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ISSN: 2090-0902
Journal of Physical Mathematics
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Guarcs in the Inside Hadronic Four-Dimensional Euclidean Space with Real Time

Eugene Kreymer*

Institute for Physics and Engineering, Donetsk, 83114, Ukraine

*Corresponding Author:
Eugene Kreymer
Institute for Physics and Engineering
Donetsk, 83114, Ukraine
Tel: (062) 311-52-27
E-mail: [email protected]

Received Date: February 28, 2015; Accepted Date: June 09, 2015; Published Date: June 16, 2015

Citation: Kreymer E (2015) Guarcs in the Inside Hadronic Four-Dimensional Euclidean Space with Real Time. J Phys Math 6:140. doi:10.4172/2090-0902.1000140

Copyright: © 2015 Kreymer E. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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The paper represents the results of the study of the four-dimensional Euclidean space with real time (E-space), where 0 ≤ ||VE|| ≤ ∞, in sub-hadronic physics. This closed space has a metric that distinguished from the Minkowski space and the results obtained in the model are different from physical law in the Minkowski space. As it follows from the model of Lagrangian Mechanics, quarks in the central-symmetric attractive potential, kinetic energy of quark diminishes while the speed grows as the quarks exchange their energy-mass with gluons possessing a zero rest mass, so that to ensure the permanent proton mass. This dependence describes the dynamical relation of constituent and current quarks masses. In the quantified motion model it has been stated, that the oscillations of the particles are cyclic, including alternating localization and translation phases, the action per cycle for a free particle equals h . The calculation of charge distribution density in proton, carried out on the basis of this model, conforms to the results of the experimental research. All relations between physical values in the E-space, mapped in the Minkowski space, correspond to the principles of SR and are Lorentz-covariant and the infinite velocity is equal to the velocity of light in the Minkowski space. These models have a transparent physical sense.


Dynamics of quarks in the proton; Euclidean invariants; Motion of quarks and gluons; Quantum cyclic motion; Charge distribution in the proton


Non-perturbative effects are of great importance for the theory of space inside hadron. Supposing a sequence of QCD problems are concentrated in the branch of occurrences that can be described through the transition from the Minkowski space M(xM0,xM1,xM2,xM3) (M-space) into the Euclidean space inside hadron via the analytical extension of the time axis onto the lower semi plane xEi0=ixM0. In this case we get the Euclidean space with the imaginary time Eim(xEi0,xE1xE2,xE3,) (Eim is space), and XEi=XM is automatically VEi=iVM and 0 ≤ ||VEi|| ≤ 1. The use of such a space has brought to great results: the QCD valuum models, lattice calculations, string theory and so on. However, e.g. QCD in lattice can now be used only for the description of a limited class of hadronic elements of the matrix. There is no common and self-congruent description of the QCD vacuum heretofore, as well as confinement occurrence and a spontaneous disturbance of the chiral invariance. In the common case the rotation group of the Euclidean space in the plane (xE0, XE) presupposes that 0 ≤ ||VE|| ≤ ∞, while Eim0 ≤ ||VEi|| ≤ 1. At the same time Eimis not even a subspace of the Euclidean space, because it is not closed in respect of the operation of composition of vectors. Thus, an infinite velocity causing non-local (instantaneous) interactions and contained in some NQCD models lies outside the frames of Eim - space. Non-local quark non-perturbative vacuum condensate plays a crucial role while creating realistic hadrons models [1]. At the same time the space correlation functions look like the curve of decreasing exponent [2] whose negative parameters include the distance of z=x-y while xEi0=const.

In correspondence with [3,4] physics of non-locality starts to be seen at the distance of λ ≈ 0, 2fm . The correlation length λ determines the spatial declining of bound gauge-invariant bilocal correlator of field gradient.

