equation

Let the moving frame of one particle i connect to the field I. Then, (9) yields:


Again neglecting quantum effects within the particle, we find by partial integration and using the normalisation <w|w>=1:


The position si of the particle is marked by ϕIs0. The direction of the field embedded into the space of cognition defines the directions s and v. f is the force acting on the particle by the variation of energy with space. We have Newton’s second law. This may be taken as a first prediction of the concept, or simply as a check for consistency.

Inserting the traveling wave solution (13) into the equation of motion (11) relates the velocity of the particle to the energy of the field acting on it:1


The last equation can be solved for v:


There is a maximum velocity vmax=c, which is reached in the limit Δe→∞.

In expression (21), only the quotient ητ appears. We may argue that the time needed to transfer information over the distance η is proportional to η. Then, this quotient can be taken as a finite constant in the sharp interface limit η→0. As τ has the dimension of a momentum, it is convenient to use τm0c with a mass m0. Defining a characteristic distance Ω¯, we set


In this setting and relating the force Δe to an absolute energy ΔE=Ω¯Δe, we see


In the limit m0→0, the velocity |v| becomes identical to the constant c=vmax. Therefore, we can call the maximum velocity the speed of massless particles: speed of light. This result can be seen as the second prediction of the concept, or, again, as a check for consistency with observations and established theories. We end up with the simple relation between the energy of the field and the momentum of its surface states:

m0v=ΔEc2+(ΔE)2m02c2;   m0cv1v2c2=ΔE,(24)

with the relativistic mass m0/1v2c2. No empirical statement about invariance of the speed of light is used. We have, however, the implicit notion of physical space behaving like an ‘ether’, the field, and the upper velocity c corresponds to the well-known hyperbolic shock in an elastic medium.

5.2 Generalised Law of Gravitation

A number of Ni<N fields ϕI(s, t), I=1,…,Ni connects one single particle i with Ni other particles j=1,…,Ni. For Ni large compared to the dimensionality D of the multi-dimensional space of cognition, it will be impossible that all fields have the same size Ωiji. We have the generalisation of (17) in the reference frame of an individual particle i with Ni attached fields:


Ei is the spatial energy of all fields connected to the particle i. Eij=EJ is the spatial energy of an individual field J connecting particle i with particle j. Ωij is the distance between points i and j. Balancing the massive energy U with the spatial energy Ei according to the principle of neutrality and defining the characteristic size Ωi:=Ni[j=1Ni1Ωij]1, we evaluate the coupling coefficient for particle i to all other particles j:

αi=96UΩicNih;   Ei=2UΩiNij=1Ni1Ωij.(26)

Up to here, Ωi and Ωij had been related to the distance between particles. There is, however, no mechanism to evaluate such a distance. At the locus of one particle, only the local spectrum of quantum fluctuations can be used to evaluate the relation between fields and particles. We note that in (26), only the relative distance ΩijΩi enters. Therefore, we can replace the evaluation of the distances by the evaluation of the local spectrum of fluctuations acting on the particle i. The apparent distance Ω˜ij at position i can be defined from the force eij by Ω˜ij=Ueij. Note that in general, Ω˜ijΩ˜ji; see also the discussion in Section 6. The characteristic distance Ωi will be replaced by the apparent distance Ω˜i=Ni[j=1NieijU]1. This apparent distance will be treated in the following as an independent variable acting like a chemical potential equalising the fluxes of quanta acting on one individual particle from different fields.

We find by partial integration, in analogy to (18) and (19), the force fij acting on particle i from the fields connecting it to particle j in accordance to Newton’s first law:


where nij is the normal vector of the field in the space of cognition, evaluated at the position of particle i.

The energy has two size dependencies: the dependence of αi on Ω˜i and the dependence of the energy of the individual field Eij on the apparent distance Ω˜ij. Both must be varied independently. One finds the force fij of particle j acting on particle i, using (26)


and the total force fi from all masses j


Ω˜i defines a marginal distance where the force vanishes. Massive particles i and j, which are distant by Ω˜ij<Ω˜i, are accelerated towards each other by the quantum fluctuations they receive from all other particles, behaving like being attracted. The masses i and j, which are distant by Ω˜ij>Ω˜i, repel each other. Ω˜i separates interactions from attractive to repulsive. Thereby, a cloud of Ni masses separates spontaneously into dense and dilute regions. In the limit Ω˜ijΩ˜i, (29) reduces to the classical Newton law of gravitation if we identify the pre-factor 2UΩ˜iNi divided by the product of the masses mi and mj with the coefficient of gravitation in the local environment of elementary mass i, Gi:


This is the final prediction of the concept. It is a mere result of quantum fluctuations in finite space and the postulate of energy conservation in the strong (quasi-local) form. The generalised law of gravitation (30) predicts repulsion of distant masses. This repulsion will increase further unbound in distance. This statement offers an explanation of the observed acceleration of expansion of the universe [17].

