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Title Page

Contents

ABSTRACT 6

1. INTRODUCTION 7

2. POLYNOMIALS OF SEVERAL VARIABLES 10

2.1. Algebras of Polynomials in several variables 10

2.2. The Accardi-Bożejko isomorphism for polynomials in several variables 11

2.3. The *-algebra Pμ,d[이미지참조] 12

2.4. State and semi-scalar product on P 16

3. THE 3-DIAGONAL JACOBI RELATION ON P 18

3.1. Orthogonal gradation 18

3.2. The symmetric Jacobi relations 22

3.3. The fundamental operators and quantum decomposition of a classical random variable 25

3.4. Symmetric states on P: statistics implies algebra 27

4. COMPLEXIFICATION OF THE MAP X : υ → Xυ[이미지참조] 28

4.1. The momentum operator associated to a symmetric classical random field 28

4.2. The Kernel operator on the *-algebra P(aj+, aj-)[이미지참조] 30

4.3. The standard CCR 32

4.4. Commutation relations for the complexified random fields 33

5. SECOND QUANTIZATION OF LINEAR MAPS VC → VC[이미지참조] 40

5.1. The adjoint of Γⓧ(A)[이미지참조] 46

References 51

초록보기

Based on quantum probability space, we recall the construction of orthogonal polynomials space in multi-dimensional case and review the quantum decomposition and we give an extension of symplectic form in multi-dimensional case. Eventually, using a certain notion of positive-definite operator, we open the problem for the extension of quantum Ornstein Uhlenbeck semigroups in the multi-dimensional case.

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