Quantum Phase Field

Abstract

In this chapter a new paradigm for the understanding of the physical world is developed based on phase-field type non-linear wave solutions. Two 1-dimensional wave fronts, right- and left-moving are combined to define the doubloon, the basic element of mass and space. The interface energy of a standard phase field defines the mass of elementary particles. The non-zero values of the phase field, the core of the doubloon, defines space in an introverted view. The sum constraint of the multi-phase-field model defines a doubloon network which is closed in itself. The model can be applied to investigate scale formation in the universe.

1 Introverted Picture of Mass and Space

How to Start?

There are so many aspects coming together to form a consistent theory, culminating in phase-field theory. From the physical point of view, this includes thermodynamics, wave mechanics, interface-driven phenomena and capillarity, and the kinetics of phase transformations. From the practical point of view, we have the numerics of the solutions of partial differential equations, and multi-physics problems. The main applications of phase-field theory to date have been engineering problems. From the mathematical point of view, we derive a set of partial differential equations using variational principles from a governing functional that has a gradient contribution acting on the field. In quantum-mechanical language, this gradient is a momentum operator, and the field is a quantum field with a discrete spectrum. If this sounds strange, then be reassured that, as before, we will go through this step by step.

Let us start from the conceptual point of view. We discuss a free-boundary problem. The free boundary is the inner boundary between two or more domains, which are characterized by phase fields. In classical phase-field theory, we call the boundary an interface. The boundary is termed “free” because it is not fixed by some external condition; it can evolve in time (see Chap. 1). This inner boundary is different from outer boundaries, i.e., the boundaries of the calculation domain, which are fixed. The inner boundary between phases evolves in space and time. This evolution is closely coupled to transport phenomena in the domains: transport of temperature, solute, and momentum, as well as displacement. These phenomena have been discussed in the previous lectures.

The object of interest in phase-field theory is the inner boundary. This is not something that we specify; it is something that will come out: it “emerges.” It shall be independent of the settings of the outer boundary in any system; if there is any outer boundary at all.

Boundaries bound “spaces”. We may distinguish between “introverted” and “extroverted” spaces. The latter means the space around an object. The extroverted space is a kind of background, existing with or without the object. This is the traditional view of space. Objects are placed in space. As an illustration see Fig. 8.1. The green of the billiard table represents space. The objects, billiard balls, are placed in this space. They interact according to Newton’s laws, momentum and energy conservation. In a real billiard game the player uses the cushions to mirror space. We may, hypothetically, push the cushions to infinity, then our play ground will be infinitely large, but balls would never come back. We may postulate periodic or “Moebius” boundary conditions. But all this becomes irrelevant if we revert the picture to an introverted space. Introverted space lives inside the objects: it is bounded by the objects. These introverted spaces are sketched by the yellow lines in Fig. 8.1. The relative position of all balls is uniquely determined by the network of inner spaces. An absolute outer space is not needed to characterize their interrelation.

Fig. 8.1

The billiard table as a representation of the extroverted view of space in traditional physics: objects are placed in space. The network of introverted spaces (yellow lines) connecting the balls reverts the view to a closed network system without outer boundaries (Ⓒ www.winsport.de with permission)

In the physical world, which is composed of particles and space, the only objects that could serve as a bound of an introverted space are elementary particles. So, if particles (zero-dimensional objects) are the boundaries, space is the 1-dimensional line-distance between particles. There is no other choice.

If there are many particles, there will be many spaces. In the quantum-phase-field concept, each space is associated with a phase field ϕ_α. From the number of elementary particles within the observable universe N, a maximum of N combinatorial 2 spaces is a number of the order of 10¹²⁰. Space will also be taken as “substantial,” i.e., it is not just “empty”: it has an energy, a negative energy related to vacuum fluctuations, as shown in Sect. 8.3. The 3D “space of cognition,” the space of our daily experience, will be reconstructed later, in Sect. 8.4. We will proceed like this: Define a system of quantum phase fields, closed in itself, without having outer boundaries.

