[Paper Review] Standard-model symmetry in complexified spacetime algebra

José B. Almeida|ArXiv.org|Jul 11, 2003

Algebraic and Geometric Analysis8 references3 citations

TL;DR

This paper demonstrates that the Standard Model's gauge symmetry group $U(1)\otimes SU(2)\otimes SU(3)$ can be embedded within complexified spacetime algebra (CSTA), a 4-dimensional geometric algebra with complex coefficients isomorphic to $\mathcal{G}_{1,4}$. Using Dirac-Pauli matrix representations, it shows that four distinct copies of the full gauge group exist within CSTA, providing a geometric unification of the Standard Model symmetries without extra dimensions.

ABSTRACT

Complexified spacetime algebra is defined as the geometric (Clifford) algebra of spacetime with complex coefficients, isomorphic $\mathcal{G}_{1,4}$. By resorting to matrix representation by means of Dirac-Pauli gamma matrices, the paper demonstrates isomorphism between subgroups of CSTA and SU(3). It is shown that the symmetry group of those subgroups is indeed $U(1) \otimes SU(2) \otimes SU(3)$ and that there are 4 distinct copies of this group within CSTA.

Motivation & Objective

• To embed the full Standard Model gauge group $U(1)\otimes SU(2)\otimes SU(3)$ within a geometric algebra framework.
• To demonstrate that complexified spacetime algebra (CSTA), isomorphic to $\mathcal{G}_{1,4}$, provides a natural geometric setting for the Standard Model symmetries.
• To show that $SU(3)$ symmetry can be realized within a 4-dimensional complexified Clifford algebra, avoiding the need for higher-dimensional or oversized algebras.
• To identify and construct four distinct geometric copies of the full gauge group within CSTA, each corresponding to different subalgebra embeddings.
• To unify the description of $U(1)$, $SU(2)$, and $SU(3)$ symmetries using matrix representations of the Dirac-Pauli algebra within a single geometric algebra framework.

Proposed method

• Uses complexified spacetime algebra (CSTA), defined as the geometric algebra $\mathcal{G}_{1,3}$ with complex coefficients, isomorphic to the real $\mathcal{G}_{1,4}$ algebra.
• Employs the standard Dirac-Pauli matrix representation of STA and extends it to CSTA by allowing complex coefficients in multivector components.
• Represents $SU(3)$ generators using $4\times4$ matrices derived from the Gell-Mann matrices multiplied by the imaginary unit $j$, embedded in the $4\times4$ matrix representation of CSTA.
• Constructs $SU(2)$ generators via the Pauli matrices $\hat{\sigma}_k$ embedded in the same matrix representation, with $U(1)$ represented by complex scalars.
• Applies right permutations of the $SU(3)$ generator matrices to generate four distinct copies of the full gauge group within the same algebraic structure.
• Verifies the group structure by confirming that the $\hat{\lambda}_a$ matrices satisfy the standard $SU(3)$ commutation relations with structure constants $f_{abc}$.

Experimental results

Research questions

• RQ1 Can the full Standard Model gauge group $U(1)\otimes SU(2)\otimes SU(3)$ be geometrically realized within a 4-dimensional complexified spacetime algebra?
• RQ2 How can $SU(3)$ symmetry be embedded in a geometric algebra framework that avoids the need for 7-dimensional or higher-dimensional spaces?
• RQ3 What is the number and nature of distinct copies of the gauge group $U(1)\otimes SU(2)\otimes SU(3)$ that can coexist within CSTA?
• RQ4 Do the matrix representations of the Gell-Mann matrices (with imaginary unit $j$) in the $4\times4$ Dirac-Pauli basis satisfy the correct $SU(3)$ algebraic relations within CSTA?
• RQ5 Can the unification of $U(1)$, $SU(2)$, and $SU(3)$ symmetries be achieved without introducing extra dimensions or auxiliary spaces?

Key findings

• The complexified spacetime algebra (CSTA) provides a geometric framework where the full Standard Model gauge group $U(1)\otimes SU(2)\otimes SU(3)$ is isomorphic to subgroups of the algebra.
• Four distinct copies of the gauge group $U(1)\otimes SU(2)\otimes SU(3)$ are explicitly constructed within CSTA through different embeddings of the $SU(3)$ generators.
• The $SU(3)$ generators are realized as $4\times4$ matrices derived from the Gell-Mann matrices multiplied by $j$, and they satisfy the standard $SU(3)$ commutation relations with structure constants $f_{abc}$.
• The $SU(2)$ symmetry is embedded via the Pauli matrices $\hat{\sigma}_k$, which are part of the standard STA matrix representation.
• The $U(1)$ symmetry is represented by complex scalars within the algebra, consistent with the complexification of coefficients in CSTA.
• The construction avoids the need for higher-dimensional geometric algebras (e.g., $\mathcal{G}_{4,3}$ or $\mathcal{G}_{7}$), providing a more compact and geometrically interpretable unification of the Standard Model symmetries.