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arXiv:1506.00541

Asymptotic Equation for Zeros of Hermite Polynomials from the Holstein-Primakoff Representation

Lucas Kocia

Published Jun 1, 2015• 0 citations
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Abstract

The Holstein-Primakoff representation for spin systems is used to derive expressions with solutions that are conjectured to be the zeros of Hermite polynomials $H_n(x)$ as $n \rightarrow \infty$. This establishes a correspondence between the zeros of the Hermite polynomials and the boundaries of the position basis of finite-dimensional Hilbert spaces.

DOI
10.48550/arxiv.1506.00541

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The Holstein-Primakoff representation for spin systems is used to derive expressions with solutions that are conjectured to be the zeros of Hermite polynomials Hn(x)H_n(x)Hn​(x) as n→∞n \rightarrow \inftyn→∞. This establishes a correspondence between the zeros of the Hermite polynomials and the boundaries of the position basis of finite-dimensional Hilbert spaces.
The fundamental position and momentum operators of quantum mechanics are unbounded, but finite rank compressions of the operators can be considered to obtain partial information on the operators and their properties. Motivated by problems in photonic quantum computing, we bring together results from quantum theory and the theory of orthogonal polynomials to show that a natural finite rank compression of the position and momentum operator representation on Fock space Hilbert space has eigenvalues given by roots of the classical Hermite polynomials. We discuss the corresponding compressed displacement operators and potential applications in approximate quantum error correction.
Operator square roots are ubiquitous in theoretical physics.They appear, for example, in the Holstein-Primakoff representation of spin operators and in the Klein-Gordon equation.Often the use of a perturbative expansion is the only recourse when dealing with them.In this paper, we show that under certain conditions, differential equations can be derived which can be used to find perturbatively inaccessible approximations to operator square roots.Specifically, for the number operator n = a † a we show that the square root √ n near n = 0 can be approximated by a polynomial in n.This result is unexpected because a Taylor expansion fails.A polynomial expression in n is possible because n is an operator, and its constituents a and a † have a non trivial commutator [a, a † ] = 1 and do not behave as scalars.We apply our approach to the zero-mass Klein-Gordon Hamiltonian in a constant magnetic field and, as a main application, the Holstein-Primakoff representation of spin operators, where we are able to find new expressions that are polynomial in bosonic operators.We prove that these new expressions exactly reproduce spin operators.Our expressions are manifestly Hermitian, which offers an advantage over other methods, such as the Dyson-Maleev representation.
The present series of papers deals with various realizations of the dynamical group 𝒮𝓅c(2d,R) of microscopic collective states for an A nucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of 𝒮𝓅c(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert space ℱc of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in ℱc, classified according to the chain 𝒮𝓅c(2d,R)⊇𝒰c(d), can be put into one-to-one correspondence with basis functions, classified according to the chain 𝔘(ν)⊇𝔘(d), in a Bargmann Hilbert space ℬ of analytic functions in ν complex variables representing ν-dimensional boson states. By equating the complex variables of ℱc and their conjugate momenta with those of ℬ, we get the non-Hermitian Dyson representation of 𝒮𝓅c(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a 𝔘(ν) group, which in the physical three-dimensional case reduces to 𝔘(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.