Krein was at the time still a young mathematician, only two years older than Naimark, but had already built a research group in functional analysis, and they worked together on some Naimark's first works on symmetric and Hermitian forms.
22.
With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an almost Hermitian manifold and is the reason why every almost complex manifold ( in particular every symplectic manifold ) has a Spin c structure.
23.
Lefschetz proved that the line bundle " L ", associated to the Hermitian form " H " is ample if and only if " H " is positive definite, and in this case " L " 3 is very ample.
24.
However, Hermitian forms have basis-independent signature in both the complex and the quaternionic case . ( The real case reduces to the symmetric case . ) A skew-Hermitian form on a complex vector space is rendered Hermitian by multiplication by, so in this case, only is interesting.
25.
However, Hermitian forms have basis-independent signature in both the complex and the quaternionic case . ( The real case reduces to the symmetric case . ) A skew-Hermitian form on a complex vector space is rendered Hermitian by multiplication by, so in this case, only is interesting.
26.
Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is "'Z "'there is a classification similar to the one above using Hermitian forms over the group ring of "'Z " '.
27.
At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts : the real part is symmetric ( orthogonal ), and the imaginary part is skew-symmetric ( symplectic ) and these are related by the complex structure ( which is the compatibility ).
28.
The CFT is called unitary if the space of states has a positive definite Hermitian form such that L _ 0 and \ bar L _ 0 are self-adjoint, L _ 0 ^ \ dagger = L _ 0 and \ bar L _ 0 ^ \ dagger = \ bar L _ 0.
29.
The complex projective orthogonal group, PO ( " n ", "'C "') should not be confused with the projective unitary group, PU ( " n " ) : PO preserves a symmetric form, while PU preserves a hermitian form PU is the symmetries of complex projective space ( preserving the Fubini Study metric ).
30.
It is an axiom of the theory that possible measurement that we can possibly do on the system and get a real number " must " be represented by some Hermitian form Q in the following sense : Measurements in quantum physics are always probabilistic, but the expected value of the measured result when we start from state \ psi is \ tfrac { Q ( \ psi, \ psi ) } { \ langle \ psi | \ psi \ rangle }.
