Shale's Paper

Core idea

Definitions from David Shale's 1962 paper "Linear Symmetries of Free Boson Fields", which characterizes which symplectic transformations are unitarily implementable in bosonic Fock space.

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Symplectic Structures

Basic Definitions

• Single particle structure Σ(H)
• Symplectic form
• Symplectic Hilbert space (K,B)
• Complex structure Λ on K
• Decomposition K = M ⊕ M

Symplectic Groups

• Symplectic group Sp(K)
• Restricted symplectic group rSp(K)

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Operator Ideals and Groups

Schatten Classes

• Hilbert–Schmidt operator
• Trace-class operator
• Determinant on I + trace-class

Perturbation Subgroups

• Shale's subgroups GL(H)₀, GL(H)₁, GL(H)₂
• Restricted general linear group rGL(H)
• Polar decomposition

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Gaussian Measures and L₂ Spaces

Segal's Gaussian Construction

• Tame function (Segal)
• Gaussian measure on a Hilbert space
• Radon–Nikodym derivative
• Jacobian X(T)

Representations on Gaussian L₂

• Segal/Shale representation 𝔘(T)
• Wiener transform W

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Fock Space Quantization

Symmetric Fock Space

• Symmetric tensor product
• Symmetric Fock space S(H)
• Creation and annihilation operators

CCR Quantization

• Weyl CCR quantization
• Fock–Cook quantization
• Duality transform D (Segal)

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Representations and Continuity

Operator Topologies

• Strong vs weak operator topology
• Weak continuity of a representation
• Projective unitary representation

Spectral Theory

• Spectral theorem for compact selfadjoint operators

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C*-Algebras and States

• C*-algebra of field observables 𝔄
• State, pure state, regular state
• Field automorphism θ(T)
• Kadison transitivity