arXiv subjects

Stan Gudder

Publications and source records attributed to Stan Gudder.

At least 55 records · Page 3Linked to original sources

arXiv2015

Wave Equations for Discrete Quantum Gravity

This article is based on the covariant causal set ($c$-causet) approach to discrete quantum gravity. A $c$-causet $x$ is a finite partially ordered set that has a unique labeling of its vertices. A rate of change on $x$ is described by a covariant difference operator and this operator acting on a wave function forms the left side of the wave equation. The right side is given by an energy term acting on the wave function. Solutions to the wave equation corresponding to certain pairs of paths in $x$ are added and normalized to form a unique state. The modulus squared of the state gives probabilities that a pair of interacting particles is at various locations given by pairs of vertices in $x$. We illustrate this model for a few of the simplest nontrivial examples of $c$-causets. Three forces are considered, the attractive and repulsive electric forces and the strong nuclear force. Large models get much more complicated and will probably require a computer to analyze.

gr-qc↗

arXiv2015

Inflation and Dirac in the Causal Set Approach to Discrete Quantum Gravity

In this approach to discrete quantum gravity the basic structural element is a covariant causal set ($c$-causet). The geometry of a $c$-causet is described by a shell-sequence that determines the discrete gravity of a universe. In this growth model, universes evolve in discrete time by adding new vertices to their generating $c$-causet. We first describe an inflationary period that is common to all universes. After this very brief cycle, the model enters a multiverse period in which the system diverges in various ways forming paths of $c$-causets. At the beginning of the multiverse period, the structure of a four-dimensional discrete manifold emerges and quantum mechanics enters the picture. A natural Hilbert space is defined and a discrete, free Dirac operator is introduced. We determine the eigenvalues and eigenvectors of this operator. Finally, we propose values for coupling constants that determine multiverse probabilities. These probabilities predict the dominance of pulsating universes.

gr-qc↗

arXiv2015

Emergence of Four Dimensions in the Causal Set Approach to Discrete Quantum Gravity

One could begin a study like the present one by simply postulating that our universe is four-dimensional. There are ample reasons for doing this. Experience, observation and experiment all point to the fact that we inhabit a four-dimensional universe. Another approach would be to show that four-dimensions arise naturally from a reasonable model of the universe or multiverse. After reviewing the causal set approach to discrete quantum gravity in Section~1, we shall discuss the emergence of four-dimensions in Section~2. We shall see that certain patterns of four arise that suggest the introduction of a 4-dimensional discrete manifold. In the later sections we shall discuss some consequences of this introduced framework. In particular, we will show that quantum amplitudes can be employed to describe a multiverse dynamics. Moreover, a natural unitary operator together with energy, position and momentum operators will be introduced and their properties studied.

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arXiv2014

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.

gr-qc↗

arXiv2014

A Unified Approach to Discrete Quantum Gravity

This paper is based on a covariant causal set (c-causet) approach to discrete quantum gravity. A c-causet is a partially ordered set $(x,<)$ that is invariant under labeling. We first consider the microscopic picture which describes the detailed structure of c-causets. The unique labeling of a c-causet $x$ enables us to define a natural metric $d(a,b)$ between comparable vertices $a,b$ of $x$. The metric is then employed to define geodesics and curvatures on $x$. We next consider the macroscopic picture which describes the growth process $x \to y$ of c-causets. We propose that this process is governed by a quantum dynamics given by complex amplitudes. Denoting the set of c-causets by $\pscript$ we show that the growth process $(\pscript ,\to)$ can be structured into a discrete 4-manifold. This 4-manifold presents a unified approach to a discrete quantum gravity for which we define discrete analogues of Einstein's field equations and Dirac's equation.

