Generalized Dirac and Klein-Gordon Equations for Spinor Wavefunctions

Transcription

1 Adv. Studies Theor. Phys., Vol. 8, 014, no. 3, HIKARI Ltd, Generalized Dirac Klein-Gordon Equations for Spinor Wavefunctions R. Huegele, Z. E. Musielak J. L. Fry Department of Physics The University of Texas at Arlington Arlington, TX 76019, USA Copyright c 013 R. Huegele, Z. E. Musielak J. L. Fry. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. Abstract A novel method is developed to derive the original Dirac equation demonstrate that it is the only Poincaré invariant dynamical equation for 4-component spinor wavefunctions. New Poincaré invariant generalized Dirac Klein-Gordon equations are also derived. In the non-relativistic limit the generalized Dirac equation gives the generalized Lévy-Leblond equation the generalized Pauli-Schrödinger equation. The main difference between the original generalized Dirac equations is that the former latter are obtained with zero nonzero phase functions, respectively. Otherwise, both equations describe free elementary particles with spin 1/, which have all other physical properties the same except their masses. The fact that the generalized Dirac equation describes elementary particles with larger masses is used to suggest that non-zero phase functions may account for the existence of three families of elementary particles in the Stard Model. This suggestion significantly differs from those previously made to account for the three families of particle physics. Keywords: Minkowski space-time, Poincaré group, spinor wave functions, Dirac Klein-Gordon equations

2 110 R. Huegele, Z. E. Musielak J. L. Fry 1 Introduction In the Special Theory of Relativity STR, the background space-time is flat endowed with the Minkowski metric. All transformations of coordinates that leave the metric invariant form a representation of the inhomogeneous Lorentz group, which is also known as the Poincaré group [1,]. Wigner [1] classified all irreducible representations irreps of the Poincaré group used them to establish classes of elementary particles that exist in this space-time []. A dynamical equation that is invariant with respect to the transformations of coordinates is a Poincaré invariant equation [3]. The Klein-Gordon KG equation [4,5], one of the fundamental Poincaré invariant equations of quantum field theories QFT, describes bosons with spin zero that are represented by scalar wave functions. Other fundamental equations of QFT are the Dirac [6] Proca [7] equations, which describe fermions with spin 1/ bosons with spin 1, respectively. The wavefunctions that represent fermions are spinors while vector wavefunctions represent bosons. To obtain the KG equation, the STR energy-momentum relationship is typically used differential operators are substituted for the energy momentum [8]. On the other h, the Dirac equation is usually derived by Dirac s method [8] or by using the transformation properties of spinors under the Lorentz group [9]. A similar procedure is also used to obtain the Proca equation. All these equations can also be formally obtained from an appropriate Poincaré invariant Lagrangian density [10]. Another method was introduced by Bargmann Wigner [11], who used the irreps of the Poincaré group to obtain the fundamental equations of QFT. A complete description of the group theoretical derivation of the Dirac equation is presented by Thaller [1]. A different approach was developed by Fry, Musielak & Chang [13], who formally derived the KG equation for free spin-zero particles using the principle of relativity the principle of analyticity. In Paper I of this series [14], we derived the Lévy-Leblond equation [15,16] for a four-component spinor wavefunction as well as the corresponding Schrödinger equation, proved that they were the only Galilean invariant four-component spinor equations with the Schrödinger phase function. The relationship between these equations the Pauli-Schrödinger equation [17] was discussed. Extensive studies of Lévy-Leblond Pauli-Schrödinger equations were performed by Fushchich Nikitin [3], who derived the equations investigated their symmetries. Moreover, a general method of constructing Galilean invariant theories was developed by de Montigny et al. [18] Niederle Nikitin [19]. A different method based on the principle of relativity the principle of analyticity was developed by Musielak Fry [0] by Fry Musielak [1]. Using different phase functions than the Schrödinger phase function consid-

3 Generalized Dirac Klein-Gordon equations 111 ered in Paper I, we derived new Galilean invariant dynamical equations, which were called the generalized Lévy-Leblond generalized Schrödinger equations []; these equations reduce to the stard Lévy-Leblond Schrödinger equations when the Schrödinger phase function was used. We demonstrated that the stard generalized equations described the same elementary particle with spin 1/ but with different masses; the mass was larger for the generalized equation. Despite that all results were obtained in the nonrelativistic regime [], we suggested that the difference in mass resulting from using different phase functions may account for the existence of three families of elementary particles in the Stard Model of particle physics [3]. One of the main goals of this paper is to verify the above suggestion in the relativistic regime. In order to achieve this goal, we first develop a novel method of deriving the Dirac equation, then use this method to search for new fundamental Poincaré invariant dynamical equations. The result of this search is the generalized Dirac equation for four-component spinor wavefunctions. To check the validity of this generalized Dirac equation, we evaluated it in the non-relativistic limit showed that the generalized Lévy-Leblond equation [] the generalized Pauli-Schrödinger equation are obtained. Moreover, we also derive a second order equation, the generalized Klein-Gordon KG equation. The new generalized Dirac Klein-Gorodon equations are obtained by using non-zero phase factors, these equations reduce to the stard Dirac Klein-Gordon equations once the phase function is set to zero. The outline of this paper is as follows: governing equations are given in section ; a novel derivation of the Dirac equation is presented in section 3; the generalized Dirac Klein-Gordon equations are derived in sections 4 5, respectively; the generalized Lévy-Leblond Pauli-Schrödinger equations are obtained in the non-relativistic limits of the generalized Dirac equations in section 6; the generalized Dirac equation the existence of three familes of elemementary particles in the Stard Model are discussed in section 7; our conclusions are presented in section 8. Governing equations The Minkowski metric can be written as ds = dt dx dy dz, where x, y z are the spatial coordinates t is the time coordinate given in natural units where the speed of light is c = 1. The group of this metric is the Poincaré group P, whose structure is given by the following semi-direct product: P = H p s T 3 + 1, where T is an invariant subgroup of space-time translations H p is a non-invariant subgroup consisting of the remaining transformations the identity transformation. In this paper, we consider the proper orthochronous group P + that is a subgroup of P.

