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PubblicazioneQuasi-Equivalence of Gaussian States and Energy Estimates for Functions of Modular Hamiltonians( 2025)To compare two Gaussian states of the Weyl-CCR algebra of a free scalar QFT we study three closely related perspectives: (i) quasi-equivalence of the GNS-representations, (ii) differences of the total energy (on some Cauchy surface), and (iii) differences between functions of the modular Hamiltonians. (For perspective (ii) we will only consider real linear free scalar quantum fields on ultrastatic spacetimes.) These three perspectives are known to be related qualitatively, due to work of Araki and Yamagami, Verch and Longo. Our aim is to investigate quantitative relations, including in particular estimates of differences between functions of modular Hamiltonians in terms of energy differences. E.g., for a suitable class of perturbations of the Minkowski vacuum state of a massive free scalar field, which have a positive energy density and a finite total energy E on some inertial time slice, the modular Hamiltonian K satisfies ‖1coshK2‖HS2≤8Em.
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PubblicazioneIntegrating evidence under uncertainty: from sensory perception to reward-guided predictions(SISSA, 2026-09-25)Adaptive behavior requires organisms to infer the state of the environment from information that is incomplete, noisy, and distributed across multiple sources and timescales. Although perception, learning, and decision-making are often studied as separate cognitive functions, they can also be viewed as manifestations of a common inferential problem: constructing and continuously updating internal models of the world to guide behavior under uncertainty. This thesis investigates how humans and rodents solve this problem in three complementary contexts. The first study examines temporal perception in tactile categorization tasks. Using behavioral experiments in rats and humans together with optogenetic manipulations of primary somatosensory cortex (vS1), we show that duration perception does not rely on a single fixed computational mechanism. Continuous tactile stimuli produce the well-established interaction between perceived intensity and perceived duration, consistent with a model in which duration is derived from the accumulation of sensory cortical activity. However, introducing salient onset and offset markers abolishes this interaction and renders duration judgments largely insensitive to manipulations of sensory cortical activity. These findings reveal the existence of at least two distinct strategies for temporal estimation whose recruitment depends on stimulus structure and sensory context. The second study investigates adaptive inference from reward-history information. Rats and humans performed a two-alternative foraging task in which reward locations followed probabilistic sequential patterns generated by a hidden Markov process. Both species successfully learned and exploited these regularities and adapted their behavior when environmental statistics changed. Across species, behavior exhibited a robust asymmetry between responses following rewarded and unrewarded outcomes, with rewarded outcomes exerting a stronger influence on subsequent choices. The third study examines how sensory evidence and reward-history information are combined during decision-making. Human participants performed a task in which tactile cues and probabilistic reward patterns jointly predicted reward location. Performance exceeded that observed when either information source was available alone, indicating that individuals integrate sensory and reward-based information rather than relying on either source independently. Psychophysical analyses further suggest that choices are guided by a unified decision variable combining current sensory evidence with beliefs derived from recent experience. Together, these studies demonstrate that adaptive behavior emerges from flexible mechanisms that integrate information across sensory signals, reward histories, and contextual factors. These findings support the view that perception, learning, and decision-making are unified by common principles of adaptive inference under uncertainty.
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PubblicazioneFourier transform of BV functions and applications( 2026)This paper investigates the relation between the Fourier transform of {\rm BV} (bounded variation) functions and their jump sets. We introduce the notion of $L^2$-jump product and obtain a weighted Plancherel identity for {\rm BV} functions. As a corollary, we get a characterization of sets of finite perimeter in terms of their Fourier transform. Moreover, we sharpen a result of Herz on the set-theoretic derivative of the Fourier transform of characteristic functions of sets. Last, we obtain sharp bounds on the quadratic discrepancy of {\rm BV} functions, and as a consequence, we generalize the classic estimates of Beck and Montgomery.