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arXiv2014

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.

gr-qc↗

arXiv2014

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.

gr-qc↗

arXiv2014

Elementary Particles and the Causet Approach to Discrete Quantum Gravity

In a previous paper, the author introduced a covariant causet ($c$-causet) approach to discrete quantum gravity. A $c$-causet is a finite partially ordered set that is invariant under labeling. The invariant labeling of a $c$-causet $x$ enables us to uniquely specify $x$ by a sequence $\brac{s_j(x)}$, $j=0,1,2,\ldots$, of positive integers called a shell sequence of $x$. A $c$-causet $x$ describes the microscopic structure of a possible universe at a particular time step. In general, $x$ represents one of many universes in a multiverse and $x$ grows by a single element at each time step. Since early stages of a universe were probably composed of elementary particles, we propose that elementary particles can be described by simple $c$-causets. Although we do not have a rigorous theory for such a description, we present our guess as to how it might appear. The shell sequence can be applied to find theoretical masses of particles and these seem to approximately agree with known masses. We point out that the causal order provides a unification of the strong and weak forces for elementary particles and also determines the geometry of $c$-causets which describes gravity. Moreover, it appears that some particles correspond to dark matter-energy.

gr-qc↗

arXiv2013

A covariant causal set approach to discrete quantum gravity

A covariant causal set (c-causet) is a causal set that is invariant under labeling. Such causets are well-behaved and have a rigid geometry that is determined by a sequence of positive integers called the shell sequence. We first consider the microscopic picture. In this picture, the vertices of a c-causet have integer labels that are unique up to a label isomorphism. This labeling enables us to define a natural metric $d(a,b)$ between time-like separated vertices $a$ and $b$. The time metric $d(a,b)$ results in a natural definition of a geodesic from $a$ to $b$. It turns out that there can be $n\ge 1$ such geodesics. Letting $a$ be the origin (the big bang), we define the curvature $K(b)$ of $b$ to be $n-1$. Assuming that particles tend to move along geodesics, $K(b)$ gives the tendency that vertex $b$ is occupied. In this way, the mass distribution is determined by the geometry of the c-causet. We next consider the macroscopic picture which describes the growth process of c-causets. We propose that this process is governed by a quantum dynamics given by complex amplitudes. At present, these amplitudes are unknown. But if they can be found, they will determine the (approximate) geometry of the c-causet describing our particular universe. As an illustration, we present a simple example of an amplitude process that may have physical relevance. We also give a discrete analogue of Einstein's field equations.

gr-qc↗

arXiv2013

An Approach to Discrete Quantum Gravity

This article presents a simplified version of the author's previous work. We first construct a causal growth process (CGP). We then form path Hilbert spaces using paths of varying lengths in the CGP. A sequence of positive operators on these Hilbert spaces that satisfy certain normalization and consistency conditions is called a quantum sequential growth process (QSGP). The operators of a QSGP are employed to define natural decoherence functionals and quantum measures. These quantum measures are extended to a single quantum measure defined on a suitable collection of subsets of a space of all paths. Continuing our general formalism, we define curvature operators and a discrete analogue of Einstein's field equations on the Hilbert space of causal sets. We next present a method for constructing a QSGP using an amplitude process (AP). We then consider a specific AP that employs a discrete analogue of a quantum action. Finally, we consider the special case in which the QSGP is classical. It is pointed out that this formalism not only gives a discrete version of general relativity, there is also emerging a discrete analogue of quantum field theory. We therefore have discrete versions of these two theories within one unifying framework.

math-ph↗

arXiv2013

Compatibility for Probabilistic Theories

We define an index of compatibility for a probabilistic theory (PT). Quantum mechanics with index 0 and classical probability theory with index 1 are at the two extremes. In this way, quantum mechanics is at least as incompatible as any PT. We consider a PT called a concrete quantum logic that may have compatibility index strictly between 0 and 1, but we have not been able to show this yet. Finally, we show that observables in a PT can be represented by positive, vector-valued measures.

quant-ph↗

arXiv2013

A Dynamics for Discrete Quantum Gravity

This paper is based on the causal set approach to discrete quantum gravity. We first describe a classical sequential growth process (CSGP) 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 (causet) and the "completed" universe is given by a path through a discretely growing chain of causets. We then quantize the CSGP by forming a Hilbert space $H$ on the set of paths. The quantum dynamics is governed by a sequence of positive operators $ρ_n$ on $H$ that satisfy normalization and consistency conditions. The pair $(H,\brac{ρ_n})$ is called a quantum sequential growth process (QSGP). We next discuss a concrete realization of a QSGP in terms of a natural quantum action. This gives an amplitude process related to the sum over histories" approach to quantum mechanics. Finally, we briefly discuss a discrete form of Einstein's field equation and speculate how this may be employed to compare the present framework with classical general relativity theory.

