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International Journal of Theoretical Physics

Playing with Numbers, with Fermions and Bosons

Francesco Antonino Raffa, Mario Rasetti

Published Oct 9, 2007• 0 citations
Publisher
DOI
10.1007/s10773-007-9573-1

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The number operator a*a, where a is an annihilation operator, plays a fundamental role in the statistics of bosons and fermions. However, it is possible for other statistics to have the same number operator. We have previously shown that for one degree of freedom there is a type of statistics having this number operator corresponding to each p which is a positive integer or ∞. Fermions are obtained when p = 1 and bosons are obtained when p = ∞. No state can have more than p particles. In order to treat many degrees of freedom it is necessary to first consider the case of one degree of freedom differently. We show here that for more degrees of freedom a similar situation occurs but for each case other than bosons and fermions there is a positive integer q, such that no state can have more than q particles, even when the number of degrees of freedom is infinite. Thus these statistics are probably not physically realizable except in an approximate way.
The bosonization method developed in nuclear physics in the last 20 years is briefly reviewed. Introduction In the last 20 years, a considerable amount of work has gone into the development of bosonization methods for highly correlated finite fermion systems. This work has been stimulated by the phenomenological successes of the interacting boson model introduced in 1974 in which atomic nuclei composed of nucleons (fermions) are treated in terms of an interacting system of bosons (fermion pairs). In this article the logic scheme of the bosonization method in nuclear physics will be briefly reviewed. The purpose here is to provide the basic references upon which the review is built and sources where further references can be found. Bosonization methods were introduced much earlier than 1974 in connection with infinite Fermi systems. The method developed in nuclear physics has some similarities with those developed in other areas of physics, but it also has major differences. In nuclear physics one maps not only operators but also states and seeks a description not only of the ground state but also of the entire excitation spectrum. Furthermore, number projection plays a very important role contrary to the case of infinite systems where number projection is not relevant. These differences, and others, will be briefly remarked upon in this article. Strongly Interacting Finite Fermion Systems Fermions and the Shell Model Quite often in physics one has to deal with a system of interacting fermions.
The sample complexity is the minimum number of copies required to learn an accurate classical description of a quantum state. Bosonic and fermionic Gaussian quantum states are families of quantum states that play a key role in quantum science and technology, from quantum optics and many-body physics to quantum chemistry, quantum computing, and quantum information theory. Despite their importance, their sample complexity had not been fully determined. We settle this open problem and show that both bosonic and fermionic Gaussian states can be learned using a number of copies that scales quadratically in the number of modes, regardless of whether the state is pure or mixed, and independently of any energy bound on the state. We derive these results by using the representation theory of Gaussian unitaries and by putting forth a generalization of the random purification channel to this setting and beyond.