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The principal goal of this paper is to pass all quantum probability formulas to the projective space associated to the complex Hilbert space of a given quantum system, providing a more complete geometrization of quantum theory. Quantum events have consecutive and conditional probabilities, which have been used in the author's previous work to clarify 'collapse' and to generalize the concept of entanglement by incorporating it into quantum probability theory. In this way all of standard textbook quantum theory can be understood as a geometric theory of projective subspaces without any special role for the zero-dimensional projective subspaces, which are also called pure states. The upshot is that quantum theory is the probability theory of projective subspaces, or equivalently, of quantum events. For the sake of simplicity the ideas are developed here in the context of a type I factor, but comments will be given about how to adopt this approach to more general von Neumann algebras., Comment: 15 pages, new abstract, new introduction, more references. many clarifications and corrections

Spot pricing is often suggested as a method of increasing demand-side flexibility in electrical power load. However, few works have considered the vulnerability of spot pricing to financial fraud via false data injection (FDI) style attacks. In this paper, we consider attacks which aim to alter the consumer load profile to exploit intraday price dips. We examine an anomaly detection protocol for cyber-attacks that seek to leverage spot prices for financial gain. In this way we outline a methodology for detecting attacks on industrial load smart meters. We first create a feature clustering model of the underlying business, segregated by business type. We then use these clusters to create an incentive-weighted anomaly detection protocol for false data attacks against load profiles. This clustering-based methodology incorporates both the load profile and spot pricing considerations for the detection of injected load profiles. To reduce false positives, we model incentive-based detection, which includes knowledge of spot prices, into the anomaly tracking, enabling the methodology to account for changes in the load profile which are unlikely to be attacks.

Short-term forecasts of energy consumption are invaluable for the operation of energy systems, including low voltage electricity networks. However, network loads are challenging to predict when highly desegregated to small numbers of customers, which may be dominated by individual behaviours rather than the smooth profiles associated with aggregate consumption. Furthermore, distribution networks are challenged almost entirely by peak loads, and tasks such as scheduling storage and/or demand flexibility maybe be driven by predicted peak demand, a feature that is often poorly characterised by general-purpose forecasting methods. Here we propose an approach to predict the timing and level of daily peak demand, and a data fusion procedure for combining conventional and peak forecasts to produce a general-purpose probabilistic forecast with improved performance during peaks. The proposed approach is demonstrated using real smart meter data and a hypothetical low voltage network hierarchy comprising feeders, secondary and primary substations. Fusing state-of-the-art probabilistic load forecasts with peak forecasts is found to improve performance overall, particularly at smart-meter and feeder levels and during peak hours, where improvement in terms of CRPS exceeds 10%.

The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting non-commutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of entire functions, where polynomials in one complex variable form a dense subspace. The complex coordinate naturally acts as an unbounded multiplication operator generating, together with its adjoint, a highly non-commutative *-algebra of operators. The Toeplitz operators are then geometrically constructed as special elements from this algebra; they are associated to the symbols from another quadratic non-commutative algebra, which is interpretable as polynomials over a plane to be quantized. Such a conceptual framework promotes interesting non-trivial conditions on the initial scalar product. These are analyzed in detail. Various illustrative examples are computed., Comment: 46 Classical Pages

In this treatise I introduce the time dependent Generalized Born's Rule for the probabilities of quantum events, including conditional and consecutive probabilities, as the unique fundamental time evolution equation of quantum theory. Then these probabilities, computed from states and events, are to be compared with relative frequencies of observations. Schrodinger's equation still is valid in one model of the axioms of quantum theory, which I call the Schrodinger model. However, the role of Schrodinger's equation is auxiliary, since it serves to help compute the continuous temporal evolution of the probabilities given by the Generalized Born's Rule. In other models, such as the Heisenberg model, the auxiliary equations are quite different, but the Generalized Born's Rule is the same formula (covariance) and gives the same results (invariance). Also some aspects of the Schrodinger model are not found in the isomorphic Heisenberg model, and they therefore do not have any physical significance. One example of this is the infamous collapse of the quantum state. Other quantum phenomena, such as entanglement, are easy to analyze in terms of the Generalized Born's Rule without any reference to the unnecessary concept of collapse. Finally, this leads to the possibility of quantum theory with other sorts of auxiliary equations instead of Schrodinger's equation, and examples of this are given. Throughout this treatise the leit motif is the central importance of quantum probability and most especially of the simplifying role of the time dependent Generalized Born's Rule in quantum theory., Comment: 117 pages, much more on entanglement, one new section, minor changes, minor errors corrected

The definition due to Accardi of a pair of complementary observables is adapted to the context of the Lie algebra $ su(2) $. We show that the pair of Pauli matrices $ A,B $ associated to the unit directions $ \alpha $ and $ \beta $ in $ \mathbb{R}^{3} $ are Accardi complementary if and only if $ \alpha $ and $ \beta $ are orthogonal if and only if $ A $ and $ B $ are orthogonal. In particular, any pair of the standard triple of Pauli matrices is complementary., Comment: 6 pages, clearer exposition, new abstract, one new reference

