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Carroll Fermions
by Eric A. Bergshoeff, Andrea Campoleoni, Andrea Fontanella, Lea Mele, Jan Rosseel
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Submission summary
| Authors (as registered SciPost users): | Andrea Campoleoni |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202312_00031v2 (pdf) |
| Date submitted: | 2024-04-16 11:16 |
| Submitted by: | Campoleoni, Andrea |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Using carefully chosen projections, we consider different Carroll limits of relativistic Dirac fermions in any spacetime dimensions. These limits define Carroll fermions of two types: electric and magnetic. The latter type transforms as a reducible but indecomposable representation of the Carroll group. We also build action principles for all Carroll fermions we introduce; in particular, in even dimensions we provide an action principle for a minimal magnetic Carroll fermion, having the same number of components as a Dirac spinor. We then explore the coupling of these fermions to magnetic Carroll gravity in both its first-order and second-order formulations.
Author comments upon resubmission
First of all we wish to thank the Referees for the useful comments. We added some clarifications according to the suggestions by Referee 2, that we detail in the following list of changes. We hope that these additional clarifications will make our paper suitable for publication.
List of changes
- Below (2.2) we defined $\Gamma_{AB}$ and below (2.3) we stressed that in our approach we do not rescale gamma matrices.
- Below (2.6) we clarified that we used $\Gamma_0$ to define a projection operator because our aim was to introduce a projection allowing us to distinguish between rotations and boosts. This was motivated by the desire to rescale fields in such a way to preserve a non-trivial action of boosts. We also modified the paragraph below (2.9) to recall how we obtained an indecomposable representation of the homogeneous Carroll group even if we worked with the usual relativistic Carroll algebra.
- The electric limit can be actually defined in a simpler way by taking the $\tilde{c} \to \infty$ limit of Dirac's Lagrangian after rescaling the fields by as $\psi = \tilde{c}^{-\frac{1}{2}} \Psi$. We stressed at the beginning of section 2.2 that our choice of starting from an off-diagonal Lagrangian was motivated by the desire to describe both an electric and a magnetic limit in a unifying framework. We also stressed in footnote 10 and in the paragraph below (2.17) that our choice of keeping only the $+$ components in the electric limit is purely conventional and that the truncation (with the corresponding reduction in the number of degrees of freedom) has been included only to avoid kinetic terms with different signs (see also the new comment below (2.15)). The latter feature has been introduced by working with a unifying Lagrangian and it is not a general feature of the electric limit.
- As stressed in our answer to point 1, in our work we do not modify the Clifford algebra. An alternative approach to Carroll fermions based on degenerate Carroll algebras, inducing a contraction of $so(1,d)$ has been considered in the literature and we mentioned it in footnote 1.
- We mentioned in footnote 12 that the truncation defining the magnetic Carroll Lagrangian naively breaks parity, thus explaining the option to add mass terms with $\Gamma_\star$. We defer to further work an analysis of possible modifications of the action of parity on Carrollian spinors that might restore parity.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2024-4-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202312_00031v2, delivered 2024-04-30, doi: 10.21468/SciPost.Report.8955
Strengths
1. Well-written and easily understandable paper.
2. Technical details are also straight forward.
Weaknesses
1. No practical reason for understanding Carroll fermions have been mentioned.
Report
Dear Editor,
The resubmission of $\textit{Carroll Fermions}$ by E.A.Bergshoeff, A.Campoleoni, A.Fontanella, L.Mele and J.Rosseel addresses most of the points raised in my first review. However, a few clarifications are still required.
1. As shown in the paper arXiv: 2109.06708, for any Carroll invariant theory, the energy density $\mathcal{E}(x)$ must satisfy the following property: $[\mathcal{E}(x),\mathcal{E}(x')]=0$. I can see this is obvious for the electric Carroll fermion. But how does this hold for magnetic ones?
2. As the authors pointed out, there is nothing special with $\psi_+$ in the electric limit, one can also do the same with $\psi_-$. But my question was, as both together can't be present in the electric Lagrangian, then it's obvious that degrees of freedom are getting halved. What is the physical reason behind this?
3. The authors have mentioned "...the purpose of having introduced the parent Lagrangian (2.10) is to unify the electric limit with the magnetic limit discussed below". However if one performs c-expansion from relativistic theory, electric and magnetic theory appear at different order, the latter comes with additional constraints as done in the paper arXiv: 2110.02319. So how can one unify the limits ?
Recommendation
Ask for minor revision