Abstract
We reexamine the minimal coupling procedure in Hestenes’ geometric algebra formulation of the Dirac equation, where spinors are identified with even elements of the real Clifford algebra of spacetime. This point of view, as we argue, leads naturally to a non-Abelian generalisation of the electromagnetic gauge potential.
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Notes
The basis vectors \(e_1,\ldots ,e_n\) can be represented by matrices, with the ordinary matrix product taking the role of the Clifford product (cf. the Dirac algebra of \(\gamma \)-matrices). However, such representation misses the geometric origin and the geometrical significance of real Clifford algebras. Moreover, any matrix calculation can be performed (more efficiently) using the algebraic rules of Clifford algebras, and a wealth of identities derived therefrom [19].
A proof consists in expanding \(v=v_1 e_1 + v_2 e_2\), \(e^{\phi e_1e_2} = \cos \phi + e_1e_2 \sin \phi \), and in using the basic identities \(e_1e_2=-e_2e_1\) and \(e_1^2=e_2^2=1\).
Magnitude of a bivector \(B=a \wedge b\) is defined to be the area of the parallelogram spanned by the vectors a and b: \(|B|=\sqrt{a^2 b^2 - (a \cdot b)^2}\).
It is perhaps worth to note that a real spinor \(\Psi \) that satisfies \(\Psi {\tilde{\Psi }} \ne 0\) is invertible, and can always be decomposed, similarly to the case in three dimensions, as \(\Psi = \sqrt{\rho }\, e^{I\beta /2} R\), where \(\rho \) is a non-negative scalar, \(\beta \) is a scalar, and R is a rotor [8, Ch. 5.4].
The two-sided prescription has, in fact, a very good geometric meaning. Since
$$\begin{aligned} U (v_1 \ldots v_r) \widetilde{U} = (U v_1 \widetilde{U}) \ldots (U v_r \widetilde{U}) \end{aligned}$$for any r, it acts on a generic multivector by transforming each of its vector constituents according to Eq. (15).
Rotors in a generic space \(V^{p,q}\) are defined as Clifford products \(U=u_k \ldots u_1\), where k is even, \(u_j^2 = \pm 1\) for all \(j=1,\ldots ,k\), and \(U \widetilde{U} = 1\). They form the group \({{\,\mathrm{Spin}\,}}^+(p,q)\) [19, Ch. 3-8].
We adopt the notation \(\langle A + i\, B\rangle \equiv \langle A\rangle + i\, \langle B\rangle \) for any two elements of a real Clifford algebra.
In fact, we are working with a Clifford bundle [31, Ch. 6]—a fiber bundle with typical fibre \({\mathcal {C}}\ell (E^{1,3})\), whose bundle maps are the isometries of \({\mathcal {C}}\ell (E^{1,3})\).
For a general introduction to gauge theories (in traditional matrix formulation) see, e.g., Ref. [1].
By not identifying \(\omega _\mu \) with \({\mathcal {A}}_\mu \) we are allowing more freedom to the form of the gauge fields than “necessary”. This might seem inappropriate, but is in fact common. The principle of local gauge invariance necessitates only gauge potentials of the form of pure gauges, i.e., it does not force the field strengths to be nonzero. And yet, we allow (or even demand) the gauge fields to have nontrivial field strengths—simple because they are observed in nature.