In other studies, a minimal Gauss model, offered in [5], is used for condensates in a non-perturbative vacuum. The parameter of nonlocality λ characterizes an average square of quarks’ impulse in the QCD vacuum. Its estimations by means of QCD in lattice have shown the following range of probable values: equation[6,7]. Eim- is homomorphic in respect of the M-space and non-local, in other words, the instantaneous interactions even at some low λ value contradict with S principles.

It gives a reason to consider that the use of merely a part of fourdimensional Euclidean space volume in the models with Eim does not allow using its potential to the full extent. The article expounds the first steps in the research of the inside hadronic four-dimensional Euclidean space with real time model E(xE0, xE1, xE2, xE3) (Е-space), where equation and its aim is to show the expedience of the studies in the Е-space as s probable prospective direction of sub-hadronic physics development. The article contains researches of the E-space properties in protons and it is presupposed that the obtained correlations have a common nature and can cover all the hadrons. Moreover it has been considered been considered that the models in the Е-space will not be an alternative for the theoretical developments in Eim, but will extend their possibilities. The following requirement is the basic condition enabling this model to exist:

Requirement 1: Space-time relations and regularities in the Е-space model mapped into the Е -space must correspond to the principles of SR and be Lorentz-covariant.

Inside Hadronic Euclidean Frames of Reference

In the Е-space no frames of reference, which are microscopic in reality, can be physically implemented. To determine the spatial coordinates the laboratory frame of reference LFR with the coordinates (xM0, xM1, xM2, xM3) has been used, where hadron rests, and dxE= dxM. The own time of particles in the LFR is admitted to be the temporal coordinate xE0, Thus the Е-space is “subsidiary” towards the Е-space.

Definition 1: Inner hadronic four-dimensional Euclidean Frame of Reference (xE0, xE1,xE2 ,xE3) EFR, is a system, where the space coordinates are indexed by the coordinates of (xM1, xM2, xM3) LFR and the own time of the particles is equal to the own time of the particles in the LFR

equation, (2.1)

Where VMi is the velocity of the i-number particle in the LFR. The transition to the other IFR is carried out by means of Lorenz transformation. The Е -space of the real particles corresponds to the Е-space upper closed cone equation and equation that ensures the execution of the causality principle. The EFR has an invariant which taking into consideration the Definition 1 is equal to

equation (2.2)

Then there is symmetry between EFR and LFR: the time of one space is the invariant of the other.

From (2.1) and (2.2) it follows that

equation, (2.3)

Where vE - is the velocity of the particle in the EFR. And, correspondingly

equation. (2.4)

If vE → ∞, then vM → 1. There is also 4-vector of velocity in the EFR


And its invariant is equal to the invariant of the corresponding relativistic 4-vector.

SO(2) Group of the rotation of plane (xE0, xE), cannot be applied in the EFR, because the existence of the infinite velocity makes the time absolute, and xE0 can take no negative values. In accordance with (2.2), E-group of position-vector rotations Е which describes the particles moving with different velocity is valid in the EFR. This group does not mix the temporal xE0 and the spatial coordinates RE((xE1,xE2,xE3). Mapping kinetic parameters of the particle in ERF observed in into LRF putting the fundamental quadratic forms

equation, where gEμν -Kronecker symbol, μ, ν=0,1,2,3 and

equation, where gMμν-metric tensor. The translation matrix

equation must involve kinematic KEM and metric GEMtransformations. With (2.2) we obtain the kinematic transformation matrix equation the metric transformation matrix ||GEI||= diag (1, -1, -1, -1). There is a distinction of properties of the studied space from the Minkowski space that emerges because of different metric: E-group of radius-vector rotations xM0 does not mix the temporal xE0 and spatial coordinates RE(xE1,xE2,xE3).

The Model of E-invariant Lagrange Mechanics Particle

4-vector energy- momentum

Lagrange function of the free particle

equation. (3.1)

The momentum of the particle

equation, (3.2)

And the kinetic energy

equation. (3.3)

This equation is valid under condition that equation.. At the same time

E2E+P2E=m2 (3.4)

From (3.4) we can make a conclusion that there is a 4-vector of energy-momentum in the EFR, and its invariant is equal to the invariant of the corresponding relativistic 4-vector and it is one more symmetry between the LFR and EFR. Translating the 4-vector of the particle in LFR through (2.3), we obtain equation E = PEM mvM. These values stay equation -invariant.

