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2.7 Quantino field theory

2.7.1 The quantino density of a single electric point charge

In section 2.6 was explained why quantino theory assumes that the electrical force and thus also magnetism and gravity are transmitted by messenger particles which propagate in all directions relative to the source charge and cause a momentum change when they hit on another charge. In this section, the previously only qualitatively described mechanism is quantitatively investigated and a field theory is developed. It starts with the calculation of the field density for the messenger particles as a function of the trajectory of a single emitting source.

In the section 2.6.2 it was mentioned that quantino theory assumes that the messenger particles called quantinos are not emitted at a constant velocity, but that their emission velocity $w$ obeys a certain probability distribution $\Gamma(w)$. Not much can be said about the form of this distribution at this point, except that the speed $w=0$ cannot occur. If this would be the case, more and more quantinos would accumulate at the location of the point charge in the course of time. If now the source would be accelerated, the resulting singularity would remain, which however would be unphysical.

In order to obtain a continuous density distribution, it is necessary to abstract from the fact that the field of quantinos actually consists of individual discrete field quanta. This type of approximation is often used in physics; for example, a gas can be seen as a continuum and it can be abstracted from the fact that it actually consists of individual molecules by defining a density. This in turn is nothing more than the number of molecules in relation to a certain volume. The choice of the reference volume is, of course, arbitrary. To get rid of this arbitrariness, one chooses the smallest possible volume, namely zero. The absolute number of molecules in a zero volume is also zero. But the ratio of number to volume, i.e. the density, is this usually not.

Figure 2.7.1.1: The emission sphere (red) detaches from the source at time $\tau$ and is keep moving straight ahead.
Such a density shall now also be found for the quantino field. First we consider that a quantino that detaches from the source at time $\tau$ and moves away at the speed $\vec{w}$, is at time $t > \tau$ at location
$$\vec{r}_q(t,\tau) = \vec{r}_s(\tau) + (\vec{w} + \dot{\vec{r}}_s(\tau))(t-\tau).$$ (2.7.1.1)
$\vec{r}_s(\tau)$ is the location of the source and $\dot{\vec{r}}_s(\tau)$ its speed at time $\tau$. If we imagine that at a certain point in time $\tau$ further quantinos are emitted in all other directions with the same velocity $w$, so we obtain a sphere whose radius increases with the velocity $w$. The center of this emission sphere is keep on moving even then, according to the following equation
$$\vec{r}_c(t,\tau) = \vec{r}_s(\tau) + \dot{\vec{r}}_s(\tau)(t-\tau),$$ (2.7.1.1)
when the speed of the source changes after emission. This means in particular that the emission spheres detach themselves from the sources when these are accelerated. Figure 2.7.1.1 illustrates this in form of an animated sketch.

With the aid of the emission sphere (2.7.1.1) it is possible to model a density $p_w$ of all quantinos ever emitted from a point charge $e$ with an emission velocity $w$. For each point $\vec{r}$ the density is at any time $t$ is given by
$$p_w(\vec{r},w,t) = \int\limits_{-\infty}^{t} p_{\tau}(\vec{r},w,t,\tau)\,\d{\tau}$$ (2.7.1.2)
with
$$p_{\tau}(\vec{r},w,t,\tau) = \frac{a_c}{4\,\pi} \Gamma(w) \frac{ \delta\left(\Vert\vec{r}-\vec{r}_c(t,\tau)\Vert- w (t-\tau)\right)}{\Vert\vec{r}-\vec{r}_c(t,\tau)\Vert^2}$$ (2.7.1.3)
The density describes the number of quantinos per space volume and speed interval at a certain time $t$. The unit is therefore $m^{-3}\,(m/s)^{-1}$. $p_{\tau}$ is the contribution to the density $p_w$ during the infinitesimal small time interval $[\tau,\tau + \d{\tau}]$. The unit of $p_{\tau}$ is therefore $1/m^4$. The $\Gamma(w)$ function is the aforementioned probability distribution, which assigns a relative probability to each emission velocity. Since the physical unit of a probability distribution is always equal to the inverse of its parameter, it follows that $\Gamma(w)$ has the unit $s/m$. The Dirac function must have in this case the unit $1/m$ according to the same logic. This means that an additional parameter $a_c$ with the unit $1/s$ is required. $a_c$ should be considered as a constant, which defines the absolute number of quantinos emitted per second by an elementary charge. It should be noted that this really means all quantinos, even those that cannot be absorbed due to their too high speed.

