PERSPECTIVE published: 01 June 2021 doi: 10.3389/fphy.2021.618392

CPTM for the and Its Extended Version Based on the Vector Representation of the

E. Marsch 1 and Y. Narita 2*

1 Extraterrestrial Physics Group, Institute for Experimental and Applied Physics, Christian Albrechts University at Kiel, Kiel, Germany, 2 Space Research Institute, Austrian Academy of Sciences, Graz, Austria

We revisit the CPT theorem for the Dirac equation and its extended version based on the vector representation of the Lorentz group. Then it is proposed that CPTM may apply to this fundamental equation for a massive a s a singlet or a doublet with isospin. The symbol M stands here for reversing the sign of the mass in the Dirac equation, which can be accomplished by operation on it with the so-called gamma-five that plays an essential role for the in the . We define the CPTM symmetry for the standard and extended Dirac equation and discuss its physical implications and some possible consequences for general relativity. Edited by: Mariana Frank, Keywords: CPT theorem, Dirac equation, chirality, mass sign reversal, beyond standard model Concordia University, Canada Reviewed by: 1. INTRODUCTION Albert Petrov, Federal University of Paraíba, Brazil Diego Restrepo, The famous CPT invariance of quantum field theory means that thereversalofthesignofthecharge University of Antioquia, Colombia (charge conjugation, C), and of the space- coordinates ( reflection P and time reversal T) *Correspondence: are fundamental physical symmetries [1–6] of any field equations, and the related Lagrangians thus Y. Narita should be invariant under these transformations. Lehnert [7] provides a concise modern review of [email protected] CPT symmetry and its proofs and possible violation. It seems that the combined CPT symmetry cannot be easily violated, but it has been put into Specialty section: question, for example recently by Heald [8]. In his conclusion “CPT symmetry has been violated in This article was submitted to relation to beta decay of mesons and for the three inverted CPT operations of that result High-Energy and Astroparticle in odd parity, and so it cannot be a fundamental symmetry of Nature. In the case of the broken CPT Physics, symmetry of beta decay for instance, the additional hidden symmetry can be provided by the a section of the journal negative mass of antimatter contained within the meson.” Frontiers in Physics Checking the equality of the masses of particles and is the adequate empirical test Received: 16 October 2020 of the validity of CPT invariance within the Standard Model (SM). In a recent letter, Larin [9] also Accepted: 19 April 2021 argued that there is a possibility of violation of CPT symmetry in the SM, yet consistent with the Published: 01 June 2021 CPT theorem, by choosing non-standard phases of the quark fields. To check this experimentally Citation: requires by an order of magnitude higher of measurements of the and anti- Marsch E and Narita Y (2021) CPTM proton mass difference than possible presently. Some recent experimental achievements in this Symmetry for the Dirac Equation and Its Extended Version Based on the field are discussed by Ulmer et al. [10]. Barenboim and Salvado [11] recently put forward ideas and Vector Representation of the Lorentz made related calculations indicating that cosmology provides nowadays the most sensitive data to Group. Front. Phys. 9:618392. test CPT violation by scrutinizing the possible neutrino-antineutrino mass splitting, and by thus doi: 10.3389/fphy.2021.618392 providing a bound on CPT violation for the absolute value of that splitting from cosmological data.

