Angular symmetry of space-time and the spinor representation of Poincaré group

In the paper, a relativistic theory is suggested; we add three independent angles to the four coordinates of the Minkowski space that define the (x, t) position of a moving local observer. These angular coordinates define the orientation of an observer under free rotations and they allow us to introduce the generators of the Poincaré group in the angular representation. Instead of a multi-component wave function of any spin-component wave function, one component wave function  (and its Lorentz transform) is introduced, depending on the four coordinates of Minkowski space and three angular coordinates. Poincaré invariant first-order linear differential equations are derived. The matrix representation of the above operator equations based on appropriate angular is equivalent to Dirac and Maxwell equations. It is predicted and proven that the number of generations of leptons is three.


INTRODUCTION
It is assumed that states with spin j and the corresponding equations describing these states possess the angular symmetry of fields and particles.Diverse formulations are used for spin and Dirac and Maxwell equations (Fushchich and Nikitin, 1994;Varadarajan, 1989); however, the dependence of angular was represented implicitly in previous formulations.Beginning with Kaluza-Klein, numerous compactified and unobservable dimensions were introduced to explain the nature of the four types of fundamental forces (electromagnetic, gravitational, strong, and weak forces).For instance, Rumer and Fet (1977) introduced frames of reference of free rotation at any point of space-time with variable metric.The position of these frames of reference is defined by angles.The equivalence between the Schrödinger operator equation and Heisenberg's matrix mechanics was proved in 1927 for operators depending on the time-space coordinates (Teschl, 2009).A similar construction (substitution matrices by operators) dealing only with the operators acting on the angular variables (which are introduced to explain the nature of spinors) is considered in the present paper.
Complete knowledge of free particle states and their behaviour can be obtained once all the unitary irreducible representations of the Poincaré group are found (Ohnuki, 1988).The relationship between the Lorentz group and Poincaré group in the angular representation and the equations for relativistic particles is focused on in this article, as well as the obtaining of generalized Lorentz group and Poincaré group.

WAVE FUNCTION
The transformation properties of a multicomponent wave function that describes the transformation of fields and states with spin under rotations of ordinary observer or the original Cartesian of the coordinate system Since the theory of relativity states it is equivalent to rotate the ordinary observer or the local observer (the state, object), then the transformation properties will be described from the point of view of the local observer.
In the Minkowski space, the states of spin particles or fields with spin can be described by using a one component wave function,

   , ,
are internal variables and fully describe the degree of freedom of the spin.Let be the entries of the corresponding rotation matrix (or the projections of the corresponding unit vectors to one another) that has the following form (Biedenharn and Louck, 1984) To the multicomponent wave function Obviously, the transformation properties of the basis  and of the amplitudes C are dual.The basis of the states with the spin j, which consists of 2j+1 functions ) ,.. , ( must have the following transformation properties (Biedenharn and Louck, 1984): Where α, β, c stand for the angles defining the new orientation of the ordinary observer, the angles define the new variables in the coordinate system of the ordinary observer, and the matrix of the Wigner functions (or the rotation matrix) D j has the following properties (Biedenharn and Louck, 1984): Here, T stands for the transposition, the over line means complex conjugation, and Sk denotes the matrix operator of the spin angular momentum j in the z-representation.For example, the transformation (Equation 1) for j = 1/2 is of the form: Where  stands for the rotation angle about the axis Xk and σ k are the Pauli matrices, , the Wigner functions satisfy the normalization conditions is the known normalizing coefficient, and the domains of the angles in the arguments of the Wigner D-function are equal to respectively.Moreover, X (i)  k stands for the basis with j = 1, and we also have The basis of spinors with transformation properties of Equation 1 consists either of the columns ) (m  or of their linear combinations.

Proof
Equation 2 for a column of the matrix j D (Biedenharn and Louck,   1984) is equivalent to the relation Spinor representation of a group of rotation is given by matrix (j) D dimension 2j + 1, the Lie algebra which is isomorphic to the Lie algebra of three-dimensional rotations SO(3), (Biedenharn and Louck, 1984).Quantities-(2j +1)-dimensional vector transformations according to the spinor representation of Equation 1 are called spinors.For matrices of even dimension, this corresponds to an irreducible representation unitary group SU(2).Irreducible unitary representations of group SO(3) always have odd dimension.