Formula (3.3) testifies to an unusual behavior in the E-space of the kinetic energy: it diminishes when the speed grows. The next unit will demonstrate that it is so because of the energy-mass exchange between quarks and gluons.

Mechanics of quark in the proton

Here we use the model where quarks are considered electrically neutral particles, and we admit that in the center of a proton there is a hypothetical source creating central-symmetrical attractive potential V(r) of strong interactions. It is considered that this simplified model will provide the possibility to determine some peculiarities of quarks motion in the proton.

The E-invariant Lagrange function of the quark in the potential V(r)

equation (3.5)

On the analogy with (3.3) the energy of the system “quark – potential V(r) »

equation (3.6)

If a particle is under the influence of power equation parallel to the velocity, that it will change the momentum as follows:

equation (3.7)

The alteration of the energy

equation . (3.8)

From which

equation. (3.9)

From the (3.9b) and (3.6) it follows that EV=0. The zero-value of EV is a result of the fact that gluons have not been taken into account. To ensure the constant proton mass, the alteration of the quark kinetic motion must be compensated by the relevant alteration of gluons energy – mass. Taking gluons into account.

LEV=LEq(vq)+LEG(vG)-V(r), where LEG(vG) is the Lagrange function for gluons.

The preserved energy of “quark – gluon – potential V(r) system makes

equation (3.10)

This equation has a solution, if vG=vq. Then in the potential V(r)

equation (3.11)

Gluon momentum is

equation (3.12)

And the energy

equation (3.13)

From Esq. (3.3) and (3.13) we can draw a conclusion, that the energy – mass of the quark translates into the energy - mass of the gluon, and their sum makes equals mq. At the same time PEG=PEq and gluons are moving along with quarks creating valon. As a result, the constituent mass of quarks includes zero rest mass. This determines the dynamical relation of constituent and current quarks’ masses. The quark mass diminishes as it approaches to the centre of a proton. This corresponds to the existing idea that quark has a minimum mass under a big transferred to it q2 momentum. There are some scientific studies devoted to the NQCD, in which gluons are described as possessors of dynamical energy - mass [8]. Contains an approximate solution of Dyson-Schwinger equation, where a propagator of non-perturbative gluon is regulated by the dynamical generated mass of a gluon. The usage of this propagator gives an opportunity to calculate sections of pp- scattering and achieve a good concord of calculations with experimental data for an effective gluon mass of 370 MeV [9], this value corresponds to mq in the nucleon. The fact that gluon has peculiarities of a massive particle is confirmed by calculations in lattice [10,11]. In the papers [12,13] different non-zero masses of gluons have also been studied. Let us examine the quark motion in the linearly increasing potential V(r) =cr. The zero orbital moment of a proton along with experimental studies of the charge distribution in proton means that the quark is vibrating along the diameter towards the center of a proton. Let us presuppose that the quark vibrates under the power of |Fz|=constant along the z axis which has a null in of the center a proton. Basing n the eq. (3.2), (3.3) and (3.9a) we obtain

equation (3.14)

Where equation is a “classic” acceleration? Then

equation (3.15)


equation (3.16)

The dependence of the quark energy on the radius is equation From the (3.6) we can draw a conclusion that equation, where rp is the radius of a proton and equation. Under the condition that equation in the coordinates (xE0, z) quark makes a circumference with a radius a−1clz. But the allowable values are xE0=[0;∞) and this formula must be specified. The half period of quark vibration is equation and to preserve xE0 in the given range of values we need to put (3.15) it in the following way:

equation, (3.17)

Where equation, [n] is the biggest whole number in equation. The digits before the root take turns depending on the alteration of [n].