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The main physical motivation of this paper is to show that apply as well to this fundamental equation for a massive fermion there is another new symmetry for the Dirac equation, namely described within the framework of special relativity. Changing mass inversion M. The symmetries of the Dirac equation are the sign of the mass in the Dirac equation can be accomplished not complete with the combined CPT symmetry, but due to by operation of the so-called gamma-five matrix that plays an the chirality (associated with the γ5 matrix) there is another essential role for chirality in the SM. After revisiting the CPT symmetry, namely mass inversion. However, by allowing the theorem for the extended Dirac equation based on the vector mass sign inversion to extend the CPT transformation into representation of the Lorentz group [13], we newly define CPTM CPTM, the maximal symmetry of the Dirac equation based on and discuss its consequences. A brief final section addresses the five gamma matrices is achieved. CPTM transformation is some related theoretical aspects of general relativity. In a short then simply proportional to the unity operator in Dirac-matrix Appendix we present some relevant matrices. terms. The fact that CPTM is proportional to the unit matrix means that the Dirac field obeys it in all its spinor 2. REVISITING THE CPT THEOREM components like a scalar field, which it should as the squared Dirac equation reproduces the Klein-Gordon equation, as will be In this section we derive and discuss the basic symmetries of the shown in Equations (23) and (28). This has no basic implications standard and extended (including isospin) Dirac equation, which for a free fermion spinor field, yet may so in the context of are charge conjugation, parity reflection and time inversion. We coupling to gauge fields or gravity. write the Dirac equation by using capital (lower) letters for We stress at the outset that this paper is essentially a the Gamma (gamma) matrices corresponding to the extended theoretical work which discusses CPTM as an extension of the [13] and (standard) [17] Dirac equation. Lack of space forbids symmetry of the Dirac equation in . According to explicitly quote here all related matrices in the Weyl to our knowledge this idea in the present algebraic form has first representation. Some of them are quoted below in the Appendix. been discussed by Marsch and Narita [12] before. We further For more on this we refer to the cited references. The extended define the CPTM symmetry for the new extended Dirac equation Dirac equation also has the usual form as proposed by Marsch and Narita [13, 14] and provide here the related detailed algebra. There it was also shown that the Ŵ i∂ m. (1) extended Dirac equation can easily be expanded to include the = SU(N) gauge fields of the standard model. A field is denoted conventionally as ψ, whereas Especially, in the context of modern cosmology and general for the extended Dirac equation we use throughout the capital relativity, an extension of the CPT theorem has been considered symbol for the quantum field. To include an electromagnetic by Bondarenko [15], which addresses the possibility of a negative gauge field A the derivative ∂ in Equation (1) is replaced by the sign of the particle mass, and related with it the mass sign- covariant derivate inversion symmetry. Therefore, in order to preserve fundamental invariance, the CPTM theorem has been proposed by him, D ∂ ieA. (2) whereby here the symbol M stands for the inversion of the = + sign of the mass m. CPTM was so defined to include besides The three basic symmetries consist of the simple operations charge-conjugation, parity reflection, and time-reversal now also of complex conjugation, named C, parity reflection or space the sign-inversion of the mass. As a result, antimatter might inversion, named P, and time inversion, named T. They have the attain a negative “gravitational mass,” and so CPTM invariance following effects on the field: C , P(x) ( x), and = ∗ = − is required and must be ensured by that generalized theory. T(t) ( t). These operations are their own inversions, i.e., = − But observationally, the gravitational behavior of antimatter is we have C C 1, P P 1, and T T 1. Furthermore, there are = − = − = − not known, and the interaction between and antimatter also three relevant matrices involved, such that we can generally might be either attractive or repulsive. Villata [16] theoretically define the symmetry operations as investigated a decade ago the behavior under CPT of the covariant equations of motion of an electron in a gravitational C C C, = and electromagnetic field and came to the conclusion that there P P P, . (3) = should be gravitational repulsion between matter and antimatter. T T T = Bondarenko [15] originally had in mind the possible application of the CPTM symmetry of the space-time solutions of These matrices (C, P, T) have like the Dirac matrices the Einstein equations to the resolution of the cosmological constant dimension 8 8 and still need to be determined by explicit × problem. He also discussed the consequences of the CPTM operation of C, P, T on the Dirac equation. The transformed fields symmetry for cosmological problems such as the values of the are identified by a calligraphic subscript. We define C CC, = vacuum energy density and cosmological constant. But he further P PP, and T TT. Correspondingly, we obtain the = = considered possible consequences of CPTM symmetry when three Dirac equations for the transformed fields reading being applied to quantum field theory and relativity. Here we aim, within a much narrower scope, at revisiting the Ŵ iD∗ C mC, = CPT theorem for the extended and original Dirac equation [17] Ŵ i∂P mP,. (4) = and are led to propose that CPTM may, by good physical reasons, (Ŵ ) i∂T mT. − ∗ =

