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Medicine show: A Calogero model with principal series states
by Tarek Anous, Jackson R. Fliss, Jeremy van der Heijden
Submission summary
| Authors (as registered SciPost users): | Tarek Anous |
| Submission information | |
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| Preprint Link: | scipost_202511_00071v1 (pdf) |
| Date submitted: | Nov. 27, 2025, 12:50 a.m. |
| Submitted by: | Tarek Anous |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The{\it Calogero model} is an interacting, $N$-particle, $\mathfrak{sl}(2,\mathbb{R})$-invariant quantum mechanics, whose Hilbert space is furnished by a tower of discrete series modules. The system enjoys both classical and quantum integrability at any $N$ and at any value of the coupling; this is guaranteed by the existence of $N$ mutually-commuting currents, one of them being the Hamiltonian. In this paper, we alter the Calogero model so that it may accommodate states in the unitary principal series irreducible representation of $\mathfrak{sl}(2,\mathbb{R})$. Doing so requires changing the domain of the quantum operators---a procedure which succeeds in preserving unitarity and $\mathfrak{sl}(2,\mathbb{R})$-invariance, but breaks the integrability of the theory. We explicitly solve the deformed model for $N=2,3$ and outline a procedure for solving the model at general $N$. We expect this deformed model to provide us with general lessons that carry over to other systems with states in the principal series, for example, interacting massive quantum field theories on de Sitter space.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
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Report
The starting point is the Calogero model of $N$ interacting particles on a line. This model exhibits SL(2,R) symmetry; however, it carries the discrete series representations only. To include states from the principal series, the authors propose a certain modification of this model. This modification is based on an analytic continuation to principial series of de Alfaro-Fubini-Furlan model, made in Ann. Phys. 367 (2016) 227 ([42] refers rather to the classical case). Such an approach seems interesting; however, the price of realising the principal series representation is that the coordinate $r$, which for the Calogero model is positive, now runs over the real axis (even for $N=2$). Thus, the physical meaning of the new model involves some explanations or at least comments on this point.
The work is quite well written; however, at times the authors could have highlighted the reviewed content a bit better (especially in the first part of the work). Finally, I think the title could be simplified (e.g. Calogero model and principal series states). In view of the above, I suggest a revision of the manuscript.
Recommendation
Ask for minor revision