The graded product of real spectral triples

Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the product of two real spectral triples is problematic: among other drawbacks, it is non-commutative, non-associative, does not transform properly under unitaries, and often fails to define a proper spectral triple. In this paper, we explain that these various problems result from using the ungraded tensor product; by switching to the graded tensor product, we obtain a new prescription where all of the earlier problems are neatly resolved: in particular, the new product is commutative, associative, transforms correctly under unitaries, and always forms a well defined spectral triple.Comment: 15 pages, no figure

Non-Associative Geometry and the Spectral Action Principle

Chamseddine and Connes have argued that the action for Einstein gravity, coupled to the SU(3)\times SU(2)\times U(1) standard model of particle physics, may be elegantly recast as the "spectral action" on a certain "non-commutative geometry." In this paper, we show how this formalism may be extended to "non-associative geometries," and explain the motivations for doing so. As a guiding illustration, we present the simplest non-associative geometry (based on the octonions) and evaluate its spectral action: it describes Einstein gravity coupled to a G_2 gauge theory, with 8 Dirac fermions (which transform as a singlet and a septuplet under G_2). This is just the simplest example: in a forthcoming paper we show how to construct more realistic models that include Higgs fields, spontaneous symmetry breaking and fermion masses.Comment: 24 pages, no figures, matches JHEP versio

A new algebraic structure in the standard model of particle physics

We introduce a new formulation of the real-spectral-triple formalism in non-commutative geometry (NCG): we explain its mathematical advantages and its success in capturing the structure of the standard model of particle physics. The idea, in brief, is to represent $A$ (the algebra of differential forms on some possibly-noncommutative space) on $H$ (the Hilbert space of spinors on that space), and to reinterpret this representation as a simple super-algebra $B=A\oplus H$ with even part $A$ and odd part $H$. $B$ is the fundamental object in our approach: we show that (nearly) all of the basic axioms and assumptions of the traditional real-spectral-triple formalism of NCG are elegantly recovered from the simple requirement that $B$ should be a differential graded $\ast$-algebra (or "$\ast$-DGA"). Moreover, this requirement also yields other, new, geometrical constraints. When we apply our formalism to the NCG traditionally used to describe the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct. In particular, these new constraints provide a novel interpretation of electroweak symmetry breaking that is geometric rather than dynamical. This formalism is more restrictive than effective field theory, and so explains more about the observed structure of the standard model, and offers more guidance about physics beyond the standard model.Comment: 30 pages, no figures, matches JHEP versio

Spinor driven cosmic bounces and their cosmological perturbations

When coupling fermions to gravity, torsion is naturally induced. We consider the possibility that fermion bilinears can act as a source for torsion, altering the dynamics of the early universe such that the big bang gets replaced with a classical non-singular bounce. We extend previous studies in several ways: we allow more general fermion couplings, consider both commuting and anti-commuting spinors, and demonstrate that with an appropriate choice of potential one can easily obtain essentially arbitrary equations of state, including violations of the null energy condition, as required for a bounce. As an example, we construct a model of ekpyrotic contraction followed by a non-singular bounce into an expanding phase. We analyze cosmological fluctuations in these models, and show that the perturbations can be rewritten in real fluid form. We find indications that spinor bounces are stable, and exhibit several solutions for the perturbations. Interestingly, spinor models do not admit a scalar-vector-tensor decomposition, and consequently some types of scalar fluctuations can act as a source for gravitational waves already at linear order. We also find that the first order dynamics are directionally dependent, an effect which might lead to distinguished observational signatures.Comment: 43 pages, 10 figure

The Wavefunction of Anisotropic Inflationary Universes With No-Boundary Conditions

We study the emergence of anisotropic (Bianchi IX) inflationary universes with no-boundary conditions in the path integral approach to quantum gravity. In contrast to previous work, we find no evidence for any limit to how large the anisotropies can become, although for increasing anisotropies the shape of the instantons becomes significantly different from Hawking's original no-boundary instanton. In all cases an inflationary phase is reached, with the anisotropies decaying away. Larger anisotropies are associated with a much larger imaginary part of the action, implying that the highly anisotropic branches of the wavefunction are heavily suppressed. Interestingly, the presence of anisotropies causes the wavefunction to become classical much more slowly than for isotropic inflationary universes. We derive the associated scaling of the WKB classicality conditions both numerically and analytically.Comment: 25 pages, 11 figure

Standard model physics and beyond from non-commutative geometry.