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For the time inversion we had to first complex conjugate the For charge conjugation again close inspection now of kinetic term, as the Hamiltonian corresponding to the Dirac Equation (7) leads to the matrix Lagrangian is real, and thus its eigenvalues must be real as well 0 i1 (see for example the related discussions in Refs. [4–6]). The above C 4 . (14) kinetic terms define the requirements to be posed on the matrices = i14 0 − C, P, T such that transformed fields as well obey the original Dirac equation [Equation (1)]. Insertion of the operations given As a result, we obtained the mathematical expressions for all three in Equation (3) and exploiting the definitions of C,P,T yields symmetry operations. Now we can state their combined effects in the famous CPT Theorem of the Standard Model. The result is 1 (C− Ŵ C)iD∗ ∗ m∗, 1 1 = CPT 0 i 4 0 i 0 (P− Ŵ P)i∂( x) m( x), . (5) 1 C P T Ŵ5CPT. 1 − = − = i 4 0 i 0 0 = (T (Ŵ ) T)i∂( t) m( t). − − − − − ∗ − = − (15) This operator transforms the field as follows Concerning charge conjugation, we compare the first of the above three equations with the complex conjugated Dirac equation CPT(x, t) Ŵ5∗( x, t). (16) reading = − −

This result is the analog to the well known result for the normal (Ŵ )∗( i)D∗ ∗ m∗, (6) − = Dirac equation as used in the SM of elementary particle physics from which we read off by comparison that [4], which reads similarly

1 CPT C− Ŵ C (Ŵ )∗. (7) ψ(x, t) γ5ψ∗( x, t). (17) = − = − − Here the three involved standard Dirac 4 4 matrices are Similarly, by inversion of the spatial and temporal variables in the × original Dirac equation [Equation (1)], we obtain 0 iσy 0 12 iσy 0 C , P 1 , T . (18) = iσy 0 = 2 0 = 0 iσy Ŵ i∂( x, t)( x, t) m( x, t), − − − = − . (8) Ŵ i∂(x, t)(x, t) m(x, t). − − = − The definition of C differs by a minus sign from the conventional definition. The product of the three CPT matrices yields the Here the derivate is ∂(x, t) (∂/∂t, ∂/∂x). Again by comparison = γ5 [ 1, 1,1,1]. The Dirac matrices in Weyl of Equation (5) with Equation (8), we find that the relations = − − representation attain the standard form quoted in the Appendix. must hold: When one replaces in Equation (5) the field by ψ and Ŵ by γ 1 1 and inserts the matrices of Equation (18) and the standard Dirac P− ŴP Ŵ, and P− Ŵ P Ŵ . (9) = − 0 = 0 gamma matrices, then one finds that the same requirements as given in Equations (7, 9, 10) hold for, and are fulfilled by, the Finally, for time inversion we obtain matrix γ. More generally, one may consider phase factors of 1 1 module unity in each of the above symmetry operations, an issue T− (Ŵ)∗T Ŵ, and T− (Ŵ )∗T Ŵ . (10) = − 0 = 0 that is discussed in detail by Lehnert [7], but we take here trivial values of unity which suffices for the present purpose and the case So the remaining task is to determine the three matrices we want to make. C, P, T for the extended Dirac equation. For parity close inspection of Equation (9) readily shows that P Ŵ , and = 0 therefore 3. EXTENSION TO CPTM SYMMETRY INVOKING MASS SIGN-INVERSION 0 i P P. (11) = i 0 CPT − It is interesting to note that the conventional operation involves the Ŵ matrix for the extended and γ for the standard For time inversion we need to know (Ŵ) , which is obtained by 5 5 ∗ Dirac equation. We find this result puzzling because each complex conjugation of Equation (A3) in the Appendix.Wethus component of the ψ and obeys the quadratic scalar obtain Ŵ Ŵ and 0∗ = − 0 Klein-Gordon (KG) equation [18], which has only the complex conjugation and space-time inversion as basic symmetries, which 0 + Ŵ∗ i . (12) means CPT, and does not involve any Dirac matrices. We suggest = − 0 that a possible solution to this problem may be found in another intrinsic symmetry of the Dirac equation. Consequently, we For time inversion close inspection of Equation (10) then define similarly to what we did before the mass sign-version suggests that symmetry as M M MM. The operation M has the = = 0 following effect on the massive-particle field, M(m) ( m), T . (13) 1= − = 0 and is of course its own inversion, i.e., giving M M− . We recall − =