Thus, a vibrating quark makes two half circumferences with z>0 and z<0, moved at equation. Figure 1 shows the graph of the quark oscillations. The calculation involves the rms radius of the proton rp=0.84fm.


Figure 1: Diametrical quark oscillations in the proton: dash line is for the model of Lagrangian Mechanics, continuous line is for the model of quantified motion: points A, C, E-Ezq v=∞, 0; points B, D-EEq=0 Ezq v= ∞, E m Eq q c=1.

Here we can show how the formula (3.6) is functioning. Under z=0 and V=0 the speed makes vEzq=∞ and EEq=0 (points A, C, E). Under equation and vEzq=0 as well as EEq=mq, as well as V=mq (points B, D). And therefore EEq-V=0.

This brings up a question: how do the oscillations of quarks provide total zero momentum in the motionless proton while they are oscillations? Under multi-particle interactions, a symmetric disposition of particles corresponds to the minimum of energy and therefore a proton possesses a spherical symmetry and that means that 3 quarks make diametric oscillations creating a space angle π and their impulses are getting balanced. This supposition correlates with analytical studies described in [14]; according to them effective fields in baryons has a Y-shaped configuration of quarks’ plane making an equilateral triangle. This conclusion has also been confirmed by calculations in lattice [15].

Models of E-Invariant Quantized Motion of Massive Particles

A peculiarity of inside hadronic E-space is that its size in the three- dimensional space is comparable to the Compton quark wave length and the maximum value of quark kinetic energy makes mq. According to the quantum mechanics the minimum quark energy in the limited space must excess its mass. This is also applicable for oscillators’ energy in the quantum field theory. Thus the wave equations cannot be applied in our case, including probability interpretation. Though the quarks’ behavior in hadrons has a casual nature and the definite metric of the E-space enables to precede straight forward to the probability characteristics.

Free scalar particle

The model is oriented towards the inside hadronic space, in which a particle cannot be free, so this part is of a methodic character.

Let us introduce the probabilistic space indexed by E-elements and defined by three quantities (Ω,Σμ) where Ω is a multitude of eve, Σ σ is algebra of Ω subsets and μ is a positive measure normalized, and μ(Ω) ≤1. If XE is the real random variable and XÅ ∈ Ω , then the distribution of XE is the probabilistic measure on Ωequation

Definition 2: The state of the particle is described by the function equation, belonging to E and selected for equation

If the functions equation describe a scalar particle then its Lagrangian will equal Ò

equation, (4.1)

From which in a usual way we can get a Klein-Gordon-Fock equation in the E - space

equation, (4.2)

Correspondingly to the (3.4)

The obvious function equation may not seem to be the solution of the (4.2), as it will give the conditional expectation value equation. Under PE=0 we obtain the nonphysical value equation. It is also impossible to use the transition to K-representation through a Fourier transformation, as the frequency kE0 and the wave vector KE not satisfy the (3.4).

Despite the time coordinate, (4.2) describes the static state. However the infinite velocity in E – space, makes it possible to transform the equation for the description of dynamic systems. Now represent equation as the product of two functions equation, each depending on just one variable. Such separation has the following physical meaning. For vE=0 the function equation will describes the localization phase, and the function equation the translation phase for vE=∞. These phases cannot exist simultaneously, and supposing the average duration of the localization phase is equation, and that of the translation phase is equation, and the average phase change occurring with τE and χE, then after every cycle of phase change we get the motion of the particle at the average velocity of equation. Such a separation is due to the infinite velocity.,

Probabilistic approach in compliance with definition 2, consider equation being the multidimensional random vector. The random projection of this vector on, for example, axis xE0 defines the probability of event equation and requires the condition equation to be commonly met. The latter condition is met for XE=0, that is equation. Accordingly, u, equation. for equation, i.e. xE0=0. As a result, we arrive at (4.2).