Theorem 1
Two spherical angles (φ stands for the azimuthal angle and  for the polar angle) are insufficient to describe the transformation properties of state with spin by using the wave function ) , (    .For non-integer j, there are no 2j + 1 functions ) ,.. , ( of the variables φ, that are transformed according to some representation of the group SU(2) by (1). ) exp(imf .Further, let us define the dependence of the basis ± of eigen functions 3 S on , for j = 1/2.To this end, consider the rotation by the angle α around X3. On the section of the sphere  = const we have

Proof
Let us now define the dependence of the basis   of the eigen functions 2 S for j = 1/2.The bases of the eigen functions


. Moreover, we use the rotation around X2 by the angle , which is equivalent to the transformation . We finally obtain the form of a basis whose existence is assumed, namely, ).
This basis is transformed according to Equation 1 up to the phase factor exp(i f/2).The generators of the rotation groups of the ordinary observer and local observer or of the vector and isovector rotations for the functions ,, are the operators of angular momentum Jk, J (n) (Biedenharn and Louck, 1984).
The rotation groups of the ordinary observer and the local observer commute with each other and the spatial and isovector rotations are realized.To be more precise, Where ijk stands for the antisymmetric tensor with 123 = 1 and the eigen functions of the operators J 2 , J (3) and J3 are Wigner Dfunctions, The operators of raising and lowering indices m,m' act by the usual rules, In z-representation, the spin operator Sp and its eigenvectors are identical to the matrix representation Jp on the basis of the spinor We refer to the operator Jp as the spin operator in the operator representation p p S   

J
. In terms of Jp, the transformation of Equation 1 acquires the form , because the action of the rotation operator The rotation of a spin state (spinor) by the angle  around the axis x2 corresponds to the transformation , it suffices to verify the relation

Theorem 2
The spatial inversion , , : I  − equivalent to the rotation in the isovector space by the angle − around the axis

Proof
Let us prove the properties.We have: The conservation law for parity is immediately related to the conservation for symmetry between left and right.The bases Any translation of coordinates xi, t does not change the angular variables , .

THE GENERALIZED SPIN LORENTZ GROUP
The Lie algebra of the Lorentz group and its generators in the coordinate representation

K L
have this form (Ohnuki, 1988): Where stands for the velocity, and p = 1, 2, 3, and Let us find the form of generators of Equation 7 satisfying the conditions that these are purely imaginary vector generators (which are Hermitean-first-order linear differential operators) and ensure the transformation properties of the basis , j =1/2,1,3/2… in accordance with the Lorentz transformations.To this end, we assume that the representation p = 3 has the Lorentz transformation on the basis ) , ( or on the basis of eigen functions of , J , J 2 (3) 3 J in the well-known diagonal form (Weinberg,   2003).
Therefore, the functions , we obtain two other generators p = 1, 2, the angular representations of the Lorentz group, The dependence of the operators Q on j is excluded by replacing j with the scalar operator J  independent of j and such that    j J  .
Consider action of the operators J, Q for j = 1 on the basis The operators Q, X, J and N with the superscripts and subscripts +, The action of the operators ), , ( is of the form: Matrix representation of Equation ( 7) is identical to the spinor representation of the Lorentz group.
Similarly, we introduce the operators (Ohnuki, 1988) with the Lie algebra of the Lorentz group which coincides with the algebra of groups SO(3) and SU(2): It is known (Ohnuki, 1988:24) that an irreducible representation of the Lorentz group is uniquely determined by eigen values of some operators LK K L , 2 2  , which correspond to two operators . Every irreducible representation of the Lie algebra is characterized by a pair of numbers
Scalar unit operator 1 and the generators of the groups ensuring the vector, isovector and Lorentz rotations of the bases in the new variables generate an generalized spin Lorentz group of 16 generators, Equations ( 4) and ( 12).
Visual model of Lie algebra and the generalized spin Lorentz group is shown in Figure 2.
In Figure 2, J (k) corresponds to Q (k) 0; Jn corresponds to Q (0) n .The operator , , : W of permutation introduces the ordinary observer and the local observer or the vector and isovector rotations.The operation W preserves the invariance of the Lie algebra of the generalized spin Lorentz group and corresponds to the transposition operation for the group representation and matrix have the same Lie algebra in Equations 6 and 13; they are preserved under the cyclic permutation of the superscripts 1, 2, 3.
The rotation of the coordinate system The extended basis is transformed as a four-dimensional vector, the basis ) , ( (2) (1) X X  is transformed as a Lorentz bivector, and all these objects together form a full system of 10 bases for j = 1.The basis differs from the first basis for j = 0 only in the dimension and Lorentz transformation Transformation operators are performed according to the Baker-Campbell-Hausdorff (Biedenharn and Louck, 1984): The operators as a bivector, and (3) J as a scalar, The Lorentz transformation of the basis is equivalent to an angular transformation and a scale transformation of the basis.
The operator B in the above Lorentz transformation does not depend on the angles , .The Lorentz transformation of the angles ,, follows from the Lorentz transformation of Equations 8 and 15 of the bases . The Lorentz transformation of Equation 15of The independence of the speed of light of the reference system implies that the shape of spin states for a massless particle with spin j that flies along the x3 is invariant under the Lorentz transformation along x3.The only eigen functions of the generator of the Lorentz group We introduce the operators in the symmetric form of the Lorentz invariant scalar product of two four-vector momentum operators P, P0/c and P (−) , P (−)  0 or P (+) , P (+) 0 .
In the equation for the wave function with arbitrary spin, we will use this operator , where Below we will use the invariant real-valued operators instead of the Lorentz invariant operators N (±) .These operators have different parity and are connected to each other by rotations around a movable axis