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that the solution of the Dirac equation depends on the mass as Here we use the abbreviation γ p γ p, with p (E(p), p). = = characteristic parameter of the particle. Still, the matrix M needs Similarly, we have p x p x, with x (t, x). The vectors u = = to be defined. One now obtains the Dirac equation in the two and v form a in the fermion rest frame and are orthogonal equivalent forms and can be normalized in a Lorentz-invariant fashion. Explicitly they read in Weyl representation as follows: uT (1,0,1,0), 1 = Ŵ i∂M mM, uT (0,1,0,1), vT (1,0, 1,0), vT (0,1,0, 1), whereby = (19) 2 1 2 Ŵ i∂( m) m( m). = = − = − − = − − index 1 and 2 refers to up and down. When we operate with the charge conjugation operator on Equation (24) and use that By comparison, we then obtain by insertion of M a condition C C and C2 1 , we find that on the matrix M that is given by the equation = ∗ = 4

1 CψP1,2 C(γ∗ p m)u∗ exp ( ip x) M− Ŵ M Ŵ . (20) 1,2 = − = + + (Cγ∗C p m)Cu∗ exp ( ip x) ψ , = + 1,2 + = ± A2,1 Consequently, M Ŵ5, since this delivers by anticommutation = CψP1,2(p, m x, t) ψA2,1(p, m x, t), (25) with the Gammas the negative sign in front of the mass ; = ± ; (m 0). We emphasize that with that definition we find after ≥ since Cu1,2∗ v2,1, whereby the spin is flipped. As expected, Equation (15) that CPTM 1 CPTM, which is an important = ∓ = 8 the particle is transformed into the solution with result. Let us briefly return to the KG equation. Formally, on can opposite spin, and vice versa, which is not shown here. Exploiting of course even in that equation, the relations Equations (9) and (10) with Gamma being replaced 2 by gamma, and using the definitions given in Equation (18), one (∂ ∂ m ) 0, (21) + = finds after some algebra that replace m by m, which is trivial but may be denoted as a “mute − PψP1,2(p, m x, t) ψP1,2(p, m x, t), mass sign-inversion.” ; = + ; Pψ (p, m x, t) ψ (p, m x, t), Consequently, could M perhaps be the sought after additional A1,2 ; = − A1,2 ; (26) Tψ (p, m x, t) ψ (p, m x, t), symmetry of the Dirac equation, which would also apply to P1,2∗ ; = ∓ P2,1 ; Tψ (p, m x, t) ψ (p, m x, t). neutral but massive neutrinos, for example? Taking the roots A1,2∗ ; = ∓ A2,1 ; of m2 of course delivers m, where the negative sign is ± Here we used the fact that in the rest frame the basis vectors conventionally not considered. As the Dirac equation results behave as follows against time and space inversion, i.e., we have from taking in Lorentz-invariant fashion the “roots” of the KG Pu u , and Pv v , which means that particles 1,2 = 1,2 1,2 = − 1,2 equation, there is may be some physical meaning to the negative have positive and antiparticles negative rest-frame parity, and mass, corresponding to a negative frequency in the particles rest moreover Tu u and Tv v , which implies a CPTM 1,2∗ = ∓ 2,1 1,2∗ = ∓ 2,1 frame. Let us first consider the operation of on simple spin flip. Concerning the mass sign-inversion one obtains with waves. As is well-known, fermion quantum fields can be Fourier M MM the result decomposed, i.e., be written as a sum of plane waves with negative = (particles) and positive (antiparticles) frequency like Mψ (p, m x, t) ψ (p, m x, t), P1,2 ; = P1,2 ; (27) MψA1,2(p, m x, t) ψA1,2(p, m x, t). φ (x, t) exp ( i(E(p)t p x)), (22) ; = ; ± = ± − We note that to obtain this result the spinors u and v are with the positive definite relativistic particle energy E(p) understood to depend implicitly