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In this paper, we explore the consequences of diffeomorphism invariance in generally covariant theories. Such theories in two and three dimensions are known to admit topological excitations called geons. It is shown by specific examples... more
In this paper, we explore the consequences of diffeomorphism invariance in generally covariant theories. Such theories in two and three dimensions are known to admit topological excitations called geons. It is shown by specific examples that a quantum state of two identical geons may not be an eigenstate of the geon exchange operator, which means that a geon may have no definite statistics. As shown before by Sorkin and as discussed further here, it may also happen, for instance, that in 3 + 1 dimensions a tensorial (spinorial) geon obeys Fermi (Bose) statistics, while in 2 + 1 dimensions an "integral-spin" geon can obey "fractional statistics". Thus ideas on spin and statistics borrowed from Poincaré invariant theories are not always valid in quantum gravity, at least without further physical inputs. Previously, it was shown by Friedman and Sorkin that pure gravity in three space dimensions may admit spinorial states. This result is extended to two dimensions where pure gravity is shown to admit "fractional spin" geons.
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There exists a class of particle-like topological excitations in generally covariant theories called geons, discussed by Friedman and Sorkin, and by these authors, and others. Here, we show by specific examples that certain of these geons... more
There exists a class of particle-like topological excitations in generally covariant theories called geons, discussed by Friedman and Sorkin, and by these authors, and others. Here, we show by specific examples that certain of these geons can be so quantized that they are characterized by no definite statistics. For instance, three-dimensional geons may be neither bosons nor fermions (nor paraparticles). It can also happen, as pointed out before by Sorkin, and as we briefly discuss here, that a tensorial (spinorial) goen obeys Fermi (Bose) statistics. Our usual conceptions about the statistics of particle species thus do not seem to be valid in generally covariant theories, at least without further physical inputs such as, perhaps, the possibility of topology change.
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Research Interests: Measure Theory, Stochastic Process, Physics, Quantum Physics, Quantum Mechanics, and 9 moreQuantum Measurement Theory, Interpretation of Quantum Mechanics, Brownian Motion, Quantum and classical statistical mechanics, Interpretations of Quantum Mechanics, Quantum Process, Consistent Histories, Quantum Probability, and Open Quantum System
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We establish, for the quantum system made up of a single free particle, the formula ΔE Δt≳(v/c) ħ, where ΔE is the precision to whichE can be ascertained in time Δt. The measurement can be carried out with zero disturbance inE itself.
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A gauge-invariant and microscopically causal theory of the sourceless electromagnetic field in a topologically arbitrary globally hyperbolic asymptotically flat background metric is proposed. The topology manifests itself in an algebra of... more
A gauge-invariant and microscopically causal theory of the sourceless electromagnetic field in a topologically arbitrary globally hyperbolic asymptotically flat background metric is proposed. The topology manifests itself in an algebra of superselected quantities, one of which is the net (apparent) charge. Relative to a particular Cauchy hypersurface, the field resolves into a 'Coulombic' part generating the above algebra and a 'radiative' part expressible in terms of photon creation and destruction operators. An appendix extends 'Hodge theory' to non-compact, but asymptotically-flat three-manifolds.
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The generalization of Stokes' theorem to non-orientable manifolds shows that a suitable topology can appear to carry net electromagnetic charge. By treating this as the origin of electric charge in nature one explains the... more
The generalization of Stokes' theorem to non-orientable manifolds shows that a suitable topology can appear to carry net electromagnetic charge. By treating this as the origin of electric charge in nature one explains the non-existence of magnetic monopoles.
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We cast the flat space theory of a scalar field in generally covariant form by introducing an auxiliary field λ. The resulting theory is couched in terms of an action integral S, and all the fields (the scalar, the space–time metric, and... more
We cast the flat space theory of a scalar field in generally covariant form by introducing an auxiliary field λ. The resulting theory is couched in terms of an action integral S, and all the fields (the scalar, the space–time metric, and λ) are dynamical in the sense of being varied freely in S. Conservation of energy–momentum emerges as a formal consequence of diffeomorphism invariance, in close analogy with the situation in ordinary general relativity.
