Skip to main content
SpringerLink
Log in

Spinor Structure and Internal Symmetries

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Spinor structure and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It is shown that tensor products of biquaternion algebras are associated with the each irreducible representation of the Lorentz group. Space-time discrete symmetries P, T and their combination PT are generated by the fundamental automorphisms of this algebraic background (Clifford algebras). Charge conjugation C is presented by a pseudoautomorphism of the complex Clifford algebra. This description of the operation C allows one to distinguish charged and neutral particles including particle-antiparticle interchange and truly neutral particles. Spin and charge multiplets, based on the interlocking representations of the Lorentz group, are introduced. A central point of the work is a correspondence between Wigner definition of elementary particle as an irreducible representation of the Poincaré group and SU(3)-description (quark scheme) of the particle as a vector of the supermultiplet (irreducible representation of SU(3)). This correspondence is realized on the ground of a spin-charge Hilbert space. Basic hadron supermultiplets of SU(3)-theory (baryon octet and two meson octets) are studied in this framework. It is shown that quark phenomenologies are naturally incorporated into presented scheme. The relationship between mass and spin is established. The introduced spin-mass formula and its combination with Gell-Mann–Okubo mass formula allows one to take a new look at the problem of mass spectrum of elementary particles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. We choose s p i n +(1,3) as a generating kernel of the underlying spinor structure. However, the group s p i n +(2,4)≃SU(2,2) (a universal covering of the conformal group SO0(2,4)) can be chosen as such a kernel. The choice s p i n +(2,4)≃SU(2,2) takes place in the Penrose twistor programme [15] and also in the Paneitz-Segal approach [17–19].

  2. Gel’fand and Yaglom [21] developed a general theory of such equations including infinite-component wave equations of Majorana type [22].

  3. Using a mapping of curvature tensor into \(\mathbb {R}^{6}\), Petrov [29] introduced his famous classification of Einstein spaces.

  4. It is interesting to note that from (25) it follows directly that the electron mass is the minimal rest mass μ 0.

  5. This topics leads to deeply developed mathematical tools related with induced representations [32].

  6. As it mentioned above (see Sections 1 and 2.1.1), in the well-known Penrose twistor programme [14, 15] a spinor structure is understood as the underlying (more fundamental) structure with respect to Minkowski space-time. In other words, space-time continuum is not fundamental substance in the twistor approach, this is a fully derivative (in spirit of Leibnitz philosophy) entity generated by the underlying spinor structure. In this context space-time discrete symmetries P and T should be considered as projections (shadows) of the fundamental automorphisms belonging to the background spinor structure.

  7. Some applications of the fundamental automorphisms to discrete symmetries of quantum field theory were considered by Rashevskii in [34] (see also his paper [36]).

  8. The space H can be understood as finite-dimensional space H n and also as infinite-dimensional space \({\boldsymbol {\mathsf {H}}}_{\infty }\) [43].

  9. In the framework of refined algebraic quantization [45, 46], the inner product of states is defined using the technique of group averaging. Group averaging uses the integral

    $${\int}_{G}\langle\phi_{1}|U(g)|\phi_{2}\rangle dg $$

    over the gauge group G, where dg is a so-called symmetric Haar measure on G, U(g) is a representation of G, and ϕ 1 and ϕ 2 are state vectors from an auxiliary Hilbert space \({\mathcal {H}}_{aux}\). Convergent group averaging gives an algorithm for construction of a complete set of observables of a quantum system [47–51]. This topics is related closely with a quantum field theory on the Poincaré group [52, 53] and also with a wavelet transform for resolution dependent fields [54, 55].