Since the quantino density can principally only be perceived locally by a receiver, it must be transformed into the reference frame of such a receiver. For this purpose, we first conclude that a quantino, which was sent out at the location $\vec{r}_s(\tau)$ at time $\tau$ has the instantaneous velocity
$$u = \left\Vert\frac{\vec{r}_d(t) - \vec{r}_s(\tau)}{t-\tau} - \dot{\vec{r}}_d(t)\right\Vert$$ (2.7.1.4)
when it reaches the receiver at time $t$. $\vec{r}_d(t)$ is in this context the trajectory of the receiver and $\dot{\vec{r}}_d(t)$ its instantaneous speed at the moment of collision with the quantino. If the reader does not immediately understand this relation, it will be discussed in more detail here.

If we want to know the density of all quantinos with the speed $u$ from the point of view of a moving receiver at time $t$, we have to filter the expression (2.7.1.2) by writing
$$p_u(u,t) = \int\limits_{0}^{+\infty} \int\limits_{-\infty}^{t} \delta\left(\left\Vert\frac{\vec{r}_d(t) - \vec{r}_s(\tau)}{t-\tau} - \dot{\vec{r}}_d(t)\right\Vert- u\right)\,p_{\tau}(\vec{r}_d(t),w,t,\tau)\,\d{\tau}\,\d{w}.$$ (2.7.1.5)
The integration over $w$ can be carried out immediately. We get
$$p_u(u,t) = \frac{a_c}{4\,\pi} \int\limits_{-\infty}^{t} \Gamma\left(\frac{\Vert\vec{r}_d(t)-\vec{r}_c(t,\tau)\Vert}{t-\tau}\right)\,\frac{\delta\left(\left\Vert\frac{\vec{r}_d(t) - \vec{r}_s(\tau)}{t-\tau} - \dot{\vec{r}}_d(t)\right\Vert - u\right)}{(t-\tau)\,\Vert\vec{r}_d(t)-\vec{r}_c(t,\tau)\Vert^2}\,\d{\tau}.$$ (2.7.1.6)

Finally, it is still necessary to integrate over all quantino speeds $u$. As described in section 2.6.2 it is important to realize that the receiver charge cannot interact with quantinos that are from its point of view too fast, that means faster than $c$. This ensures that the speed of light is universally constant, regardless the relative speed between source and receiver.

The effective quantino density, i.e. the subjectively by the receiver perceptible, is therefore
$$p_e(t) = \int\limits_0^c \,p_u(u,t)\,\d{u} = \int\limits_{-\infty}^{+\infty} \,\intfunc_{0}^{c}(u)\,\,p_u(u,t)\,\d{u}.$$ (2.7.1.7)
The function $\intfunc$ is by the way the very handy interval function .

After the integration remains
$$p_e(t) = \frac{a_c}{4\,\pi} \int\limits_{-\infty}^{t} \Gamma\left(\frac{\Vert\vec{r}_d(t)-\vec{r}_c(t,\tau)\Vert}{t-\tau}\right)\,\frac{\intfunc_{0}^{c}\left(\left\Vert\frac{\vec{r}_d(t) - \vec{r}_s(\tau)}{t-\tau} - \dot{\vec{r}}_d(t)\right\Vert\right)}{(t-\tau)\,\Vert\vec{r}_d(t)-\vec{r}_c(t,\tau)\Vert^2}\,\d{\tau}.$$ (2.7.1.8)
The effective quantino density now depends only on $t$ and describes the number of quantinos that the receiver can perceive in a very small volume element in the area of his own location at the time $t$. It applies both to arbitrarily moved, especially accelerated sources, and to arbitrarily moved, accelerated receivers.

