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Classical-Quantum dualities
Full text linkIn quantum field theory, the generating functional is the functional Fourier transform of e^iS, with S the classical action. Therefore, applying the inverse transform, a classical object, e^iS, can be expressed in terms of a path integral over quantum objects. This suggests that it may be possible to formulate a theory where classical mechanics is seen as the ''exact'' result and quantum mechanics is recovered in a ''quantum limit'', dual to the usual classical limit.
The aim of this thesis is to explore this idea using a Hilbert space formulation for classical mechanics. In particular, we analyze the Koopman-von Neumann operatorial formulation of classical mechanics and derive from it a path integral representation. Then, we review the Wigner-Weyl (WW) formalism, usually applied in the formulation of quantum mechanics in phase space, and use it to reformulate classical mechanics in the quantum Hilbert space.
Using the WW formalism, we derive two possible realizations of the quantum limit.
First, we show that the role of Planck's constant h can be reversed because the limit h->0 can be interpreted as a limit in which the classical algebraic structure reduces to the quantum non-commutative structure. This suggests to interpret, at a dynamical level, the limit h->0 as a classical-quantum interface and to to consider the two theories, classical and quantum, on equal footing.
Then, we show that applying the WW formalism to the classical Liouville equation, which governs the dynamics of classical statistical ensembles, it is possible to consider a ''local quantum approximation'' where the classical dynamics reduces to the quantum one if the states are sufficiently localized. This gives an alternative quantization procedure, where quantum dynamics is derived using a limit process
Circuit Complexity in presence of a defect
No full textWe study circuit complexity for the ground state of a harmonic chainwith defect in 1+1 dimensions, choosing as a reference state,
the ground state of the homogeneous chain. By employing the covariance matrix for-malism, we compute numerically C2 complexity and
extract its divergence pattern in the continuum limit. We find that, upon a suitable choice of the coordinates,
C2 complexity displays a logarithmic divergence. Finally, we compare our results with the existing ones for the entanglement entropy
of half chain and the holographic complexity in the presence of a defect
Investimenti in tecnologie digitali: processi di trasformazione per diventare imprese 4.0
No full textIl seguente elaborato si pone l'obiettivo di indagare il fenomeno industria 4.0, focalizzando l'attenzione sulle principali tecnologie digitali e sui processi di trasformazione adottati dalle imprese per diventare 4.0. Sono inoltre presentati un'analisi statistica del grado di diffusione delle tecnologie digitali nelle imprese manufatturiere italiane e il caso Sirmax, multinazionale veneta che ha intrapreso vari progetti di trasformazione digitale nel reparto operations
Stochastic block model with k communities: a spectral algorithm with optimal recovery
Full text linkThis thesis discusses a spectral method to solve the Community Detection
problem in a Stochastic Block Model (SBM) with k communities. Firstly, we
analyze the easier case of a dense graph with k=2 communities and we
highlight the fundamental ideas of the spectral method. Then, we study a
sparse SBM with k=2 communities: due to the low edge densities, the problem
requires a different approach, and we prove the correctness of a spectral
method under optimal assumptions. Finally, the main result of this thesis
deals with a sparse SBM with k>2 communities: we extend the method
developed for k=2 adding several steps and we prove its correctness and the
optimality on the assumptions