Symmetries and currents of the ideal and unitary Fermi gases

The maximal algebra of symmetries of the free single-particle Schroedinger equation is determined and its relevance for the holographic duality in non-relativistic Fermi systems is investigated. This algebra of symmetries is an infinite dimensional extension of the Schroedinger algebra, it is isomorphic to the Weyl algebra of quantum observables, and it may be interpreted as a non-relativistic higher-spin algebra. The associated infinite collection of Noether currents bilinear in the fermions are derived from their relativistic counterparts via a light-like dimensional reduction. The minimal coupling of these currents to background sources is rewritten in a compact way by making use of Weyl quantisation. Pushing forward the similarities with the holographic correspondence between the minimal higher-spin gravity and the critical O(N) model, a putative bulk dual of the unitary and the ideal Fermi gases is discussed.Comment: 67 pages, 2 figures; references added, minor improvements in the presentation, version accepted for publication in JHE

Covariant quantum mechanics as a tool for quantum-gravity phenomenology

Covariant quantum mechanics (CQM) is a background-indipendent reformulation of quantum mechanics in which the time coordinate is treated as a dynamical degree of freedom, rather than an external evolution parameter. CQM was originally conceived and developed as a basic relational quantum formalism for discussing some conceptual issues associated with quantum gravity in a simplified setting. However, in recent times, a few papers have fruitfully employed it to incorporate quantum-gravity effects such as spacetime noncommutativity into simple phenomenological models. In this Ph.D thesis, I explore in more detail this possibility, providing a first systematic investigation of CQM as a tool for quantum-gravity phenomenology, rather than a toy model of full quantum gravity. In particular, starting from the ordinary CQM of a single relativistic particle, I build generalized models for the description of free quantum particles propagating on noncommutative or curved spacetimes. The present work is theoretical in character. I mainly focus on the development and characterization of a CQM-based framework suitable for dealing with generic spacetime noncommutativity or metric, and study simple examples

Supersymmetric Origins of the Properties of sech-Pulses and sine-Gordon Solitons

In this thesis, we show that the members of a class of reflectionless Hamiltonians, namely, Akulin\u27s Hamiltonians, are connected via a supersymmetric (SUSY) chain. While the reflectionless property in question (vanishing reflection coefficients at all values of the spectral parameter, e.g. energy) has been mentioned in the literature for over two decades, the enabling algebraic mechanism was previously unknown. We show that the supersymmetric connection of the Akulin\u27s Hamiltonians to a potential-free Hamiltonian is the origin of this property. As the first application for our findings, we show that the SUSY decomposition of Akulin\u27s Hamiltonians explains a well-known effect in laser physics: when a two-level atom, initially in the ground state, is subjected to a laser pulse of the form V(t) = (n&hbar/&tau)/cosh(t/&tau), with n an integer and &tau the pulse duration, it remains in the ground state after the pulse has been applied, for any choice of the laser detuning. The second application concerns the sine-Gordon equation: we demonstrate that the first member of the Akulin\u27s chain is related to the L-operator of the Lax pair for the one-soliton solution of the sine-Gordon equation: its reflectionless nature is now explained by supersymmetry

The Einstein-Jordan conundrum and its relation to ongoing foundational research in local quantum physics

We demonstrate the extraordinary modernity of the 1924/25 “Einstein-Jordan fluctuation conundrum”, a Gedankenexperiment which led Jordan to his quantization of waves published as a separate section in the famous Born- Heisenberg-Jordan 1926 “Dreimännerarbeit”. The thermal nature of energy fluctuations caused by the restriction of the QFT vacuum to a subvolume remained unnoticed mainly because it is not present in QM. In order to understand the analogy with Einstein’s fluctuation calculation in a thermal black body system, it is important to expose the mechanism which causes a global vacuum state to become impure on a localized subalgebra of QFT. The present work presents the fascinating history behind this problem which culminated in the more recent perception that “causal localization” leads to thermal manifestations. The most appropriate concept which places this property of QFT into the forefront is “modular localization”. These new developments in QFT led to a new access to the existence problem for interacting quantum fields whose solution has remained outside the range of renormalized perturbation theory. It also clarifies open problems about the relation of particles and fields in particular about the incompletely understood crossing property. Last not least it leads to a constructive understanding of integrable versus non-integrable QFTs..