Universal quantum measurements

We introduce a family of operations in quantum mechanics that one can regard as "universal quantum measurements" (UQMs). These measurements are applicable to all finite dimensional quantum systems and entail the specification of only a minimal amount of structure. The first class of UQM that we consider involves the specification of the initial state of the system—no further structure is brought into play. We call operations of this type "tomographic measurements", since given the statistics of the outcomes one can deduce the original state of the system. Next, we construct a disentangling operation, the outcome of which, when the procedure is applied to a general mixed state of an entangled composite system, is a disentangled product of pure constituent states. This operation exists whenever the dimension of the Hilbert space is not a prime, and can be used to model the decay of a composite system. As another example, we show how one can make a measurement of the direction along which the spin of a particle of spin s is oriented (s = 1/2, 1,...). The required additional structure in this case involves the embedding of CP^1 as a rational curve of degree 2s in CP^2s

Microcanonical distributions for quantum systems

The standard assumption for the equilibrium microcanonical state in quantum mechanics, that the system must be in one of the energy eigenstates, is weakened so as to allow superpositions of states. The weakened form of the microcanonical postulate thus asserts that all quantum states giving rise to the same energy expectation value must be realised with equal probability. The consequences that follow from this assertion are investigated. In particular, a closed-form expression for the density of states associated with any system having a nondegenerate energy spectrum is obtained. The result is applied to a variety of examples, for which the behaviour of the state density, as well as the relation between energy and temperature, are determined. Numerical studies indicate that the density of states converges to a distribution when the number of energy levels approaches infinity

Twistor cosmology and quantum space-time

The purpose of this paper is to present a model of a quantum space-time in which the global symmetries of space-time are unified in a coherent manner with the internal symmetries associated with the state space of quantum-mechanics. If we take into account the fact that these distinct families of symmetries should in some sense merge and become essentially indistinguishable in the unified regime, our framework may provide an approximate description of or elementary model for the structure of the universe at early times

Informed traders

This article is available open access through the publisher’s website at the link below. Copyright @ 2008 The Royal Society.An asymmetric information model is introduced for the situation in which there is a small agent who is more susceptible to the flow of information in the market than the general market participant, and who tries to implement strategies based on the additional information. In this model market participants have access to a stream of noisy information concerning the future return of an asset, whereas the informed trader has access to a further information source which is obscured by an additional noise that may be correlated with the market noise. The informed trader uses the extraneous information source to seek statistical arbitrage opportunities, while at the same time accommodating the additional risk. The amount of information available to the general market participant concerning the asset return is measured by the mutual information of the asset price and the associated cash flow. The worth of the additional information source is then measured in terms of the difference of mutual information between the general market participant and the informed trader. This difference is shown to be non-negative when the signal-to-noise ratio of the information flow is known in advance. Explicit trading strategies leading to statistical arbitrage opportunities, taking advantage of the additional information, are constructed, illustrating how excess information can be translated into profit

Quantum phase transitions without thermodynamic limits

Accepted versio

Stable-1/2 Bridges and Insurance

We develop a class of non-life reserving models using a stable-1/2 random bridge to simulate the accumulation of paid claims, allowing for an essentially arbitrary choice of a priori distribution for the ultimate loss. Taking an information-based approach to the reserving problem, we derive the process of the conditional distribution of the ultimate loss. The "best-estimate ultimate loss process" is given by the conditional expectation of the ultimate loss. We derive explicit expressions for the best-estimate ultimate loss process, and for expected recoveries arising from aggregate excess-of-loss reinsurance treaties. Use of a deterministic time change allows for the matching of any initial (increasing) development pattern for the paid claims. We show that these methods are well-suited to the modelling of claims where there is a non-trivial probability of catastrophic loss. The generalized inverse-Gaussian (GIG) distribution is shown to be a natural choice for the a priori ultimate loss distribution. For particular GIG parameter choices, the best-estimate ultimate loss process can be written as a rational function of the paid-claims process. We extend the model to include a second paid-claims process, and allow the two processes to be dependent. The results obtained can be applied to the modelling of multiple lines of business or multiple origin years. The multi-dimensional model has the property that the dimensionality of calculations remains low, regardless of the number of paid-claims processes. An algorithm is provided for the simulation of the paid-claims processes

A Twistor Description of Six-Dimensional N=(1,1) Super Yang-Mills Theory

We present a twistor space that describes super null-lines on six-dimensional N=(1,1) superspace. We then show that there is a one-to-one correspondence between holomorphic vector bundles over this twistor space and solutions to the field equations of N=(1,1) super Yang-Mills theory. Our constructions naturally reduce to those of the twistorial description of maximally supersymmetric Yang-Mills theory in four dimensions.Comment: 15 pages, typos fixed, published versio

A New Proposal for the Picture Changing Operators in the Minimal Pure Spinor Formalism

Using a new proposal for the "picture lowering" operators, we compute the tree level scattering amplitude in the minimal pure spinor formalism by performing the integration over the pure spinor space as a multidimensional Cauchy-type integral. The amplitude will be written in terms of the projective pure spinor variables, which turns out to be useful to relate rigorously the minimal and non-minimal versions of the pure spinor formalism. The natural language for relating these formalisms is the Cech-Dolbeault isomorphism. Moreover, the Dolbeault cocycle corresponding to the tree-level scattering amplitude must be evaluated in SO(10)/SU(5) instead of the whole pure spinor space, which means that the origin is removed from this space. Also, the Cech-Dolbeault language plays a key role for proving the invariance of the scattering amplitude under BRST, Lorentz and supersymmetry transformations, as well as the decoupling of unphysical states. We also relate the Green's function for the massless scalar field in ten dimensions to the tree-level scattering amplitude and comment about the scattering amplitude at higher orders. In contrast with the traditional picture lowering operators, with our new proposal the tree level scattering amplitude is independent of the constant spinors introduced to define them and the BRST exact terms decouple without integrating over these constant spinors.Comment: 56 pages, typos correcte

Twistor methods for AdS5

We consider the application of twistor theory to five-dimensional anti-de Sitter space. The twistor space of AdS$_5$ is the same as the ambitwistor space of the four-dimensional conformal boundary; the geometry of this correspondence is reviewed for both the bulk and boundary. A Penrose transform allows us to describe free bulk fields, with or without mass, in terms of data on twistor space. Explicit representatives for the bulk-to-boundary propagators of scalars and spinors are constructed, along with twistor action functionals for the free theories. Evaluating these twistor actions on bulk-to-boundary propagators is shown to produce the correct two-point functions.Comment: 24 pages, 4 figures. v2: typos fixed, published versio

N=8 Superspace Constraints for Three-dimensional Gauge Theories

We present a systematic analysis of the N=8 superspace constraints in three space-time dimensions. The general coupling between vector and scalar supermultiplets is encoded in an SO(8) tensor W_{AB} which is a function of the matter fields and subject to a set of algebraic and super-differential relations. We show how the conformal BLG model as well as three-dimensional super Yang-Mills theory provide solutions to these constraints and can both be formulated in this universal framework.Comment: 34 + 10 pages; added references, minor correction