gr-qc↗

arXiv2012

Causal set approach to discrete quantum gravity

We begin by describing a sequential growth model in which the universe grows one element at a time in discrete time steps. At each step, the process has the form of a causal set and the "completed" universe is given by a path consisting of a discretely growing chain of causal sets. We then introduce a quantum dynamics to obtain a quantum sequential growth process (QSGP) which may lead to a viable model for discrete quantum gravity. A discrete version of Einstein's field equation is derived and a definition for discrete geodesics is proposed. A type of QSGP called an amplitude process is introduced. An example of an amplitude process called a complex percolation process is studied. This process conforms with general principles of causality and covariance. We end with some detailed quantum measure calculations for a specific percolation constant.

gr-qc↗

arXiv2012

An Einstein equation for discrete quantum gravity

The basic framework for this article is the causal set approach to discrete quantum gravity (DQG). Let $Q_n$ be the collection of causal sets with cardinality not greater than $n$ and let $K_n$ be the standard Hilbert space of complex-valued functions on $Q_n$. The formalism of DQG presents us with a decoherence matrix $D_n(x,y)$, $x,y\in Q_n$. There is a growth order in $Q_n$ and a path in $Q_n$ is a maximal chain relative to this order. We denote the set of paths in $Q_n$ by $Ω_n$. For $ω, ω'\inΩ_n$ we define a bidifference operator $\varbigtriangledown_{ω, ω'}^n$ on $K_n\otimes K_n$ that is covariant in the sense that $\varbigtriangledown_{ω, ω'}^n$ leaves $D_n$ stationary. We then define the curvature operator $\rscript_{ω, ω'}^n=\varbigtriangledown_{ω, ω'}^n-\varbigtriangledown_{ω', ω}^n$. It turns out that $\rscript_{ω, ω'}^n$ naturally decomposes into two parts $\rscript_{ω, ω'}^n=\dscript_{ω, ω'}^n+\tscript_{ω, ω'}^n$ where $\dscript_{ω, ω'}^n$ is closely associated with $D_n$ and is called the metric operator while $\tscript_{ω, ω'}^n$ is called the mass-energy operator. This decomposition is a discrete analogue of Einstein's equation of general relativity. Our analogue may be useful in determining whether general relativity theory is a close approximation to DQG.

gr-qc↗

arXiv2012

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 $Ω$ in $\pscript$ to obtain $Ω=Ω^{\rmant}\cupΩ^{\rmmix}\cupΩ^{\rmmat}$. We introduce a further classification $Ω^{\rmmix}=Ω_{\rmm}^{\rmmix}\cupΩ_{\rma}^{\rmmix}$ into matter-antimatter parts. Approximate classical probabilities and quantum propensities for these various classifications are considered. Some conjectures and unsolved problems are presented.

gr-qc↗

arXiv2011

Discrete Quantum Gravity Is Not Isometric

We show that if a discrete quantum gravity is not classical, then it cannot be generated by an isometric dynamics. In particular, we show that if the quantum measure μ (or equivalently the decoherence functional) is generated by an isometric dynamics, then there is no in- terference between events so the system describing evolving universes is classical. The result follows from a forbidden configuration in the path space of causal sets.

gr-qc↗

arXiv2011

Models for Discrete Quantum Gravity

We first discuss a framework for discrete quantum processes (DQP). It is shown that the set of q-probability operators is convex and its set of extreme elements is found. The property of consistency for a DQP is studied and the quadratic algebra of suitable sets is introduced. A classical sequential growth process is "quantized" to obtain a model for discrete quantum gravity called a quantum sequential growth process (QSGP). Two methods for constructing concrete examples of QSGP are provided.

gr-qc↗

arXiv2011

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.

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