  10. Recall that a superposition of the state vectors forms an irreducible unitary representation U (quantum elementary particle system) of the group \(\mathbf {spin}_{+}(1,3)\simeq \text {SL}(2,\mathbb {C})\) which acts in the Hilbert space \({\mathsf {H}}_{\infty }\). At the reduction of the superposition we have \(U\rightarrow \boldsymbol {\tau }_{l\dot {l}}\) and \({\mathsf {H}}_{\infty }\rightarrow \text {Sym}_{(k,r)}\). For example, in the case of electron we have two spin states: the state 1/2 described by the representation τ 1/2,0 on the spin-1/2 line and the state -1/2 described by τ 0,1/2 on the dual spin-1/2 line. The representations τ 1/2,0 and τ 0,1/2 act in the spaces Sym(1,0) and Sym(0,1), respectively. The superposition of these two spin states leads to a unitary representation U m,+,1/2 of the orbit \(\boldsymbol {O}^{+}_{m}\) which acts in the Hilbert space \({\mathsf {H}}^{m,+,1/2}_{\infty }{\simeq {\boldsymbol {\mathsf {H}}}^{S}_{2}}\otimes {\boldsymbol {\mathsf {H}}}_{\infty }\). At the reduction we have \(U^{m,+,1/2}\rightarrow \boldsymbol {\tau }_{1/2,0}\) or \(U^{m,+,1/2}\rightarrow \boldsymbol {\tau }_{0,1/2}\) and \({\mathsf {H}}^{m,+,1/2}_{\infty }\rightarrow \text {Sym}_{(1,0)}\) or \({\mathsf {H}}^{m,+,1/2}_{\infty }\rightarrow \text {Sym}_{(0,1)}\).

  11. Of course, in the case of charge quadruplet (for example, Δ-quadruplet of the spin 3/2) we have four values −1, 0, 1, 2.

  12. At this moment it is not possible to enumerate all the superselection rules for \({\boldsymbol {\mathsf {H}}}^{S}\otimes {\boldsymbol {\mathsf {H}}}^{Q}\otimes {\boldsymbol {\mathsf {H}}}_{\infty }\).

  13. At this point we do not use the quark structure of F 1/2, since this structure is a derivative construction of SU(3)-symmetry. The quark scheme in itself is a reformulation of SU(3) group representations in terms of tensor products of the vectors of fundamental representations \(\text {Sym}^{0}_{(1,0)}\) and \(\text {Sym}^{0}_{(0,1)}\). So, quarks u, d, s are described within \(\text {Sym}^{0}_{(1,0)}\), and antiquarks \(\overline {u}\), \(\overline {d}\), \(\overline {s}\) within \(\text {Sym}^{0}_{(0,1)}\). Quarks and antiquarks have fractional charges Q and hypercharges Y. The each hadron supermultiplet can be constructed from the quarks and antiquarks in the tensor space \(\mathbb {C}^{k,r}\) which corresponds to a standard representation of SU(3). The space \(\mathbb {C}^{k,r}\) is a tensor product of k spaces \(\mathbb {C}^{3}\) and r spaces \(\overset {\ast }{\mathbb {C}}^{3}\). The quark composition of a separate particle, belonging to a given supermultiplet of SU(3), is constructed as follows. I-basis is constructed from the eigenvectors of Q and Y in the space of irreducible representation of the given supermultiplet. These basis vectors present particles of the supermultiplet, the each of them belongs to \(\mathbb {C}^{k,r}\) and, therefore, is expressed via the polynomial on basis vectors e 1, e 2, e 3 of \(\mathbb {C}^{3}\) and basis vectors \(\tilde {e}_{1}\), \(\tilde {e}_{2}\), \(\tilde {e}_{3}\) of \(\overset {\ast }{\mathbb {C}}^{3}\) with the degree k+r. The substitution of e 1, e 2, e 3, \(\tilde {e}_{1}\), \(\tilde {e}_{2}\), \(\tilde {e}_{3}\) by u, d, s, \(\overline {u}\), \(\overline {d}\), \(\overline {s}\) leads to a quark composition of the particle. It is assumed that quarks and antiquarks have the spin 1/2 (however, spin is an external parameter with respect to SU(3)-theory). Hence it follows that a maximal spin of the particle, consisting of k quarks and r antiquarks, is equal to (k+r)/2. When k+r is odd we have fermions and bosons when k+r is even.

    In this connection it is interesting to note that k+r tensor products of \(\mathbb {C}^{2}\) and \(\overset {\ast }{\mathbb {C}}_{2}\) biquaternion algebras in (11), which generate the underlying spinor structure, lead to a fermionic representation of s p i n +(1,3) when k+r is odd and to a bosonic representation when k+r is even (see spin-lines considered in the Section 2.1.1). Due to the difference between dimensions of basic constituents in tensor products (n=2 for spinors and n=3 for quarks) which define spinor and quark structures, we can assume that spinors are more fundamental than quarks.

