Multiple Fourier Integral into Applied to Modeling Neutrons Beta Decay
Курнышев Б. С. Multiple Fourier Integral into Applied to Modeling Neutrons Beta Decay // Молодой ученый. 2012. №9. С. 7-12.
According to modern concepts the neutron is not point. He has a structure.
The modern conception of a structure of neutron were developed based on experimental data and quantum chromodynamics. Considered reliably established that neutron consists of three quarks. The lifetime of neutron in the free state about 15 minutes. Thanks to the laws of conservation of energy, electric charge, baryon and lepton quantum numbers, neutron decays in the form of so-called beta-decay into a proton, an electron and an electron antineutrino. At present, according to scientific publications, there is not a mathematical model to calculate the physical process of the neutron decay. That is, this process cannot be calculated in details into time and in space.
This article presents an original approach to computer modeling of neutron beta decay. Into model is applied the 4-multiple Fourier integral.
We may derive the equations of motion in the decay process from the principle of least action. Only for consideration the motion within the structure of the neutron, need to know the spatial and temporal distribution of the density of the Lagrange function. Let us assume that - a density distribution of the Lagrange function within structure of the neutron, where - the coordinate of the space-time. Fourier integral, generated by the function , if for function exists the integral
by definition is
The combination of functions of the next form (3) in the modern theory of information is called a pair of the functions:
With the help of the metric tensor can determine the metric of the space-time in the structure of the neutron. In the general case should be imparity . Then the phase of the exponential functions in (3) will be equal to
By Euler formula the exponential function in (3) can be represented as real and imaginary parts:
From the (3), (5) get:
Moreover the functions may be decomposed into components
Then under the CPT-theorem and to (1) we can introduce condition
that is from (14) and (9) … (12) can get
In this way, can be solved by well-known problem of the baryon asymmetry.
According to (14) the functions (9), (11) are taken as mathematical description of the structure substance. Also according to (14) the functions (10), (12) are taken as mathematical description of the structure antimatter. Analogically, the function on the left in (13) is taken as mathematical description of the structure substance, and function on the right in (13) is taken as mathematical description of the structure antimatter. Thus, the inversion in the coordinates and time this is replacement of substance on antimatter as in the theorem about CPT-invariance.
The upper limits of integration in (9) … (12) have the infinite values. It means that the application of the integrals (9) … (12) to the solution of the input problem requires additional conditions that can exclude these infinite value. Otherwise there is a contradiction. Indeed, according to the original submission, the density of the Lagrange function corresponds each space-time point. But to pinpoint her in (7) together with (9), (10), it is necessary in (9), (10) exactly define the spectrum of density of the Lagrange function.
In turn, the spectrum of the density of the Lagrange function may be accurately determined by using (8), (11), (12). But the limits of integration in (11), (12) must be in this case from zero to infinity. Thus, to accurately determine the density of the Lagrange function in a one space-time point it is necessary determine the distribution of the density of the Lagrange function from to on all four coordinates. And this applies to every space-time point.
As a consequence, the value of the density of the Lagrange function can not be exactly determined in principle. The only solution in such a situation - the restriction of the upper limits of integration in (13), (9) … (12).
Then from (13), (9) … (12) get
According to theory of the waves there are the links between these deviations:
Similar ratios as in (21) there is in the quantum theory. It is about uncertainty principle.
I may say that the whole presented here approach were based on the ratios (21). These ratios allows to get rid of the infinities. The ratios (21) show us priceless physical and mathematical properties of the wave processes and of Fourier integrals, their inextricable link.
They have the following interpretation. Spectrum the waves of the density of the Lagrange function may be roughly in the same phase near a some point. As a result the wave amplitude in sum will be bigger. But the waves will be cancel each other a distance on account of the difference of the phases. Thus, non-zero density of the Lagrange function will be concentrated "near" given point ( "near" is determined by the ratio of the wavelength and of spatial distance up to this point; if the distance is comparable with the wavelength, then applicable "near"). It turns out that the space-time region in which waves approximately in one and the same phase, is inversely proportional to the width of the spectrum in this region.
In each case values of the root mean square deviation, of course, to be justified. For example, the width of deviation of coordinates for the neutron should be not less than his physical size. When modeling the neutron beta decay the width of deviation of coordinates should be not less, than physical size of hydrogen atom and so on.
Note that according to (14) the functions (17), (19) are taken as mathematical description of the structure particles. Also according to (14) the functions (18), (20) are taken as mathematical description of the structure antiparticles. Analogically, the function on the left in (16) is taken as mathematical description of the structure particles, function on the right in (16) is taken as mathematical description of the structure antiparticles. Thus, the inversion in the coordinates and time this is replacement of particles by antiparticles as in theorem about CPT-invariance.
Model experiment can be carried out by means of (17) … (20) and (21) with limits of integration initially under formulas (22), (23) - see below. Limits of integration can calculated as follows.
The rest mass of the neutron this is energy which is defined as , where - the rest mass of a neutron, - the speed of light in vacuum. This energy corresponds to value of , where - Planck constant, - frequency. Hence, in particular, it follows that s-1. This is called the characteristic number in the elementary particle physics (in order of magnitude).
can be seen as the root mean square deviation of coordinates of neutron.
The value of
can be seen as the root mean square deviation of the components.
Similar considerations can also be applied to the observable universe.
can be taken as the root mean square deviation of coordinates in the large-scale structure of our universe.
The root mean square deviation of the components of 4-vector of large-scale structure of the universe is specified as
Note that in this approach
This is known in physics dimensionless number (in order of magnitude).
So that the suggested approach is confirmed by astronomical observations.
The calculations were made using following values of the fundamental constants:
So, finally the system of functions
gives a mathematical description of neutron beta decay. In (27) - the root mean square deviation of the components for antineutrino, - the root mean square deviation of coordinates for hydrogen atom, - the components of the wave 4-vector, - the coordinate of the space-time.
This approach is based on the properties of the wave processes and of Fourier integrals.
The main conclusion is that quarks, neutron, antineutron, proton, antiproton, electron, positron, antineutrino, neutrino, the atoms and others particles may be represented as a continuous spectrum of waves of the density of the Lagrange function.
The process of neutron decay may be observed by means of the computer modeling (27).
Into process decay of neutron can be observed the structure of quarks and others (proton, electron, antineutrino, hydrogen atom, perhaps photons).
Into process formations of the neutron should be observed the formation structure of antineutrons.
By this approach the problem of the asymmetry baryons in the universe may be solved.
Creation of a computer program according to (17) … (20), (21), (24), (25) may solve the problem singular state of our universe.