Complex spacetime

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.

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 that we extend 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 WWII, 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.

There have been attempts to formulate the Dirac equation in complex spacetime by analytic continuation.

See also

• Four vector
• Hilbert space
• Twistor space
• Spherical basis
• Riemann–Silberstein vector

References

Further reading

• Hubert F. M. Goenner (2014). "On the History of Unified Field Theories Part II (ca. 1930 — ca. 1965)". Living Reviews in Relativity. 17 (5).

• Walter, Scott (1999), "The non-Euclidean style of Minkowskian relativity" (PDF), in J. Gray (ed.), The Symbolic Universe: Geometry and Physics 1890-1930, Oxford University Press, pp. 91–127