THE GENERALIZED DIRAC EQUATION
Consider the generalized Dirac equation for a particle with arbitrary spin for для . ., .. 1/2,1,3/2.= j written either in terms of N (1) or in terms of N (2) , k = 1 or k = 2: 18) This is the first-order linear differential equation for wave functions C b   . Consider the set of the eigenvectors of the operators , consisting of (2j +1)(2j +1) of functions and let me be the mass of the particle and h be the Planck's constant.Equation 11readily shows that the generalized Dirac equation reduces the staircase equations for the . This explains the fact that if j > 1/2 then particles or fields are composite.For example, for j = 3/2, presence of the physical modes is obligatory for the Rarita -Schwinger fields with spin 1/2.
The matrix representation of equations ( 18) on the basis (5) with j = 1/2, k = 2 coincides with the following Dirac equations in spinor representation (Fushchich and Nikitin, 1994) 19, it suffices to calculate the action of the operators N (1) ,N (2) on the basis vectors of Equation 5, In order to simplify the above equations we have used the following identities: The generalized Dirac equation is invariant under inversion of P  , I  and the solution Ψ = Ψ C for a charge-conjugate particle satisfies the complex conjugate Equation 18: Where A is the vector-potential, q/h is the charge of a particle.For example, if j = 1/2, then 1974).
For spin j = ½, there exists 2 = 2j + 1 Lorentz-invariant states . The generalized Weyl equations for the rightor left-handed neutrino have the form (Akhiezer, 1959).For the anti neutrino and neutrino, the balance between left and right orientations is violated because of asymmetry of equations .

THE GENERALIZED MAXWELL EQUATIONS
The Maxwell equations describing the state of the vector field E, H, have the following form: Where I = (I1, I2, I3) is the density of electric current and I0 is the density of electric charge.Let Consider the generalized Maxwell's equations which imply the standard Maxwell equations for E and H which are not just a new representation, but a tool for an expanded description of the states of electromagnetic fields with spin.To this end, we introduced the wave function Ψ.The Lorentz-invariant of the wave function ΨC for charges and currents has the following form: Let us describe the state  on the basis of eigen functions of and consider all three cases 0 J , J . This Ψ is composed of 3=2j +1, (j=1) Lorentz-invariants of the electromagnetic field written in the form of the two complex- the right and the left vector states, respectively.
The Lorentz-invariance of the Ψ (+) implies the well-known Lorentz transformations of the fields (Pauli, 1991): correspond to the projections of the spin (1, 0,−1) on the x3 axis.We expand similarly )/(8 The mixed state is the sum of the right and the left vector states, currents and fields:

Proposition 4
The spin (the energy, the Poynting vector) of the mixed state is the sum of the spins (the energy, the Poynting vector) of the right and the left states.
Proof follows from the identity, similarly for Q (3) :