2 Formal Definition of Quantum Phase Fields

We define (and you have all the background from the previous lectures) a set of a large number of phases ϕ_α. The phases form a system that is closed in itself, forming a “universe.” It is a system without outer boundaries. We do this in analogy to the multi-phase-field system from Chap. 6:

$$\displaystyle \begin{aligned} \sum_\alpha \phi_\alpha = 1. {} \end{aligned} $$

(8.1)

Here, ϕ_α is simply termed the “phase” since up to now no “space” has been defined. Each phase ϕ_α is an element of the universe that is different from all other elements ϕ_β. Therefore, an interface between these phases is formed: a fermionic particle, as we will see. The phases incorporate the basic elements of physical matter, “mass” and “space.” We will associate them with the conserved quantity: energy H. The previous statement that “the phases incorporate mass and space” is important: the phases (our phase fields) ϕ_α are not defined in space, as in classical or quantum field theories, but rather they define space! Massive elementary particles are not “placed into space,” they “connect spaces.” We will derive this step by step.

The energy functional of the system in quantum mechanics $\hat H$ is an operator (as indicated by the hat). This is acting on the wave function of the system |w〉. In our case, the system—though this sounds quite highbrow—is the universe; |w〉 is the wave function of the universe. The only ingredient of the theory, up to now, is energy $H=\langle w| \hat H |w\rangle$ as the expectation value of $\hat H$. Energy is conserved (first law of thermodynamics).¹ Furthermore, we will postulate that H = 0, i.e., there is no net energy: all energetic states, positive and negative, have to sum up to 0. The argument for this is simple: there is no evidence regarding where a finite energy should come from (see also the Wheeler–DeWitt theory [7]).

We will allow changes $d \hat H$, that the zero-energy state, the state of “nothing,” separates into positive and negative energetic states, the state of “something.” This may be related to the Big Bang as the origin of our universe, if you will. Changes are related to a non-conserved quantity, entropy: the second law of thermodynamics. We will expand $\hat H$, in the changes $d \hat H$ with respect to the phases ϕ_α. $\hat H$ thereby is itself a function of all fields {ϕ_β}, $\hat H = \hat H(\{\phi _\beta \})$, and, as a reminder, all fields are connected by the sum constraint (8.1):

$$\displaystyle \begin{aligned} \hat H = \sum_\alpha \int_0^1 d\phi_\alpha \frac{\partial \hat H(\{\phi_\beta\})}{\partial \phi_\alpha}. {} \end{aligned} $$

(8.2)

The integral runs over the definition range of the phase fields from 0 to 1, meaning yes or no, existing or not-existing, and we allow the fields to vary between these bounds, i.e., that they are diffuse, as is usual in phase-field theory.

The phases will be used to indicate different states of energy. This means that they will be used to derive relations between the different states of energy. We will allow a BIG number of fields connecting a BIG number of different states of energy, both positive and negative. An important factor in this regard is that positive and negative states are not just “mirrored” states like matter and anti-matter in traditional physics: they are topologically different, as we will discuss in closer detail when we discuss the ordering scheme of space and mass in Sect. 8.4. For now we simply state, that there are negative energy states E_α associated to a phase α, and positive states U_αβ associated to junctions between phases α and β.

There is no fundamental space. We repeat the argument: where should it come from?

“Space” is introduced as a distance Ω_α, defined by the inverse of the negative state of energy E_α < 0 of the field ϕ_α:

$$\displaystyle \begin{aligned} \Omega_\alpha = - \tilde \alpha \frac{hc}{48 E_\alpha}, {} \end{aligned} $$

(8.3)

where h is Planck’s constant, c is the speed of light, and $\tilde \alpha$ is a positive and dimensionless coupling parameter to be determined. It will be proven self-consistently in Sect. 8.3 that this definition leads to a 1D line coordinate s_α specific to each phase, ϕ_α = ϕ_α(s_α). Having this line coordinate we can treat ϕ_α(s_α) as a classical phase field in 1D.

There is no proof that this is the only possible construction to define our universe, no rigorous argument against concepts of “parallel universes” or “multiverses,” or 10- or 21-dimensional spaces with strings and branes embedded, as postulated by several researchers (see, e.g., [14] for 10 dimensions and [18] for 21 dimensions). You will find almost every number of dimensions for space–time postulated from 4 to 21, maybe even higher, since the original work of Kaluza and Klein [15, 16]. A whole discipline, called “mathematical physics,” is searching for a “unitarian description of all physical phenomena” in multidimensional manifolds with appropriate geometrical measures.

We will approach this differently. The present theory introduces time and space as auxiliary coordinates: the coordinates are not “fundamental.” It is shown that the theory, which is based on a set of principal statements, is consistent with these statements when formulated in these auxiliary coordinates. It is “self-consistent,” and it resembles our cognition of the physical world.

