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 *ϕ*_{I} 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 $\frac{\partial}{\partial s}$ and $\frac{\partial}{\partial t}$ are applied to the wave function |*w*> or the field *ϕ*_{I}, respectively.

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

$${u}_{I}=\frac{4U\eta}{{\pi}^{2}}\left[{\mathrm{(}\frac{\partial {\varphi}_{I}}{\partial s}\mathrm{)}}^{2}-\frac{1}{{c}^{2}}{\mathrm{(}\frac{\partial {\varphi}_{I}}{\partial t}\mathrm{)}}^{2}+\frac{{\pi}^{2}}{{\eta}^{2}}\mathrm{|}{\varphi}_{I}\mathrm{(}1-{\varphi}_{I}\mathrm{)}|\right]\mathrm{.}$$(8)

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

$$\begin{array}{c}\mathrm{(}1-2{\varphi}_{I}\mathrm{)}\frac{4U\eta}{{\pi}^{2}}[\frac{\partial {\varphi}_{I}}{\partial s}<w\left|\frac{\partial}{\partial s}\right|w>\\ -\frac{1}{{c}^{2}}\frac{\partial {\varphi}_{I}}{\partial t}<w\left|\frac{\partial}{\partial t}\right|w>]=0.\end{array}$$(9)

The force *e*_{I} related to the volume of field *I* is defined as

$${e}_{I}{\mathrm{|}}_{{\varphi}_{I}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}=\frac{4U\eta}{{\pi}^{2}}{\varphi}_{I}^{2}<w\left|\frac{{\partial}^{2}}{\partial {s}^{2}}-\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}}\right|w>\mathrm{.}$$(10)

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 ${\varphi}_{I}^{2}$ in (10) by the function $m\mathrm{(}\varphi \mathrm{)}=\frac{1}{4}\left\{\mathrm{(}2\varphi -1\mathrm{)}\mathrm{}\sqrt{\varphi \mathrm{(}1-\varphi \mathrm{)}}+\frac{1}{2}\mathrm{arcsin}2\varphi -1\right\},$ 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 *ϕ*_{1}=1−*ϕ*_{2}=*ϕ*, the equation of motion (6) read, with Δ*e*=*e*_{1}−*e*_{2}, ${m}_{\varphi}=\frac{\partial m}{\partial \varphi},$ and $\tau =\frac{4}{{\pi}^{2}}\tilde{\tau}:$

$$\begin{array}{c}\tau \frac{\partial}{\partial t}\varphi =\tau v\frac{\partial}{\partial s}\varphi =U\left[\eta \frac{{\partial}^{2}\varphi}{\partial {s}^{2}}\mathrm{(}1-\frac{{v}^{2}}{{c}^{2}}\mathrm{)}+\frac{{\pi}^{2}}{\eta}\mathrm{(}\varphi -\frac{1}{2}\mathrm{)}\right]\text{\hspace{0.17em}}\\ +{m}_{\varphi}\Delta e\mathrm{.}\end{array}$$(11)

I have transformed the time derivative of the field $\frac{\partial}{\partial t}\varphi $ into the moving frame with velocity *v*, $\frac{\partial}{\partial t}=v\frac{\partial}{\partial s}$ and used the Euler–Lagrange relation

$$\frac{\delta}{\delta \varphi}{\displaystyle {\int}_{-\infty}^{+\infty}\text{d}s}{\displaystyle {\int}_{0}^{+\infty}\text{d}t\to \frac{\partial}{\partial \varphi}-\frac{\partial}{\partial t}\frac{\partial}{\partial {\varphi}_{t}}-\frac{\partial}{\partial s}\frac{\partial}{\partial {\varphi}_{s}}}\mathrm{.}$$(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 (*s*_{1}<*s*_{2}, *s*_{1}<*s*<*s*_{2})

$$\varphi \mathrm{(}s\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}=\frac{1}{2}\left[\mathrm{sin}\frac{\pi \mathrm{(}{s}_{1}+s+vt\mathrm{)}}{{\eta}_{v}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\mathrm{sin}\mathrm{(}\frac{\pi \mathrm{(}{s}_{2}+s-vt\mathrm{)}}{{\eta}_{v}}\mathrm{)}\right]\mathrm{,}$$(13)

where ${\eta}_{v}=\eta \sqrt{1-\frac{{v}^{2}}{{c}^{2}}}$ 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. *s*_{1} and *s*_{2} are the spatial coordinates of the particles in the quasi-static picture related to the distance Ω=|*s*_{1}−*s*_{2}|. Figure 1 depicts the solution for two fields where the particles travel with velocity *v* and have finite extension *η*_{v}. 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.

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

Continuing the analysis of (13), one easily proves

$${\frac{\partial}{\partial s}\varphi |}_{\text{left}}={-\frac{\partial}{\partial s}\varphi |}_{\text{right}}=\frac{\pi}{{\eta}_{v}}\sqrt{\varphi \mathrm{(}1-\varphi \mathrm{)}}\mathrm{,}$$(14)

$${\frac{{\partial}^{2}}{\partial {s}^{2}}\varphi |}_{\text{left}}={\frac{{\partial}^{2}}{\partial {s}^{2}}\varphi |}_{\text{right}}=\frac{{\pi}^{2}}{{\eta}_{v}^{2}}\mathrm{(}\frac{1}{2}-\varphi \mathrm{)}\mathrm{,}$$(15)

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

$${\int}_{-\infty}^{+\infty}\text{d}s\frac{4U}{{\pi}^{2}}\left[{\eta}_{v}{\mathrm{(}\frac{\partial \varphi}{\partial s}\mathrm{)}}^{2}+\frac{{\pi}^{2}}{{\eta}_{v}}\mathrm{|}\widehat{\varphi}\mathrm{(}1-\varphi \mathrm{)}|\right]}=2U\mathrm{.$$(16)

## 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 Ω_{I} for field *I*. According to Casimir [16], we have to compare quantum fluctuations in the box with discrete spectrum *p* and frequency ${\omega}_{p}=\frac{\pi cp}{2{\Omega}_{I}}$ to a continuous spectrum. This yields the negative energy *E*_{I} of the field *I*:

$${E}_{I}=\alpha \frac{hc}{4{\Omega}_{I}}\left[{\displaystyle \sum _{p\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{\infty}}p-{\displaystyle {\int}_{1}^{\infty}p\text{d}p}\right]=-\alpha \frac{hc}{48{\Omega}_{I}}\mathrm{,}$$(17)

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 *p*→*pe*^{−ϵp} .

## 4.2 Multidimensional Interpretation

As stated at the beginning, the present concept has no fundamental space. The distance Ω_{I} 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 $\frac{\partial {\varphi}_{I}}{\partial s}.$ 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’.

Figure 2: Scheme of a number of seven fields connected by three particles. The particles have an uncertainty *η*_{v} 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.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.