J J J
. The generalized Maxwell equations describing just right state or only left state with spin 1 have the following form: We rewrite Equations 21 as a pair of complex-conjugate Equations 22 for the wave functions either . Each of these equations describes just the right or the left vector spin-1 states: we obtain the following identity: The generalized Maxwell Equation 24 for states with zero projection of spin and its analogue 2) is the real one, then CL = CR and Equations 22 have the following form: The transformation properties of Equation 15 of the basis , (2) (1)

X X
and the E, H of electric and magnetic fields are dual to each other, so that ; it is composed from D-Wigner functions for spin j = 1, after averaging over the angle γ (Figure 3).Vector is the set of its projections.

Theorem 3
The generalized Dirac equation N (1) Ψ = −mec/(2h) Ψ for a particle with mass me, spin j = 1, and zero projection of the spin on any axis Xi and its analogue on any axis X (k) for the amplitudes on the basis are equivalent to the Maxwell and Proca equations, respectively, for the 1-spin particle of mass me (heavy vector virtual photon).

Proof
We represent the solution as a sum of Lorentz invariants Zero values of the spin projection and its counterpart are equivalent to the real values of the amplitudes of E,H, A,A0.The generalized Dirac equation splits into two equations:

/c
The Maxwell equations imply the continuity equation for Is which coincides with the Lorentz calibration of the vector potential of magnetic field, and the first London equation, which describes the Meissner effect of Equation 25, (London and London, 1935), where Λ is the London penetration depth, A is the vector potential, Is is the superconducting component of the electric current.
, written as the Proca equation (Ginzburg, 1979), where  is the d'Alembert operator.The above equations can describe the superconductivity phenomena because the acquisition of mass by photons is associated with the losses of long-range interactions.This can reduce the energy losses by radiation.

RESULTS AND DISCUSSION
The reasons for the above proposed generalizations are related to the fact that fields E, H are not sufficient enough to describe the electromagnetic fields with spin; this is because the spin part is closely related to the wave function Ψ.By virtue of this, we introduce the wave function which is equal to the sum of the left and right vector states with spin 1.The contribution of the left and right states We propose the following mechanism for the transition of a conductor to superconducting state.The lowtemperature superconductivity corresponds to a spontaneous transition to the state with zero spin projection and its analogues without changing the electromagnetic field E,H.
The high-temperature superconductivity can be considered as a transition to the state with spin 1. Besides, state  must possess an analogue of spin 1, whose projection to X (3) axis is equal to Our conjecture is the following: there exists just three generations of leptons (electrons, muons, tau-leptons), because the generalized spin group possesses at the same time just the three different spinor representations of the Lorentz group of Equation 7. The three representations of the Poincaré group derived by cyclic permutations of superscripts p = 1,2,3 correspond to the three representations of the Dirac equation.From a mathematical point of view, they are equivalent.The spinor representation of Poincaré group p = 3 associated with the first (stable in the decay) generation of leptons (electron) as well as the generator of the Lorentz transformation is independent of the angles φ,.We conjecture that there exists just three colors of quarks (red, green, blue), since the generalized spin Lorentz group possesses at the same time just the three different transposed spinor representations of the Lorentz group.
Minkowski space (non-locally isotropic in the presence of particles and fields) has three additional independent dimensions    , , , which fully describe the degree of freedom of the spin.
Note that physical systems normally can be represented precisely in terms of purely complex generalized spin Lorentz groups, but not just by the Lorentz group in the spinor representation, due to the existence of an additional degree of freedom.Generalized spin Lorentz groups consist of the three Lorentz groups in spinor representations, and the three transposed Lorentz groups in spinor representations.
Similarly, spin of the particle, electromagnetic field (in state E, H, A) can be regarded as a consequence of the presence of internal degree of freedom (analog of spin 1 for isovector space) of Minkowski space.