Substituting $d\phi _\alpha = \frac {\partial \phi _\alpha }{\partial s_\alpha } ds_\alpha$ and introducing the forces $\hat h_\alpha = \frac {d \hat H}{ds_\alpha }$ yields

$$\displaystyle \begin{aligned} \hat H = \sum_{\alpha=1}^N \int_{-\infty}^\infty ds_\alpha \frac {\partial \phi_\alpha}{\partial s_\alpha} \frac {\partial \hat H(\{\phi_\beta\})}{\partial \phi_\alpha} = \sum_{\alpha=1}^N \int_{-\infty}^\infty ds_\alpha \hat h(\{\phi_\beta\}). {} \end{aligned} $$

(8.4)

Space emerges, i.e., it is created by variations in the phases dϕ_α: compare the “diffuseness of phase field.” We relate the line coordinate s_α to the distance Ω_α, defined by the inverse of the energy quantum E_α (see (8.3)), by the integral

$$\displaystyle \begin{aligned} \int_{-\infty}^{+\infty} ds_\alpha \phi_\alpha = \Omega_\alpha = - \tilde \alpha \frac{hc}{48 E_\alpha}. {} \end{aligned} $$

(8.5)

This is easily be seen from the solution of ϕ_α(s_α), Fig. 8.2. Phases and space are complementary objects, defined by phase fields. In the present concept, the phase field generates space instead of living in a space. But we need to swallow a toad: space s_α is a 1D line coordinate. There is no evidence of a 3D space (besides our daily cognition, which might be in error!).

Fig. 8.2

Two doublons in a periodic setting. Each doublon is formed by a right-moving and a left-moving soliton. The velocity is proportional to the energy difference between adjacent doublons

To construct the energy operator, or Hamiltonian $\hat H$, we use the standard form of the Ginzburg–Landau functional in 2D Minkowski notation:²

$$\displaystyle \begin{aligned} \hat H = \sum_{\alpha=1}^N \int_{-\infty}^{+\infty}ds \frac {4 U\eta}{\pi^2} \left\{\left(\frac \partial{\partial s} \phi_\alpha\right)^2 - \frac 1{c^2}\left(\frac \partial{\partial t} \phi_\alpha\right)^2 + \frac {\pi^2} {\eta^2} |\phi_\alpha (1-\phi_\alpha)|\right\}. {}\end{aligned} $$

(8.6)

You will recognize the similarity with the free-energy model with a double-obstacle potential from Chap. 2. In addition to the gradient contribution in space, we include a gradient contribution in time. The time derivative accounts for dissipation, i.e., that in contrast to gradients in space, which have a positive energy penalty, gradients in time are favored energetically: the system shall evolve and not stagnate. We may treat this philosophically or simply state that this ansatz ensures relativistic invariance due to Lorentz contraction of the interface width (see [20] for details). U is a positive energy quantum to be associated with the positive rest-mass of elementary particles, the equivalent of the surface energy in the traditional phase-field model.³ The gradient contributions of the Hamiltonian, Eq. (8.6), $\frac \partial {\partial s}$ and $\frac 1c \frac \partial {\partial t}$, shall be understood as operators acting on a quantum mechanical wave function |w〉. This allows to evaluate the expectation value of the energy for an actual state of the system. The wave function in “quasi-static approximation” will be explicitly constructed below in Sect. 8.3. The phase-field equation is written down:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde \tau \frac\partial{\partial t} \phi_\alpha = -\frac{\delta}{\delta\phi_\alpha} \int_{0}^{+\infty}dt \langle w| \hat H|w\rangle. {} \end{array} \end{aligned} $$

(8.7)

Since gradients of time are considered, we have to integrate over time to have a well-defined functional derivative, as discussed in Chap. 2. This equation has two parts: (i) a non-linear wave equation for the phase field; and (ii) a linear Schrödinger-type equation for quantum-mechanical excitations. This procedure is not new; it can be traced back to the so-called de Broglie–Bohm double-solution program [1, 2, 5, 6]. It is fully accepted as an alternative interpretation of quantum mechanics compared to the prevailing so-called “Copenhagen interpretation.” For more explanation, see [17].

Now, we separate the expectation value of the energy functional (8.6) into three different contributions. These are distinguished by whether the differential operators $\frac {\partial }{\partial s}$ and $\frac {\partial }{\partial t}$ are applied to the wave function |w〉 or the field ϕ_α.