In the theory of Toeplitz quantization of algebras, as developed by the second author, coherent states are defined as eigenvectors of a Toeplitz annihilation operator. These coherent states are studied in the case when the algebra is the generically non-commutative Manin plane. In usual quantization schemes one starts with a classical phase space, then quantizes it in order to produce annihilation operators and then their eigenvectors and eigenvalues. But we do this in the opposite order, namely the set of the eigenvalues of the previously defined annihilation operator is identified as a generalization of a classical mechanical phase space. We introduce the resolution of the identity, upper and lower symbols as well as a coherent state quantization, which in turn quantizes the Toeplitz quantization. We thereby have a curious composition of quantization schemes. We proceed by identifying a generalized Segal-Bargmann space SB of square-integrable, anti-holomorphic functions as the image of a coherent state transform. Then SB has a reproducing kernel function which allows us to define a secondary Toeplitz quantization, whose symbols are functions. Finally, this is compared with the coherent states of the Toeplitz quantization of a closely related non-commutative space known as the paragrassmann algebra., Comment: 24 pages, minor revisions, final version

In this note we quantize the free $ * $-algebra generated by finitely many variables, which is a new example of the theory of Toeplitz quantization of $ * $-algebras as developed previously by the author. This is achieved by defining Toeplitz operators with symbols in that non-commutative free $ * $-algebra. These are densely defined operators acting in a Hilbert space. Then creation and annihilation operators are introduced as special cases of Toeplitz operators, and their properties are studied., Comment: 10 pages. Portions of this work previously appeared in Section 8 of arXiv:1308.5454v2, but not in the final version arXiv:1308.5454v4. That Section was incomplete and erroneous. This new article fixes and completes that work. For the Proceedings of Operator Algebras, Toeplitz Operators and Related Topics, (OATORT) held in Boca del Rio, Veracruz, Mexico in November, 2018

The author has introduced in a recent paper a new class of operators, called co-Toeplitz operators, with symbols in a co-algebra. This is the categorical dual to Toeplitz operators which have symbols in an algebra. The mapping from a symbol to its co-Toeplitz operator gives a quantization scheme, called co-Toeplitz quantization. A new, quite simple particular case of co-Toeplitz quantization is introduced in this note. Examples are given in order to show some of its properties., Comment: 6 pages; submitted to the Proceedings of the Workshop on Geometric Methods in Physics (WGMP) XXXVII, Bialowieza, Poland, 2018

The Dunkl operators associated to a necessarily finite Coxeter group acting on a Euclidean space are generalized to any finite group using the techniques of non-commutative geometry, as introduced by the authors to view the usual Dunkl operators as covariant derivatives in a quantum principal bundle with a quantum connection. The definitions of Dunkl operators and their corresponding Dunkl connections are generalized to quantum principal bundles over quantum spaces which possess a classical finite structure group. We introduce cyclic Dunkl connections and their cyclic Dunkl operators. Then we establish a number of interesting properties of these structures, including the characteristic zero curvature property. Particular attention is given to the example of complex reflection groups, and their naturally generalized siblings called groups of Coxeter type., Comment: New example using Cuntz algebras, final version, 44 pages

We define co-Toeplitz operators, a new class of Hilbert space operators, in order to define a co-Toeplitz quantization scheme that is dual to the Toeplitz quantization scheme introduced by the author in the setting of symbols that come from a possibly non-commutative algebra with unit. In the present dual setting the symbols come from a possibly non-co-commutative co-algebra with co-unit. However, this co-Toeplitz quantization is a usual quantization scheme in the sense that to each symbol we assign a densely defined linear operator acting in a fixed Hilbert space. Creation and annihilation operators are also introduced as certain types of co-Toeplitz operators, and then their commutation relations provide the way for introducing Planck's constant into this theory. The domain of the co-Toeplitz quantization is then extended as well to a set of co-symbols, which are the linear functionals defined on the co-algebra. A detailed example based on the quantum group (and hence co-algebra) $SU_q(2)$ as symbol space is presented., Comment: Error in Prop, 9.2 corrected, re-write of Section 9, An Example: SU_q(2); 40 pages

This note is a follow-up to a recent paper by the author. Most of that theory is now realized in a new setting where the vector space of symbols is not necessarily an algebra nor is it equipped with an inner product, although it does have a conjugation. As in the previous paper one does not need to put a measure on this vector space. A Toeplitz quantization is defined and shown to have most of the properties as in the previous paper, including creation and annihilation operators. As in the previous paper this theory is implemented by densely defined Toeplitz operators which act in a Hilbert space, where there is an inner product, of course. Planck's constant also plays a role in the canonical commutation relations of this theory., Comment: 12 pages

Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group $SU_q(2)$ is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck's constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum) objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck's constant and a Hilbert space where natural, densely defined operators act., Comment: 35 pages, a new title and a new example using the quantum group $SU_q(2)$

We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which are defined as Toeplitz operators. This paper extends results of the author for the finite dimensional case., Comment: 30 pages