On the Relation between Fermions and Bosons

Morikazu Toda

Kalyanapuram Rangachari Parthasarathy, Kalyan B. Sinha

Michael G. Jabbour, Nicolas J. Cerf

Bosonic and fermionic statistics are well known to give rise to antinomic behaviors, most notably boson bunching vs fermion antibunching. Here, we establish a fundamental relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer. The bosonic and fermionic transition probabilities appear together in a single equation which constrains their values, hence expressing a boson–fermion complementarity that is independent of the details of the interferometer. For two particles in any interferometer, for example, it implies that the average between the bosonic and fermionic probabilities must coincide with the probability obeyed by classical particles. Crucially, this fundamental relation also provides a heretofore unknown mathematical identity connecting the squared moduli of the permanent and determinant of an arbitrary complex matrix, hence extending an identity by Muir dating from the nineteenth century.

Raymond A. Sorensen

A correspondence is described relating many-body states of bosons and fermions. This leads to simple connections among the total angular momentum J values allowed for j shells containing several identical bosons or fermions.

Виктор Павлович Маслов

K. Taniguchi, Atsushi Kajiyama, Y. Miyanishi

The relation between the hermitian type boson expansion theory and the boson-fermion expansion theory recently proposed as the alternative to it is discussed. It is shown that the apparent difference between some of the expansion coefficients in both theories is due to arbitrariness about the definition of the collective state in the fermion space and that one can remove the difference by the appropriate unitary transformation.