On Group Property of Hassani Transforms and Relativity Principle in Hassani Kinematics

In the present paper, based on the ideas of Algerian physicist M.E.Hassani, the generalized Hassani spatio-temporal transformations in real Hilbert space are introduced. The original transformations, introduced by M.E.Hassani, are the particular cases of the transformations, introduced in this paper. It is proven that the classes of generalized Hassani transforms do not form a group of operators in the general case. Further, using these generalized Hassani transformations as well as the theory of changeable sets and universal kinematics, the mathematically strict models of Hassani kinematics are constructed and the performance of the relativity principle in these models is discussed.

last fact, proves that tachyons (superluminal particles) can not exist and so the superluminal reference frames do not exist also. Note that using the A. Sfarti final conclusion we may try to "refute" not only the results of M.E. Hassani [9], but also all results of tachyon theory (this theory is a quite popular direction of theoretical physics). It should be emphasized that the considerations of A. Sfarti are based on the assumption that the velocity of particle should change by a continuous way. But A. Sfarti had not taken into account the following two hypotheses, which are also reasonable: -Under some (unknown at the present time) conditions, the velocity of some particle may change instantly (skippingly) or (similar hypothesis) tachyons may be obtained from the tardyons during some (unknown at the present time) particle transformations. -Tachyons may exist in Nature preassigned and they may interact with tardyons.
Thus, we have seen that the main conclusion of A. Sfarti does not rebut the existence of tachyons as well as superluminal reference frames. As we will see further, the Hassani transforms are, at least, mathematically correct objects. So we are not forbidden to investigate and to generalize them as well as to consider universal kinematics, generated by them.
Considerations similar to Safari's calculations are well known in physics. The founders of tachyon theory (O.-M. Bilaniuk, E. Recami, E. Sudarshan, etc) are famous physicists theorists, who knew the special relativity theory well. So, these considerations were well known to them. Hence to finish this discussion we cite one brilliant quote from the paper of E. Recami [5, p. 12] (see also [11, p. 65 "... together with special relativity the conviction that the light speed in vacuum was the upper limit of any speed started to spread over the scientific community the early-century physicists, being led by the evidence that ordinary bodies cannot overtake that speed. They behaved in a sense like Sudarshan's (1972) imaginary demographer studying the population patterns of the Indian subcontinent: ≪ Suppose a demographer calmly asserts that there are no people North of the Himalayas, since none could climb over the mountain ranges! That would be an absurd conclusion. People of central Asia are born there and live there: They did not have to be born in India and cross the mountain range. So with faster-than-light particles. ≫ " In Section 2 we introduce the generalized Hassani transforms over Hilbert space. In Section 3 we prove that the introduced classes of generalized Hassani transforms do not form a group of operators in the general case. In Section 4 we construct the generalized Hassani kinematics, based on generalized Hassani transforms and discuss the performance of the relativity principle in these kinematics.