4 11 R. Huegele, Z. E. Musielak J. L. Fry The semi-direct product structure of the Poincaré group guarantees that a given wavefunction ψx, t must transform like an irrep of the invariant subgroup T Thus a necessary condition that a wave described by ψ represents an elementary particle [13] is i μ ψx, t = k μ ψx, t, 1 where μ = / x μ = /c t, k μ =ω/c, k x, k y, k z. This equation must be satisfied for each component of a multi-component wavefunction. Since in this paper, we consider elementary particles with spin 1/, their wavefunctions are spinors. With the assumption that ψ is a spinor, Eq. 1 must be modified to account for different components of the spinor wavefunction. Eq. 1 is not invariant, therefore, we seek an invariant version that remains linear in its derivatives while invariant to all Poincaré transformations. This can be achieved by multiplying both sides of Eq. 1 by an arbitrary, constant matrix A μ, which in general has dimension n n to be consistent with the n-component wavefunction see below. The result is ia μ μ ψx, t = A μ k μ ψx, t. Defining B μ ia μ B c A μ k μ, we obtain the first order differential equation of the following form [B μ μ + B c ] ψx, t =0. 3 We shall now determine conditions that the matrices B μ B c must satisfy in order for the above equation to be Poincaré invariant. The requirement is that the obtained dynamical equations for ψ remain the same in all inertial frames of reference that exist in Minkowski space-time, as required by the principle of relativity. It must be noted that we seek an equation whose solutions have eigenvalues corresponding to spin 1/ for the second Casimir operator of the Poincaré group []. We also seek solutions that correspond to unitary irreps with spin 1/ that are given by the covering group of P +, which is a group of 4 4 matrices [,1]. Thus, a wavefunction for a unitary spin 1/ irrep must have at least 4 components, so n =4. 3 Novel derivation of Dirac equation 3.1 Conditions for invariance of the first order equation A space-time point in the coordinates of one observer x ν is related to the same point in the coordinates of another observer x ν by a linear transformation x ν =Λ ν μx μ + b ν. 4

5 Generalized Dirac Klein-Gordon equations 113 Under the linear transformation a differential operator transforms like the wavefunction transforms like μ =Λρ μ ρ, 5 Tψx, t =ψ x,t. 6 Moreover, since we require ψ = ψ, the wavefunctions can be related by a phase factor e iφx,t ψx, t =ψ x,t. 7 The principle of relativity requires that if a dynamical equation such as Eq. 3 is true for one observer, then there exists another equation of the same form for all other observers who have a coordinate system that is translated, rotated, boosted relative to the first observer. An arbitrary second observer would have an equation that would feature objects such as B μ, B c, μ, which may have been changed by the coordinate transformation such as [ B μ μ + ] B c ψ x,t =0. 8 Substituting the transformation rules of the differential operator see Eq. 5 the wavefunction see Eq. 6 into Eq. 7 produces a first order equation that is now written in terms of the original coordinate system x μ [ B μ Λ ρ μ ρ + B c ] e iφx,t ψx, t =0. 9 The phase factor can be commuted through the differential operator the result is Λ ρ μ ρ e iφx,t = e iφx,t Λ ρ μ ρ + i ρ φ. 10 Commuting the phase factor through to the left side of the transformed first order equation dividing it out, we obtain [ B μ Λ ρ μ ] ρ + i ρ φ+b c ψx, t =0. 11 For the first order equation to be invariant, it is required that the transformed first order equation is the same as the original first order equation. Equating terms of like differential powers then generates a set of conditions on the matrices B μ B c that must be met for the dynamical equation to be invariant. The conditions are: B μ = B β Λ μ β, 1 B c = B c + ib β Λ μ β μφ. 13

