Abstract
We analyze the causal action principle for a system of relativistic fermions composed of massive Dirac particles and neutrinos. In the continuum limit, we obtain an effective interaction described by a left-handed, massive \({\mathrm{SU}}(2)\) gauge field and a gravitational field. The off-diagonal gauge potentials involve a unitary mixing matrix, which is similar to the Maki-Nakagawa-Sakata matrix in the standard model.
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Notes
- 1.
To be precise, this results holds for operators in the class \({\mathcal {P}}^f\) (see [F10, Definition 2.7]) if the fermionic operator is rescaled such that its trace is independent of f. In the formulation with local correlation matrices (see [F13, Sect. 3.2]) and under the trace constraint, the canonical embedding \(\mathbb {C}^f \hookrightarrow \mathbb {C}^{f+1}\) allows one to regard a system of f particles as a special system of \(f+1\) particles. Since varying within the set of \(f+1\)-particle systems gives more freedom, it is obvious that the action decreases if f gets larger.
- 2.
The C++ program class_commute and its computational output as well as the Mathematica worksheets were included as ancillary files to the arXiv submission arXiv:1211.3351 [math-ph].
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Finster, F. (2016). The Continuum Limit of a Fermion System Involving Neutrinos: Weak and Gravitational Interactions. In: The Continuum Limit of Causal Fermion Systems. Fundamental Theories of Physics, vol 186. Springer, Cham. https://doi.org/10.1007/978-3-319-42067-7_4
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DOI: https://doi.org/10.1007/978-3-319-42067-7_4
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