also on the mass m, which means 2 2 = p m and p. It can then readily be seen that that u( m) v(m) and vice versa, since they are the eigenvectors + − = of the Dirac equation in the rest frame and correspond to the CPTMφ (m x, t) φ∗ ( m x, t) φ (m x, t). (23) ± ; = − ; − − = ± ; eigenvalues E m. Here we used further that operation ± = ± with M γ on the basis vectors yields Mu v , and Therefore, the plane waves are eigenfunctions of CPTM. = 5 1,2 = − 1,2 Moreover, the relativistic four-momentum P i∂ obeys the thus Mv1,2 u1,2. Apparently, the spin is not flipped in 1 = = − equation (CPTM) PCPTM P, which means it commutes this case. As a result, the solutions in Equation (24) simply are − = with the CPTM operation, since M has no effect on the particle the eigenfunctions of the mass sign-inversion operator with the energy depending on the square of the mass m. trivial eigenvalue plus one. Conventionally, in the SM the plus To substantiate our suggestion we present a few more sign corresponds to particles with negative frequency waves and instructive calculations for the individual symmetry operations. the minus sign to antiparticles with positive frequency waves. For the sake of convenience we provide them only for the This remains true also for the extended Dirac equation describing standard Dirac equation. For the extended one we simply have a massive fermion doublet. The fundamental sense of the sign of to replace the lower case with the upper gamma matrices, and m is also obvious from the polarization spinors in Equation (24). similarly for C, P, T, and M as defined above. By insertion it is Obviously, the effects of the four symmetry transformations easily validated that the particle and antiparticle solutions of the yield subtle individual results. However, if we finally combine free Dirac equation can be written them in their product CPTM, we obtain a much simpler result for the general Dirac spinor field ψ (p, m x, t) (γ p m)u exp ( ip x), P ; = + − (24) ψ (p, m x, t) (γ p m)v exp ( ip x). CPTM ψ(m x, t) CPTM ψ ψ∗( m x, t). (28) A ; = − + ; = = − ; − −

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In particular for the spinor wave functions in Equation (24) transforms symmetrically to the CPTM transformation if both we finally obtain CPTM ψ (p, m x, t) ψ (p, m x, t). the inertial mass m and the gravitational mass m change P,A ; = A,P∗ ; i g In conclusion, we suggest to include the mass sign-inversion the sign under the CPTM transformation. Note that the four- operator M as intrinsic symmetry of the Dirac equation in the acceleration (l.h.s.) is anti-symmetric. Equation (31) implies CPT theorem, which yields the extended CPTM theorem and the that the gravitational interaction between ordinary matter rather concise result in Equation (28). The operation of γ5 has of (represented by positive inertial mass) and antimatter (in case it course no effect on the scalar plane wave and particle momentum is represented by negative gravitational mass) is repulsive, which as discussed above, but effectively merely acts on the polarization according to Villata [16] could be one of the possible mechanisms vectors of the Dirac equation. Importantly, it plays a key role of large-scale structure formation such as the void regions in 1 in chirality through the projection operator P (14 γ5), galaxy clusters (Mpc or larger). ± = 2 ± which permits to decompose the spinor field ψ into its left- and right-chiral spinor components. 5. SUMMARY AND CONCLUSION