6 114 R. Huegele, Z. E. Musielak J. L. Fry The equation is invariant for phase functions of the form φx, t =ζ μ x μ + ζ c, 14 where ζ μ, ζ c are scalar functions of the transformation parameters a μ, v i, θ i potentially any number of other parameters that have yet to be introduced. Using the replacement μ φ = ζ μ, the condition for invariance see Eq. 13 becomes B c = B c + ib β Λ μ β ζ μ Matrices that satisfy the conditions for invariance The matrices B μ B c must satisfy the conditions see Eqs 1 15 in order to form a first order differential equation that is invariant under the linear transformation ˆΛ. From this point on the linear transformation will be a Poincare or inhomogeneous Lorentz transformation. This transformation can be separated into rotations boosts to ease calculations. The rotations matrices for 4-component spinors are: R θj = cos θ + ɛ jklγ k γ l sin θ, 16 where γ j γ 0 form a basis for the 4 4 spinor matrices θ j is the rotation about the j-axis in the k l-plane. The boost matrices for 4-component spinors are: S vj = cosh η + iγ jγ 0 sinh η, 17 where η is the boost angle. The boost angle is related to the velocity by tanh η = β = v/c where c = 1 is the speed of light in natural units. The Dirac representation is typically chosen whenever explicit representation of the gamma matrices γ μ is required 0 iσ γ j j = iσ j, γ 0 =, I where σ j are the stard Pauli matrices i 1 0 σ x =, σ y =, σ z =, i In this representation the Minkowski metric η μν has signature +. The covariant gamma matrices are related to the contravariant form by γ μ = η μν γ ν = {γ 0, γ j }. The gamma matrices satisfy the Clifford algebra defined by the anti-commutators {γ μ,γ ν } =δ μν I. 0

7 Generalized Dirac Klein-Gordon equations Invariant equations for the phase function φ =0 Before dealing with the more general case we consider the consequences of assuming the phase function φx, t = 0. This case requires ζ μ = ζ c = 0 simplifies the condition see Eq. 15 to B c = B c. Now, we apply rotations, which constrains the matrices to the form pi qi eσ B t = B j j fσ = j si ti gσ j hσ j, 1 where e, f, g, h, p, q, s, t are unknown scalar constants. We re-write condition see Eq. 13 with the boost transformation S, obtain Λ β μb μ = B β = SB β S 1, Applying a Lorentz boost along the z-direction in the above condition, the RHS becomes S z B β S 1 z = cosh η iγ zγ 0 sinh η B β cosh η + iγ zγ 0 sinh η = B β cosh η + γ zγ 0 B β γ z γ 0 sinh η +i B β γ z γ 0 γ z γ 0 B β sinh η cosh η = 1 B β + γ z γ 0 B β 1 γ z γ 0 cosh η + B β + γ z γ 0 B β γ z γ 0 + i B β γ z γ 0 γ z γ 0 B β sinh η. 3 While the LHS of Eq. is Λ t μ Bμ = B t cosh η ib z sinh η, 4 for β = t a boost in the z-direction. Alternatively the LHS is Λ z μ Bμ = B z cosh η + ib t sinh η, 5 for β = z while Λ x μb μ = B x Λ y μb μ = B y. Here we used the following identities sinh η = 1 cosh η + 1, 6 sinh η cosh η = 1 sinh η, 7 cosh η = 1 cosh η 1. 8

8 116 R. Huegele, Z. E. Musielak J. L. Fry The RHS of Eq. is equal the LHS when the following constraints: {B t,γ z γ 0 } =0,{B z,γ z γ 0 } =0,[B x,γ z γ 0 ] = 0, [B y,γ z γ 0 ] = 0 are obeyed. The above procedure may be repeated for boosts in the x y directions to produce a set of constraints similar to those given above. Taken together the constraints are equivalent to requiring the matrices B μ obey the Clifford algebra: {B μ,b ν } =δ μν I. We now return to the condition on B c = B c. Application of a Lorentz boost gives B c = S z B c Sz 1 = cosh η iγ zγ 0 sinh η B c cosh η + iγ zγ 0 sinh η = B c cosh η + γ zγ 0 B c γ z γ 0 sinh η +i B c γ z γ 0 γ z γ 0 B c sinh η cosh η, 9 which leads to [B c,γ z γ 0 ] = 0. However, application of rotations gives B c = R z B c Rz 1 = cos θ γ xγ y sin θ B c cos θ + γ xγ y sin θ = B c cos θ γ xγ y B c γ x γ y sin θ +B cγ x γ y γ x γ y B c sin θ cos θ, 30 leads to the constraint [B c,γ x γ y ]=0. This above reasoning can be repeated for x- y-directions, to produce similar constraints. Under these conditions the matrix B c is constrained to B c = a where a remains a free parameter. 3.4 Dirac equation 0 I Combining all the above results, we obtain [ 0 σ j σ j 0 j + 0 I t + a, 31 0 I ] ψx, t =0, 3 which is a Poincaré invariant equation for 4-component spinor state functions. This equation satisfies the principles of relativity analyticity but we must also insure that the state functions transform like irreducible representations of the Poincaré group. This means that the state functions must obey the eigenvalue equations i t ψ = ωψ i j ψ = k j ψ, with j = 1, 3

