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arXiv:0809.5098 · Physical Review A

Polymer quantization, singularity resolution, and the 1∕r2 potential

G. Kunstatter, Jorma Louko, Jonathan Ziprick

Published Mar 5, 2009• 40 citations
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Abstract

We present a polymer quantization of the $\ensuremath{-}\ensuremath{\lambda}∕{r}^{2}$ potential on the positive real line and compute numerically the bound state eigenenergies in terms of the dimensionless coupling constant $\ensuremath{\lambda}$. The singularity at the origin is handled in two ways: first, by regularizing the potential and adopting either symmetric or antisymmetric boundary conditions; second, by keeping the potential unregularized but allowing the singularity to be balanced by an antisymmetric boundary condition. The results are compared to the semiclassical limit of the polymer theory and to the conventional Schr\"odinger quantization on ${L}_{2}({\mathbb{R}}_{+})$. The various quantization schemes are in excellent agreement for the highly excited states but differ for the low-lying states, and the polymer spectrum is bounded below even when the Schr\"odinger spectrum is not. We find, as expected, that for the antisymmetric boundary condition the regularization of the potential is redundant: the polymer quantum theory is well defined even with the unregularized potential and the regularization of the potential does not significantly affect the spectrum.

DOI
10.1103/physreva.79.032104

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We present a polymer quantization of the \ensuremath−\ensuremathλ∕r2\ensuremath{-}\ensuremath{\lambda}∕{r}^{2}\ensuremath−\ensuremathλ∕r2 potential on the positive real line and compute numerically the bound state eigenenergies in terms of the dimensionless coupling constant \ensuremathλ\ensuremath{\lambda}\ensuremathλ. The singularity at the origin is handled in two ways: first, by regularizing the potential and adopting either symmetric or antisymmetric boundary conditions; second, by keeping the potential unregularized but allowing the singularity to be balanced by an antisymmetric boundary condition. The results are compared to the semiclassical limit of the polymer theory and to the conventional Schr"odinger quantization on L2(R+){L}_{2}({\mathbb{R}}_{+})L2​(R+​). The various quantization schemes are in excellent agreement for the highly excited states but differ for the low-lying states, and the polymer spectrum is bounded below even when the Schr"odinger spectrum is not. We find, as expected, that for the antisymmetric boundary condition the regularization of the potential is redundant: the polymer quantum theory is well defined even with the unregularized potential and the regularization of the potential does not significantly affect the spectrum.
We apply a singularity-resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on R\mathbb{R}R with the singular $\ensuremath{-}1/|x|$ potential. On an equispaced lattice, the resulting eigenvalue problem is identical to a finite-difference approximation of the Schr\"odinger equation. We find numerically that the antisymmetric sector has an energy spectrum that converges to the usual Coulomb spectrum as the lattice spacing is reduced. For the symmetric sector, in contrast, the effect of the lattice spacing is similar to that of a continuum self-adjointness boundary condition at x=0x=0x=0, and its effect on the ground state is significant even if the spacing is much below the Bohr radius. Boundary conditions at the singularity thus have a significant effect on the polymer quantization spectrum even after the singularity has been regularized.
"Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.
The Stone-von Neumann theorem makes canonical quantization unique only after one imposes strong continuity of the Weyl translation groups. We show that this continuity assumption cannot be certified by any finite-resolution interrogation of the Weyl relations. A finite protocol probes only finitely many phase-space displacements and bounded expectation values, thereby determining a protocol algebra CSC_SCS​ generated by a finitely generated subgroup of Weyl translations, but not the regularity of an extension to the full Weyl algebra. Using the polymer representation as a canonical non-regular representation, we prove that for every regular Weyl representation, every normal state, and every finite family of bounded protocol observables, a polymer state reproduces the same expectation values to arbitrary prescribed accuracy. Thus finite agreement with Schrödinger quantum mechanics does not operationally certify the regularity hypothesis entering uniqueness. The result does not assert physical equivalence between regular and polymer quantizations, it identifies where additional input, such as Hamiltonian dynamics, energy regularity, semiclassicality, or a continuum limit, is required to distinguish or select representations. In this precise sense, non-regular polymer kinematics are not excluded by finite Weyl data alone, a point of direct relevance to quantum-gravity motivated quantizations.
We investigate the robustness of singularity avoidance mechanisms in nonrelativistic quantum mechanics on the discretized real line when lattice points are allowed to approach a singularity of the classical potential. We consider the attractive Coulomb potential and the attractive scale-invariant potential, on an equispaced parity-noninvariant lattice and on a nonequispaced parity-invariant lattice, and we examine the energy eigenvalues by a combination of analytic and numerical techniques. While the lowest one or two eigenvalues descend to negative infinity in the singular limit, we find that the higher eigenvalues remain finite and form degenerate pairs, close to the eigenvalues of a theory in which a lattice point at the singularity is regularized either by Thiemann's loop quantum gravity singularity-avoidance prescription or by a restriction to the odd-parity sector. The approach to degeneracy can be reproduced from a nonsingular discretized half-line quantum theory by tuning a boundary condition parameter. The results show that Thiemann's singularity-avoidance prescription and the discretized half-line boundary condition reproduce quantitatively correct features of the singular limit spectrum apart from the lowest few eigenvalues.
In this work, we analyze the corrections obtained on a homogeneous one-dimensional Bose gas within the high densities limit by means of the non-regular representation of quantum mechanics introduced by the polymer quantization scheme. Thus, starting from the Bogoliubov formalism, we analyze the ground expectation value of the polymer momentum operator in terms of semiclassical states in order to obtain an analytic expression for the ground state energy of the N-body system, which allows us to solve the pathological behavior commonly associated with the one-dimensional Bose–Einstein condensation through the introduction of finite size effects characterized by the contribution of the polymer corrections. We also discuss the speed of sound in our polymer version of the Bose gas and the corresponding relative shift induced by the introduction of a minimum length parameter. Finally, we investigate the emergent superfluid behavior in our polymer model by implementing an appropriate Landau’s criterion. In this case, we are able to consequently analyze the changes in the critical velocity which defines the limit between the superfluid-condensate regions, thus deducing that the polymer length acts as a kind of pseudo-potential which induces a dissipationless flow associated with the superfluid phase even in the absence of self-interactions.
Schrodinger operators with very singular potentials fail to produce reasonable spectra, because the usual boundary conditions are insufficient to make them self-adjoint. If one regularizes a singular potential at a length , the bound state energies diverge as → 0. A meaningful spectrum can be restored by picking a self-adjoint extension of the operator, by renormalizing the theory, or by allowing nonunitary time evolution. We show that renormalizations of the 1/r potential fall into two classes: those that are asymptotically equivalent to self-adjoint extensions, and those that reduce to treatments of delta functions. We also apply the apparatus of self-adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and one-dimensional quantum mechanics. Lines composed upon reading a thesis abstract by S. Gopalakrishnan Think a moment, dear reader, of that long-legged fly, The sprung meniscus underfoot, And what asymptotes invisible to well-trained eyes Still break the surface at its root. How a singular potential picks tight-bounded energy Apart till it no longer recognizes Its own mirrored face, its extensions, or symmetry; Unless one cautiously renormalizes. You have tried in these pensive pages to show the stuff Such theory’s made of, its quiddity When logic is applied to the diamond rough, And quantum notes ring purely. May Apollo’s clarity pierce through your reasoning And self-adjointed truth and fact Elevate your thought to that sense so pleasing When truth recovers nature’s pact.