Applying the differential operators to the phase-field components and using the normalization of the wave function 〈w|w〉 = 1 yields the force u_α $[\frac {\mathrm {J}}{\mathrm {m}}]$ related to the gradient of the fields α:

$$\displaystyle \begin{aligned} \begin{array}{rcl} u_{\alpha} = \frac {4U\eta}{\pi^2}\left[\left(\frac {\partial \phi_\alpha}{\partial s}\right)^2 -\frac 1{c^2}\left(\frac {\partial \phi_\alpha}{\partial t}\right)^2 +\frac{\pi^2}{\eta^2} |\phi_\alpha(1-\phi_\alpha)|\right]. {} \end{array} \end{aligned} $$

(8.8)

This contribution relates to the interface energy in the conventional 3 dimensional phase field application. In 1 dimensions it is a force quantum, related to gradients of the phase field or to values of the phase-field 0 < ϕ_α < 1.

The mixed contribution, when one of the operators $\frac {\partial }{\partial s}$ and $\frac {\partial }{\partial t}$ is applied to the field ϕ and one to the wave function |w〉, describes the correlation between the field and the wave function. It shall be set to 0 in the quasi-static limit. In this limit, we keep the field static for the evaluation of the quantum-mechanical force. Then, we take this force for the determination of the time evolution of the field. A coupled solution has not been worked out to date:

$$\displaystyle \begin{aligned} 0= (1-2\phi_\alpha) \frac {4U\eta}{\pi^2} \Bigm[ \frac {\partial \phi_I}{\partial s} \langle w|\frac {\partial}{\partial s}|w\rangle - \frac 1{c^2}\frac {\partial \phi_\alpha }{\partial t}\langle w|\frac {\partial}{\partial t}|w\rangle \Bigm]. {} \end{aligned} $$

(8.9)

It is shown in [20] that this Eq. (8.9) is consistent with Newton’s second law of acceleration. Finally, we apply the momentum operators $\frac {\partial }{\partial s}$ and $\frac 1c \frac {\partial }{\partial t}$ to the wave function |w〉, which yields the force e_α $[\frac {\mathrm {J}}{\mathrm {m}}]$:

$$\displaystyle \begin{aligned} e_\alpha =\frac{4U\eta}{\pi^2} \phi_\alpha^2 \langle w|\frac {\partial^2}{\partial s^2}-\frac 1{c^2}\frac {\partial^2}{\partial t^2}|w\rangle. {} \end{aligned} $$

(8.10)

This contribution applies to the bulk energy of the phase field ϕ_α = 1. We will explicitly evaluate this after the structure of the solutions of the fields is discussed. For a steadily moving field with velocity v, one transforms the phase-field equation into the moving frame traveling with this velocity $\frac {\partial }{\partial t}= v \frac {\partial }{\partial s}$. Inserting Eqs. (8.8)–(8.10) into (8.7), we find:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde \tau \frac\partial{\partial t} \phi_\alpha & =&\displaystyle -\frac{\delta}{\delta\phi_\alpha} \int_{0}^{+\infty}dt \langle w| \hat H|w\rangle \\ & =&\displaystyle U \left[\eta \frac {\partial^2\phi_\alpha}{\partial s^2}\left(1-\frac{{v}^2}{c^2}\right) + \frac{\pi^2}\eta \left(\phi_\alpha-\frac{1}{2}\right)\right] + m_{\phi_\alpha}\Delta e, {} \end{array} \end{aligned} $$

(8.11)

where: Δe = e_α − e_β is the difference in the volume force (the equivalent of the Gibbs free-energy difference) between two fields, Eq. (8.10); and $m_{\phi _\alpha }$ is the appropriate coupling function. As a first result, we see that the interface width η contracts with velocity: $\eta _v= \eta \sqrt {1-\frac {v^2}{c^2}}$, Lorentz contraction. It collapses for v → c, and velocities v > c are forbidden.

The structure of the field is well known to us from Chap. 2: the minimum solution of Eq. (8.11) is the “soliton.” A periodic solution for two fields is displayed in Fig. 8.2. One field bounded by two solitonian waves, one right-moving and one left-moving, is called a “doublon.” This is the basic element of physical space ϕ_α = 1, bounded by the junction to the other field, which will be interpreted as an elementary particle later.