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PubblicazioneRepresentations of Atomic Environments and effects on Body-Ordering and Electrostatics on Machine-Learning Potentials(SISSA, 2026-10-07)Machine learning interatomic potentials (MLIPs) achieve DFT-level accuracy by replacing rigid functional forms with neural network architectures that allow for much more effective regression at the cost of less intuitive representations. This thesis explores how representations of atomic environments used by MLIPs behave in two regimes: at short range in how body-ordered expansions converge with respect to their basis functions, and at long range, in improving modeling the electronic response of systems sensitive to electrostatic interactions. We determined that Atomic Cluster Expansions (ACE) are only able to recover the DFT dimer curve if the self-interactions inherent in their efficient tensor product construction are filtered with a purification operator. Moreover, we highlight that the canonical many-body expansion has a non-monotonic and non-convergent scaling with higher body orders in trimers and systems with symmetric interactions. We also studied the ad hoc modeling of long range electrostatics through physically-motivated charge flow models, namely charge equilibration (QEq). We diagnosed how QEq-based treatments that parameterize through local atomic descriptors fail to accurately capture the correct dielectric response as diagnosed by the bare susceptibility $\bm{\chi}_0$, and extended the capabilities of our in-house code PANNA to include the split charge equilibration (SQE) model. Phonon dispersions from trained SQE models show signatures of finite longitudinal optical-transverse optical splitting in MgO where short range and charge equilibration counterparts fail to manifest, suggesting that the SQE model has the necessary ingredients to capture finite screening in ionic crystals. Both results show that physically-grounded constraints on MLIP representations recover behavior that unconstrained flexibility misses: recovering the canonical expansion through the purification methods as well as imposing restraints in charge flow through the bond topology in SQE.
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PubblicazioneReflection Positivity, Magnetic Lattice Laplacians and Emergent Phases in Lattice Fermionic Systems(SISSA, 2026-09-25)Emergent phenomena in interacting fermionic systems often arise as collective many-body effects whose formation can be understood by studying the interplay between geometry and symmetry. Understanding such phenomena from first principles remains one of the central challenges of Mathematical Physics, particularly in regimes where perturbative methods fail and non-trivial phases emerge through strongly correlated dynamics. This thesis investigates a class of lattice fermionic systems exhibiting \textit{Spontaneous Symmetry Breaking}, \textit{Flux ordering} and \textit{Topological Order}. Although the models considered belong to different physical frameworks, including Euclidean lattice field theories formulated in terms of Grassmann variables and Quantum Lattice Systems described by operators acting on Hilbert spaces, they share a common mathematical structure. In each case, geometric features encoded by lattice Dirac operators, magnetic fluxes, and discrete gauge fields interact with Reflection Positivity properties that allow for a rigorous non-perturbative analysis. A central role throughout the thesis is indeed played by \textit{Reflection Positivity} and one of its most striking consequences: the \textit{Chessboard Estimates}. These methods provide powerful tools for converting geometric information into quantitative bounds on energies, correlation functions, and phase stability. The first part of the thesis, based on \cite{fabbri2026chirallongrangeordereuclidean}, is devoted to lattice Gross–Neveu models in two dimensional Euclidean Spacetime. After a Hubbard–Stratonovich transformation, the fermionic theories are mapped to effective bosonic systems for which Reflection Positivity can be established. Combining Reflection Positivity methods with Peierls' Argument, we prove the existence of Chiral Long-Range Order for a class of lattice regularizations and obtain quantitative bounds relating the order parameter to the minimizers of the effective potential. The second part, based on \cite{GP}, focuses on fermions coupled to dynamical $\mathbb{Z}_2$-gauge fields. We prove the stability of the $\pi$-flux phase under gauging by showing that monopole excitations possess a strictly positive energy cost. These results imply the emergence of effective Dirac fermions at low energies and provide a rigorous characterization of the semimetallic phase of the model. Finally, we investigate the topological phase obtained by introducing a fermionic mass gap in the $\pi$-flux phase, following \cite{bachmann2026anyonspifluxphasefermionic}. We prove the existence of an almost four-fold degenerate low-energy sector on the torus, separated from the rest of the spectrum by a uniform gap, and construct quasi-local loop and string operators whose algebra exhibits the characteristic toric-code braiding phases. This provides a rigorous realization of topological order emerging from interacting fermions coupled to a dynamical gauge field. Taken together, these results illustrate how Reflection Positivity methods and spectral properties of the Magnetic Laplacian can be combined to obtain rigorous information on the phase diagram of interacting fermionic systems. They reveal a common mechanism underlying symmetry breaking, flux stabilization, and topological order across a variety of lattice models.