  14. The physical sense of the unitary field is unknown (see [62]). The field \({Z^{b}_{a}}\) is not one and the same for the all supermultiplets of SU(3)-theory. However, Z-fields of different supermultiplets are distinguished by only two real parameters. Z-field of type (60) takes place also at the SU (6)/SU(3)-reduction in the flavor-spin SU (6)-theory. In some sense, Z-field can be identified with a nonlocal quantum substrate in the decoherence theory [65]. In this context Z-field can be understood as a mathematical description of the decoherence process (localization) of the particles, that is, it is a reduction of the initial quantum substrate into localized particles at the given energy level.

References

  1. Barut, A.O., Raczka, R.: Theory of Group Representations and Applications. PWN, Warszawa (1977)

    Google Scholar 

  2. Michel, L.: Invariance in quantum mechanics and group extension. Group Theoretical Concepts and Methods in Elementary Particle Physics, pp 135–200. Gordon & Breach, New York (1964)

    Google Scholar 

  3. Coleman, S.: Mandula, J.: All Possible Symmetries of the S Matrix. Phys. Rev 159, 1251–1256 (1967)

    Article  ADS  MATH  Google Scholar 

  4. Pais, A.: Dynamical Symmetry in Particle Physics. Rev. Mod. Phys. 38, 215–255 (1966)

    Article  ADS  Google Scholar 

  5. Penrose, R.: The twistor programme. Rep. Math. Phys. 12, 65–76 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  6. Penrose, R., MacCallum, M.A.H.: Twistor theory: an approach to the quantization of fields and space-time. Phys. Rep. 6, 241–316 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  7. Joos, E., Zeh, H.D., Kiefer, C., Giulini, D.J.W., Kupsch, J., Stamatescu, I.-O.: Decoherence and Appearence of a Classical World in Quantum Theory. Springer-Verlag, Berlin (2003)

    Book  Google Scholar 

  8. Zurek, W.H.: Decoherence, Einselection, and the Quantum Origins of the Classical. Rev. Mod. Phys. 75, 715 (2003). arXiv:quant-ph/0105127 (2001)

  9. Zurek, W.H.: Decoherence and the transition from quantum to classical – REVISITED. arXiv:quant-ph/0306072 (2003)

  10. Wigner, E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149–204 (1939)

    Article  MathSciNet  Google Scholar 

  11. Gell-Mann, M., Ne’eman, Y.: The Eightfold Way. Benjamin, New York (1964)

    Google Scholar 

  12. Gel’fand, I.M., Minlos, R.A., Shapiro, Z.Ya.: Representations of the Rotation and Lorentz Groups and their Applications. Pergamon Press, Oxford (1963)

    MATH  Google Scholar 

  13. Varlamov, V.V.: CPT Groups of Higher Spin Fields. Int. J. Theor. Phys. 51, 1453–1481 (2012). arXiv:1107.4156 [math-ph] (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Penrose, R.: Structure of space-time. Benjamin, New York-Amsterdam (1968)

    Google Scholar 

  15. Penrose, R., Rindler, W.: Spinors and space-time. Vol. 2. Spinor and twistor methods in space-time geometry. Cambridge Monographs on Mathematical Physics, 2nd edn. Cambridge University Press, Cambridge and New York (1988)

    Google Scholar 

  16. van der Waerden, B.L.: Die Gruppentheoretische Methode in der Quantenmechanik. Springer, Berlin (1932)

    Book  Google Scholar 

  17. Paneitz, S.M., Segal, I.E.: Analysis in space-time bundles. I. General considerations and the scalar bundle. J. Funct. Anal. 47, 78–142 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Paneitz, S.M., Segal, I.E.: Analysis in space-time bundles. II. The spinor and form bundles. J. Funct. Anal. 49, 335–414 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Paneitz, S.M.: Analysis in space-time bundles. III. Higher spin bundles. J. Funct. Anal. 54, 18–112 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bhabha, H.J.: Relativistic Wave Equations for the Elementary Particles. Rev. Mod. Phys. 17, 200–216 (1945)

    Article  MathSciNet  ADS  Google Scholar 

  21. Gel’fand, I.M., Yaglom, A.M.: General relativistic-invariant equations and infinite-dimensional representations of the Lorentz group. Zh. Ehksp. Teor. Fiz. 18, 703–733 (1948)