3 Volume Energy of One Doublon

From the doublon solution in Fig. 8.2, we can see that the field forms a 1D box with fixed walls and size Ω_α for the field α. This is a standard exercise for a quantum problem: the particle in a box potential. The difference here is that the particle is massless, i.e., the dispersion relation is linear in momentum p instead of quadratic in p as for the massive particles. According to o Casimir [3], we have to compare quantum fluctuations in the box with discrete spectrum p and frequency $\omega _p= \frac {\pi c p }{2\Omega _I}$ to a continuous spectrum. This yields the negative energy E_α of the field α:

$$\displaystyle \begin{aligned} E_\alpha = \tilde\alpha \frac {h c}{4\Omega_\alpha} \left[ \sum_{p=1}^\infty p - \int_1^{\infty} p dp \right] = - \tilde\alpha \frac {h c}{48\Omega_\alpha}, {} \end{aligned} $$

(8.12)

where $\tilde \alpha$ is a positive, dimensionless coupling coefficient. I have used the Euler–Maclaurin formula in the limit 𝜖 → 0 after renormalization p → pe^−𝜖p. Since all parameters are positive, we see that “space” is accounted for by negative energy, scaling inversely proportional to the size of the doublon, which proves (8.3) for self-consistency. This energy scales like the energy of the gravitational field in Newtonian mechanics. Therefore, the force, as the derivative of the energy (8.12) with respect to space, can be associated with a gravitational attraction between the junctions between different doublons. The junctions can thus be interpreted as massive objects: elementary particles. They are associated with positive energy U [Eq. (8.8)] corresponding to the interface energy of a traditional phase field, while the bulk energy of the doublon is negative. In contrast to Newtonian mechanics and in agreement with general relativity (see “gravitational waves” [9]), the attraction is a wave phenomenon with time-dependent action. The coefficient $\tilde \alpha$ can be determined from the measured gravitational constant on Earth. Applying the theory to all masses of the visible universe gives a prediction for repulsive gravitation at ultra-long distances and an explanation of the expansion of the visible universe [20, 21].

4 Multidimensional Interpretation

As stated at the beginning, the present concept has no fundamental space. The distance Ω_α is intrinsic to one individual doublon α. Different doublons are connected in junctions due to the sum constraint (8.1): a multiple junction in the multi-phase-field terminology defines an elementary particle in the quantum-phase-field concept. The position of one particle related to an individual component of the field is determined by the steep gradient $\frac {\partial \phi _\alpha }{\partial s}$. In a multiple junction, where many fields intersect, the junction couples many different directions related to the different doublons. We shall consider the junction (or particle) as a “very small volume,” in physics terminology, this is a zero-dimensional object. Since half-sided solitons are spinor-type objects belonging to the 3D SU(2) symmetry group, we may order the incoming and outgoing doublons of a junction in 3D Euclidean space. We will call this space the “space of cognition,” since our cognition orders all physical objects in 3D space (Fig. 8.3 sketches this picture). Individual doublons form a “doublon network.” Each doublon is expanded along a 1D line coordinate and bound by two end points, described by gradients of the field, right- and left-moving solitons. Due to the constraint (6.6), the coordinates of different fields have to be synchronized within the junctions of small but finite size η. The constraint (6.6) also dictates that there are no “loose ends”; the body is closed, in itself forming a “universe.”

Fig. 8.3

Network of six doublons ϕ_α with four vertices, the particles P_i. You may recognize the analogy to “introverted spaces” in Fig. 8.1

Finishing this lecture let me remind you: In the language of network theory, the junctions of a network are called vertices (in Latin the plural of vertex, a single junction or knot). The connections between vertices, are called “edges”. Such a network can be embedded into a 3-dimensional vector space, or higher (but not lower). The doublon network, as we call it in the context of this lecture, does not define a vector space by itself! This means that empty spaces inside the meshes of the network are not accessible, as they would in a vector space. They have no physical reality. Only points on the doublons can be attributed by a local coordinate, related to the line distance to a junction or vertex. This is important, since vertices, which are not connected by edges, are “invisible:” there is no physical space between them where light, information, action etc. could be transmitted through. It will be a future task to embed the doublon network into a four-dimensional space-time where the doublons form geodesic trajectories between particles.

A last comment relates to the topology of positive and negative states of energy. The positive states U relate to particles, junctions between the doublons: each particle connects all incoming and outgoing doublons. In contrast, one doublon connects exactly two particles. This defines the network structure, see also above. Furthermore, positive and negative energy states cannot simply annihilate without changing the topology of the whole system: once separated into MANY positive and negative states, the system evolves irreversibly.

