Discrete Quantum Gravity

We discuss the causal set approach to discrete quantum gravity. We begin by describing a classical sequential growth process in which the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set and the "completed" universe is given by a path through a discretely growing chain of causal sets. We then introduce a method for quantizing this classical formalism to obtain a quantum sequential growth process which may lead to a viable model for a discrete quantum gravity. We also give a method for quantizing random variables in the classical process to obtain observables in the corresponding quantum process. The paper closes by showing that a discrete isometric process can be employed to construct a quantum sequential growth process.Comment: 19 pages which includes 3 figures created in LaTe

Examples of quantum integrals

We first consider a method of centering and a change of variable formula for a quantum integral. We then present three types of quantum integrals. The first considers the expectation of the number of heads in $n$ flips of a "quantum coin". The next computes quantum integrals for destructive pairs examples. The last computes quantum integrals for a (Lebesgue)^2 quantum measure. For this last type we prove some quantum counterparts of the fundamental theorem of calculus.Comment: I would like to submit this article for posting on Archiv

Markov random fields and Markov chains on trees

We consider probability measures on a space S(^A) (where S and A are countable and the σ-field is the natural one) which are Markov random fields with respect to a given neighbour relation ~ on A. In particular, we study the set G(II) of Markov random fields corresponding to a given Markov specification II, i.e. to a consistent family of "Markov" conditional probability distributions associated with the finite subsets of A. First, we review the relation between II and G(II). We consider also the representation of II by a family of interaction functions associated with the simplices of the graph (A,~) , together with some related problems. The rest of the thesis is concerned with the case where (A,~) is a tree. We define Markov chains on and consider their relation to the wider class of Markov random fields. We then derive analytical methods for the study of the set M(II) of Markov chains in G(II). These results are applied to homogeneous Markov specifications on regular infinite trees. Finally, we consider Markov specifications which are either attractive or repulsive with respect to a total ordering on S. For these we obtain quite strong results, including an exact condition for G(II) to contain precisely one element. We thereby generalise results obtained by Preston and Spitzer for binary S

From metaphysical principles to dynamical laws

My thesis in this paper is: the modern concept of laws of motion—qua dynamical laws—emerges in 18th-century mechanics. The driving factor for it was the need to extend mechanics beyond the centroid theories of the late-1600s. The enabling result behind it was the rise of differential equations. In consequence, by the mid-1700s we see a deep shift in the form and status of laws of motion. The shift is among the critical inflection points where early modern mechanics turns into classical mechanics as we know it. Previously, laws of motion had been channels for truth and reference into mechanics. By 1750, the laws lose these features. Instead, now they just assert equalities between functions; and serve just to entail (differential) equations of motion for particular mechanical setups. This creates two philosophical problems. First, it’s unclear what counts as evidence for the laws of motion in the Enlightenment. Second, it’s a mystery whether these laws retain any notion of causality. That subverts the early-modern dictum that physics is a science of causes

Capping the Tax Exclusion of Employer-Sponsored Health Insurance: Is Equity Feasible?

Explores the feasibility of capping tax exemptions on premiums paid for employer-sponsored insurance without creating inequities by firm size, employees' age, and type of coverage -- by taxing benefits based on actuarial value or adjusting premiums

The Universe as a Quantum Computer

This article presents a sequential growth model for the universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a $c$-causet. A $c$-causet is defined to be a causet that is independent of its labeling. We characterize $c$-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a $c$-causet has precisely two $c$-causet offspring. It follows that there are $2^n$ $c$-causets of cardinality $n+1$. This enables us to classify $c$-causets of cardinality $n+1$ in terms of $n$-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the $n$-bits by $n$-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-1 qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events.Comment: 23 pages, 1 figur

Remembering with and without Memory: A Theory of Memory and Aspects of Mind that Enable its Experience

This article builds on ideas presented in Klein (2015a) concerning the importance of a more nuanced, conceptually rigorous approach to the scientific understanding and use of the construct “memory”. I first summarize my model, taking care to situate discussion within the terminological practices of contemporary philosophy of mind. I then elucidate the implications of the model for a particular operation of mind – the manner in which content presented to consciousness realizes its particular phenomenological character (i.e., mode of presentation). Finally, I discuss how the model offers a reconceptualization of the technical language used by psychologists and neuroscientists to formulate and test ideas about memory

Labeled Causets in Discrete Quantum Gravity

We point out that labeled causets have a much simpler structure than unlabeled causets. For example, labeled causets can be uniquely specified by a sequence of integers. Moreover, each labeled causet processes a unique predecessor and hence has a unique history. Our main result shows that an arbitrary quantum sequential growth process (QSGP) on the set of labeled causets "compresses" in a natural way onto a QSGP on the set of unlabeled causets. The price we have to pay is that this procedure causes an "explosion" of values due to multiplicities. We also observe that this procedure is not reversible. This indicates that although many QSGPs on the set of unlabeled causets can be constructed using this method, not all can, so it is not completely general. We close by showing that a natural metric can be defined on labeled and unlabeled causets and on their paths.Comment: 17 pages, 1 figure. arXiv admin note: text overlap with arXiv:1305.518