The canonical commutation relations and the local geometry of symplectic spaces = 정준교환관계와 사교공간의 국소 기하학

다국어 초록 (Multilingual Abstract)

While the Darboux theorem implies there are no local symplectic invariants, many results in quantization suggest there is a necessity to make local choices on symplectic manifolds. We study how representations of the canonical commutation relations arise as a description of local symplectic geometry. As a result, a new family of irreducible representations is obtained. While analytic problems remain, this family unifies known families, extends the parameters describing equivalent representations, and exhibits topologically nontrivial configurations of representations. The unifying framework is provided geometrically, by a partition of the complex Lagrangian Grassmannian induced by complex conjugation.

국문 초록 (Abstract)

다르부의 정리에 의해, 사교공간은 국소적으로 불변량을 가지지 않는다는 사실이 잘 알려져 있다. 하지만 많은 양자화 문제의 해법들에 의하면, 사교공간의 국소적인 구조들을 선택해야할 필요성이 제기된다. 이 논문에서 우리는 정준교환관계의 표현이 사교공간의 국소적인 성질을 기술하는 방법으로서 어떻게 나타나는지 연구한다. 그 결과로 정준교환관계의 기약표현들의 새로운 모임을 얻는다. 해석학적인 문제들이 남아있지만, 이 모임은 기존에 알려진 표현들의 모임들을 취합하고, 동형인 표현들의 매개집합을 확장하며, 위상적으로 자명하지 않은 표현들의 배열이 존재함을 보여준다. 서로 다른 표현들의 모임을 취합하는 구조가 기하학적으로 주어진다는 것도 주목할 점이다.

목차 (Table of Contents)

• 1 Introduction 1
• 2 Symplectic vector spaces and their complexification 9
• 2.1 Symplectic vector spaces 9
• 2.2 Darboux bases 12
• 2.3 Subspaces of symplectic vector spaces 16
• 2.4 Compatible complex structures 22
• 2.5 Complex Lagrangian subspaces 26
• 2.6 Real projections 32
• 2.7 The partition of the complex Lagrangian Grassmannian 36
• 3 Representation theory of the Heisenberg group 49
• 3.1 Translations in symplectic vector spaces 50
• 3.2 Universal enveloping algebras 53
• 3.3 Hilbert spaces and unitary operators 58
• 3.4 Unbounded operators and adjoints 60
• 3.5 Direct integral decompositions of strongly continuous unitary representations 62
• 3.6 Differentiation and exponentiation of representations on Hilbert spaces 66
• 4 Construction of representations 69
• 4.1 Transverse pairs of complex Lagrangian subspaces 71
• 4.2 Bilinear forms 73
• 4.3 Construction of representations 76
• 4.4 Construction of representations on Hilbert spaces 81
• 5 Reconstruction of known representations 85
• 5.1 Maximal compact subgroups 86
• 5.2 Siegel upper half planes and compatible complex structures 90
• 5.3 Representations from new parameters 93
• 5.4 Schrodinger representation 96
• 5.5 Lion-Vergne’s family 97
• 5.6 Fock-Segal-Bargmann space 99
• 5.7 Grossmann-Daubechies’ family 100
• 5.8 Satake’s family 101
• 5.9 Mumford’s family 103

참고문헌 (Reference)

1 N. S. Poulsen, "On the canonical commutation relations", 32 112 { 122, 1973

2 J . Von Neumann , On rings of operators, "Reduction theory , Annals of Mathematics", Vol . 50 , No . 2 401-485, 1949

3 R. Lipsman, "Representation theory of almost connected groups", Vol 42 No 2 453-467, 1972

4 R. T. Moore ,, "Exponentiation of operator Lie algebras on Banach spaces", vol . 71 , 903 - 908, 1965

5 R. Howe, "On the role of the Heisenberg group in harmonic analysis", Vol 3 . No 2 ., 1980

6 F. I. Mautner ,, "Unitary Representations of Locally Compact Groups II , Annals of Mathematics", Vol . 52 , No . 3 528 - 556 (, 1950

7 I. Daubechies, "Coherent states and projective representation of the linear canonical transformations", 21 , 1377, 1980

8 J. R. Klauder and I. Daubechies, "Quantum mechanical path integrals with Wiener measures for all polynomial hamiltonians", Vol 52 , No . 14 , 1161-1154, 1984

9 P. L. Robinson and J. H. Rawnsley, "The metaplectic representation , Mpc strucures and geometric quantization , Memoirs of the American Mathematical Society", Vol 81 No 410, 1989