    MathSciNet  Google Scholar 

  22. Majorana, E.: Teoria relativistica di particelle con momento intrinseco arbitrario. Nuovo Cimento 9, 335–344 (1932)

    Article  Google Scholar 

  23. Varlamov, V.V.: General Solutions of Relativistic Wave Equations. Int. J. Theor. Phys. 42(3), 583–633 (2003). arXiv:math-ph/0209036 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Varlamov, V.V.: Relativistic wavefunctions on the Poincaré group. J. Phys. A: Math. Gen. 37, 5467–5476 (2004). arXiv:math-ph/0308038 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Varlamov, V.V.: Maxwell field on the Poincaré group. Int. J. Mod. Phys. A. 20(17), 4095–4112 (2005). arXiv:math-ph/0310051 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Varlamov, V.V.: Relativistic spherical functions on the Lorentz group. J. Phys. A: Math. Gen. 39, 805–822 (2006). arXiv:math-ph/0507056 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Varlamov, V.V.: Spherical functions on the de Sitter group. J. Phys. A: Math. Theor. 40, 163–201 (2007). arXiv:math-ph/0604026 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Varlamov, V.V.: General Solutions of Relativistic Wave Equations II: Arbitrary Spin Chains. Int. J. Theor. Phys. 46(4), 741–805 (2007). arXiv:math-ph/0503058 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Petrov, A.Z.: Einstein Spaces. Pergamon Press, Oxford (1969)

    Book  MATH  Google Scholar 

  30. Varlamov, V.V.: About Algebraic Foundations of Majorana-Oppenheimer Quantum Electrodynamics and de Broglie-Jordan Neutrino Theory of Light. Annales de la Fondation Louis de Broglie 27, 273–286 (2002). arXiv:math-ph/0109024 (2001)

    MathSciNet  Google Scholar 

  31. Rumer, Yu.B.: Spinorial Analysis. URSS, Moscow (2010). [in Russian]

    Google Scholar 

  32. Mackey, G.: Induced Representations of Groups and Quantum Mechanics. Benjamin, New York (1968)

    MATH  Google Scholar 

  33. Schouten, J.A.: On the geometry of spin spaces. Indag. Math. 11, 3,4,5 (1949)

    Google Scholar 

  34. Rashevskii, P.K.: The Theory of Spinors. Uspekhi Mat. Nauk. 10, 3–110 (1955). English translation in Amer. Math. Soc. Transl. (Ser. 2). 6, 1 (1957)

    Google Scholar 

  35. Varlamov, V.V.: Discrete Symmetries and Clifford Algebras. Int. J. Theor. Phys. 40(4), 769–805 (2001). arXiv:math-ph/0009026 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rashevskii, P.K.: About Mathematical Foundations of Quantum Electrodynamics. Uspekhi Mat. Nauk. 13, 3–110 (1958)

    MathSciNet  Google Scholar 

  37. Varlamov, V.V. Group Theoretical Interpretation of the CPT-theorem. Mathematical Physics Research at the Cutting Edge. In: Benton, C.V. (ed.), pp. 51–100. Nova Science Publishers, New York (2004). math-ph/0306034 (2003)

  38. Varlamov, V.V.: Universal Coverings of Orthogonal Groups. AACA 14, 81–168 (2004). arXiv:math-ph/0405040 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  39. Varlamov, V.V.: The CPT Group in the de Sitter Space. Annales de la Fondation Louis de Broglie 29, 969–987 (2004). arXiv:math-ph/0406060 (2004)

    MathSciNet  Google Scholar 

  40. Varlamov, V.V.: CPT groups for spinor field in de Sitter space. Phys. Lett. B 631, 187–191 (2005). arXiv:math-ph/0508050 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Varlamov, V.V.: CPT groups of spinor fields in de Sitter and anti-de Sitter spaces. arXiv:1401.7723 [math-ph] (2014), to appear in Adv. Appl. Clifford Algebras

  42. Varlamov, V.V.: Cyclic structures of Cliffordian supergroups and particle representations of s p i n +(1,3). AACA 24, 849–874 (2014). arXiv:1207.6162 [math-ph] (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)

    MATH  Google Scholar 

  44. Dirac, P.A.M.: The principles of quantum mechanics. Clarendon Press, Oxford (1958)

    MATH  Google Scholar 

  45. Ashtekar, A., Lewandowski, L., Marolf, D., Mourão, J., Thiemann, T.: Quantization of diffeomorphism invariant theories of connections with local degrees of freedom. J. Math. Phys. 36, 6456–6469 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. Marolf, D.: Refined algebraic quantization: Systems with a single constraint. arXiv:gr-qc/9508015 (1995)

