Superconducting Pairing Symmetry on Geometric-Algebra Foundations via the Scalar-Projection Method
Abstract
We formulate the low-energy theory of superconducting pairing in the geometric algebra Cl3,0. Conventional BCS mean-field theory and its extensions (s-, p-, d-wave, etc.) have algebraic counterparts in an eight-dimensional real multivector space. The construction is a two-level projection: matrix blocks are represented as Clifford multivectors, and their grade-0 projected geometric products extract the scalar blocks from which Hamiltonians are built. The result is an algebraic reconstruction, physically equivalent to the standard complex-matrix formulation. In this reconstruction, the dichotomy of pairing channels acquires an algebraic root: it follows from the fermionic identity B†(V)=−B†(VT), and the Fierz rearrangement is realized by the map Φ:G↦ Ge31 (with e312=−1). This yields a kernel/matrix dictionary: at the effective-kernel level grades 0⊕3 correspond to spin singlets and grades 1⊕2 to triplets; at the pairing-matrix level the singlet plane is span{e2,e31}. The BCS limit has a double grade identity: kernel grade 0 (scalar glue) and vertex grade 2, a bivector (iσ2↔ e31). We state three algebraic results: (i) a decomposition theorem for BdG quantum geometry with a null-texture criterion, applied to the chiral-state controversy in CsV3Sb5; (ii) a Fierz sign duality in the four-dimensional Minkowski algebra, with a chirality-reversal extension to two-node Weyl-semimetal models; (iii) an error bound and condition-number structure for tomographic reconstruction under a linear-response probe model. Illustrations include a glue-generation mechanism, a one-loop algebraic renormalization flow whose feeding constant coincides in magnitude with the su(2) Casimir, closed-form μ=0 inter-node spectra, and a UTe2 two-phase analysis.
Related
- Slave-boson Formalism for Superconducting Pairing at Strong Coupling
- Generalized conditions for odd-frequency pairing in superconducting systems
- Competition between on-site and off-site pairing phases and surface spectra in superconducting wallpaper fermion systems
- Influence of pairing and correlations on transport properties in three-terminal superconducting nanostructure
- Asymmetric Little--Parks Oscillations as a Probe of Coupled Pairing Channels in Surface-Superconducting \texorpdfstring{PtBi2}{PtBi2}
BibTeX
@misc{dai2026superconducting,
title = {{Superconducting Pairing Symmetry on Geometric-Algebra Foundations via the Scalar-Projection Method}},
author = {Youping Dai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33806},
eprint = {2609.33806},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33806},
}
RIS
TY - UNPB
TI - Superconducting Pairing Symmetry on Geometric-Algebra Foundations via the Scalar-Projection Method
AU - Youping Dai
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33806
UR - https://arxiv.org/abs/2609.33806
AB - We formulate the low-energy theory of superconducting pairing in the geometric algebra Cl_3,0. Conventional BCS mean-field theory and its extensions (s-, p-, d-wave, etc.) have algebraic counterparts in an eight-dimensional real multivector space. The construction is a two-level projection: matrix blocks are represented as Clifford multivectors, and their grade-0 projected geometric products extract the scalar blocks from which Hamiltonians are built. The result is an algebraic reconstruction, physically equivalent to the standard complex-matrix formulation. In this reconstruction, the dichotomy of pairing channels acquires an algebraic root: it follows from the fermionic identity B^†(V)=−B^†(V^T), and the Fierz rearrangement is realized by the map Φ:G↦ Ge_31 (with e_31^2=−1). This yields a kernel/matrix dictionary: at the effective-kernel level grades 0⊕3 correspond to spin singlets and grades 1⊕2 to triplets; at the pairing-matrix level the singlet plane is span{e_2,e_31}. The BCS limit has a double grade identity: kernel grade 0 (scalar glue) and vertex grade 2, a bivector (iσ_2↔ e_31). We state three algebraic results: (i) a decomposition theorem for BdG quantum geometry with a null-texture criterion, applied to the chiral-state controversy in CsV_3Sb_5; (ii) a Fierz sign duality in the four-dimensional Minkowski algebra, with a chirality-reversal extension to two-node Weyl-semimetal models; (iii) an error bound and condition-number structure for tomographic reconstruction under a linear-response probe model. Illustrations include a glue-generation mechanism, a one-loop algebraic renormalization flow whose feeding constant coincides in magnitude with the su(2) Casimir, closed-form μ=0 inter-node spectra, and a UTe_2 two-phase analysis.
ER -