5 Symmetry Breaking in Condensed Matter and Elementary Particle Physics

“Symmetry breaking” is a fundamental concept in materials physics, see the consideration about “phases” and the “order parameter” in Chap. 1. The term “symmetry” here relates to solid materials’ crystal symmetry, or the lack of symmetry in the amorphous state, liquid or gas. A particular case is magnetic phase transformation from the ferromagnetic to the paramagnetic phase. In the ferromagnetic state we see spontaneous magnetization of the magnetization $\pm \vec M$, which is a vector in 3-dimensional space and can have + or − direction. This is the “symmetry broken state”. In contrast, the paramagnetic phase, which is the stable phase above the Curie temperature T_c. It is called the “symmetric state” with vanishing spontaneous magnetization $\vec M =0$.

In the context of elementary particle physics, the concept of “symmetry breaking” was first introduced by Goldstone [11], referring to the theory of superconductivity, i.e. a phase transformation in condensed matter. Very soon, the publication “Broken Symmetry” by Goldstone et al. [12] appeared, which established the concept of symmetry breaking in elementary particle physics, so-called Goldstone modes. Only two years later, the work of Higgs [13] and Englert and Brout [10] appeared, which was awarded the Nobel prize in 2013 after the so-called “Higgs boson” had been confirmed experimentally [4]. In light of the present concept, the Higgs field, which couples to fermions to give them their mass, can be identified with the phase field as we call it in materials science: An order parameter field in space and time with broken symmetry. The underlying mathematical formalism, Lagrange and Hamiltonian functions are almost identical. The new contribution of the “quantum phase field” is the explicit solution of the minimum energy solution of this field in 1 dimension; the doublon, as detailed. Doublons then are connected to form a doublon network in 3+1 dimensional space-time by the sum constraint of a multi phase-field, Eq. (8.1).

$$\displaystyle \begin{aligned} \sum_\alpha \phi_\alpha = 1. \end{aligned}$$

In conclusion: phase-field theory has two different roots (see Chap. 1): one in thermodynamics and one in wave mechanics. Thermodynamical principles of energy conservation and entropy maximization set the stage. The wave-mechanical picture of phase fields rests on the theory of solitons. Quantum-phase-field theory has roots in de Broglie and Bohm’s interpretation of matter as a wave phenomenon; it combines the soliton solution of a non-linear wave equation, the phase-field equation, with a Schrödinger-type linear wave equation in a finite domain. The non-linear wave equation is thus derived from the thermodynamic theory of phase transformation: Ginsburg–Landau theory. We may use other potentials than the double well or double obstacle, as explained earlier. We may use higher gradients in ϕ. The general structure, however, is consistent, having three different contributions, as in every phase-field model: gradient, potential, and bulk (see Chap. 2).

Where is the bulk part in the Hamiltonian (8.6)? The bulk part arises from applying the gradient operator to the wave function! It is defined from the solution of Schroedinger equation in the box formed by the doublon. In the concept, and in “reality” (I think), there is no given “space,” no bulk in which objects, “elementary particles,” are placed. The elementary particles, which in the phase-field interpretation are junctions or knots between doublons, are defined by gradients of the phase fields. They are the endpoints of the doublons and form the bounds of space. Space and mass are two sides of the same coin. This all may seem “quite philosophical” to the student, or simply “confusing.” But it should not discourage you from doing the exercise below [calculating the energy of “space” from (8.10)]. We conclude with the following statements:

• Energy is substantial and conserved.

• There is no fundamental space, no fundamental time.

• Space, a one-dimensional distance, emerges from variation of energy.

• Mass is attributed by positive energy, space by negative energy.

• Two half-sided solitons form the doublon, which is the primitive object of the concept, defining mass and space.

• The doublon belongs to the 3D S(U2) group of spinors. Doublons span the 3D space of our cognition.

• The concept is relativistically invariant and makes predictions at the scale of the universe (see “further reading”).

6 Exercises

Exercise

Derive the “energy of space” (8.12): $E_\alpha = - \tilde \alpha \frac {h c}{48\Omega _\alpha }$ defined by vacuum fluctuation in the 1D box formed by a doublon of size Ω_α.

Further reading

• Hydrodynamic basis of quantum mechanics and its relation to Phase Field: [19]

• Expansion of the universe by repulsive gravitational action on ultra-long distances: [20, 21].