  47. Higuchi, A.: Quantum linearization instabilities of de Sitter spacetime. II. Class. Quant. Grav. 8, 1983–2004 (1991)

    Article  ADS  Google Scholar 

  48. Giulini, D., Marolf, D.: A Uniquiness Theorem for Constraint Quantization. Class. Quant. Grav. 16, 2489–2505 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. Marolf, D., Morrison, I.A.: Group Averaging for de Sitter free fields. Class. Quant. Grav. 26, 235–265 (2009)

    Google Scholar 

  50. Shvedov, O.Yu.: On Correspondence of BRST-BFV, Dirac and Refined Algebraic Quantization of Constrained Systems. Ann. Phys. 302, 2–21 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. Varlamov, V.V.: Group averaging for de Sitter free fields in terms of hyperspherical functions. Theor. Math. Phys. 164(3), 1230–1236 (2010). arXiv:1002.3436 [gr-qc] (2010)

    Article  MATH  Google Scholar 

  52. Gitman, D.M., Shelepin, A.L.: Fields on the Poincaré Group: Arbitrary Spin Description and Relativistic Wave Equations, vol. 40, pp 603–684 (2001). arXiv:hep-th/0003146 (2000)

  53. Gitman, D.M., Shelepin, A.L.: Field on the Poincaré group and quantum description of orientable objects. Eur. Phys. J. C. 61, 111–139 (2009). arXiv:0901.2537 [hep-th] (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Altaisky, M.V.: Quantum field theory without divergences. Phys. Rev. D. 81, 125003 (2010). arXiv:1002.2566 [hep-th] (2010)

    Article  ADS  Google Scholar 

  55. Altaisky, M.V., Kaputkina, N.E.: Continuous Wavelet Transform in Quantum Field Theory. Phys. Rev. D. 88, 025015 (2013). arXiv:1304.7177 [hep-th] (2013)

    Article  ADS  Google Scholar 

  56. Pauli, W.: Zur Quantenmechanik des magnetischen Elektrons. Z. Phys. 43, 601–623 (1927)

    Article  ADS  MATH  Google Scholar 

  57. Heisenberg, W.: Über den Bau der Atomkerne. Z. Phys. 77, 1–11 (1932)

    Article  MathSciNet  ADS  Google Scholar 

  58. Kemmer, N.: The charge-dependence of nuclear forces. Proc. Cambr. Phil. Soc. 34, 354–364 (1938)

    Article  ADS  Google Scholar 

  59. Wick, G.G., Wigner, E.P., Wightman, A.S.: Intrinsic Parity of Elementary Particles. Phys. Rev. 88, 101 (1952)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  60. Gell-Mann, M.: Symmetries of Baryons and Mesons. Phys. Rev. 125, 1067–1084 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. Ne’eman, Y.: Derivation of strong interactions from a gauge invariance. Nucl. Phys. 26, 222–229 (1961)

    Article  MathSciNet  Google Scholar 

  62. Rumer, Yu.B., Fet, A.I.: Theory of Unitary Symmetry. Nauka, Moscow (1970). [in Russian]

    Google Scholar 

  63. Okubo, S.: Note on Unitary Symmetry in Strong Interactions. Progr. Theor. Phys. 27, 949–966 (1962)

    Article  ADS  MATH  Google Scholar 

  64. Okubo, S., Ryan, C.: Quadratic mass formula in SU(3). Nuovo Cimento 34, 776–779 (1964)

    Article  Google Scholar 

  65. Zurek, W.H.: Decoherence, Einselection, and the Existential Interpretation (the Rough Guide). Phil. Trans. Roy. Soc. Lond. A 356, 1793–1820 (1998). arXiv:quant-th/9805065 (1998)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Siberian State Industrial University, Kirova 42, Novokuznetsk, 654007, Russia

    V. V. Varlamov

Authors
  1. V. V. Varlamov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Varlamov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Varlamov, V.V. Spinor Structure and Internal Symmetries. Int J Theor Phys 54, 3533–3576 (2015). https://doi.org/10.1007/s10773-015-2596-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-015-2596-0

Keywords

Search

Navigation