References
Baez, J.C., Muniain, J.P.: Gauge Fields, Knots, and Gravity. World Scientific, Singapore (1994)
Boi, L.: Clifford geometric algebras, spin manifolds, and group actions in mathematics and physics. Adv. Appl. Clifford Algebras 19, 611 (2009)
Brauer, R., Weyl, H.: Spinors in n dimensions. Am. J. Math. 57, 425–449 (1935)
Chevalley, C.: The Algebraic Theory of Spinors. Columbia University Press, New York (1954)
Clifford, W.K.: Applications of Grassmann’s extensive algebra. Am. J. Math. 1(4), 350–358 (1878)
Coquereaux, R.: Clifford algebras, spinors and fundamental interactions: twenty years after. Adv. Appl. Clifford Algebras 19, 673 (2009)
Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. A Math. Phys. Eng. Sci. 117(778), 610–624 (1928)
Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2007)
Doran, C., Lasenby, A., Gull, S.: States and operators in the spacetime algebra. Found. Phys. 23(9), 1239–1264 (1993)
Doran, C., Lasenby, A., Gull, S., Somaroo, S., Challinor, A.: Spacetime algebra and electron physics. Adv. Imaging Electron Phys. 95, 271–386 (1996). arXiv:quant-ph/0509178
Feynman, R.P.: Quantum Electrodynamics (Frontiers in Physics). Westview Press, Boulder (1998)
Francis, M.R., Kosowski, A.: The construction of spinors in geometric algebra. Ann. Phys. 317(2), 383–409 (2005). arXiv:math-ph/0403040
Greiner, W.: Relativistic Quantum Mechanics. Wave Equations. Springer, Berlin (2000)
Hamilton, W.R.: On quaternions, or on a new system of imaginaries in algebra. Philos. Mag. 25(3), 489–495 (1844)
Hestenes, D.: Real spinor fields. J. Math. Phys. 8(4), 798–808 (1967)
Hestenes, D.: Local observables in quantum theory. Am. J. Phys. 39, 1028–1038 (1971)
Hestenes, D.: Observables, operators, and complex numbers in the Dirac theory. J. Math. Phys. 16, 556–572 (1975)
Hestenes, D.: Space-time structure of weak and electromagnetic interactions. Found. Phys. 12, 153–168 (1982)
Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Springer, Berlin (1987)
Hestenes, D.: Spacetime physics with geometric algebra. Am. J. Phys. 71, 6 (2003)
Hestenes, D.: Gauge gravity and electroweak theory. In: Kleinert, H., Jantzen, R.T., Ruffini, R. (eds.) Proceeding of the 11th Marcel Grossmann Meeting on General Relativity. World Scientific (2008). arXiv:0807.0060
Hestenes, D.: Space-Time Algebra, 2nd edn. Springer, Berlin (2015)
Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980)
Lachieze-Rey, M.: Spin and Clifford algebras, an introduction. Adv. Appl. Clifford Algebras 19, 687 (2009)
Lasenby, A.N., Hobson, M.P.: Scale-invariant gauge theories of gravity: theoretical foundations. J. Math. Phys. 57, 092505 (2016)
Lasenby, A., Doran, C., Gull, S.: Gravity, gauge theories and geometric algebra. Philos. Trans. R. Soc. A 356, 487–582 (1998)
Lounesto, P.: Clifford Algebras and Spinors, 2nd edn. Cambridge University Press, Cambridge (2001)
Radford, C.J.: Localized solutions of the Dirac–Maxwell equations. J. Math. Phys. 37, 4418 (1996). arXiv:hep-th/9510065
Ramond, P.: Field Theory: A Modern Primer, 2nd edn. Westview Press, Boulder (1997)
Rausch de Traubenberg, M.: Clifford Algebras in Physics. Adv. Appl. Clifford Algebras 19, 869 (2009)
Rodrigues Jr., W.A., Capelas de Oliveira, E.: The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach. Springer, Berlin (2007)
Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984)
Acknowledgements
The author would like to acknowledge stimulating discussions with Leslaw Rachwal, Josef Schmidt, Jan Vysoký, and Alejandro Perez. This work was supported by the Czech Science Foundation, grant number GA ČR 19-15744Y.
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This article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe Kähler, Rafal Ablamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G. Stacey Staples, Wei Wang.
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Zatloukal, V. Real Spinors and Real Dirac Equation. Adv. Appl. Clifford Algebras 32, 45 (2022). https://doi.org/10.1007/s00006-022-01236-w
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DOI: https://doi.org/10.1007/s00006-022-01236-w