Unitarity Restoration in the Presence of Closed Timelike Curves

A proposal is made for a mathematically unambiguous treatment of evolution in the presence of closed timelike curves. In constrast to other proposals for handling the naively nonunitary evolution that is often present in such situations, this proposal is causal, linear in the initial density matrix and preserves probability. It provides a physically reasonable interpretation of invertible nonunitary evolution by redefining the final Hilbert space so that the evolution is unitary or equivalently by removing the nonunitary part of the evolution operator using a polar decomposition.Comment: LaTeX, 17pp, Revisions: Title change, expanded and clarified presentation of original proposal, esp. with regard to Heisenberg picture and remaining in original Hilbert spac

The Irreducible Spine(s) of Undirected Networks

Using closure concepts, we show that within every undirected network, or graph, there is a unique irreducible subgraph which we call its "spine". The chordless cycles which comprise this irreducible core effectively characterize the connectivity structure of the network as a whole. In particular, it is shown that the center of the network, whether defined by distance or betweenness centrality, is effectively contained in this spine. By counting the number of cycles of length 3 <= k <= max_length, we can also create a kind of signature that can be used to identify the network. Performance is analyzed, and the concepts we develop are illurstrated by means of a relatively small running sample network of about 400 nodes.Comment: Submitted to WISE 201

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

A tensor-based morphometry analysis of regional differences in brain volume in relation to prenatal alcohol exposure

Reductions in brain volumes represent a neurobiological signature of fetal alcohol spectrum disorders (FASD). Less clear is how regional brain tissue reductions differ after normalizing for brain size differences linked with FASD and whether these profiles can predict the degree of prenatal exposure to alcohol. To examine associations of regional brain tissue excesses/deficits with degree of prenatal alcohol exposure and diagnosis with and without correction for overall brain volume, tensor-based morphometry (TBM) methods were applied to structural imaging data from a well-characterized, demographically homogeneous sample of children diagnosed with FASD (n = 39, 9.6–11.0 years) and controls (n = 16, 9.5–11.0 years). Degree of prenatal alcohol exposure was significantly associated with regionally pervasive brain tissue reductions in: (1) the thalamus, midbrain, and ventromedial frontal lobe, (2) the superior cerebellum and inferior occipital lobe, (3) the dorsolateral frontal cortex, and (4) the precuneus and superior parietal lobule. When overall brain size was factored out of the analysis on a subject-by-subject basis, no regions showed significant associations with alcohol exposure. FASD diagnosis was associated with a similar deformation pattern, but few of the regions survived FDR correction. In data-driven independent component analyses (ICA) regional brain tissue deformations successfully distinguished individuals based on extent of prenatal alcohol exposure and to a lesser degree, diagnosis. The greater sensitivity of the continuous measure of alcohol exposure compared with the categorical diagnosis across diverse brain regions underscores the dose dependence of these effects. The ICA results illustrate that profiles of brain tissue alterations may be a useful indicator of prenatal alcohol exposure when reliable historical data are not available and facial features are not apparent

Quantum vacuum fluctuations and dark energy

It is shown that the curvature of space-time induced by vacuum fluctuations of quantum fields should be proportional to the square of Newton's constant $G$. This offers a possible explanation for the success of the approximation $G m^6 c^2 h^{-4}$ for the dark energy density, with $m$ being a typical mass of elementary particles.Comment: Changed conten

Unitarity of Quantum Theory and Closed Time-Like Curves

Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTCs) are subject to a non-unitary evolution $X$. Recently, a prescription has been proposed, which restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the composition of two or more non-unitary evolutions. We propose an alternative method by which unitarity of the evolution may be regained, by extending $X$ to a unitary evolution on a larger (possibly indefinite) inner product space. The proposal removes the ambiguity noted by Jacobson in assigning expectation values to observables localised in regions spacelike separated from the CTC region. We comment on the physical significance of the possible indefiniteness of the inner product introduced in our proposal.Comment: 13 pages, LaTeX. Final revised paper to be published in Phys Rev D. Some changes are made to expand our discussion of Anderson's Proposal for restoring unitarit

Information Metric on Instanton Moduli Spaces in Nonlinear Sigma Models

We study the information metric on instanton moduli spaces in two-dimensional nonlinear sigma models. In the CP^1 model, the information metric on the moduli space of one instanton with the topological charge Q=k which is any positive integer is a three-dimensional hyperbolic metric, which corresponds to Euclidean anti--de Sitter space-time metric in three dimensions, and the overall scale factor of the information metric is (4k^2)/3; this means that the sectional curvature is -3/(4k^2). We also calculate the information metric in the CP^2 model.Comment: 9 pages, LaTeX; added references for section 1; typos adde

Topological censorship for Kaluza-Klein space-times

The standard topological censorship theorems require asymptotic hypotheses which are too restrictive for several situations of interest. In this paper we prove a version of topological censorship under significantly weaker conditions, compatible e.g. with solutions with Kaluza-Klein asymptotic behavior. In particular we prove simple connectedness of the quotient of the domain of outer communications by the group of symmetries for models which are asymptotically flat, or asymptotically anti-de Sitter, in a Kaluza-Klein sense. This allows one, e.g., to define the twist potentials needed for the reduction of the field equations in uniqueness theorems. Finally, the methods used to prove the above are used to show that weakly trapped compact surfaces cannot be seen from Scri.Comment: minor correction

Unitarity and Causality in Generalized Quantum Mechanics for Non-Chronal Spacetimes

Spacetime must be foliable by spacelike surfaces for the quantum mechanics of matter fields to be formulated in terms of a unitarily evolving state vector defined on spacelike surfaces. When a spacetime cannot be foliated by spacelike surfaces, as in the case of spacetimes with closed timelike curves, a more general formulation of quantum mechanics is required. In such generalizations the transition matrix between alternatives in regions of spacetime where states {\it can} be defined may be non-unitary. This paper describes a generalized quantum mechanics whose probabilities consistently obey the rules of probability theory even in the presence of such non-unitarity. The usual notion of state on a spacelike surface is lost in this generalization and familiar notions of causality are modified. There is no signaling outside the light cone, no non-conservation of energy, no ``Everett phones'', and probabilities of present events do not depend on particular alternatives of the future. However, the generalization is acausal in the sense that the existence of non-chronal regions of spacetime in the future can affect the probabilities of alternatives today. The detectability of non-unitary evolution and violations of causality in measurement situations are briefly considered. The evolution of information in non-chronal spacetimes is described.Comment: 40pages, UCSBTH92-0

Quantum limits on phase-shift detection using multimode interferometers

Fundamental phase-shift detection properties of optical multimode interferometers are analyzed. Limits on perfectly distinguishable phase shifts are derived for general quantum states of a given average energy. In contrast to earlier work, the limits are found to be independent of the number of interfering modes. However, the reported bounds are consistent with the Heisenberg limit. A short discussion on the concept of well-defined relative phase is also included.Comment: 6 pages, 3 figures, REVTeX, uses epsf.st