Generalized Hassani Transforms over Hilbert Space
Generalized Hassani transforms for special case.
In the works of M.E. Hassani (see, for example, [9]) it is proposed an interesting version of the generalized Lorentz transforms for a special case, when two inertial frames are moving along the -axis in three-dimensional space and the directions of corresponding axises " " and " " are parallel: • ∈ R is the velocity of inertial reference frame l ′ , which moves relatively the fixed inertial reference frame l. • ( , , , ) are the (space-time) coordinates of any point M in the fixed frame l, • ( ′ , ′ , ′ , ′ ) are the coordinates of the same point M in the moving frame l ′ , • (·) is an arbitrary real function of real variable, possessing the following properties: where is a positive real constant, which has the physical content of the speed of light in vacuum.
First of all, we note that is enough to restrict ourselves to the functions (·) defined on [0, ∞) and to consider the expression (| |) instead of ( ) in (1). Also instead of functions, which satisfy two first conditions (2) we will consider class of functions, satisfying the more weak conditions.
Denote by ϒ the class of functions : [0, ∞) → R, satisfying the following conditions: For any function ∈ ϒ we use the following notation: According to the conditions (3), we have, D * [ ] ̸ = ∅, and moreover, Then for | | ∈ D * [ ] we can introduce the following (space-temporally) coordinate transforms: Therefore, we have introduced the generalized Hassani transforms for the same special case as for transforms (1). In the case we obtain the classical Lorentz transforms and in the case, where the function satisfies two first conditions (2) we obtain the Hassani transforms (1).
is named the velocity of the v-determined coordinate transform operator .
. Any vector n ∈ B 1 (H 1 ) generates the following orthogonal projectors, acting in ℳ (H): ]︃ Recall, that an operator ∈ ℒ (H) is referred to as unitary on H, if and only if ∃ −1 ∈ ℒ (H) and ∀ ∈ H ‖ ‖ = ‖ ‖. Let U (H 1 ) be the set of all unitary operators over the space and a ∈ ℳ (H) we introduce the following operators, acting in ℳ (H): Under the additional conditions dim (H) = 3, = 1 the right-hand part of the formula (8) is equivalent to the same part of the formula (28b) from [13, page 43]. That is why, in this case we obtain the classical Lorentz transforms for inertial reference frame in the most general form (with arbitrary orientation of axes). Now we introduce the following classes of operators: that is the set of all bijective operators ∈ ℒ (ℳ (H)) such, that: In the case H = R 3 the group of operators O + (H, ) coincides with with the full Lorentz group, being considered in [15]. In the case H = R 3 the group of operators P + (H, ) coincides with the famous Poincare group [12,Remark 2.19.1].
Case 1 : In this case from the first two equalities of (30) we obtain the equality 2 = −2 , that is = − , which contradicts to (25), because, according to (25), the both numbers and must be positive.
In this case from the first two equalities of (31) we obtain the equality = − . And substituting this value of into the third equality of (31) we obtain the equalitỹ︀ = 0, which contradicts to (25), because, according to (25), the number̃︀ must be positive.
In this case from the first two equalities of (32) we obtain the equality = .
And substituting this value of into the third equality of (32) we obtain the equalitỹ︀ = 0, which contradicts to (25), because, according to (25), this number must be positive.

Generalized Hassani Kinematics and Relativity Principle
In this section we are going to to construct universal kinematics, based on generalized Hassani transforms and demonstrate that this kinematics does not satisfy of the relativity principle in the general case. Further we use the system of notations and definitions from the theory of changeable sets and universal kinematics [14, 17-20, etc] (the most complete and detailed explanation of these theories can be found in [12]). Let (H, ‖·‖ , ⟨·, ·⟩) be a real Hilbert space with dim (H) ≥ 1, ℬ be any base changeable set such, that Bs(ℬ) ⊆ H and Tm(ℬ) = (R, ≤), where ≤ is the standard order in the field of real numbers R and be a function from the class ϒ, (see (3)). Applying the results of [12,20], to the classes of operators P (H, [ ]) and P + (H, [ ]) we can introduce the following universal kinematics: providing transition from some reference frame l = ( , [ℬ]) ∈ ℒ (ℱ) to all other frames m ∈ ℒ (ℱ), is different for different frames l. Moreover, assume that there does not exist the number ∈ (0, ∞) such that ( ) = ( ) (∀ ∈ (0, ∞)). Then, taking into account Remark 4 and Properties 1, we can prove that the set UP(l) coincides with the starting class of transforms only for some (but not all) reference frames, for example for the frame l 0,ℬ = (I, I [ℬ]) = (I, ℬ) ∈ ℒ (ℱ). But, the principle of relativity is only one of the experimentally established facts, which must not be satisfied when we exit out of the light barrier or may be satisfied only approximatively with the great accuracy even in subluminal case. In this regard it is useful to consider the kinematics UH (H, ℬ, ) in the case, where the continuous function satisfies the following conditions: 1) ( ) > (∀ ∈ [0, ∞)); 2) − 1 < ( ) < for 0 ≤ < − , where is the speed of light in vacuum and , 1 ∈ (0, ) are some small positive numbers. In this case we obtain the kinematics, which may be arbitrarily close to classical special relativity one and in which the hard-transversal light barrier may be overcome by means of the continuous change of the velocity of particle. And in this kinematics the principle of relativity is satisfied only approximatively. The possibility of violation the relativity principle is discussed in the physical literature (see for example [21][22][23][24][25][26][27][28]).