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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

This paper continues the study of paragrassmann algebras begun in Part I with the definition and analysis of Toeplitz operators in the associated holomorphic Segal-Bargmann space. These are defined in the usual way as multiplication by a symbol followed by the projection defined by the reproducing kernel. These are non-trivial examples of spaces with Toeplitz operators whose symbols are not functions and which themselves are not spaces of functions., Comment: 18 pages, continues Part I in arXiv:1204.1033v3, minor updates, final version

Paragrassmann algebras are given a sesquilinear form for which one subalgebra becomes a Hilbert space known as the Segal-Bargmann space. This Hilbert space as well as the ambient space of the paragrassmann algebra itself are shown to have reproducing kernels. These algebras are not isomorphic to algebras of functions so some care must be taken in defining what "evaluation at a point" corresponds to in this context. The reproducing kernel in the Segal-Bargmann space is shown to have most, though not all, of the standard properties. These quantum spaces provide non-trivial examples of spaces which have a reproducing kernel but which are not spaces of functions., Comment: 28 pages, new title and abstract, more clarifications, more corrections, Part II forthcoming

A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space $E$, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.

We present various relations among Versions A, B and C of the Segal-Bargmann transform. We get results for the Segal-Bargmann transform associated to a Coxeter group acting on a finite dimensional Euclidean space. Then analogous results are shown for the Segal-Bargmann transform of a connected, compact Lie group for all except one of the identities established in the Coxeter case. A counterexample is given to show that the remaining identity from the Coxeter case does not have an analogous identity for the Lie group case. A major result is that in both contexts the Segal-Bargmann transform for Version C is determined by that for Version A., Comment: 17 pages

We apply a special case, the restriction principle (for which we give a definition simpler than the usual one), of a basic result in functional analysis (the polar decomposition of an operator) in order to define $C_{\mu, t}$, the $C$-version of the Segal-Bargmann transform, associated to a finite Coxeter group acting in $\mathbb{R}^N$ and a given value $t>0$ of Planck's constant, where $\mu$ is a multiplicity function on the roots defining the Coxeter group. Then we immediately prove that $C_{\mu, t}$ is a unitary isomorphism. To accomplish this we identify the reproducing kernel function of the appropriate Hilbert space of holomorphic functions. As consequences we prove that the Segal-Bargmann transforms for Versions $A$, $B$ and $D$ are also unitary isomorphisms, though not by a direct application of the restriction principle. The point is that the $C$-version is the the only version where a restriction principle, in our definition of this method, applies directly. This reinforces the idea that the $C$-version is the most fundamental, most natural version of the Segal-Bargmann transform., Comment: 27 pages. Major revision. New title and new abstract. The central result now is centrally placed. The secondary results come after that

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal-Bargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Sa\"id and {\O}rsted and independently by Soltani) and the Dunkl heat kernel, due to R\"osler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal-Bargmann analysis on Euclidean space. Hall's results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in $\mu$-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal-Bargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall's Version C generalized Segal-Bargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal-Bargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal-Bargmann kernels for Versions A and C., Comment: Major rewrite. New normalizations. More explanations and references. 37 pages

Referring to quantum mechanics, Einstein used to say "The old one does not play dice." And this is true since the probability of quantum mechanics is not the classical probability of games such as dice. Historically this was the first example of a non-classical probability theory, which we introduce in this expository article using undergraduate linear algebra. There is a short appendix on qubits. Knowledge of quantum mechanics is not required., Comment: 14 pages. Expository article for undergraduate students. In Spanish. Many clarifications. Final version

This note explains how the two measures used to define the $\mu$-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald functions arise in this theory. Also we comment on why it is plausible that only one measure will not work. We follow Bargmann's approach by imposing a condition sufficient for the $\mu$-deformed creation and annihilation operators to be adjoints of each other. While this note uses elementary techniques, it reveals in a new way basic aspects of the structure of the $\mu$-deformed Segal-Bargmann space., Comment: Submitted to the Proceedings of the 11th Workshop on Noncommutative Harmonic Analysis and Applications in Probability held in Bedlewo, Poland in August, 2008

We present an explanation of how the $\mu$-deformed Segal-Bargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echevarria and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of Segal-Bargmann analysis. This explanation relates the $\mu$-deformed Segal-Bargmann transforms to the generalized Segal-Bargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the $\mu$-deformed Segal-Bargmann transform can be understood as Hall type transforms. In particular, we define a $\mu$-deformation of Hall's "Version C" generalized Segal-Bargmann transform which is then shown to be a $\mu$-deformed convolution with a $\mu$-deformed heat kernel followed by analytic continuation. Our results are generalizations and analogues of the results of Hall., Comment: Minor errors corrected. Some more clarifications. Final version

Both direct and reverse log-Sobolev inequalities, relating the Shannon entropy with a $\mu$-deformed energy, are shown to hold in a family of $\mu$-deformed Segal-Bargmann spaces. This shows that the $\mu$-deformed energy of a state is finite if and only if its Shannon entropy is finite. The direct inequality is a new result, while the reverse inequality has already been shown by the authors but using different methods. Next the $\mu$-deformed energy of a state is shown to be finite if and only if its Dirichlet form energy is finite. This leads to both direct and reverse log-Sobolev inequalities that relate the Shannon entropy with the Dirichlet energy. We obtain that the Dirichlet energy of a state is finite if and only if its Shannon entropy is finite. The main method used here is based on a study of the reproducing kernel function of these spaces and the associated integral kernel transform., Comment: Accepted for publication in Infinite Dimensional Analysis, Quantum Probability and Related Topics