4. CPTM SYMMETRY IN GENERAL We have shown that the CPT theorem of the standard RELATIVITY Dirac equation also holds for the extended version, which describes a massive fermion doublet obeying the isospin SU(2) It is of interest to study the CPTM transformation in the symmetry [13]. This result was stated in Equation (16). The astronomical or cosmological applications even though the combined multiplied matrices of CPT result in Ŵ5 which is CPTM symmetry is derived for the flat Minkowski space-time. the key matrix for the definition of chirality in the Dirac In the gravitational theory (in the frame of general relativity), the equation. Moreover, Ŵ5 (or γ5ψ for the standard Dirac charge conjugation C is ineffective, and parity and time reversal equation) also provides a solution to it after mass sign reversal act as the PT transformation, dx dx [16]. The energy- (named M), which we suggest to be adopted as another → − momentum tensor Tν transforms anti-symmetrically under the intrinsic symmetry of the Dirac equation. Related to it we CPTM transformation. For a point mass at x0, for example, the therefore propose the symmetry transformation M Ŵ5M. ν = energy-momentum tensor T transforms as It should be multiplied from the right to CPT, which seems incomplete, but when being upgraded to CPTM it simply is ν 1 (4) dx dx proportional to the unity operator. Therefore, for the standard m δ (x x0) dτ √g − dτ dτ as well as extended Dirac equation we obtain CPTM = ν 1 CPTM, respectively CPTM 1 CPTM, which yields the 1 (4) dx dx 4 = 8 ( m) δ (x x0)− − dτ (29) fundamental relation Equation (28) for the standard Dirac → √g − − dτ dτ equation, and similarly where g det(gν) is the determinant of the metric gν in = | | absolute value. The metric tensor g , the Ricci curvature tensor CPTM (m x, t) CPTM ∗( m x, t) (32) ν ; = = − ; − − Rν , and the scalar curvature R all transform symmetrically under the CPTM transformation in the same way as the CPT for the extended Dirac equation describing a massive fermion transformation (assuming the metric is already given or mass- doublet. We recall that the eigenfunctions of the CPTM operator independent). The Christoffel symbol is anti-symmetric to the are just the plane waves [Equation (22)], which represent CPTM transformation. Using the symmetry properties above, the the wave functions of massive free particles and antiparticles Einstein equation in relativistic quantum theory. With these results we have achieved the main purpose of this short theoretical paper. It 1 Rν gν R 8πGTν (30) emphasizes that by the introduction of the mass inversion − 2 = − operator M the CPT theorem can be simplified to the above forms for the CPTM theorem, which do not involve Dirac is violated in that the l.h.s. (which represents the space-time matrices any more but just the related unit matrix, whereas property) is symmetric to the CPTM transformation while the in the usual form of the CPT theorem [4–7] the γ matrix r.h.s. (which represents the source of gravity) is anti-symmetric. 5 appears explicitly, whereby its physical meaning in this context is The Einstein equation is CPTM symmetric in vacuum Tν 0. = according to our knowledge not discussed at all. We speculated If, however, the gravitational constant G changes the sign together [12] about this possible modification of the CPT theorem with the CPTM transformation, the symmetry of the Einstein previously, although we did not evaluate this rigorously as in the equation is restored for a non-zero energy-momentum tensor. present paper. The particle equation of motion (represented by the One natural field of application seems to us the state of geodesic equation) unusual non-baryonic matter in the universe. The extension of CPT symmetry to CPTM symmetry may be applied to d2xλ m dx dxν g Ŵλ astrophysical and cosmological contexts and studies, as we 2 ν (31) dτ = − mi dτ dτ already discussed in the introduction. For example, CPTM