9 Generalized Dirac Klein-Gordon equations 117 see Eq. 1. The requirement can be used to remove the free parameter a. Multiplying Eq. 3 by i moving the time derivative to the LHS yields i 0 I t ψx, t = i [ 0 σ j σ j 0 j + a 0 I ] ψx, t. 33 Clearly, the operator on the right h side is the Hamiltonian for psi to transform as a unitary spin 1/ irrep it must be Hermitian. The terms are all Hermitian under the condition that the constant a is imaginary or that ia is real. Now we substitute the eigenvalues for their operators, obtain ω 0 I ψx, t =k j 0 σ j σ j 0 ψx, t ia 0 I ψx, t. 34 Substituting the bispinor ψx, t = φ η, 35 for the wavefunction in Eq. 34, we obtain the following pair of linked equations: ωφ = k j σ j η iaφ, 36 ωη = k j σ j φ iaη. 37 The four momentum p μ contains the eigenvalues of energy momentum p μ =ω, k j. There always exists some frame of reference where the momentum is zero k j = 0 the energy equals the rest mass, so that ω = ω 0 where ω 0 is called the invariant frequency. When these values are substituted into the pair of equations the only remaining free parameter a can be determined in terms of the invariant frequency ia = ±ω 0 Eq. 34 can be written as i 0 I t ψx, t = [ i 0 σ j σ j 0 j ± ω 0 0 I ] ψx, t. 38 By identifying the wave mass with the rest mass, the derived equation becomes the Dirac equation [6]. Note that the above derivations were performed with the phase function φ = 0, consequently this is the only Poincare invariant firstorder differential equation for φ = 0. It is also important to point out that our derivation of the Dirac equation is new significantly different than typical derivations shown in QFT textbooks [8,9].

10 118 R. Huegele, Z. E. Musielak J. L. Fry 4 Generalized Dirac equation We now address the more general case when the phase function is other than φ = 0. Since φ is not present in the condition for B μ see Eq. 13, there is no impact on the results obtained for B μ in the previous section. The matrix B c however is constrained by Eq. 15, where the presence of φ introduces the function ζ μ θ i,v j. Now the problem is to determine the value of the functions ζ μ θ i,v j in addition to the matrix B c. First notice that the condition given by Eq. 15 can be simplified by the substitution of condition given by Eq. 1 the use of the result B μ = γ μ. This gives B c = B c + ib β Λ μ β ζ μ = B c + ib μ ζ μ = B c + iγ μ ζ μ. 39 By setting θ j = 0 v j = 0 it is clear that ζ μ 0, 0 = 0. To find the rotational dependence we move the terms containing B c to the LHS apply a z-rotation. The result is B c B c =B c + γ 1 γ B c γ 1 γ sin θ 3 B c γ 1 γ γ 1 γ B c sin θ 3 cos θ 3, 40 where we used the following identities sin θ 3 = sin θ 3 cos θ 3, cos θ 3 = cos θ 3 sin θ 3, 41 1 cos θ 3 = sin θ 3. 4 Any 4 4 complex matrix can be represented using the matrices I, γ μ, σ μν, γ 5 γ μ, γ 5 as a basis where γ 5 γ j = σ μν = i [γμ,γ ν ], 43 γ 5 = iγ 0 γ 1 γ γ 3 = i σ j 0 0 σ j 0 I γ 5 γ 0 = i 0 I 44, 45 are all given in the Dirac basis. If a matrix is required to be Hermitian, then the basis can be restricted to I, γ μ, σ μν, γ 5 γ μ excluding the anti-hermitian γ 5.

11 Generalized Dirac Klein-Gordon equations 119 Since the matrix B c must be Hermitian, it can be written in a basis composed of the Hermitian matrices where B c = ai + b μν σ μν + c μ γ μ + d μ γ 5 γ μ, 46 c μ γ μ = c 0 γ 0 c 1 γ 1 c γ c 3 γ 3, 47 with a, b i, c μ, d μ are undetermined constants. Using this basis we can calculate B c + γ x γ y B c γ x γ y B c γ x γ y γ x γ y B c. We do this for the ai term first obtain ai + γ 1 γ aiγ 1 γ =0, 48 aiγ 1 γ γ 1 γ ai =0. 49 Since these quantities vanish, a is allowed to remain a free parameter; this guarantees that the Dirac equation is obtained as a special case of φ =0. For the b μν σ μν term, we find b μν σ μν + γ 1 γ b μν σ μν γ 1 γ = i b 10 γ 0 γ 1 b 0 γ 0 γ b 1 + b 1 γ 1 γ +b 3 γ γ 3 b 31 γ 1 γ 3, 50 b μν σ μν γ 1 γ γ 1 γ b μν σ μν = i b 10 γ 0 γ b 3 γ 1 γ 3 b 31 γ γ It is apparent that all but the b 30 term must equal zero. For the c μ γ μ term, we have c μ γ μ + γ x γ y c μ γ μ γ x γ y = c 1 γ 1 c γ, 5 c μ γ μ γ x γ y γ x γ y c μ γ μ = c 1 γ +c γ The above relationships will survive since they are coefficients of individual γ-matrices, which appear in γ m uζ m u. The d μ γ 5 γ μ term produces d μ γ 5 γ μ + γ 1 γ d μ γ 5 γ μ γ 1 γ = id 1 γ 0 γ γ 3 +id γ 0 γ 1 γ 3, 54 d μ γ 5 γ μ γ 1 γ γ 1 γ d μ γ 5 γ μ =id 1 γ 0 γ 1 γ 3 +id γ 0 γ However, these terms must vanish to satisfy invariance of the first order equation. Thus, we conclude that d 1 = 0 d =0. Putting the above results together, we have γ x γ y B c γ x γ y = c 0 γ 0 c 1 γ 1 c γ + c 3 γ 3, 56

