arXiv subjects

Lorenzo Guglielmi

Publications and source records attributed to Lorenzo Guglielmi.

3 recordsLinked to original sources

arXiv2026

The Haar measure of the $p$-adic rotation group $\textrm{SO}(3)_p$ via nautical angles

We study the explicit construction of the Haar measure on the compact $p$-adic rotation group $\textrm{SO}(3)_p$ by nautical (Cardano) parametrization. Exploiting its topological group isomorphism with $\mathbb{H}_p^\times/\mathbb{Q}_p^\times$ of $p$-adic quaternions modulo scalars, we derive the corresponding change of variables formulas and compute the associated Jacobian in the $p$-adic setting, which we combine with the known Haar measure on the multiplicative group of $p$-adic quaternions $\mathbb{H}_p^\times$. This yields an explicit formula for the normalized Haar measure on $\textrm{SO}(3)_p$ in nautical coordinates, with a factorized density in the three angles. Our construction provides a concrete tool suited for applications of non-Archimedean models where an explicit angular description of invariant integration is required.

math-ph↗

arXiv2026

The tensor product of p-adic Hilbert spaces

In the framework of quantum mechanics over a quadratic extension of the ultrametric field of p-adic numbers, we introduce a notion of tensor product of p-adic Hilbert spaces. To this end, following a standard approach, we first consider the algebraic tensor product of p-adic Hilbert spaces. We next define a suitable norm on this linear space. It turns out that, in the p-adic framework, this norm is the analogue of the projective norm associated with the tensor product of real or complex normed spaces. Eventually, by metrically completing the resulting p-adic normed space, and equipping it with a suitable inner product, we obtain the tensor product of p-adic Hilbert spaces. That this is indeed the correct p-adic counterpart of the tensor product of complex Hilbert spaces is also certified by establishing a natural isomorphism between this p-adic Hilbert space and the corresponding Hilbert-Schmidt class. Since the notion of subspace of a p-adic Hilbert space is highly nontrivial, we finally study the tensor product of subspaces, stressing both the analogies and the significant differences with respect to the standard complex case. These findings should provide us with the mathematical foundations necessary to explore quantum entanglement in the p-adic setting, with potential applications in the emerging field of p-adic quantum information theory.

math-ph↗

arXiv2018

Symplectic invariants for parabolic orbits and cusp singularities of integrable systems with two degrees of freedom

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type.

math.SG↗