Under the simplifying assumption that the probability distribution $\Gamma(w)$ is approximately linear in the range up to the speed of light and closely beyond, the effective quantino density simplifies to
$$p_e(t) = \frac{a_c\,\Gamma_1}{4\,\pi} \int\limits_{-\infty}^{t} \frac{\intfunc_{0}^{c}\left(\left\Vert\frac{\vec{r}_d(t) - \vec{r}_s(\tau)}{t-\tau} - \dot{\vec{r}}_d(t)\right\Vert\right)}{(t-\tau)^2\,\Vert\vec{r}_d(t)-\vec{r}_c(t,\tau)\Vert}\,\d{\tau},$$ (2.7.1.9)
since
$$\Gamma(w) \approx \Gamma_1\,w$$ (2.7.1.10)
applies here. This is sufficient for some first fundamental considerations.

2.7.2 Quantino density of an oscillating dipole

2.7.2.1 Calculation of the effective quantino density

In this section, the formula (2.7.1.9) derived in the previous section is analyzed by using an oscillating dipole consisting of two elementary charges. For reasons of simplicity, it is assumed that both are resting at the origin of the coordinates. At $t=0$, however, the negative charge shall begin to oscillate around the origin in direction of the z-axis (in Figure 2.7.2.2.1; up and down). For the speed of the negative charge, the equation
$$\dot{\vec{r}}_{s}^{(-)}(t) = l\,\omega\,\vec{e}_z\,\intfunc_{0}^{\infty}(t)\, \cos(\omega t)$$ (2.7.2.1.1)
applies. To keep the calculation simple, it is assumed that the displacement $l$ is so small that it can be neglected. In fact, this approximation is usually very well fulfilled in practice, since the oscillations of the electromagnetic field often spread over hundreds of kilometers into space, but the displacement of the charges usually only reaches fractions of millimeters.

Based on this approximation, formula
$$\vec{r}_s^{(-)}(\tau) = l\,\vec{e}_z\,\intfunc_{0}^{\infty}(t)\, \sin(\omega t) \approx 0$$ (2.7.2.1.2)
applies. If we use this in formula (2.7.1.1), we obtain the equation of the emission spheres
$$\vec{r}_c^{(-)}(t,\tau) = l\,\omega\,\vec{e}_z\,\intfunc_{0}^{\infty}(\tau)\, \cos(\omega \tau)(t-\tau).$$ (2.7.2.1.3)
This in turn can be inserted in equation (2.7.1.9) and we get
$$p_e^{(-)}(t) = \frac{a_c\,\Gamma_1}{4\,\pi} \int\limits_{-\infty}^{t} \frac{\intfunc_{0}^{c}\left(\frac{\Vert\vec{r}_0 + \vec{v}\,\tau\Vert}{t-\tau}\right)}{(t-\tau)^2\,\Vert\vec{r}_0 + \vec{v}\,t-l\,\omega\,\vec{e}_z\,\intfunc_{0}^{\infty}(\tau)\, \cos(\omega \tau)(t-\tau)\Vert}\,\d{\tau}$$ (2.7.2.1.4)
with $\vec{r}_d(t) = \vec{r}_0 + \vec{v}\,t$ the equation of motion of the receiver. The remaining part of this section is concerned with simplifying formula (2.7.2.1.4).

