In mathematics and mathematical physics, complex
spacetime extends the traditional notion of
spacetime described by real-valued space and
time coordinates to complex-valued space and
time coordinates.
The notion is entirely mathematical with no
physics implied, but should be seen as a tool,
for instance, as exemplified by the Wick rotation.
== Real and complex spaces ==
=== 
Mathematics ===
The complexification of a real vector space
results in a complex vector space (over the
complex number field).
To "complexify" a space means extending ordinary
scalar multiplication of vectors by real numbers
to scalar multiplication by complex numbers.
For complexified inner product spaces, the
complex inner product on vectors replaces
the ordinary real-valued inner product, an
example of the latter being the dot product.
In mathematical physics, when we complexify
a real coordinate space Rn we create a complex
coordinate space Cn, referred to in differential
geometry as a "complex manifold".
The space Cn can be related to R2n, since
every complex number constitutes two real
numbers.
A complex spacetime geometry refers to the
metric tensor being complex, not spacetime
itself.
=== Physics ===
The Minkowski space of special relativity
(SR) and general relativity (GR) is a 4-dimensional
"pseudo-Euclidean space" vector space.
The spacetime underlying Einstein's field
equations, which mathematically describe gravitation,
is a real 4-dimensional "Pseudo-Riemannian
manifold".
In QM, wave functions describing particles
are complex-valued functions of real space
and time variables.
The set of all wavefunctions for a given system
is an infinite-dimensional complex Hilbert
space.
== History ==
The notion of spacetime having more than four
dimensions is of interest in its own mathematical
right.
Its appearance in physics can be rooted to
attempts of unifying the fundamental interactions,
originally gravity and electromagnetism.
These ideas prevail in string theory and beyond.
The idea of complex spacetime has received
considerably less attention, but it has been
considered in conjunction with the Lorentz–Dirac
equation and the Maxwell equations.
Other ideas include mapping real spacetime
into a complex representation space of SU(2,
2), see twistor theory.In 1919, Theodor Kaluza
posted his 5-dimensional extension of general
relativity, to Albert Einstein, who was impressed
with how the equations of electromagnetism
emerged from Kaluza's theory.
In 1926, Oskar Klein suggested that Kaluza's
extra dimension might be "curled up" into
an extremely small circle, as if a circular
topology is hidden within every point in space.
Instead of being another spatial dimension,
the extra dimension could be thought of as
an angle, which created a hyper-dimension
as it spun through 360°.
This 5d theory is named Kaluza–Klein theory.
In 1932, Hsin P. Soh of MIT, advised by Arthur
Eddington, published a theory attempting to
unifying gravitation and electromagnetism
within a complex 4-dimensional Reimannian
geometry.
The line element ds2 is complex-valued, so
that the real part corresponds to mass and
gravitation, while the imaginary part with
charge and electromagnetism.
The usual space x, y, z and time t coordinates
themselves are real and spacetime is not complex,
but tangent spaces are allowed to be.For several
decades after publishing his general theory
of relativity in 1915, Einstein tried to unify
gravity with electromagnetism, to create a
unified field theory explaining both interactions.
In the latter years of World War II, Einstein
began considering complex spacetime geometries
of various kinds.In 1953, Wolfgang Pauli generalised
the Kaluza–Klein theory to a six-dimensional
space, and (using dimensional reduction) derived
the essentials of an SU(2) gauge theory (applied
in QM to the electroweak interaction), as
if Klein's "curled up" circle had become the
surface of an infinitesimal hypersphere.
In 1975, Jerzy Plebanski published "Some Solutions
of Complex Einstein Equations".There have
been attempts to formulate the Dirac equation
in complex spacetime by analytic continuation.
== See also ==
Construction of a complex null tetrad
Four vector
Hilbert space
Twistor space
Spherical basis
Riemann–Silberstein vector
