Let us split our Hilbert space into two isomorphic pieces with respect to the above decomposition of the complex structure, W ⊕ ˜W , and complexify the real Hilbert space, naturally we have W ⊗ C ⊕ ˜W ⊗ C.
Large N limit of SO(N) gauge theory of fermions and bosons
If A has a real structure, we can start with a representation of AR on a real Hilbert space HR and then complexify.
The Range of United K-Theory
When K = R, we may also view X as a holomorphic (i.e. complex analytic) vector ﬁeld by complexifying it.
Convergence versus integrability in Poincare-Dulac normal form
More generally, any Coxeter group can be seen as a complex reﬂection group, by complexifying the reﬂection representation.
Explicit presentations for exceptional braid groups
It is necessary to complexify 10H , just as in a supersymmetric theory.
Grand Unification with and without Supersymmetry
It is easy to see that again there is a need to complexify the Higgs ﬁelds, by arguments similar to the case of 10H .
Grand Unification with and without Supersymmetry
Complexifying the Yukawa sector, in turn, leads to the interesting possibility of invoking a Peccei-Quinn symmetry, thus allowing for the axion as a candidate dark matter.
Grand Unification with and without Supersymmetry
Examples are real algebraic subvarieties of Rn which complexify to complex algebraic subvarieties of Cn .
Recent developments in mathematical Quantum Chaos
Note, the symplectic structure ωY is not a part of the original data, and appears only after we complexify the original phase space M .
Quantization via Mirror Symmetry
Our proof of theorem 4 will involve complexifying a and R.
L^p bounds for higher rank eigenfunctions and asymptotics of spherical functions
We do not need to complexify to do this, and so shall think of (Ξ, Y ) as real corespect to (cid:101)X along the submanifold {Ξ = 0} gives (cid:101)X φH (0, Y ) = X f (Y ), where X is the image of (cid:101)X under pro jection to the Y co-ordinate.
L^p bounds for higher rank eigenfunctions and asymptotics of spherical functions
The asymptotic (55) then follows by complexifying R and deforming the contour of integration in (56) in a small ball about e, so that in geodesic normal co-ordinates about e there is a smaller ball in which it is a plane on which Dφh is real and negative deﬁnite.
L^p bounds for higher rank eigenfunctions and asymptotics of spherical functions
A similar situation also arises in the passage from quantum to classical mechanics (the correspondence principle) when one uses the Feynman-Kac device to convert Feynman path integrals to Wiener space integrals by complexifying time, whence the problem reduces to that of small noise behavior of diffusions.
Small Noise Asymptotics for Invariant Densities for a Class of Diffusions: A Control Theoretic View (with Erratum)
We complexify our orbifolds by complexifying the real hyperbolic three-space.
Chen-Ruan orbifold cohomology of the Bianchi groups
By complexifying the variables, we can change the corresponding BPS equations into an l × l(l ≥ 2) system of nonlinear elliptic equations.
A Sharp Existence Theorem for Vortices in the Theory of Branes
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