5.3 Size of the Voids in the Universe

In order to derive an estimate of the marginal, or characteristic distance Ω˜i, I assume that hydrogen and neutrons are the dominant elements in the universe. Taking the mass of the universe M≈1052 kg [18] with the mass of the hydrogen atom mhu≈1.66 10−27 kg and the mass of the neutron mnu, we find the number of masses visible from the earth NE and the characteristic distance Ω˜E based on the measured gravitational coefficient on earth GE6.671011m3kg s2,

NE=Mu;   Ω˜E=GEM2c21024m.(31)

This numerical value of Ω˜E corresponds well to the size of the so-called ‘voids’ [19]. The voids are regions in the universe that are nearly empty of masses; masses at the rim of one void repel each other so that no mass enters one void by ‘gravitational’ forces.

6 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, 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<1, on the other hand, it is related to a one-dimensional metric that distinguishes different values of the field. The particles have a small but finite size η where several fields coincide. Here, the wave functions of different fields have to be superposed non-locally. Outside the particle, the wave function collapses into a single field wave function, which carries, however, the probabilistic quantum information of the particle to the particle at the opposite end of the field. The non-local region of the particle hereby may be extremely small, as discussed by Zurek [21]. The expression ‘quasi-local’ shall emphasize that we have a non-local theory with highly localised quantum states. A detailed quantum mechanical description of this mechanism is far beyond the scope of this work. We might, however, relate the existence of separated volume regions of the fields to ‘hidden variables’ in Bohm’s interpretation of quantum mechanics [22], [23]: individual energy quanta, emitted from one particle into the volume of one field, already ‘know’ the particle where they can be received, as one field component connects two distinct particles only. The quantum–statistical process of where to emit is attributed to the particles only. This interpretation of the exchange mechanism may also be related to Wheeler–Feynman’s absorber theory of light [24], or Cramer’s transactional interpretation of quantum mechanics [25], which connects the emission of a light quantum to a unique future event of absorption. I leave closer interpretation to future work. Within the quasi-local region of one particle, an ‘action at a distance’ in the sense of the Einstein-Podolsky-Rosen paradox exists: entangled quantum states (for a recent discussion, see [26], in German). The particles exchange energy with the field by an exchange flux for which a continuity equation in the classical sense must hold: generalised Newton’s equation (27). It is hereby unnecessary to ‘know’ the actual energy content of any state of the body of the universe, except the homogeneous initial state (without space, time, and energy). Any future state must have the same energy if no energy is created or destroyed. The mechanism of transferring action between massive bodies in the present concept is emitting quanta into the field, or receiving quanta from the field. This happens in the quasi-local environment of one individual particle. According to the definition of space by the spectrum of quantum fluctuations, decreasing and increasing the spectrum of fluctuation means contraction and elongation of space, respectively. It is evident, however, that this change of length will not happen instantaneously. There will be fluctuations of quanta within the field, which I assume to dissipate with the speed of light. In other words, action between bodies is transferred with the speed of light. There will be a ‘delay’ of action. We might argue that the dependence of the apparent size of a field in the case of accelerated particles Ωij≠Ωji on the position of the observer and the direction of acceleration is complementary to the gravitational time dilatation in general relativity [27]. Here, more detailed investigations are necessary in future work, too.

In light of the present concept, ‘dark matter’ loses its mystery. We simply relax the idea that all particles and fields have to be connected to all others. Particles that do not have a connecting field to the observer are ‘invisible’, as there is no space through which the light could travel. However, they will be detectable by their influence on massive objects that they are connected to and that have a direct connection to the local observer.

Finally, let me try an estimate of the size η of one particle. Comparing the energetic and spatial constants of the expressions for the volume of a field in (10) and (17), we read, using the numerical value of the number of particles in the observable universe NE from (31),


The proportionality constant is of order 1 depending on the volume integration over the particle, which is done here only for the special case of two connecting fields. Despite the large uncertainties in several ingredients to determine the actual value of η, it can be concluded that the size of a junction that relates to one single elementary particle ηsingle, like a neutrino, must be considered as ‘point-like’, far below Planck’s length. A triplet of three quarks in a confined state, however, defines a two-dimensional object. This means that the number of connected fields N in (32) must be related to an area proportional to η2. The radius of this area ηtriple is estimated to be comparable to the size of a neutron:


7 Conclusion

The new monistic concept of matter treats energy, ordered by a set of quantum-phase-fields, as the only existing substance. The dual elements of matter, mass and space, are described by volume- and gradient-energy contributions of the fields, respectively. The concept is based on the statement that energy can neither be created nor destroyed – the first law of thermodynamics. The origin of the universe is treated as a spontaneous decomposition of the symmetric state of 0 energy (‘nothing’) into ‘matter’, mass and space, by the demand of entropy production – the second law of thermodynamics. The time evolution of the fields dictates the time dependence of the Hamiltonian and the wave function. The wave function |w> 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 do 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 Ω˜E. 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 Ω˜E 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.


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.


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  • 1

    Note that the solution (13) implies η(1v2c2)2s2ϕπ2η(ϕ12)=0. 

About the article

Received: 2016-04-20

Accepted: 2016-10-27

Published Online: 2016-12-23

Published in Print: 2017-01-01

Citation Information: Zeitschrift für Naturforschung A, Volume 72, Issue 1, Pages 51–58, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784, DOI: https://doi.org/10.1515/zna-2016-0270.

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©2016, Ingo Steinbach, published by De Gruyter.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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