arXiv subjects

Ilaria Svampa

Publications and source records attributed to Ilaria Svampa.

7 recordsLinked to original sources

arXiv2026

A dynamical approach to Schur's Theorem

A classical result of Schur of 1904 shows that an abstract group with finite central quotient has finite derived subgroup. Schur's Theorem has many important consequences and generalizations, which have been extensively investigated in the literature. We develop a new dynamical interpretation of Schur's Theorem for locally compact groups, using the notion of topological entropy of Adler, Konheim and McAndrew. We first consider groups with compact central quotient, introduced and called $\mathsf{Z}$-groups by Grosser and Moskowitz in the 1960s, proving that if $G$ is a connected group such that $G/Z(G)$ is compact and with continuous endomorphisms of finite topological entropy, then also $\overline{[G,G]}$ is compact and with continuous endomorphisms of finite topological entropy. The connectedness assumption is essential, since its absence allows us to construct a profinite group as counterexample. Furthermore, we study the Heisenberg groups $\mathbb{H}_n(R)$ on certain locally compact rings $R$ as a framework in which a dynamical Schur-Type Theorem persists, even though the central quotient need not be compact. In particular, we find new formulas for the $p$-rank of these Heisenberg groups.

math.GR↗

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

Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates

In the context of $p$-adic quantum mechanics, we investigate composite systems of $p$-adic qubits and $p$-adically controlled quantum logic gates. We build on the notion of a single $p$-adic qubit as a two-dimensional irreducible representation of the compact $p$-adic special orthogonal group SO(3)$_p$. We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3)$_p$ mod $p^k$. Then, we tackle the problem of $p$-adic qubit composition and entanglement, fundamental for a $p$-adic formulation of quantum information processing. We classify the representations of SO(3)$_p$ mod $p$, and analyse tensor products of two $p$-adic qubit representations lifted from SO(3)$_p$ mod $p$. We solve the Clebsch-Gordan problem for such systems, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems. For $p=3$, we construct a set of gates from $4$-dimensional irreducible representations of SO(3)$_p$ mod $p$ that we prove to be universal for quantum computation.

quant-ph↗

arXiv2024

Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces

We determine the Haar measure on the compact $p$-adic special orthogonal groups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting the machinery of inverse limits of measure spaces, for every prime $p>2$. We characterise $\mathrm{SO}(d)_p$ as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each $\mathrm{SO}(d)_p$. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on $\mathrm{SO}(d)_p$. Our results pave the way towards the study of the irreducible projective unitary representations of the $p$-adic rotation groups, with potential applications to the recently proposed $p$-adic quantum information theory.

math-ph↗

arXiv2024

Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups

We provide a general expression of the Haar measure $-$ that is, the essentially unique translation-invariant measure $-$ on a $p$-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the $p$-adic special orthogonal groups in dimension two, three and four (for every prime number $p$). In particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$ and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain $p$-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the $p$-adic special orthogonal groups, with potential applications in $p$-adic quantum mechanics and in the recently proposed $p$-adic quantum information theory.

math-ph↗

arXiv2023

Geometry of the $p$-adic special orthogonal group $SO(3)_p$

We derive explicitly the structural properties of the $p$-adic special orthogonal groups in dimension three, for all primes $p$, and, along the way, the two-dimensional case. In particular, starting from the unique definite quadratic form in three dimensions (up to linear equivalence and rescaling), we show that every element of $SO(3)_p$ is a rotation around an axis. An important part of the analyis is the classification of all definite forms in two dimensions, yielding a description of the rotation subgroups around any fixed axis, which all turn out to be abelian and parametrised naturally by the projective line. Furthermore, we find that for odd primes $p$, the entire group $SO(3)_p$ admits a representation in terms of Cardano angles of rotations around the reference axes, in close analogy to the real orthogonal case. However, this works only for certain orderings of the product of rotations around the coordinate axes, depending on the prime; furthermore, there is no general Euler angle decomposition. For $p=2$, no Euler or Cardano decomposition exists.

math.NT↗

arXiv2022

An approach to $p$-adic qubits from irreducible representations of $SO(3)_p$

We introduce the notion of $p$-adic quantum bit ($p$-qubit) in the context of the $p$-adic quantum mechanics initiated and developed by Volovich and his followers. In this approach, physics takes place in three-dimensional $p$-adic space rather than Euclidean space. Based on our prior work describing the $p$-adic special orthogonal group, we outline a programme to classify its continuous unitary projective representations, which can be interpreted as a theory of $p$-adic angular momentum. The $p$-adic quantum bit arises from the irreducible representations of minimal nontrivial dimension two, of which we construct examples for all primes $p$.

math-ph↗