12 10 R. Huegele, Z. E. Musielak J. L. Fry B c γ x γ y γ x γ y B c = c 1 γ +c γ The remaining terms must vanish thus provide a means of eliminating some of the free parameters of Eq. 46. Inserting these results into the LHS of Eq. 40, we have iγ μ ζ μ = B c B c = c 1 γ 1 c γ sin θ 3 c 1 γ +c γ 1 sin θ 3 cos θ 3 = c 1 γ 1 c γ 1 cos θ 3 +c 1 γ c γ 1 sin θ The γ-matrices are linearly independent so the coefficients of each matrix must vanish independently. For a z-rotation we conclude that ζ 3 θ 3 = 0 ζ 0 θ 3 = 0 ζ 1 = ic 1 1 cos θ 3 +ic sin θ 3, 59 ζ = ic 1 cos θ 3 ic 1 sin θ Similar expressions can be produced for the dependence of ζ 1 ζ on the angles θ 1 θ. All the free parameters of B c in Eq. 46 will vanish except for the c μ terms a. The phase function φ will also depend on the boost parameters v j. To find the functional dependence of φ on v j, we apply boosts in Eq. 15 attempt to solve for φ. Equation 15 is simplified by substituting in Eq. 1 to get B c = B c + ib β Λ μ β ζ μ = B c + ib β ζ β. 61 Applying a Lorentz boost along the z-direction produces B c S z B c S 1 z = B c cosh η iγ zγ 0 sinh η B c cosh η + iγ zγ 0 sinh η = B c B c cosh η γ zγ 0 B c γ z γ 0 sinh η i B c γ z γ 0 γ z γ 0 B c sinh η cosh η. 6 With the remaining free parameters B c = ai + c ν γ ν, we calculate B c γ 3 γ 0 γ 3 γ 0 B c = c 0 γ 3 + c 3 γ 0, 63

13 Generalized Dirac Klein-Gordon equations 11 Then Eq. 6 can be written as γ 3 γ 0 B c γ 3 γ 0 = c 0 γ 0 + c 1 γ + c γ c 3 γ iγ μ ζ m u = i [γ 0 ζ 0 γ 1 ζ 1 γ ζ γ 3 ζ 3 ], [ = γ 0 c 0 sinh η +ic 3 sinh η cosh η ] [ +γ 3 c 3 sinh η +ic 0 sinh η cosh η ], 65 the z-boost dependence of ζ is ζ 1 = 0 ζ = 0, which gives ζ 0 =ic 0 sinh η +c 3 sinh η cosh η, 66 ζ 3 =ic 3 sinh η c 0 sinh η cosh η. 67 Equations show how the phase function φ depends upon a boost in the z-direction. This procedure can be repeated for boosts in the x- y-directions with similar results. After performing boosts rotations in the condition given by Eq. 39, the only remaining free parameters for B c from Eq. 46 are a c μ. No other restriction is available to eliminate these free parameters so they will appear in the first order equation. These free parameters are allowed because of the added flexibility in the state function that is afforded by the freedom to select any phase function φ to offset non-invariant terms generated by the transformation of the constant matrix B c. The free parameters c μ a will appear in the first order equation making it more general than the Dirac equation given by Eq. 38. Thus we have a generalized Dirac equation [γ μ μ + ai + c μ γ μ ] ψx, t =0, 68 which satisfies the principles of relativity analyticity. We call this new fundamental Poincaré invariant equation the generalized Dirac equation because the original Dirac equation see Eq. 38 is obtained from it as a special case when the phase function φ is set to zero. To be more specific, the stard results are obtained when the phase function φ = 0 requires ζ μ = 0 which in turn requires the free parameters to vanish c μ =0. Other generalized Dirac equations were published in the literature [4-8], however, they are completely different than Eq. 68. Kruglov [7,8] generalized a Dirac equation to account for a particle with two mass states, the resulting generalized Dirac equation was a second order differential equation. As such the equation resembles rather the Klein-Gordon equation