Separating the variables it is necessary to take into account that τE and ÷ must be a 4 – vectors: equation, and equationwhere

equation (4.3)

From this it follows that equations for each phase of the i-cycle are to be solutions of (4.1)



equation (4.5)

Equation (4.4a) has the following solution equation. As attractive potential V(r) ≥ 0 equally affects the particle as well as the antiparticle, according to (3.6)equation and correspondingly, KE ≥ 0. Considering that xE0 ≥0 from Definition 2 it follows that C1 =1, C2 = 0 , and there remains the decreasing exponent. The probability density equationpossesses necessary properties: the densities are not negative and the integral of the densities over all values of xE0 equals unity. The mathematical expectation of the localization phase duration

equation. (4.7)

For several cycles, segments xE0 form the simplest stream with no aftereffect. For the free particle in the translation phase the displacement vector of the particle xE and vector kÅ are co-directed and (4.4b) has the following solution

equation, (4.8)

Where equation is the unit vector. Probability density equation i.e. probability density is also positive and the mathematical expectation of the particle displacement in the translation phase is

equation . (4.9)

As has been assumed ,τE and χE are the components of E-vector and with (2.2)

equation, (4.10)

where τMl is the average cycle duration in LRF. E-invariant is a value

equation, (4.11)

which equals quantum of action .

On the grounds of (4.10) we consider the cycle duration in LRF xM0l to be the two-dimensional random vector with random coordinates xE0 and xE , distributed by the exponential law. Then xM0l is also distributed by the exponential law equation with the average value equation. From equality equation it follows that xMl is also distributed by the exponential lawequation and equation. Using equations (2.3b) and (4.9) we obtain

equation (4.12)

and the average velocity of equation. The value equation is also relativistically covariant, and equals to relativistic Lagrangian accurate to a coefficient and changes from equation to 0.

Thus the motion of the particle in E-space is discreet, consists of alternating translation and localization phases and the resultant action for every cycle equals a quantum of motion. The averaged graph of free – particle motion in equationis a random step function with the average step length τE and the average step height χE .

The average duration of free – particle cycle in LRF quantizes time xM0l into intervals with the average value equation dependent only on particle mass. And homogeneity is not violated.

Free spinor particle

Spinor function equation also should be a solution to the Dirac equation in E and describe two phases of motion. To derive the Dirac equation model in E we need to take into account that equation. The sense of such E-space partition is in the fact that in it the rotation is only possible in equationand consequently only bispinors have effect. Let us factorize (4.2)

equation. (4.13)

Matrices equation satisfy the relation equation, where gEμν – the Kronecker symbol, and equal

equation (4.14)

Function equation should describe two phases of motion

equation. (4.15)

For the localization phase together with (4.5) we obtain

equation, (4.16)

where equation - bispinor with kE>0 and xE0 ≥ 0 .

Equation for the translation phase is

equation (4.17)

and in compliance with (4.8) the solution is equation. Then

equation, (4.18)

When movement is along axis x3

equation (4.19)

and the space of bispinor ψtr (x3) is a proper space of the diagonal matrix σ3 with positive and negative helicity and there may be only a discrete transition between these subspaces. The duration of localization phases and the extent of translation phases are defined by formulae (4.7) and (4.11).

All the features of the quantum theory of the scalar particle are valid for spinors as well. But in the latter case we have a new detail of helicity. In E, the helicity of massive fermions is only observed in the translation phase, and it is a “good” quantum number, whereas in M the helicity of massive fermions with a nonzero mass can’t be a quantum number characterizing the particle, since it can be inverted by appropriate Lorentz transformations. Nevertheless, in nature, there exist left and right fermions that are quite different particles and this is seen in E.