Frontiers in Physics | www.frontiersin.org 5 June 2021 | Volume 9 | Article 618392 Marsch and Narita CPTM Symmetry promotes the notion that negative mass states or particles are DATA AVAILABILITY STATEMENT one of the likely candidates for dark energy, as was suggested by Petit and d’Agostini [19]. It is pointed out that negative mass The original contributions presented in the study are included particles may exist in the de Sitter space-time like for the inflation in the article/supplementary material, further inquiries can be phase of the early universe, an issue discussed by Mbarek and directed to the corresponding author/s. Paranjape [20]. Another application would be massive neutrinos. Combination of the Majorana-type mass with the Dirac-type AUTHOR CONTRIBUTIONS mass in the neutrino field can, after Yanagida [21] and Mohapatra and Senjanovic[´ 22], generate neutrino states with negative mass. EM: concept development, calculations, and manuscript writing. Other future applications remain to be determined, and the YN: discussion, calculations, literature review, and manuscript consequences for quantum field theory beyond the SM still need writing. Both authors contributed to the article and approved the to be investigated. submitted version.

REFERENCES 15. Bondarenko S. CPTM discrete symmetry, quantum wormholes and cosmological constant problem. Universe. (2020) 6:121. 1. Pauli W. The connection between spin and statistics. Phys Rev. (1940) 58:716– doi: 10.3390/universe6080121 22. doi: 10.1103/PhysRev.58.716 16. Villata M. CPT symmetry and antimatter gravity in general relativity. EPL. 2. Streater RF, Wightman AS. PCT, Spin and Statistics, and All That. Princeton, (2011) 94:200001. doi: 10.1209/0295-5075/94/20001 NJ: Princeton University Press (2000). 17. Dirac PAM. The quantum theory of the electron. Proc R Soc Lond Ser A. (1928) 3. Lüders G, Dan K. Proof of the TCP theorem. Ann Phys. (1957) 2:1–15. 117:610–24. doi: 10.1098/rspa.1928.0023 doi: 10.1016/0003-4916(57)90032-5 18. Klein O. Elektrodynamik und Wellenmechanik vom Standpunkt des 4. Schwartz MD. and the Standard Model. Cambridge, Korrespondenzprinzips. Z Phys. (1926) 42:407–42. doi: 10.1007/BF014 UK: Cambridge University Press (2014). 00205 5. Kaku M. Quantum Field Theory, A Modern Introduction. New York, NY: 19. Petit JP, d’Agostini G. Negative mass hypothesis in cosmology and Oxford University Press (1993). the nature of dark energy. Astrophys Space Sci. (2014) 354:611–615. 6. Peskin ME, Schroeder DV. An Introduction to Quantum Field Theory. doi: 10.1007/s10509-014-2106-5 Reading, MA: Addison-Wesley Publishing Company (1995). 20. Mbarek S, Paranjape MB. Negative mass bubbles in de Sitter space- 7. Lehnert R. CPT symmetry and its violation. Symmetry. (2016) 8:114. time. Phys Rev D. (2014) 90:101502(R). doi: 10.1103/PhysRevD.90.1 doi: 10.3390/sym8110114 01502 8. Heald G. CPT Violation and CPTM Symmetry. Online preprint on 21. Yanagida T. Horizontal symmetry and masses of neutrinos. Prog Theor Phys. ResearchGate. Available online at: https://www.researchgate.net/publication/ (1980) 64:1103–5. doi: 10.1143/PTP.64.1103 338826193_CPT_Violation_and_CPTM_Symmetry (2020). 22. Mohapatra RN, Senjanovic´ G. Neutrino mass and spontaneous 9. Larin SA. A possibility of CPT violation in the Standard Model, EPL. (2020) parity nonconservation. Phys Rev Lett. (1980) 44:912–5. 129:21003. doi: 10.1209/0295-5075/129/21003 doi: 10.1103/PhysRevLett.44.912 10. Ulmer S, Smorra C, Mooser A, Franke K, Nagahama H, Schneider G, et al. 23. Pauli W. Zur Quantenmechanik des magnetischen elektrons. Z Phys. (1927) High-precision comparison of the -to-proton charge-to-mass ratio. 43:601–23. doi: 10.1007/BF01397326 Nature. (2015) 524:196–9. doi: 10.1038/nature14861 11. Barenboim G, Salvado J. Cosmology and CPT violating neutrinos. Eur Phys J Conflict of Interest: The authors declare that the research was conducted in the C. (2017) 77:766. doi: 10.1140/epjc/s10052-017-5347-y absence of any commercial or financial relationships that could be construed as a 12. Marsch E, Narita Y. Fermion unification model based on the intrinsic potential conflict of interest. SU(8) symmetry of a generalized Dirac equation. Front Phys. (2015) 3:82. doi: 10.3389/fphy.2015.00082 Copyright © 2021 Marsch and Narita. This is an open-access article distributed 13. Marsch E, Narita Y. Dirac equation based on the vector under the terms of the Creative Commons Attribution License (CC BY). The representation of the Lorentz group. Eur Phys J Plus. (2020) 135:782. use, distribution or reproduction in other forums is permitted, provided the doi: 10.1140/epjp/s13360-020-00798-7 original author(s) and the copyright owner(s) are credited and that the original 14. Marsch E, Narita Y. Connecting in the Dirac equation the Clifford algebra of publication in this journal is cited, in accordance with accepted academic practice. Lorentz invariance with the of SU(N) gauge symmetry. Symmetry. No use, distribution or reproduction is permitted which does not comply with these (2021) 13:475. doi: 10.3390/sym13030475 terms.

