From Null Modes to Gauge Generators: A Bordered Faddeev--Jackiw Theory of Constraint Chains
Abstract
A null mode of the bordered Faddeev--Jackiw matrix is not, in general, a gauge generator. We determine exactly what the residual kernel encodes and identify the additional condition required for genuine gauge symmetry. Writing f(m)=f(0)&B −BT&0 and Γ=NTB, we prove dimker f(m)=dimkerΓT+dimkerΓ−rank(ΠMΠ), where M=BTf(0)+B and Π projects onto kerΓ. The two kernel contributions have distinct meanings: the first consists of null directions of f(0) tangent to the constraint surface, while the second counts independent first-class combinations. We then show that Γ is not the Dirac constraint matrix: it is the primary--secondary block of the reduced constraint matrix, with M providing the secondary--secondary block. For a residual null mode with nonzero multiplier component, we derive the associated two-step chain and prove that it closes to an exact gauge symmetry precisely when the chain contraction vanishes in the effective ring modulo the constraint ideal. A four-variable counterexample shows that the Faddeev--Jackiw stopping condition can hold while the system has no gauge freedom, and that the chain criterion rejects the false generators. Mechanical examples then exhibit the criterion and show that parameter strata can carry distinct gauge and dynamical content that is erased by premature cancellation.
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BibTeX
@misc{chanlopez2026from,
title = {{From Null Modes to Gauge Generators: A Bordered Faddeev--Jackiw Theory of Constraint Chains}},
author = {E. Chan-López},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27966},
eprint = {2609.27966},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27966},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Null Modes to Gauge Generators: A Bordered Faddeev--Jackiw Theory of Constraint Chains
AU - E. Chan-López
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27966
UR - https://arxiv.org/abs/2609.27966
AB - A null mode of the bordered Faddeev--Jackiw matrix is not, in general, a gauge generator. We determine exactly what the residual kernel encodes and identify the additional condition required for genuine gauge symmetry. Writing f^(m)=f^(0)&B −B^T&0 and Γ=N^TB, we prove dimker f^(m)=dimkerΓ^T+dimkerΓ−rank(ΠMΠ), where M=B^Tf^(0)+B and Π projects onto kerΓ. The two kernel contributions have distinct meanings: the first consists of null directions of f^(0) tangent to the constraint surface, while the second counts independent first-class combinations. We then show that Γ is not the Dirac constraint matrix: it is the primary--secondary block of the reduced constraint matrix, with M providing the secondary--secondary block. For a residual null mode with nonzero multiplier component, we derive the associated two-step chain and prove that it closes to an exact gauge symmetry precisely when the chain contraction vanishes in the effective ring modulo the constraint ideal. A four-variable counterexample shows that the Faddeev--Jackiw stopping condition can hold while the system has no gauge freedom, and that the chain criterion rejects the false generators. Mechanical examples then exhibit the criterion and show that parameter strata can carry distinct gauge and dynamical content that is erased by premature cancellation.
ER -