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The Riemann Curvature Tensor and Higgs Scalar Field within CAM Theory

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Abstract

The composition algebra based methodology (CAM) (Wolk in Pap Phys 9:090002, 2017, Phys Scr 94:025301, 2019, Adv Appl Clifford Algebras 27(4):3225, 2017, J Appl Math Phys 6:1537, 2018, Phys Scr 94:105301, 2019) has previously been shown to generate the pre-Higgs Standard Model Lagrangian. In this paper the symmetry of general covariance is incorporated into CAM. The Riemann curvature tensor thereby arises, from which gravity-field Lagrangians are constructed. A Higgs-like scalar field coupled to the spacetime metric tensor also manifests.

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Notes

  1. The Cayley–Dickson composition algebras \(\left\{ \mathbb {K},{{{\mathbb {K}}}^{\prime }}\right\} \), with the Hurwitz algebras \(\mathbb {K}=\left\{ \mathbb {R},\mathbb {C},{{\mathbb {H}}},\mathbb {O}\right\} \) and \({{\mathbb {K}}}^{\prime }\) being the unique split versions of \(\mathbb {K}\) [14]. The only Cayley–Dickson algebras which are composition algebras (algebras \(\mathbb {A}\) such that for any two elements the norm of their product equals the product of their norms [30, 45]: \(\left\| xy\right\| =\left\| x\right\| \left\| y\right\| \;\forall x,y\in \mathbb {A}\)) reside in 1, 2, 4 and 8 dimensions, corresponding to \(\mathbb {K}=\left\{ \mathbb {R},\mathbb {C},\mathbb {H},\mathbb {O}\right\} \) [30, 45] and their split versions \(\mathbb {K}^{\prime }=\left\{ \mathbb {C}^{\prime },\mathbb {H}^{\prime },\mathbb {O}^{\prime }\right\} \) [12, 48]. Only the \(\mathbb {K}\) algebras are division algebras (composition algebras without zero divisors) [12, 48]. However, since the non-division, composition algebra \(\mathbb {O}^{\prime }\) was also needed along with the \(\mathbb {K}\) algebras in developing \(SU(3)_{c}\) the term CAM is the appropriate designation for the formalism in general.

  2. Ref. [59], pp. 91–92.

  3. Also termed the Riemann curvature operator \(\mathscr {R}_{\mu \nu }\) with Levi-Civita connection: \(\left( \mathscr {R}_{\mu \nu },\nabla ^{\varvec{g}}\right) \rightarrow \left[ \nabla _{\mu },\nabla _{\nu }\right] \) (Ref. [34], Eq. (10.2)).

  4. For example, See Ref. [66], Sec. 2.2 regarding SU(2) force field operators.

  5. See Ref. [64] and citations therein on \(\mathcal {Z}\).

  6. Ref. [59], Eq. (6.6.1).

  7. A consideration separate from our knowledge of the necessary form of the Lagrangian shows that use of \(g^{\nu \omega }\) in Eq. (23) is correct. In the U(1) formalism the electromagnetic potential field \(A_{\mu }\) was involved in the analogous operator equation [63]. Since the tensor \(g_{\mu \nu }\) is the gravitational potential field in analogy with \(A_{\mu }\) [6, 24, 39, 50, 59], we may have conjectured that \(g_{\mu \nu }\) would be the entity involved in the operation, just as \(A_{\mu }\) was for U(1).

  8. See, e.g., Ref. [59], Sec. 7.1 & 12.1–12.4; Ref. [6], Ch. X & XII.

  9. At this stage in the development of SU(2) we had \(G_{\mu \nu }\) and had \(H_{\mu \nu }\) and \(F_{\mu \nu }\) for the respective U(1) and SU(3) gauge groups [63, 64, 66], thereafter constructing the force field Lagrangians with Eq. (31).

  10. Ref. [34], Sec. 11.2.

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    Brian Jonathan Wolk

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Wolk, B.J. The Riemann Curvature Tensor and Higgs Scalar Field within CAM Theory. Adv. Appl. Clifford Algebras 30, 4 (2020). https://doi.org/10.1007/s00006-019-1033-5

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