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APPENDIX: EXTENDED DIRAC MATRICES 0 IN WEYL REPRESENTATION Ŵ i − . (A3) = + 0 The standard Dirac matrices can be found and looked up in any The involved Sigma matrices and their characteristics were textbook [4, 5]. But we first quote them here for completeness in already quoted above. The new Delta matrix corresponds to the the Weyl representation metric in Minkowski space-time and is defined as 1 σ 0 2 0 10 0 0 γ 1 , σ , (A1) = 2 0 0  0 10 0  − − , (A4) = 0 0 1 0 with the well-known three-vector of the Pauli [23] matrices σ  −  =  00 0 1  (σx, σy, σz). According to the new derivations by Marsch and  −  Narita [13], the Dirac matrices for the extended Dirac equation 2 which yields 14. Delta has the important property that including isospin are based on the generalized 4 4 spin matrices, = 2 1 × ± ∓, which guarantees that Ŵj 8, and thus reading = = − Ŵ2 3 1 , and consequently that the component matrices = − 8 Ŵ anticommute, and therefore obey the Clifford algebra. The 0 10 0 j ± matrices Ŵ and Ŵ are given by the expressions  1 0 00  5 0 ± ± , x = 0 00 i  −  14 0 0 i  0 0i0  Ŵ5 − , Ŵ0 , (A5)   = 0 14 = i 0 0 0 1 0 − ±  0 0 0i  which obeys (Ŵ )2 1 and mutually anticommutes with ± , 0 8 y = 10 00 =  ±  the other four Gamma matrices by its definition. The spin or  0 i 0 0   −  spinorial rotation operator is given by 0 00 1 ±  0 0 i 0  1 + 0 ± − , (A2) S . (A6) z = 0i0 0 = 2 0 −    100 0   ±  It is interesting to note that charge exchange leads to spin inversion, i.e., flips the spin, namely with the commutator [±, ∓] 0. Also, ± ± 2i±. = × = 1 By complex conjugation of the Sigma matrices in Equation (A2), C− SC S, (A7) we can see that they obey ( ) . The Gamma matrices = − ± ∗ = − ∓ are defined in the following way which means these two operators anticommute.

Frontiers in Physics | www.frontiersin.org 7 June 2021 | Volume 9 | Article 618392