The orthogonal group of a Form Hilbert space

In 1913, É. Cartan discovered that the special orthogonal groupSO(k) has a ‘two-valued’ representation (i.e. a projective representation) on a complex vector spaceSof dimension 2n, wherek= 2nor 2n+ 1. The projective representation in question lifts to a true representation of the double cover Spin (k) ofSO(k). We restrict attention to the casek= 2n. Under the action of Spin (2n),Sbreaks up into 2 irreducible subspaces:The vectors inSare calledspinors(relative toSO(2n)), those inS+orS−are calledhalf-spinors(4).

Download Full-text

SYMPLECTIC SPACE OR ORTHOGONAL SPACE OF n QUBITS

International Journal of Quantum Information ◽

10.1142/s0219749911008064 ◽

2011 ◽

Vol 09 (06) ◽

pp. 1449-1457

Author(s):

JIAN-WEI XU

Keyword(s):

Hilbert Space ◽

Orthogonal Group ◽

Symplectic Group ◽

Orthonormal Basis ◽

Mathematical Structure ◽

Symplectic Space ◽

Orthogonal Space ◽

Spin Flip

In Hilbert space of n qubits, we introduce symplectic space (n odd) or orthogonal space (n even) via the spin-flip operator. Under this mathematical structure we discuss some properties of n qubits, including homomorphically mapping local operations of n qubits into symplectic group or orthogonal group, and proving that the generalized "magic basis" is just the biorthonormal basis (i.e. the orthonormal basis of both Hilbert space and the orthogonal space). Finally, a demonstrated example is given to discuss the application in physics of this mathematical structure.

Download Full-text

Ergodic Subgroups of the Orthogonal Group on a Real Hilbert Space

Annals of Mathematics ◽

10.2307/1970001 ◽

1957 ◽

Vol 66 (2) ◽

pp. 297 ◽

Cited By ~ 4

Author(s):

I. E. Segal

Keyword(s):

Hilbert Space ◽

Orthogonal Group ◽

Real Hilbert Space

Download Full-text

The Finsler geometry of groups of isometries of Hilbert Space

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700028202 ◽

1987 ◽

Vol 42 (2) ◽

pp. 196-222 ◽

Cited By ~ 6

Author(s):

C. J. Atkin

Keyword(s):

Hilbert Space ◽

Lie Algebra ◽

Orthogonal Group ◽

Finsler Geometry ◽

Operator Norm ◽

Full Description ◽

Infinite Dimensional ◽

Infinite Dimensional Hilbert Space ◽

Finsler Structure ◽

Groups Of Isometries

AbstractThe paper deals with six groups: the unitary, orthogonal, symplectic, Fredholm unitary, special Fredholm orthogonal, and Fredholm symplectic groups of an infinite-dimensional Hilbert space. When each is furnished with the invariant Finsler structure induced by the operator-norm on the Lie algebra, it is shown that, between any two points of the group, there exists a geodesic realising this distance (often, indeed, a unique geodesic), except in the full orthogonal group, in which there are pairs of points that cannot be joined by minimising geodesics, and also pairs that cannot even be joined by minimising paths. A full description is given of each of these possibilities.

Download Full-text