Real structure explained

\sigma:{C}\to{C}

, with

\sigma(z)={\barz}

, giving the "canonical" real structure on

{C}

, that is

{C}={R} ⊕ i{R}

.

The conjugation map is antilinear:

\sigma(λz)={\barλ}\sigma(z)

and

\sigma(z1+z2)=\sigma(z1)+\sigma(z2)

.

Vector space

\sigma:V\toV

. A real structure defines a real subspace

VR\subsetV

, its fixed locus, and the natural map

VRR{C}\toV

is an isomorphism. Conversely any vector space that is the complexificationof a real vector space has a natural real structure.

One first notes that every complex space V has a realification obtained by taking the same vectors as in the original set and restricting the scalars to be real. If

t\inV

and

t0

then the vectors

t

and

it

are linearly independent in the realification of V. Hence:

\dimRV=2\dimCV

Naturally, one would wish to represent V as the direct sum of two real vector spaces, the "real and imaginary parts of V". There is no canonical way of doing this: such a splitting is an additional real structure in V. It may be introduced as follows.[1] Let

\sigma:V\toV

be an antilinear map such that

\sigma\circ\sigma=idV

, that is an antilinear involution of the complex space V. Any vector

v\inV

can be written

{v=v++v-

}\,, where

v+={1\over{2}}(v+\sigmav)

and

v-={1\over{2}}(v-\sigmav)

.

Therefore, one gets a direct sum of vector spaces

V=V+V-

where:

V+=\{v\inV|\sigmav=v\}

and

V-=\{v\inV|\sigmav=-v\}

.

Both sets

V+

and

V-

are real vector spaces. The linear map

K:V+\toV-

, where

K(t)=it

, is an isomorphism of real vector spaces, whence:

\dimRV+=\dimRV-=\dimCV

.

The first factor

V+

is also denoted by

VR

and is left invariant by

\sigma

, that is

\sigma(VR)\subsetVR

. The second factor

V-

is usually denoted by

iVR

. The direct sum

V=V+V-

reads now as:

V=VRiVR

,

i.e. as the direct sum of the "real"

VR

and "imaginary"

iVR

parts of V. This construction strongly depends on the choice of an antilinear involution of the complex vector space V. The complexification of the real vector space

VR

, i.e.,

VC=VRRC

admits a natural real structure and hence is canonically isomorphic to the direct sum of two copies of

VR

:

VRRC=VRiVR

.

It follows a natural linear isomorphism

VRRC\toV

between complex vector spaces with a given real structure.

A real structure on a complex vector space V, that is an antilinear involution

\sigma:V\toV

, may be equivalently described in terms of the linear map

\hat\sigma:V\to\barV

from the vector space

V

to the complex conjugate vector space

\barV

defined by

v\mapsto\hat\sigma(v):=\overline{\sigma(v)}

.[2]

Algebraic variety

For an algebraic variety defined over a subfield of the real numbers,the real structure is the complex conjugation acting on the points of the variety in complex projective or affine space.Its fixed locus is the space of real points of the variety (which may be empty).

Scheme

For a scheme defined over a subfield of the real numbers, complex conjugationis in a natural way a member of the Galois group of the algebraic closure of the base field.The real structure is the Galois action of this conjugation on the extension of thescheme over the algebraic closure of the base field.The real points are the points whose residue field is fixed (which may be empty).

Reality structure

In mathematics, a reality structure on a complex vector space V is a decomposition of V into two real subspaces, called the real and imaginary parts of V:

V=VRiVR.

Here VR is a real subspace of V, i.e. a subspace of V considered as a vector space over the real numbers. If V has complex dimension n (real dimension 2n), then VR must have real dimension n.

The standard reality structure on the vector space

Cn

is the decomposition

Cn=RniRn.

In the presence of a reality structure, every vector in V has a real part and an imaginary part, each of which is a vector in VR:

v=\operatorname{Re}\{v\}+i\operatorname{Im}\{v\}

In this case, the complex conjugate of a vector v is defined as follows:

\overlinev=\operatorname{Re}\{v\}-i\operatorname{Im}\{v\}

This map

v\mapsto\overlinev

is an antilinear involution, i.e.

\overline{\overlinev}=v,\overline{v+w}=\overline{v}+\overline{w},and \overline{\alphav}=\overline\alpha\overline{v}.

Conversely, given an antilinear involution

v\mapstoc(v)

on a complex vector space V, it is possible to define a reality structure on V as follows. Let
\operatorname{Re}\{v\}=1
2

\left(v+c(v)\right),

and define

VR=\left\{\operatorname{Re}\{v\}\midv\inV\right\}.

Then

V=VRiVR.

This is actually the decomposition of V as the eigenspaces of the real linear operator c. The eigenvalues of c are +1 and -1, with eigenspaces VR and

i

 VR, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.

See also

References

Notes and References

  1. Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29.
  2. Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29.