Spinors in Hilbert SpaceInfinite-dimensional Clifford algebras and their Fock representations originated in the quantum mechanical study of electrons. In this book the authors give a definitive account of the various Clifford algebras over a real Hilbert space and of their Fock representations. A careful consideration of the latter's transformation properties under Bogoliubov automorphisms leads to the restricted orthogonal group. From there, a study of inner Bogoliubov automorphisms enables the authors to construct infinite- dimensional spin groups.Apart from assuming a basic background in functional analysis and operator algebras, the presentation is self-contained with complete proofs, many of which offer a fresh perspective on the subject. The book will therefore appeal to a wide audience of graduate students and researchers in mathematics and mathematical physics. |
Common terms and phrases
A+[V adjoint algebra map anticommutation antilinear automorphism of C(V Bogoliubov automorphism C+(V C+[V central linear functional Clifford algebra C[V Clifford map Clifford relations commutes complete orthonormal system complex Clifford algebra complex Hilbert space creators and annihilators cyclic decomposition define denote dimension elements even-dimensional fact finite rank Fock representation Fock space follows grading automorphism Hilbert-Schmidt HJ(V homomorphism implies inequivalent infinite-dimensional inner product space irreducible isometric isomorphism J-linear J-vacuum ker g ker g+I kernel left regular representation lies linear span Neumann algebra nonzero norm Note odd-dimensional orthogonal transformation orthonormal basis orthonormal system parity precisely Proof rank and ker real Hilbert space real inner product regular representation result scalar self-adjoint spin group subspace Theorem trace-class transformation g unique unit vector unitarily equivalent unitary implementer unitary isomorphism unitary operator unitary structure universal mapping property vN Clifford algebra von Neumann algebra whence
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