Non-commutative differential geometry (NCG) extends Riemannian geometry and yields a striking reinterpretation of the standard model of particle physics (SM) as gravity on a `non-commutative' manifold. The basic idea behind NCG is to shift focus away from topological spaces and manifolds, to instead focus on the algebra of functions defined over them (``the algebra of coordinates"). This simple idea allows one to explore geometries where one has only the algebra and there is no classical notion of the underlying space whatsoever. In particular, this idea extends to the case in which the input algebra is non-commutative. In this thesis we go a step further: we propose a simple reformulation of the input data corresponding to NCG, which naturally extends to describe geometries which may also be non-associative. The content of our reformulation is as follows: In the traditional approach to NCG, one replaces the usual geometric data of manifolds and metrics {M,g} with `spectral' data held in so called `spectral triples' T = {A,H,D}, consisting of an input algebra A, and a Dirac operator D represented on a Hilbert space H. We show that the data held in a spectral triple may be `fused' together into a larger `fused' algebra B. In this way the various elements of NCG are unified together into a single more fundamental object, while their seemingly unrelated geometric axioms and conditions are re-expressed simply and naturally as the intrinsic properties of ΩB. This approach naturally extends to describe non-associative spaces in the sense that ΩB need not be associative. When ΩB has more general associativity properties, then appropriate generalizations of the associative NCG axioms derive readily from the intrinsic properties of ΩB, allowing for the construction of a wide range of non-associative geometries which we showcase in this work. While our formulation naturally extends to describe non-associative NCG, it also elucidates many aspects of the associative formalism. We show that asking for ΩB to be associative imposes new constraints beyond those traditionally imposed by the NCG axioms. These new constraints resolve a long-standing problem plaguing the NCG construction of the SM, by precisely eliminating from the action a collection of 7 unwanted terms that previously had to be removed by an extra, non-geometric, assumption. We also explain how this same reformulation yields a new perspective on the symmetries of a NCG, which arise simply and naturally as the automorphisms of ΩB. Applying this perspective to the NCG traditionally used to describe the SM we find, instead, an extension of the SM by an extra U(1) B-L gauge symmetry, and a single extra complex scalar field σ, which is a singlet under the SM gauge group, but has B-L=2. The σ field has cosmological implications, and offers a similar solution to the discrepancy between the observed Higgs mass and the NCG prediction as that proposed elsewhere in the literature

Clinical evaluation of the Life Support for Trauma and Transport (LSTAT) platform

INTRODUCTION: The Life Support for Trauma and Transport (LSTAT™) is a self-contained, stretcher-based miniature intensive care unit designed by the United States Army to provide care for critically injured patients during transport and in remote settings where resources are limited. The LSTAT contains conventional medical equipment that has been integrated into one platform and reduced in size to fit within the dimensional envelope of a North Atlantic Treaty Organization (NATO) stretcher. This study evaluated the clinical utility of the LSTAT in simulated and real clinical environments. Our hypothesis was that the LSTAT would be equivalent to conventional equipment in detecting and treating life-threatening problems. METHODS: Thirty-one anesthesiologists and recovery room nurses compared the LSTAT with conventional monitors while managing four simulated critical events. The time required to reach a diagnosis and treatment was recorded for each simulation. Subsequently, 10 consenting adult patients were placed on the LSTAT after surgery for postoperative care in the recovery room. Questionnaires about aspects of LSTAT functionality were completed by nine nurses who cared for the patients placed on the LSTAT. RESULTS: In all of the simulations, there was no clinically significant difference in the time to diagnosis or treatment between the LSTAT and conventional equipment. All clinicians reported that they were able to manage the simulated patients properly with the LSTAT. Nursing staff reported that the LSTAT provided adequate equipment to care for the patients monitored during recovery from surgery and were able to detect critical changes in vital signs in a timely manner. DISCUSSION: Preliminary evaluation of the LSTAT in simulated and postoperative environments demonstrated that the LSTAT provided appropriate equipment to detect and manage critical events in patient care. Further work in assessing LSTAT functionality in a higher-acuity environment is warranted