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".

During the late 1860's, James Clerk Maxwell's ideas about electromagnetism gradually became more mathematically complex. His spacetime geometry contained two imaginary dimensions to accommodate the electric potential E and the magnetic field H, plus another imaginary dimension for the gravitational potential V, all three being mathematically orthogonal to real Euclidean space. In his discussion of the findings of the great electromagnetic experimentalist Michael Faraday, these comprised six spatial dimensions (three real and three imaginary).

"I am getting converted to Quaternions, and have put some in my book, in a heretical form..." — James Clerk Maxwell, correspondence with Prof. Lewis Campbell, 19 Oct 1872

In his scientific description of electromagnetism, Maxwell used what he called a "heretical form" of quaternion algebra, which explicitly separated the three imaginary dimensions from the real part. He stated emphatically that tensors and vectors were inadequate mathematical tools to correctly encapsulate the electromagnetic fields and forces. He also quietly discussed with colleagues how one might detect and measure "non-observable" or "hidden" spatial dimensions, which he conceived of as "storing energy", both kinetic and potential, in the elastic fabric of space itself.

"The peculiarity of our space is that of its three dimensions, none is before or after another. As is x, so is y, and so is z. If you have 4 dimensions, this becomes a puzzle. For first, if three of them are in our space, then which three? Also, if we lived in space of m dimensions, but were only capable of thinking n of them, then first, which n? Second, if so, things would happen requiring the rest to explain them, and so we should either be stultified or made wiser. I am quite sure that the kind of continuity which has four dimensions all co-equal, is not to be discovered by merely generalising Cartesian space equations." — James Clerk Maxwell, in correspondence with C.J. Monro, Esq., 15 Mar 1871

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 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, that is, 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 4d spacetime quantities are complex-valued, while the spatial parts of these quantities are ordinary 3d vectors. The usual space x, y, z and time t coordinates themselves are real.

During the final decade of his life, Einstein (and his assistant Ernst G. Straus) struggled to comprehend the implications of unifying electromagnetism and gravity in a spacetime with six-dimensional (3r+3i) complex geometry. Their main difficulty was eliminating the peculiar negative kinetic energy or reverse-time "ghosts" that their theory produced. Einstein considered such "states of negative norm" to be physically impossible, although the mathematics of his unified field theory appeared to be thoroughly robust in every other respect.

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.

"I really do not yet know, whether this new system of [complex] equations has anything to do with physics. What justly can be claimed only is that it represents a consequent generalization of the gravitational equations for empty space." — Albert Einstein, correspondence with Erwin Schrödinger, 6 Mar 1947

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 infinite-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.