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arXiv:0707.0273 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum gravity and the Coulomb potential

Viqar Husain, Jorma Louko, Oliver Winkler

Published Oct 3, 2007• 40 citations
arXivHTMLPDF

Abstract

We apply a singularity-resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on $\mathbb{R}$ 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 $x=0$, 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.

DOI
10.1103/physrevd.76.084002

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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∣\ensuremath{-}1/|x|\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.
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 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.
"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.
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.
A rather nonstandard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation, has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schr\"odinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schr\"odinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schr\"odinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle, and a simple cosmological model.
The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space E3\mathbb{E}_{3}E3​ where the potential is singular but invariant under the continuous inhomogeneous orthogonal group IO(3)\mathcal{I}O(3)IO(3). The invariance under the translation subgroup is compared to the corresponding unitary transformation in the Schrödinger representation of quantum mechanics. This scenario is well known but serves as a reference point for the other scenarios. (ii) Next, the discrete potential equation as a partial difference equation in a three-dimensional lattice space is studied. In this arena the potential is non-singular but invariance under IO(3)\mathcal{I}O(3)IO(3) is broken. This is the usual picture of lattice theories and numerical approximations. (iii) Next we study the six-dimensional continuous phase space. Here a phase space representation of quantum mechanics is utilized. The resulting potential is singular but possesses invariance under IO(3)\mathcal{I}O(3)IO(3). (iv) Finally, the potential is derived from the discrete phase space representation of quantum mechanics, which is shown to be an \emph{exact} representation of quantum mechanics. The potential function here is both non-singular and possesses invariance under IO(3)\mathcal{I}O(3)IO(3), and this is proved via the unitary transformations of quantum mechanics in this representation.
This thesis addresses the representations of the Weyl algebra of quantum mechanics with an explicitly Born reciprocal structure and the dynamics of physical systems in these representations. In this context, we discuss both the regular modular representations based on Aharonov's modular variables, as well as the irregular modular polymer representations inspired by the spin network representation in loop quantum gravity. \n \nWe introduce the modular space as the quantum configuration space corresponding to the modular representation and treat it as an alternative to the classical configuration space. The modular representations have a built-in Abelian gauge symmetry. We discuss the singular limits in which the modular representation converges to the Schrodinger and momentum representations, and show that these limits require two distinct fixings of the modular gauge symmetry. \n \nIn order to explore the propagation of quantum states in the modular space, we construct a Feynman path integral for the harmonic oscillator explicitly as the transition amplitude between two modular states. The result is a functional integral over the compact modular space, which shows novel features of the dynamics such as winding modes, an Aharonov-Bohm phase, and a new action on the modular space. We compare the stationary trajectories that extremize this modular action to the phase space solution of Hamilton's equations in the Schrodinger representation and find a new translation symmetry. We also identify the other symmetries of the harmonic oscillator in the modular action and show their correspondence to the classical case. \n \nWe generalize the relationship between the classical Hamiltonian function and the modular action by postulating a new modular Legendre transform. For demonstration we apply this transformation on the Kepler problem and reformulate it in terms of modular variables. \n \nWe then switch our focus to the representations of the Weyl algebra that violate the assumptions of the Stone-von Neumann theorem. In a similar fashion to the polymer representations studied as a toy model for quantum gravity, we polymerize the modular representation to obtain an inequivalent representation called the modular polymer (MP) representation. \n \nThe new MP representation lacks both position and momentum operators, therefore the Hamiltonian of the harmonic oscillator is regularized with the Weyl operators and the non-separable MP Hilbert space is split into superselection sectors. The solutions to the Schrodinger equation fall into two categories depending on whether the ratio between the modular scale and the polymeric regularization scale is a rational or irrational number. We find a discrete and finite energy spectrum in each superselection sector in the former case, and a continuous, bounded one in the latter.