First, we get rid of the interval function above the fraction line by taking advantage of the fact that the lower limit is irrelevant, since the argument of the interval function can never be smaller than zero due to the calculation of the absolute value and because of $t\geq\tau$. The upper limit is reached if $\tau$ is
$$\tau_c = \frac{c^2\,t + \vec{r}_0 \vec{v} - \sqrt{(\vec{r}_0 \vec{v})^2 + c^2 (\vec{r}_0 + \vec{v}\,t)^2 - r_0^2 v^2}}{c^2 - v^2}$$ (2.7.2.1.5)
and the condition $v < c$ is fulfilled. By the way, for $v \ll c$ the equation (2.7.2.1.5) can be replaced by the somewhat simpler approximation
$$\tau_c \approx \frac{(c\,r_0 - \vec{r}_0\,\vec{v})(c\,t - r_0)}{c^2\,r_0}.$$ (2.7.2.1.6)


Equation (2.7.2.1.5) can now be used to transform equation (2.7.2.1.4) into
$$p_e^{(-)}(t) = \frac{a_c\,\Gamma_1}{4\,\pi} \int\limits_{-\infty}^{\tau_c} \frac{1}{(t-\tau)^2\,\Vert\vec{r}_0 + \vec{v}\,t-l\,\omega\,\vec{e}_z\,\intfunc_{0}^{\infty}(\tau)\, \cos(\omega \tau)(t-\tau)\Vert}\,\d{\tau}.$$ (2.7.2.1.7)
In the next step, the initially required condition is used that the displacement $l$ is very small. For this purpose, we expand into a Taylor series about $l$ and stop after the term of first order. We get
$$p_e^{(-)}(t) = \frac{a_c\,\Gamma_1}{4\,\pi\,\Vert\vec{r}_0 + \vec{v}\,t\Vert} \int\limits_{-\infty}^{\tau_c} \frac{1}{(t-\tau)^2} \left( 1 + \frac{l\,\omega \vec{e}_z(\vec{r}_0 + \vec{v}\,t)\intfunc_{0}^{\infty}(\tau)\cos(\omega \tau)(t-\tau)}{\Vert\vec{r}_0 + \vec{v}\,t\Vert^2}\right)\d{\tau}.$$ (2.7.2.1.8)
Further rearranging yields
$$\begin{eqnarray} p_e^{(-)}(t) & = & \frac{a_c\,\Gamma_1}{4\,\pi} \frac{1}{\Vert\vec{r}_0 + \vec{v}\,t\Vert} \int\limits_{-\infty}^{\tau_c} \frac{1}{(t-\tau)^2} \d{\tau} + \\ & & \frac{a_c\,\Gamma_1}{4\,\pi} \frac{l\,\omega\,\vec{e}_z\,(\vec{r}_0 + \vec{v}\,t)}{\Vert\vec{r}_0 + \vec{v}\,t\Vert^3} \intfunc_{0}^{\infty}(\tau_c)\, \int\limits_{0}^{\tau_c} \frac{\cos(\omega \tau)}{t-\tau} \d{\tau}. \end{eqnarray} $$ (2.7.2.1.9)
These integrals can be solved. It follows
$$p_e^{(-)}(t) = \frac{a_c\,\Gamma_1}{4\,\pi} \frac{1}{\Vert\vec{r}_0 + \vec{v}\,t\Vert} \frac{1}{(t-\tau_c)} + \frac{a_c\,\Gamma_1}{4\,\pi} \frac{l\,\omega\,\vec{e}_z\,(\vec{r}_0 + \vec{v}\,t)}{\Vert\vec{r}_0 + \vec{v}\,t\Vert^3} \intfunc_{0}^{\infty}(\tau_c)\,\varrho(t,\tau_c)$$ (2.7.2.1.10)
with
$$\varrho(t,\tau_c) = \cos(\omega\,t) \big(\mathrm{Ci}(\omega\,t) - \mathrm{Ci}(\omega\,(t - \tau_c))\big) + \sin(\omega\,t)\big(\mathrm{Si}(\omega\,t) - \mathrm{Si}(\omega\,(t - \tau_c))\big).$$ (2.7.2.1.11)

