An Isometric Dynamics for a Causal Set Approach to Discrete Quantum Gravity

We consider a covariant causal set approach to discrete quantum gravity. We first review the microscopic picture of this approach. In this picture a universe grows one element at a time and its geometry is determined by a sequence of integers called the shell sequence. We next present the macroscopic picture which is described by a sequential growth process. We introduce a model in which the dynamics is governed by a quantum transition amplitude. The amplitude satisfies a stochastic and unitary condition and the resulting dynamics becomes isometric. We show that the dynamics preserves stochastic states. By "doubling down" on the dynamics we obtain a unitary group representation and a natural energy operator. These unitary operators are employed to define canonical position and momentum operators.Comment: 18 pages, 1 figur

Discrete Quantum Processes

A discrete quantum process is defined as a sequence of local states $\rho_t$, $t=0,1,2,...$, satisfying certain conditions on an $L_2$ Hilbert space $H$. If $\rho =\lim\rho_t$ exists, then $\rho$ is called a global state for the system. In important cases, the global state does not exist and we must then work with the local states. In a natural way, the local states generate a sequence of quantum measures which in turn define a single quantum measure $\mu$ on the algebra of cylinder sets \cscript. We consider the problem of extending $\mu$ to other physically relevant sets in a systematic way. To this end we show that $\mu$ can be properly extended to a quantum measure \mutilde on a "quadratic algebra" containing \cscript. We also show that a random variable $f$ can be "quantized" to form a self-adjoint operator \fhat on $H$. We then employ \fhat to define a quantum integral \int fd\mutilde. Various examples are givenComment: 29 page

A Matter of Matter and Antimatter

A discrete quantum gravity model given by a quantum sequential growth process (QSGP) is considered. The QSGP describes the growth of causal sets (causets) one element at a time in discrete steps. It is shown that the set \pscript of causets can be partitioned into three subsets \pscript = (\rmant)\cup (\rmmix)\cup (\rmmat) where \rmant is the set of pure antimatter causets, \rmmat the set of pure matter causets and \rmmix the set of mixed matter-antimatter causets. We observe that there is an asymmetry between \rmant and \rmmat which may explain the matter-antimatter asymmetry of our physical universe. This classification of causets extends to the set of paths $\Omega$ in \pscript to obtain \Omega =\Omega ^{\rmant}\cup\Omega ^{\rmmix}\cup\Omega ^{\rmmat}. We introduce a further classification \Omega ^{\rmmix}=\Omega_{\rmm}^{\rmmix}\cup\Omega_{\rma}^{\rmmix} into matter-antimatter parts. Approximate classical probabilities and quantum propensities for these various classifications are considered. Some conjectures and unsolved problems are presented.Comment: 22 pages, including 1 figur

Spooky Action at a Distance

This article studies quantum mechanical entanglement. We begin by illustrating why entanglement implies action at a distance. We then introduce a simple criterion for determining when a pure quantum state is entangled. Finally, we present a measure for the amount of entanglement for a pure state.Comment: A survey of entanglement for students and general reader. 13 page

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

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