On the Equivalence of Fermionic and Bosonic Descriptions in Two Dimensions

Dongsheng Ge, Giuseppe Policastro

Abstract We consider the circuit complexity of free bosons and free fermions in 1+1 dimensions. Motivated by the results of [1, 2, 3] who found different behavior in the complexity of free bosons and fermions, in any dimension, we consider the 1+1 dimensional case where, thanks to the bosonisation equivalence of the Hilbert spaces, we can consider the same state from both the bosonic and the fermionic perspectives. This allows us to study the dependence of the complexity on the choice of the set of gates, which explains the discrepancy. We study the effect in two classes of states: i) bosonic-coherent / fermionic- gaussian states; ii) states that are both bosonic- and fermionic-gaussian. We consider the complexity relative to the ground state. In the first class, the different complexities can be related to each other by introducing a mode-dependent cost function in one of the descriptions. The differences in the second class are more important, in terms of the structure of UV divergencies and the overall behavior of the complexity.

Bent E. Sørensen

Bent Sørensen

K. Ziegler

A modified Gross-Neveu model with n=0 (two fermion and two boson) components is considered in two space dimensions. This model is equivalent, in terms of perturbation theory, to the Gross-Neveu model with a 2n-component fermion field, combined with the replica trick n to 0. Using the universality hypothesis, the author demonstrates that the model is related to the statistics of polymer chains. This result extends earlier investigations, obtained in a strong coupling expansion.

Piotr Garbaczewski

For original paper see Commun. Math. Phys., vol.104, p.457-70, 1986. The author links the recently proposed unification of the boson and fermion stochastic calculus with the general problem of boson-fermion equivalence (duality, reciprocity, etc.) for quantum fields. Even if via the Fock construction the common Fock space for bosons and fermions can be introduced, it still does not allow for the unrestricted boson-fermion equivalence for field theory models. All local fermion field theory models thus have boson equivalents (violating the weak local commutativity condition for space dimension three). The reverse statement is not valid: not all boson models admit a pure fermion reconstruction.

Bent Sørensen