Neutral spin or particle in the strong potential

If the particle is affected by the attractive potential which in the general case equals V(x0,xE), then (4.13) will take the form

equation. (4.20)

If potential V(xE) works then in the localization phase

equation. (4.21)

The solution to this equation is

equation. (4.22)

The average duration of the localization phase is

equation. (4.23)

The equation for the translation form will take the form

equation (4.24)

and the solution

equation. (4.25)

The average extent of the translation phase is

equation. (4.26)

With the quantized motion for V(r)=cr (3.9) takes the following form

equation. (4.27)

Equation (4.27b) proves that while the translation phase is on when xE0=const, there are instant nonlocal interactions in E. However, when mapped in M they take place with speed c.

Application of the Model of Quantized Motion of Quarcs to Determine Some Properties of Quarks in Protons

Quantized motion of quarks

The calculation of the quantized motion of quarks has been done on the basis of the IVC (Figure 1) on the assumption that the quark moves along the axis z which passes though the centre of the proton, parameters of motion τViand equation being of average value. The following data are used in the calculation: root-mean-square radius of the proton rp=0.84 fm and equation, averaged constituent mass u and d of quarks 0.33Gev. This mass is included into the calculation as Compton wave-length of a quark equation. The motion of a quark is divided into deceleration and acceleration portions. The initial point for the calculation (point A) is chosen at the beginning of the deceleration portion when a quark has passed through the centre of a proton and at point equation the quark localization phase starts. The acceleration portion starts with the translation phase at point B when vEz=0 and the end of the translation phase coordinate xE0 has become more than 0.84fm.

The quark deceleration in the second half-period of oscillation starts also with the localization phase at point C for vEz=∞ and z<0 and the calculation is done in the way similar to the first half-period. Here the following peculiarity is disclosed: the coordinates of the beginning of the second oscillation (0.08fm 0.32fm) are close to the accepted coordinates of the beginning of the first oscillation (0.0, 0.3fm).

Charge distribution in the proton

Central-symmetric motion of quarks (Section 3.1) makes it possible to confine to the calculation of the charge distribution for one quark considering that its charge equals the charge of a proton. The calculation is done on the assumption that V(r) =cr and the charge distribution is defined by the probability of the quark being at a given point of radius r=|Z| and this probability must be determined from the M-space "viewpoint"|

equation, where equation. As the calculations show that the second oscillation practically repeats the first oscillation the parameters of the first oscillation are accepted as the calculation basis. The calculation of the charge density has been done under the condition that the charge is located in the spherical layer with a unit thickness which has radius r. After the approximation by the exponential function the equation for the charge density calculation is obtained equation for validity factor R2=0.85. The calculated charge distribution along the radius is equation (Figure 2).


Figure 2: Electric charge distribution in the proton: continuous line is for calculation data, dashlineis for experimental data.

For the comparison the experimental data for the electric formfactor of the proton have been used which are usually described by dipole approximation equation [16] for the preset square of 4-momentum q2. This dependence gives the experimental value of charge density equation and that of the distribution of a charge along the radius equation (Figure 2).

Graph jc(r) systematically exceeds je(r). It is connected with the fact that definitional domain jc(r) equals 0<r<0.85 fm and the box under jc(r) equals ≈1 . Definitional domain jc(r) equals 0<r<∞ and the box under this curve on the section 0<r<0.9fm equals 0.6.


It is stated that in the E-space model, Radius-vector rotations group does not mix temporal and spatial coordinates;- kinetic energy diminishes when the speed grows. This determines the existence of constituent and current quarks and describes the dynamic relation of their masses; - to describe quantum movement in the E-space, wave equations cannot be applied. The application of the random function theory has shown that the quarks’ movement consists of localization and translation phases;- helicity of massive fermions can be observed only during translation phase and is a “good” quantum number;- an infinite velocity and non-local interactions connected with it while mapping in the M-space does not upset the RS-principles: the maximum interaction transmission velocity and the maintenance of causality principle; -the proton charge calculation result plausibly agrees with the experimental data; the four-dimensional values in the E-space are the 4-vector with scalar invariants which have analogies in the M-space;- the E-invariant models have a transparent physical content and are no alternative for the existing QCD methods, but expand their possibilities.


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