14 1 R. Huegele, Z. E. Musielak J. L. Fry than the Dirac equation. Actually, the equation has the additional first order parameter found in the Dirac equation, therefore, it can be considered as a sum of Dirac Klein-Gordon equations. The generalized Dirac equation described by Nozari [9] follows from a generalized uncertainty principle exists in the context of spacetime with an assumed minimal distance on the order of the Planck length. This equation is not Poincaré invariant thus its meaning is different than Eq. 68. More discussion relevant to the comparison of our results to those obtained previously will be given in section 7. 5 Generalized Klein-Gordon equation Having derived the generalized Dirac equation, we may now use it to obtain the generalized Klein-Gordon equation. However, before this formally done, first we write the generalized Dirac equation in its alternate form. We multiply Eq. 68 by iγ 0, use the momentum operator ˆp i = i i, energy operator ˆε = i t, separate the time derivative from the space derivative. The result is ˆεψ = Hψ =[α i ˆp i +ãβ + c 0 I + α i c i ] ψ, 69 where α i = γ 0 γ i, β = γ 0,ã = ia, c 0 = ic 0, c i = ic i. In this form the RHS can be identified as the Hamiltonian H acting on the wavefunction ψ. Since the eigenvalues of H must be real thus H must be Hermitian. Thus, α i β are Hermitian the Hermiticity of the Hamiltonian is satisfied if ã c μ are real. Now, a second order equation such as the Klein-Gordon equation [4,5] can be constructed by applying the operators of the generalized Dirac equation to the wavefunction a second time. Starting with Eq. 69 making substitutions ã = m 0, c 0 = ε, c j = p j, ˆp j = i j, we obtain the following first order equation i t ψ =[ iα j j + m 0 β ε + α j p j ] ψ. 70 After applying the operators to the wavefunction a second time, the equation becomes t ψ = j ψ + m 0 ψ + p j ψ + ε ψ i p j j ψ m 0 εβψ +i εα j j ψ ε p j α j ψ, 71 which is the generalized Klein-Gordon equation. The terms are all Hermitian the equation is Poincaré invariant.

15 Generalized Dirac Klein-Gordon equations 13 6 Non-relativistic limits of generalized Dirac equation 6.1 Generalized Pauli-Schrödinger equation In order to consider the non-relativistic limit, we replace the natural units in the generalized Dirac equation by the physical units. The resulting equation shows explicitly the speed of light c, which is required to take the limit. The generalized Dirac equation in its physical units can be written as i t ψ = H =[cα i ˆp i + cãβ + c c 0 I + cα i c i ] ψ. 7 Introducing the electromagnetic four potential A μ =A 0 x,a j x, which can be incorporated into the generalized Dirac equation via the minimal coupling p μ p μ e c Aμ Π μ, 73 where Π μ is the kinetic momentum e is the electron charge. Then, the generalized Dirac equation with the electromagnetic potential is i t ψ = [cα i ˆp i e ] c A i + ea 0 + cãβ + c c 0 I + cα i c i ψ. 74 The four-component spinor ψ can be decomposed into a pair of two-component spinors φ χ φ ψ =. 75 χ Inserting explicit representations for the matrices α i, β, I, we write the generalized Dirac equation as i t φ χ = c σi Π i χ σ i Π i φ +c c 0 φ χ + ea 0 φ χ + c c i σi χ σ i φ + cã φ χ. 76 The non-relativistic limit of the generalized Dirac equation requires making some assumptions about the value of the free parameters ã c μ. In the stard Dirac equation ã = m 0 c, where m is the mass of the particle c is the speed of light. We shall assume this value for ã in order to interpret the third term on the RHS as a rest mass energy. The similarity of the fifth term to the first term is suggestive of a momentum interpretation while the similarity between the fourth term left h side term suggests an energy interpretation. We shall thus set c c 0 = ε, where the negative sign is chosen

16 14 R. Huegele, Z. E. Musielak J. L. Fry to match the sign of ε as it will appear on the left. Now if we assume m 0 c is the largest energy, the spinor state function may be further split into two parts φ χ = φ χ exp [ im 0 c t ]. 77 The generalized Dirac equation is now φ σi Π i t = c i χ φ 0 + ea χ σ i Π i φ 0 m χ 0 c χ φ σi χ ε + c c χ i. 78 σ i φ If we consider the second component of the equation when kinetic potential energies are small compared to the rest mass energy, i.e., when i χ/ t m 0 c χ, εχ m 0 c χ, ea 0 χ m 0 c χ then we find that χ = σ j Π j + c j φ. 79 m 0 c This relation can be substituted into the first component Eq. 78, as a result, the generalized Pauli-Schrödinger equation is obtained [ 1 i t φ = p j e m 0 c A e hσ j j + c j m 0 c B j + σ ] jɛ jkl k c l + ea 0 ε φ. 80 m 0 Note that for a constant c j the curl term vanishes leaving one difference from the original Pauli-Schrödinger equation [17]. Here the c j term contributes to the time rate of change of φ in the same way as the momentum does. 6. Generalized Lévi-Leblond equation Returning to the first component equation of 78, we use the energy eigenoperator ˆε = i t the eigen-equation ˆεψ = εψ, set the electromagnetic potential charge to zero, we obtain the first half of the generalized Lévy- Leblond equation. Doing the same with equation 79, we get the second half of the generalized Lévy-Leblond equation. Finally, we can write the generalized Lévy-Leblond equation [] in its full form as ε + εφ = σ j p j + c j χ, 81 m 0 χ = σ j p j + c j φ. 8 Derivation of this generalized Lévy-Leblond equation was already done by Huegele et al. [], who used the principle of relativity the principle of analyticity, as well as the extended Galilei group. They showed that the equation is Galilean invariant discussed its possible applications.