Conclusion
In this paper, spinors and their transformation properties are described as the properties of the rotation of local observer, but not in terms of rotation of the ordinary (meter) observer.This permutation of the local and ordinary observers agrees with the general theory of relativity.To this end, we introduce one-component wave functions are independent, but their transformation properties are bound between them.For an arbitrary integer or halfinteger spin j, the Poincaré group in angular (spinor) representation is described explicitly.
The matrix representation of the algebra ) iJ , , , ( (-) (-) (3) Q Q J coincides with the Poincaré group in spinor representation.A matrix representation of 16 generators of the generalized spin Lorentz group on the basis of j = 1/2 coincides with 16 basic elements of the Clifford algebra formed by the Dirac gamma-matrices (Flügge, 1974).Momentum operator in the Lie algebra of the angular variables of the Poincaré group is always a complex operator.Analogs of discrete operator P  (reverse the orientation of space) are included in the generalized spin Lorentz group as continuous operator  (Ohnuki, 1988).The elements in each component are characterized by whether or not they reverse the orientation of space and/or time.Generalized spin Lorentz group and the Lorentz group O(1, 3) have four connected components.
Also are derived in uniform manner the generalized Dirac and Maxwell equations for Ψ with arbitrary spin j in terms of scalar product of two four-vector momentum operators (generators the Poincaré group) in the angular representation and coordinate representation.The matrix representations of the corresponding operator equations for spin j = 1/2, 1 particles coincide with the Dirac, Maxwell, or Proca equations.
These generalized Dirac and Maxwell equations imply the following conclusions: (1) Free fixed electron and positron have analogues of the spin projection ±1/2, /2 ± = J (1)   along X (1) axis because the term, depending on the momentum P in (18), k=2, vanishes.In the standard representation (Weinberg, 2003), this determines our choice of the basis: (2) Neutrinos and antineutrinos, left and right photons moving along x 3 are considered as states with spin projections on the axis x 3 ; analogues of the spin projections on the axis of X (3) are equal to ±1/2 and ±1, respectively.For neutrinos and antineutrinos, we have .The Lorentz transformation along x 3 does not change form  for neutrino, antineutrino and photons, since  is an eigen function of generator Lorentz transformations Q (3)  3 .

Figure 1 .
Figure 1.Orientation of the local observer in relation to the ordinary observer.

.
The expansion of  with respect to the basis corresponds  to the multicomponent wave function C, under the Lorentz transformations.
integer is the eigen functions of the operators angular moment of L that depends only on two angles  ,

.
Making a rotation in the section of the sphere 0   by the angle β around X2, we have

.
This enables one to decompose the entire basis into two equal groups of bases in all cases except for .The difference between left and right with weight m is evaluated as the mean value of the operator ) 3 (

Q
be the generators of the Lorentz group for arbitrary spin in the angular representation.The Lorentz transformation is of the form impose the space-time isotropy condition on the generators of the Lorentz group, namely,

Figure 2 .
Figure 2. Visual model of Lie algebra and the generalized spin Lorentz group. k k around the xk axis and the Lorentz transformation corresponding to the velocity v = |v|xn are of the form the vector E of the electric field, the vector H of the magnetic field, and the Umov-Poynting vector [EH],(Pauli W.,  1991).Therefore, there exists an analogy between the Lorentz transformation of Equation 16 of the angles the angles of the polarized light, where the Umov-Poynting vector is directed along the vector basis of the massless field.For j =1/2 let us show that the neutrinos are purely left handed particles as the momentum operator in the Lie algebra of the angular variables, as they have the same Lie algebra of the Poincaré group, so that spinor representations.To prove Equation right and left vector states are collinear or anticollinear to the Poynting vector.The proof follows from the identities and the density of the electromagnetic energy 22)Each of the Equations 22 agrees with the Maxwell equations either for the right components ER,HR, IR, I0R or for the left components.This implies that the total components Maxwell equations.Maxwell Equation 20 follow from Equations 21, 10 and identity 23 for j = 1, where G(x1, x2, x3) is an arbitrary vector:

Figure 3 .
Figure 3. Visual model of the electric, magnetic fields: Maxwell Equation20.Model of the electric field, magnetic field and the field of magnetic vector potential is represented of the Maxwell equations.The spin of electromagnetic field E, H, Ψ is equal to the sum of spins of the left and the right states of the field.Any electromagnetic field E,H related to a stationary state can possess a spin.The generalized Maxwell equation admits spin states, but does not describe completely their variation.The spin of electromagnetic fields E,H corresponds to either the right or left of the vector.Spins of the left and right vector states E,H are collinear and anticollinear to the Poynting vector.
the position and the orientation of a local observer in the Minkovski space-time determined by the three Euler angles :