17 research outputs found
The Hamiltonian H=xp and classification of osp(1|2) representations
The quantization of the simple one-dimensional Hamiltonian H=xp is of
interest for its mathematical properties rather than for its physical
relevance. In fact, the Berry-Keating conjecture speculates that a proper
quantization of H=xp could yield a relation with the Riemann hypothesis.
Motivated by this, we study the so-called Wigner quantization of H=xp, which
relates the problem to representations of the Lie superalgebra osp(1|2). In
order to know how the relevant operators act in representation spaces of
osp(1|2), we study all unitary, irreducible star representations of this Lie
superalgebra. Such a classification has already been made by J.W.B. Hughes, but
we reexamine this classification using elementary arguments.Comment: Contribution for the Workshop Lie Theory and Its Applications in
Physics VIII (Varna, 2009
Wigner quantization of some one-dimensional Hamiltonians
Recently, several papers have been dedicated to the Wigner quantization of
different Hamiltonians. In these examples, many interesting mathematical and
physical properties have been shown. Among those we have the ubiquitous
relation with Lie superalgebras and their representations. In this paper, we
study two one-dimensional Hamiltonians for which the Wigner quantization is
related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the
Hamiltonian H = xp, is popular due to its connection with the Riemann zeros,
discovered by Berry and Keating on the one hand and Connes on the other. The
Hamiltonian of the free particle, H_f = p^2/2, is the second Hamiltonian we
will examine. Wigner quantization introduces an extra representation parameter
for both of these Hamiltonians. Canonical quantization is recovered by
restricting to a specific representation of the Lie superalgebra osp(1|2)
Quantum state transfer in spin chains with q-deformed interaction terms
We study the time evolution of a single spin excitation state in certain
linear spin chains, as a model for quantum communication. Some years ago it was
discovered that when the spin chain data (the nearest neighbour interaction
strengths and the magnetic field strengths) are related to the Jacobi matrix
entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect
state transfer takes place. The extension of these ideas to other types of
discrete orthogonal polynomials did not lead to new models with perfect state
transfer, but did allow more insight in the general computation of the
correlation function. In the present paper, we extend the study to discrete
orthogonal polynomials of q-hypergeometric type. A remarkable result is a new
analytic model where perfect state transfer is achieved: this is when the spin
chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The
other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk
polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn
polynomials and q-Racah polynomials) do not give rise to models with perfect
state transfer. However, the computation of the correlation function itself is
quite interesting, leading to advanced q-series manipulations
Wigner Quantization of Hamiltonians Describing Harmonic Oscillators Coupled by a General Interaction Matri
In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix. The quantization of a Hamiltonian describing such a system has been done in the canonical case. In this paper, we take a more general approach and look at the system as a Wigner quantum system. Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations. Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n). We determine the spectrum of the considered Hamiltonian in specific representations of these Lie superalgebras and discuss the results in detail. We also make the connection with the well-known canonical case
Angular momentum decomposition of the three-dimensional Wigner harmonic oscillator
In the Wigner framework, one abandons the assumption that the usual canonical
commutation relations are necessarily valid. Instead, the compatibility of
Hamilton's equations and the Heisenberg equations are the starting point, and
no further assumptions are made about how the position and momentum operators
commute. Wigner quantization leads to new classes of solutions, and
representations of Lie superalgebras are needed to describe them. For the
n-dimensional Wigner harmonic oscillator, solutions are known in terms of the
Lie superalgebras osp(1|2n) and gl(1|n). For n=3N, the question arises as to
how the angular momentum decomposition of representations of these Lie
superalgebras is computed. We construct generating functions for the angular
momentum decomposition of specific series of representations of osp(1|6N) and
gl(1|3N), with N=1 and N=2. This problem can be completely solved for N=1.
However, for N=2 only some classes of representations allow executable
computation
The Berry-Keating Hamiltonian and the Local Riemann Hypothesis
The local Riemann hypothesis states that the zeros of the Mellin transform of
a harmonic-oscillator eigenfunction (on a real or p-adic configuration space)
have real part 1/2. For the real case, we show that the imaginary parts of
these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2
projected onto the subspace of oscillator eigenfunctions of lower level. This
gives a spectral proof of the local Riemann hypothesis for the reals, in the
spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.Comment: 9 pages, no figures; v2 included more mathematical background, v3 has
minor edits for clarit
Analytically solvable Hamiltonians for quantum systems with a nearest neighbour interaction
We consider quantum systems consisting of a linear chain of n harmonic
oscillators coupled by a nearest neighbour interaction of the form ( refers to the position of the th oscillator). In principle,
such systems are always numerically solvable and involve the eigenvalues of the
interaction matrix. In this paper, we investigate when such a system is
analytically solvable, i.e. when the eigenvalues and eigenvectors of the
interaction matrix have analytically closed expressions. This is the case when
the interaction matrix coincides with the Jacobi matrix of a system of discrete
orthogonal polynomials. Our study of possible systems leads to three new
analytically solvable Hamiltonians: with a Krawtchouk interaction, a Hahn
interaction or a q-Krawtchouk interaction. For each of these cases, we give the
spectrum of the Hamiltonian (in analytic form) and discuss some typical
properties of the spectra
General covariant xp models and the Riemann zeros
We study a general class of models whose classical Hamiltonians are given by
H = U(x) p + V(x)/p, where x and p are the position and momentum of a particle
moving in one dimension, and U and V are positive functions. This class
includes the Hamiltonians H_I =x (p+1/p) and H_II=(x+ 1/x)(p+ 1/p), which have
been recently discussed in connection with the non trivial zeros of the Riemann
zeta function. We show that all these models are covariant under general
coordinate transformations. This remarkable property becomes explicit in the
Lagrangian formulation which describes a relativistic particle moving in a 1+1
dimensional spacetime whose metric is constructed from the functions U and V.
General covariance is maintained by quantization and we find that the spectra
are closely related to the geometry of the associated spacetimes. In
particular, the Hamiltonian H_I corresponds to a flat spacetime, whereas its
spectrum approaches the Riemann zeros in average. The latter property also
holds for the model H_II, whose underlying spacetime is asymptotically flat.
These results suggest the existence of a Hamiltonian whose underlying spacetime
encodes the prime numbers, and whose spectrum provides the Riemann zeros.Comment: 34 pages, 3 figure