17 Generalized Dirac Klein-Gordon equations 15 7 Generalized Dirac equation families of elementary particles We now examine motions of a free particle described by the generalized Dirac equation see Eq. 78. Let us look for stationary states ψx that have no time dependence, so the wavefunction is separable from the stationary state such that ψx, t =ψxexp[ iεt]. 83 Under this condition the generalized Dirac equation is a time independent equation of the stationary state εψx =[ iα j j + m 0 β ε + α j p j ] ψx. 84 Splitting the four-component spinor into a pair of two-component spinors using explicit representations for the matrices α β, we write Eq. 84 as the following pair of equations εφ = σ j p j χ + m 0 φ εφ + σ j p j χ, 85 εχ = σ j p j φ m 0 χ εχ + σ j p j φ. 86 States with a defined momentum p j φ φ0 = exp [i ˆp χ j x j ], 87 χ 0 can be substituted into Eqs Moreover, we replace the operator ˆp j with its eigenvalues p j, obtain ε + ε m 0 φ 0 σ j p j + p j χ 0 =0, 88 σ j p j + p j φ 0 +ε + ε + m 0 χ 0 =0. 89 This system of equations admits non-trivial solution only when which can be simplified to ε + ε m 0 ε + ε + m 0 [σ j p j + p j ] =0, 90 ε + ε m 0 p j + p j =0. 91 Comparison of this energy-momentum relationship to that of Special Theory of Relativity STR shows the presence of two additional terms ε p j

18 16 R. Huegele, Z. E. Musielak J. L. Fry in the former. If ε = 0 p j = 0, then the stard STR energy-momentum relationship, which also underlines the stard Dirac equation see Eq. 68, is recovered. To reconcile both relationships, we introduce E = ε + ε P j = p j + p j, recognize the fact that both relationships describe the same elementary particle with spin 1/ but with different masses. With m>m 0 being a mass of the more massive particle, we write the resulting energymomentum relationship for the generalized Dirac equation as E m P j =0, 9 which, as expected, is consistent with STR. The main obtained result is that the stard generalized Dirac equations describe the same elementary particle with spin 1/ but with different masses; the mass m in the generalized Dirac equation is always larger than the mass m 0 in the stard Dirac equation. Since the stard Dirac equation is derived with the phase function φ = 0 the generalized Dirac equation corresponds to φ = 0, we may draw an important conclusion, namely, that non-zero phase functions automatically account for the same particles but with different masses. A significant physical consequence is that the existence of the three currently known families of elementary particles in the Stard Model of particle physics [] may be theoretically accounted for by choosing different phase factors. To the best of our knowledge this is a new idea, which requires comparison to the previously suggested solutions of this problem. 8 Comparison to the previous work The existence of three families of elementary particles in the Stard Model [] confirmed by high energy physics experiments is one of the main longsting problems in modern physics. A number of interesting ideas were proposed to explain the origin of the three families [3-31] but to the best of our knowledge the problem remains still unsolved. Barut his coworkers [4,5] obtained a second-order dynamical equation describing elementary particles with two mass states suggested that this approach gives a unified description of electrons muons. Similar equation was considered analyzed by Kruglov [6]. However, in his more recent work Kruglov extended Barut s work by obtaining a generalized Dirac equation for a 0-component [7] 16-component [8] wavefunctions showed that such functions can represent fermions with two mass states; it must be pointed out that Kruglov s work described in [7] is based on the assumption that Lorentz invariance is violated. In theoretical models with extra dimensions, the Dirac equation is also considered its three zero modes introduced by some background fields of

