Complex spacetime

In 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 multiplication by real numbers to include multiplication by complex numbers. 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 Minkowski space of Special and General Relativity (GR) is a 4-dimensional "pseudo-euclidean" vector space. The spacetime underlying Einstein's Field Equations, which mathematically describe gravitation, is a 4-dimensional "Reimannian manifold", in which the four dimensions of space and time (x, y, z, ict) are regarded as spatial distances, measured in metres (or light-seconds), and fundamentally interchangeable.

In 1919, Theodor Kaluza posted his 5-dimensional extension of General Relativity to Einstein, who was deeply impressed at the natural way in which the equations of electromagnetism emerged from Kaluza's five-dimensional maths. In 1926, Oskar Klein suggested that Kaluza's extra dimension might be "curled up" into an infinitesimal 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°.

For several decades after publishing his General Theory of Relativity in 1915, Einstein tried to unify gravitational field theory with Maxwell's electromagnetism, to create a unified field theory which could explain the gravitational and electromagnetic fields in terms of a universal "gravito-electro-magnetic" field. He was attempting to consolidate the fundamental forces and the elementary particles into a single uniform field theory. His last few scientific papers reveal his valiant struggles to unify GR and QM in Kaluza's 5-dimensional spacetime.

"Scientifically, I am still lagging because of the same mathematical difficulties which make it impossible for me to affirm or contradict my more general relativistic field theory [...]. I will not be able to finish it [the work]; It will be forgotten and at a later time arguably must be re-discovered. It happened this way with so many problems." — Albert Einstein, correspondence with Maurice Solovine, 25 Nov 1948

In 1932, Hsin P. Soh of MIT, advised by Arthur Eddington, published a theory unifying gravitation and electromagnetism within a complex 4-dimensional Reimannian geometry, having real coordinates which were "associated with mass (gravitation), and the imaginary part with charge (electromagnetism)."

In the latter years of WWII, Einstein began considering complex spacetime geometries of various kinds. He eventually settled on a framework with real coordinates for spatial distance, and imaginary coordinates for the electric, magnetic and gravitational force fields, the same 6D complex space used by Maxwell 80 years earlier in his original equations for unifying the electric and magnetic fields (subsequently "simplified" by Oliver Heaviside, so lost to history). In 1945, Einstein published his six-dimensional unified theory of gravitation in complex spacetime.

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 hyper-sphere.

Although the wavefunctions and particles of quantum mechanics (QM) are thought of as inhabiting the exact-same Minkowski space as described by GR, the (configuration) state space in QM is actually a multi-dimensional, complex (Hilbert) vector space. Imaginary numbers fall from every equation, and behind all the mathematics of QM is Euler's formula, "the most remarkable formula in mathematics" according to Richard Feynman:

$e^{ix}=\cos x+i\sin x$

The exact-same equation defines Kaluza-Klein's extra "curled up" spatial dimension, and it is the basis of the complex numbers, and of the unitary group U(1), also known as the circle group, which is the most fundamental symmetry in the universe.