The effective quantino density of the resting positive charge is obtained by setting $l=0$ in the upper solution. We get
$$p_e^{(+)}(t) = \frac{a_c\,\Gamma_1}{4\,\pi} \frac{1}{\Vert\vec{r}_0 + \vec{v}\,t\Vert} \frac{1}{t-\tau_c}.$$ (2.7.2.1.12)
Finally, both densities are combined to the effective total quantino density
$$ p_e(t) = p_e^{(+)}(t) - p_e^{(-)}(t) = -\frac{a_c\,\Gamma_1}{4\,\pi} \frac{l\,\omega\,\vec{e}_z\,(\vec{r}_0 + \vec{v}\,t)}{\Vert\vec{r}_0 + \vec{v}\,t\Vert^3} \intfunc_{0}^{\infty}(\tau_c)\,\varrho(t,\tau_c).$$ (2.7.2.1.13)
By the way, the unit of this expression is $1/m^3$, as one can easily verify.

2.7.2.2 Evaluation of the result

Figure 2.7.2.2.1: The effective quantino density of the dipole at an oscillation frequency of 25kHz from the point of view of a stationary receiver. The values of the x- and y-axis are in meters. The total duration of the animation is 500 $\mu s$
In this section the effective total quantino density of the dipole, i.e. the formula (2.7.2.1.13), is analysed. It is obvious at first glance that the equation (2.7.2.1.13) contains an interval function that makes the entire expression disappear if the argument, namely $\tau_c$, given by the formula (2.7.2.1.5), is smaller than $0$.

If we look at $\tau_c$ we see that this is a function that depends only on the time $t$, the speed of light $c$, the speed of the receiver $\vec{v}$, and its location $\vec{r}_0$ at the time the dipole oscillation starts. An interesting question is now, at which time $t$ this expression becomes zero, i.e. for which $t$
$$\frac{c^2\,t + \vec{r}_0 \vec{v} - \sqrt{(\vec{r}_0 \vec{v})^2 + c^2 (\vec{r}_0 + \vec{v}\,t)^2 - r_0^2 v^2}}{c^2 - v^2} = 0$$ (2.7.2.2.1)
applies. By rearranging we get
$$(c^2\,t + \vec{r}_0 \vec{v})^2 = (\vec{r}_0 \vec{v})^2 + c^2 (\vec{r}_0 + \vec{v}\,t)^2 - r_0^2 v^2.$$ (2.7.2.2.2)
Expanding of the square terms and a subsequent reduction leads to
$$c^4\,t^2 = c^2\,r_0^2 + c^2\,v^2\,t^2 - r_0^2 v^2.$$ (2.7.2.2.3)
After some reshaping, factorization and summarizing we finally get the simple equation
$$t = \frac{r_0}{c}.$$ (2.7.2.2.4)

It is remarkable that this result no longer depends on $\vec{v}$ but only on the distance $r_0$ of the receiver to the source. That means, it is irrelevant at all, with which speed the receiver moves! The density fluctuation always needs the same time $r_0/c$ to reach the receiver!

This is also evident in the examples shown in Figures 2.7.2.2.1, 2.7.2.2.2 and 2.7.2.2.3. They show the temporal development of the effective quantino density at a dipole frequency of $25\,kHz$. This corresponds to an electromagnetic wave below the long wave range. Due to the high value of the speed of light, such waves have wavelengths several kilometers long, which makes this frequency well suited for representations.

It should be noted that the effective quantino density is not yet synonymous with an electromagnetic wave, because the latter it is vector field. However, the effective quantino density already contains many interesting properties, which makes it to an important intermediate step.

The visual analysis starts with figure 2.7.2.2.1. It shows the temporal development of the effective quantino density multiplied by the square ot the distance between source and receiver from the point of view of a resting receiver. It can be seen how an oscillation begins to spread at the time $0$. After about $340\,\mu s$ the entire drawing area is filled. This means that the wave has moved around $100\,km$ within this period, which corresponds approximately to the speed of light. By looking at the animation it is furthermore noticeable that the sign of the wave reverts on the x-axis. This is not surprising, because the dipole is oscillating, which means that the charge shift is time-varying and on one side always dominates the charge that is inferior on the other. This is not surprising, because the dipole is oscillating, which means that the charge shift is time-varying and on one side always the charge dominates that is weaker on the other side. As one can also recognize, the oscillation on the x-axis itself is completely eliminated. However, close above and below the x-axis the sign is constantly changing. The sign change itself runs along the axis at the speed of light. It is immediately clear that this is an effect that is closely related to TEM waves (transverse electromagnetic waves).