19 Generalized Dirac Klein-Gordon equations 17 nontrivial topology are typically identified with the three generations of elementary particles [30,31]. A new mechanism for the origin of the three fermion generations was recently proposed by Kaplan Sun [3], who considered fifth dimensional space-time as a topological insulator suggested that the three generations of leptons quarks correspond to surface modes in their theory. These are interesting theoretical suggestions, however, their main problem is that so far there is no experimental evidence for the existence of any extra dimensions. Our novel results presented in this paper suggest that there is another way to account for the three families of elementary particles in the Stard Model [3,33], namely, by considering non-zero phase functions in the relativistic models. If this is indeed the case, then it would be possible, at least in principle, to determine theoretically a number of families of elementary particles by finding how many physically meaningful phase functions are allowed. However, such investigation is out of the scope of the present paper. 9 Conclusion We developed a method that allowed us to search for new Poincaré invariant dynamical equations describing free elementary particles, which are represented by four component spinor wavefunctions. The original Dirac equation was obtained shown to be the only Poincaré invariant dynamical equation for 4-component spinor wavefunctions when the phase function is φ = 0. We also considered non-zero phase functions demonstrated that they give new Poincaré invariant generalized Dirac Klein-Gordon equations, which reduce to the original Dirac Klein-Gordon equations when the phase function φ is set to zero. These are important results as they show that other fundamental Poincaré invariant equations do exist in Minkowski space-time. To validate our generalized Dirac equation, we derived the previously obtained generalized Lévy-Leblond equation a new generalized Pauli-Schrödinger equation by taking the non-relativistic limits of the generalized Dirac equation. We also demonstrated that both generalized equations could be reduced to the stard Lévy-Leblond Pauli-Schrödinger equations by taking the phase function to be zero. Our results clearly show that the main difference between the original generalized Dirac equations is that they are obtained with zero non-zero phase functions, respectively, that these equations describe free elementary particle with spin 1/, which have all other physical properties the same except their masses. The fact that the generalized Dirac equation describes elementary particles with larger masses is used to suggest that non-zero phase functions may account for the existence of three families of elementary particles in the Stard Model. This suggestion significantly differs from those

20 18 R. Huegele, Z. E. Musielak J. L. Fry previously made to account for the three families of particle physics. ACKNOWLEDGEMENTS. Z.E.M. acknowledges the support of this work by the Alexer von Humboldt Foundation by The University of Texas at Arlington through its Faculty Development Program. References [1] E.P. Wigner, Ann. Math., , 149. [] Y.S. Kim M.E. Noz, Theory Applications of the Poincaré Group, Reidel, Dordrecht, [3] W.I. Fushchich A.G. Nikitin, Symmetries of Equations of Quantum Mechanics Allerton Press, Allerton, NY, [4] O. Klein, Z. Phys., , 895. [5] W. Gordon, Zeits. für Phys , 117; , 11. [6] P.A.M. Dirac, Proc. Roy. Soc. London A117, ; A , 351. [7] A. Proca, Le J. de Phys. et le Radium , 347. [8] W. Greiner, Relativistic Quantum Mechanics, Springer-Verlag, Berlin, Heidelberg, [9] L.H. Ryder, Quantum Field Theory Cambridge Uni. Press, New York, [10] N.A. Doughty, Lagrangian Interaction Addison-Wesley, New York, [11] V. Bargmann E. Wigner, Proc. Nat. Acad. Sci. U.S. 34, [1] B. Thaller, The Dirac Equation Springer-Verlag, Berlin, Heidelberg, New York, [13] J.L. Fry, Z.E. Musielak Trei-wen Chang, Ann. Phys., , 197. [14] R. Huegele, Z.E. Musielak J.L. Fry, J. Phys. A: Math. Theor., 45 01, [15] J.M. Lévy-Leblond, J. Math. Phys., , 776. [16] J.M. Lévy-Leblond, Comm. Math. Phys., , 86.

21 Generalized Dirac Klein-Gordon equations 19 [17] W. Pauli, Zeitschr. Phys., 43, [18] M. de Montigny, J. Niederle A.J. Nikitin, J. Phys. A: Math. Gen., , [19] J. Niederle A.J. Nikitin, J. Phys. A: Math. Theor., 4 009, 10507; J. Niederle A.J. Nikitin, J. Phys. A: Math. Theor.,4 009, [0] Z.E. Musielak J.L. Fry, Ann. Phys., , 96. [1] J.L. Fry Z.E. Musielak, Ann. Phys., , 668. [] R. Huegele, Z.E. Musielak J.L. Fry, Adv. Studies Theor. Phys., 7 013, [3] G.L. Kane, Modern Elementary Particle Physics, Addison-Wesley, Reading, MA, [4] A.O. Barut, Phys. Lett., 73B 1978, ; A.O. Barut, Phys. Rev. Lett., , 151. [5] A.O. Barut, P. Cordero G.C. Ghirardi, Phys. Rev., 18, 1969, 1844; A.O. Barut, P. Cordero G.C. Ghirardi, Nuovo Cim., A , 36. [6] S.I. Kruglov, Int. J. Mod. Phys., A16 001, 495; S.I. Kruglov, Ann. Found. Louis de Broglie, 9 004, [7] S.I. Kruglov, Eur. J. Theor. Phys., 3 006, 11. [8] S.I. Kruglov, arxiv: v [gr-qc] 15 Oct 01. [9] K. Nozari, Chaos, Solitons Fractals, 3 006, 30. [30] S. Aguilar D. Singleton, Phys. Rev., D73006, [31] C. Beasley, J.J. Heckman C. Vafa, J. High Energy Phys., , 058. [3] D.B. Kaplan S. Sun, Phys. Rev. Lett., , [33] A. Bettini, Introduction to Elementary Particle Physics, Cambridge University Press, Cambridge, 008. Received: October 1, 013