Andreas Boukas
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View article: THE CHARACTERISTIC FUNCTION OF THE CUBE OF A GAUSSIAN RANDOM VARIABLE
THE CHARACTERISTIC FUNCTION OF THE CUBE OF A GAUSSIAN RANDOM VARIABLE Open
Using the spectral resolution of the multiplication operator on the Schwartz class of L2(R,C), we compute the characteristic function of the cube of a Gaussian random variable.
View article: HOLOMORPHIC FUNCTIONAL CALCULUS APPROACH TO THE CHARACTERISTIC FUNCTION OF QUANTUM OBSERVABLES
HOLOMORPHIC FUNCTIONAL CALCULUS APPROACH TO THE CHARACTERISTIC FUNCTION OF QUANTUM OBSERVABLES Open
We show how Cauchy’s Integral Formula and the ideas of Dunford’s Holomorphic Functional Calculus (for unbounded operators) can be used to compute the Vacuum Characteristic Function (Quantum Fourier Transform) of quantum random variables de…
View article: On the Diagonalizability and Factorizability of Quadratic Boson Fields
On the Diagonalizability and Factorizability of Quadratic Boson Fields Open
We provide a necessary and sufficient condition on the coefficient
\nmatrices A,C for the diagonalizability of quadratic fields of the form,
\n\\[X =\\sum_{i,j=1}^n (A_{i,j}a^+_i a^+_j+\\overline{A}_{i,j}a_i a_j+C_{i,j}a^+_i a_j)\\]
\nwher…
View article: Spectral Theorem Approach to the Characteristic Function of Quantum Observables
Spectral Theorem Approach to the Characteristic Function of Quantum Observables Open
We compute the resolvent of the anti-commutator operator $XP+PX$ and of the quantum harmonic oscillator Hamiltonian operator $\frac{1}{2}(X^2+P^2)$. Using Stone's formula for finding the spectral resolution of an, either bounded or unbound…
View article: The $n$-dimensional quadratic Heisenberg algebra as a "non--commutative" $\rm{sl}(2,\mathbb{C})$
The $n$-dimensional quadratic Heisenberg algebra as a "non--commutative" $\rm{sl}(2,\mathbb{C})$ Open
We prove that the commutation relations among the generators of the quadratic Heisenberg algebra of dimension $n\in\mathbb{N}$, look like a kind of \textit{non-commutative extension} of $\hbox{sl}(2, \mathbb{C})$ (more precisely of its uni…
View article: The n-Dimensional Quadratic Heisenberg Algebra as a “Non–Commutative” sl(2,C)
The n-Dimensional Quadratic Heisenberg Algebra as a “Non–Commutative” sl(2,C) Open
We prove that the commutation relations among the generators
\nof the quadratic Heisenberg algebra of dimension $n\\in\\mathbb{N}$, look like a kind of non-commutative extension of sl(2,C) (more precisely of its unique 1–
\ndimensional cen…
View article: Von Neumann's Minimax Theorem for Continuous Quantum Games
Von Neumann's Minimax Theorem for Continuous Quantum Games Open
The concept of a classical player, corresponding to a classical random\nvariable, is extended to include quantum random variables in the form of self\nadjoint operators on infinite dimensional Hilbert space. A quantum version of\nVon Neuma…
View article: Normally Ordered Disentanglement of Multi-Dimensional Schrödinger Algebra Exponentials
Normally Ordered Disentanglement of Multi-Dimensional Schrödinger Algebra Exponentials Open
We derive a normally ordered disentanglement or splitting formula for exponentials of the infinite-dimensional Schrödinger Lie algebra generators.As an application we compute the vacuum characteristic function of a quantum random variable …
View article: On Infinite Stochastic and Related Matrices
On Infinite Stochastic and Related Matrices Open
We study the Lie structure of the set of infinite matrices associated with bounded operators on ℓ∞ with the property that their row sums are constant.That includes, in particular, infinite row stochastic and zerorow-sum matrices.We also co…
View article: Probability distributions and orthogonal polynomials associated with the one-parameter Fibonacci group
Probability distributions and orthogonal polynomials associated with the one-parameter Fibonacci group Open
Starting from F , the matrix generating Fibonacci numbers, we find the one-parameter Lie group generated by F 2 .The matrix elements of the group provide "special functions" identities that include special relationships for Fibonacci and L…
View article: Structure and decompositions of the linear span of generalized stochastic matrices
Structure and decompositions of the linear span of generalized stochastic matrices Open
We study the topological properties of the Lie group of invertible constant row sum matrices and the structure and Levi decomposition of the derived Lie algebra of constant row sum matrices and of the Lie algebra of constant, and in partic…
View article: Contractions and central extensions of Quantum White Noise Lie algebras
Contractions and central extensions of Quantum White Noise Lie algebras Open
We show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN, with the convolution type renormalization δnt − s = δsδt − s of the n≥ 2 powers of the Dirac delta function, can be obtained through a contraction of the Reno…