Figure 2.7.2.2.2: The quantino density from the point of view of a receiver that is moving upwards (z-direction).
Figure 2.7.2.2.3: The quantino density from the view of a receiver that is moving to the right (x-direction).

How does the situation appear for a moving receiver? The figures 2.7.2.2.2 and 2.7.2.2.3 illustrate it. The left side shows the effective quantino density from the perspective of a receiver moving in the z-direction, while the right side shows the situation from the perspective of a receiver that is moving in the x-direction. The speed of both observers is the same in both figures, namely $c/5$.

It is obvious as well in the Figures 2.7.2.2.2 and 2.7.2.2.3 that the wave propagates at the speed of light, because the outer ring forms a perfect circle that does not reach a moving observer sooner or later than a stationary one. However, there are also clear differences to Figure 2.7.2.2.1. For example, both animations show a noticeable Doppler effect and the dipole itself seems to move, because the situation is presented from the perspective of a moving receiver.

As can be clearly seen in the figures 2.7.2.2.1, 2.7.2.2.2 and 2.7.2.2.3, the oscillation not only always moves at the same speed $c$, also the shape of the oscillation is preserved. The attentive reader may wonder what happens to the quantinos, which are moving slower than $c$? Shouldn't the oscillation become diffuse and blurred? In fact, this is not the case, since the slower quantinos neutralize each other in terms of their effect.

To show this even more clearly, a dipole is analysed, in which the oscillation is switched off again after exactly one period. Mathematically this can be modeled by limiting the upper bound of the interval function in equation (2.7.2.1.1) with $2\,\pi/\omega$. A following calculation of the corresponding effective quantino density is completely identical up to equation (2.7.2.1.8). Only at formula (2.7.2.1.9) differences arise, because it is here now also necessary to pay attention to the upper limit.

In general, for any function $f$, the following relationship applies

$$\int\limits_{-\infty}^{\tau_c}\,f(\tau)\,\intfunc_{0}^{2\,\pi/\omega}(\tau)\,\d{\tau} = \intfunc_{0}^{2\,\pi/\omega}(\tau_c)\,\int\limits_{0}^{\tau_c}\,f(\tau)\,\d{\tau} + \intfunc_{2\,\pi/\omega}^{\infty}(\tau_c)\,\int\limits_{0}^{2\,\pi/\omega}\,f(\tau)\,\d{\tau}.$$ (2.7.2.2.5)
If this is applied to the calculation, the effective total quantino density
$$ p_e(t) = -\frac{a_c\,\Gamma_1}{4\,\pi} \frac{l\,\omega\,\vec{e}_z\,(\vec{r}_0 + \vec{v}\,t)}{\Vert\vec{r}_0 + \vec{v}\,t\Vert^3} \left(\intfunc_{0}^{2\,\pi/\omega}(\tau_c)\,\varrho(t,\tau_c)+ \intfunc_{2\,\pi/\omega}^{\infty}(\tau_c)\,\varrho\left(t,2\,\pi/\omega\right)\right)$$ (2.7.2.2.6)
follows. The figures 2.7.2.2.4 and 2.7.2.2.5 show the temporal development of the density, one from the perspective of a stationary and one from the perspective of a receiver moving to the right. The parameters correspond to the values already used for the other animations.

Figure 2.7.2.2.4: The effective quantino density of a dipole oscillating for only one period from the point of view of a stationary receiver.
Figure 2.7.2.2.5: The effective quantino density of a dipole oscillating for only one period from the perspective of a receiver moving to the right.

Both figures show that the oscillation propagates at the speed of light and only at the speed of light. The slower quantinos only cause a kind of slowly decaying dipole field in the immediate vicinity of the dipole.