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Research Interests: Mathematics, Field Theory, Set Theory, Physics, Quantum Gravity, and 15 moreGeneral Relativity, Quantum Mechanics, Quantum Field Theory, Medicine, Gravity, Physical sciences, Space Time, Classical Limit, Length scale, Spacetime, Lorentzian Beta Kenmotsu Manifold, Coarse Grained Soil, Curse of Dimensionality, Partial Order, and Causal Structure
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The Kaluza-Klein idea assimilates gauge fields to gravity by comprehending gauge-transformations as diffeomorphism (“coordinate transformations”). Although this can be done for any choice of gauge group, the original U(1)-theory has the... more
The Kaluza-Klein idea assimilates gauge fields to gravity by comprehending gauge-transformations as diffeomorphism (“coordinate transformations”). Although this can be done for any choice of gauge group, the original U(1)-theory has the distinction of requiring only a single fundamental field: it is nothing but 5-dimensional gravity with a specific choice of metric and topology to represent the vacuum. Since you have already heard this theory described, I will sketch it only in enough detail to fix notation.
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Why should we be interested in the effects of spacetime topology, and specifically in the microscopic spatial topology I shall discuss in these lectures? One reason is just that it might be there, in which case it is bound to be... more
Why should we be interested in the effects of spacetime topology, and specifically in the microscopic spatial topology I shall discuss in these lectures? One reason is just that it might be there, in which case it is bound to be important. Another reason is that topology — in particular the topology of fields defined on ordinary Minkowski space — has proved to be important in contexts closely related to that of quantum gravity. Thus, such topology enters essentially into the structure of “Skyrmeons” (models for baryons such as the nucleon), and also of the magnetic-monopole solutions in non-abelian gauge theories.
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We model a black hole spacetime as a causal set and count, with a certain definition, the number of causal links crossing the horizon in proximity to a spacelike or null hypersurface Σ. We find that this number is proportional to the... more
We model a black hole spacetime as a causal set and count, with a certain definition, the number of causal links crossing the horizon in proximity to a spacelike or null hypersurface Σ. We find that this number is proportional to the horizon’s area on Σ, thus supporting the interpretation of the links as the “horizon atoms” that account for its entropy. The cases studied include not only equilibrium black holes but ones far from equilibrium.
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ABSTRACT We describe a Markov Chain Monte Carlo algorithm which can be used to generate naturally labeled n-element posets at random with a probability distribution of one’s choice. Implementing this algorithm for the uniform... more
ABSTRACT We describe a Markov Chain Monte Carlo algorithm which can be used to generate naturally labeled n-element posets at random with a probability distribution of one’s choice. Implementing this algorithm for the uniform distribution, we explore the approach to the asymptotic regime in which almost every poset takes on the three-layer structure described by Kleitman and Rothschild (KR). By tracking the n-dependence of several order-invariants, among them the height of the poset, we observe an oscillatory behavior which is very unlike a monotonic approach to the KR regime. Only around n = 40 or so does this “finite size dance” appear to give way to a gradual crossover to asymptopia which lasts until n = 85, the largest n we have simulated.
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A sleeping bag for camping use which serves alternately as a temporary garment on a cold morning after awaking, and until the user has started a fire. The bag is formed with a pair of foldable sleeve sections each joined to the body of... more
A sleeping bag for camping use which serves alternately as a temporary garment on a cold morning after awaking, and until the user has started a fire. The bag is formed with a pair of foldable sleeve sections each joined to the body of the bag below the normal elbow position of the wearer and a pair of foldable leg sections, each joining to the body of the bag below the normal knee position of the wearer, with pockets externally fastened to the bag for retention of the arm and leg sections when the sleeper is encased in the bag. An elastic loop is fixed to each of the two lower corners of the bag. The bag is fitted with a conventional front slide type fastener extending vertically from the crotch to the neck section and formed with a helmet section for enclosing the head of the wearer.
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Why should we be interested in the effects of spacetime topology, and specifically in the microscopic spatial topology I shall discuss in these lectures? One reason is just that it might be there, in which case it is bound to be... more
Why should we be interested in the effects of spacetime topology, and specifically in the microscopic spatial topology I shall discuss in these lectures? One reason is just that it might be there, in which case it is bound to be important. Another reason is that topology — in particular the topology of fields defined on ordinary Minkowski space — has proved to be important in contexts closely related to that of quantum gravity. Thus, such topology enters essentially into the structure of “Skyrmeons” (models for baryons such as the nucleon), and also of the magnetic-monopole solutions in non-abelian gauge theories.