arXiv subjects

Samuel Shuman

Publications and source records attributed to Samuel Shuman.

4 recordsLinked to original sources

arXiv2026

Polymer quantum mechanics on compact configuration spaces

"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 "Schr\"odinger" 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 Schr\"odinger 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.

quant-ph↗

arXiv2024

The Geometry of Tangent Spaces on Causal Sets

In this paper, we expand on previous work describing partial derivatives and metric component estimators to define tangent spaces on causal sets. Partial derivative operators are the basis vectors of the tangent space, and the metric defines the inner product. First, we use partial derivatives of the metric components to define the connection and partial derivatives of the connection to define the curvature. Numerical results show that both of these approach the expected values for a flat spacetime as density increases. Then we used the connection to define parallel transport and geodesics.

gr-qc↗

arXiv2024

Partial Derivatives on Causal Sets

We will discuss two approaches to estimating partial derivatives and the metric components; one utilizing past work describing a causal set $\Box$ operator, and one using a construction from linear algebra called the Moore-Penrose inverse. After running numerical tests on a causal diamond in $\mathbb{M}^2$, we find that the approach using the Moore-Penrose inverse is significantly more accurate. Despite the large variances in the method using the $\Box$ operator, there is reason to believe both approaches should become more accurate at higher densities.

gr-qc↗

arXiv2023

Path Sums for Propagators in Causal Sets

A major challenge in Causal Set research is that theories need only to match general relativity and quantum field theory in the appropriate limits. This means that there should be many different ways to calculate a scalar field propagator in a causal set that match the known limits, but may give significantly different results on the small scale. In this work, we explore under what conditions a path sum will correspond to a scalar field propagator in such a way that it matches the known value in the continuum limit. A family of solutions for the path sum is found and is verified numerically in a few specific cases.

gr-qc↗