Exact operator bosonization of finite number of fermions in one space dimension

Abstract

Abstract:We derive an exact operator bosonization of a finite number of fermions in one space dimension. The fermions can be interacting or noninteracting and can have an arbitrary hamiltonian, as long as there is a countable basis of states in the Hilbert space. In the bosonized theory the finiteness of the number of fermions appears as an ultraviolet cut-off. We discuss implications of this for the bosonized theory. We also discuss applications of our bosonization to one-dimensional fermion systems dual to (sectors of) string theory such as LLM geometries and c=1 matrix model.

Bosonization of nonrelativistic fermions on a circle: Tomonaga’s problem revisited

“…In the context of the recent discussion of baby universes in string theory black holes, we have pointed out that the O(N) piece in our bosonized hamiltonian provides a simple model for understanding the origin of two different kinds of nonperturbative O(e −N ) corrections to the partition function. We may recall here that in the application of our bosonization to the half-BPS sector of N = 4 super Yang-Mills theory [1,2], our bosonic oscillators turned out to create single-particle giant graviton states from the AdS 5 × S 5 ground state. It would be intersting to investigate whether our bosonic oscillators also have a natural interpretation in the baby universe context.…”

“…In this paper we have used the tools recently developed by us [1] for an exact bosonization of a finite number N of non-relativistic fermions to discuss the classic Tomonaga problem. We have shown that the standard cubic effective hamiltonian for a massless relativistic boson arises in a systematic large-N and low-energy limit.…”

“…A new set of unconstrained bosonic operators was introduced in [1], N of them for N fermions. In effect, this set of bosonic operators provides the independent degrees of freedom in terms of which the above constraint is solved.…”

“…In effect, this set of bosonic operators provides the independent degrees of freedom in terms of which the above constraint is solved. Let us denote these operators by σ k , k = 1, 2, · · · , N and their conjugates, σ † k , k = 1, 2, · · · , N. As we shall see shortly, these operators will turn out to be identical to the σ k 's used in [1]. The action of σ † k on a given fermion state |f 1 , · · · , f N is rather simple.…”

“…In Sec 2 we summarize the work of [1] 3 on the exact bosonization of a finite number N of nonrelativistic fermions. The presentation here is different and simpler, though completely equivalent to that in [1]. The merit of this presentation is that it makes the bosonization rules simpler and graphical, making applications of these rules very easy.…”