Talk:Complex spacetime

Style note
Note to self (and collaborators): KISS —keep it simple! Aim to inform, educate, and entertain (a little). This article isn't a mathematical physics textbook, nor a research paper aimed at pro physics dudes. Link everything that's technical, math, or physics terminology. Minimise jargon. Be kind to teens, curious non-physicists, and yer Mom. Zen mind, beginner's mind. Keep it as simple as possible, but no simpler. Conversational tone, not a college lecture. It should be intelligible to anyone with a decent high-school science education. Imagine you're chatting with a bright, interested teenager. Show your enthusiasm. Have fun. Be bold! Rjowsey (talk) 05:01, 9 June 2015 (UTC)

What's the metric (and geometry)?
Real Minkowski space is characterized by the Minkowski metric, given by the line element dx² + dy² + dz² - c² dt². So, what's the metric for complex Minkowski space? We need references on this. If its line element is |dx|² + |dy|² + |dz|² - c² |dt|² then that's equivalent to a 6+2 dimensional metric obtained by taking the real and imaginary parts of each of the coordinates separately. If, on the other hand, it is ½ ((dx² + dy² + dz² - c² dt²) + (dx² + dy² + dz² - c² dt²)*), then it's equivalent to a 4+4 dimensional metric in which Re(x), Re(y), Re(z) and Im(t) are space-like and Im(x), Im(y), Im(z) and Re(t) are time-like.

If the line element is just dx² + dy² + dz² - c² dt², then that contains the above-mentioned 4+4 metric as its real part, and a symplectic form as its imaginary part. The situation is analogous to how the Hilbert space inner product is decomposed, which forms the basis of the Kahler manifold. Related links to Kahler manifolds might be relevant to this article, if any can be found.

These are two different geometries and only the 4+4 one has a Euclidean 4D geometry contained in it, which is relevant if this topic is linked in any way to Euclideanization. Different possibilities, including the two just mentioned as special cases, are discussed in this arXiv reference https://arxiv.org/pdf/1601.03862.pdf. — Preceding unsigned comment added by 2603:6000:AA4D:C5B8:222:69FF:FE4C:408B (talk) 04:35, 3 December 2020 (UTC)

Complex Dimensional Analysis
Excerpts from conversations on the talk page for the Dimensional Analysis article which provide some relevant background information for this topic:

Regarding original research:
 * Complex Dimensional Analysis is a toolkit which is proving extremely useful to theoretical/particle physicists seeking to resolve the profound impasses of complexification and the unification of GR with QM. As for my material being "original research", you'll note that every statement of bare fact has been carefully referenced to published and/or peer-reviewed work, i.e. "reliable, published sources", as required by the Wikipedia policy. Kindly let me know if anything I've written needs further citations — I'll dig them out forthwith. Wherever I have re-packaged or re-interpreted established physics, e.g. the Planck Units, I have done so (I hope) in a completely transparent manner, such that a 9th-grader could follow the mathematical logic. In fact, such students are my best proof-readers and style critics; adults tend not to tell me when they don't have a clue what I'm gabbing on about. The gifted young people whom I tutor are the next generation of scientists, and regularly use Wikipedia for clarification of classwork, lab experiments and lectures. They appreciate that I am going to the bother of writing out my somewhat "unconventional" understanding of fundamental physics; in fact, it was they who encouraged me in this direction. I'll try cutting to the chase (complex DA → 6D special relativity) a bit faster, so you'll see where all this deep dimensional reduction is heading. Thanks for the feedback. Rjowsey (talk) 05:05, 1 June 2015 (UTC)

Regarding concerns as to the relevance and applicability of Deep Dimensional Reduction:
 * I've given a bit more thought to your problem(s) with the L/T material, and I think (hope) I've understood the issue(s) you might be having with the framework. I've been using this 2-dimensional space/time matrix (or fabric) for a while now, so its usefulness is profoundly obvious to me, and to the youngsters I'm tutoring. Perhaps if I explain how I'm using this math framework, its value might be more obvious to you. When teaching the fundamentals of physics, I introduce kids to the Planck units by having them substitute Maxwell's L3/T2 into the SI dimensions of all the Planck quantities. They can knock that off in about 15 minutes. Then I suggest they try arranging their "2D" units into a (log-log) space/time matrix, according to indices, which takes them about 10 minutes, and they're quite delighted at how simple it all is. Then I show them how to multiply and divide, demonstrating with a couple basic equations, e.g. F = ma, and E = mc2. Then I let 'em rip, discovering more patterns and the relationships between quantities for themselves, e.g. find all the ways quantities can multiply or divide to get units of force, energy, or power. They love it! And they learn the Planck units in about 1/10th the time compared to the "textbook" approach. Then we do the same process with the principal electromagnetic quantities. They're quickly able to see the relationship of potentials to field strength, etc. They can intuitively grasp the concept of rates of flow, flux, current, etc, and relate shifting across cells, or downwards, with differentiation, gradients and charge density. They enjoy discovering the fundamental expressions relating the quantities, e.g. V x I = P, even a few of Maxwell's equations, by themselves. Because they're curious, and interested, and they're learning by finding patterns, using simple mathematical tools they built for themselves! By generalising into a 4D volume (tessaract) based on quaternions, i.e. 1 dimension of real space, 2 dimensions of imaginary space, and 1 dimension of imaginary time (ct), they learn how to do vector projections from flat Minkowski space onto 3 imaginary planes. From there we can proceed easily into a discussion about special relativity, time dilation and Lorentz contraction, with the aid of a purpose-built simulator (animated graphics software). Then we delve into General Relativity, using the same animated 4D computer graphics, so absent any mind-numbing tensors, metrics, Kronecker deltas, Christoffel gammas and partial differential equations. From there we can introduce the Dirac equation (using a bispinor wavefunction simulator), thence to phase and quantum spin. KISS! It's a very natural, effortless progression, and they end up with a really solid grasp of foundational principles. These kids use Wikipedia constantly, so they badgered me to write something about the math framework they'd learned, since they found it so much more straight-forward than the standard textbook approach. I hope that helps allay your concerns about the relevance of this L/T spacetime fabric to the DA article. I'm definitely not pushing any kind of wingnut "theory" — this is just a simplified math framework which has proved really useful. If you feel that complex DA doesn't fit in here, or is irrelevant to DA, perhaps it would be better to migrate it into a new article. They say you learn best when trying to teach something; it's even more true when one is trying to write about it! BTW, thanks for the additions and corrections here and there, much appreciated. :D Rjowsey (talk) 02:29, 2 June 2015 (UTC)

Special Relativity in 5D hyperspace (6D spacetime)

 * Regarding special relativity, here's an animation in 6D hyperspace showing a test particle of Planck mass being accelerated to the speed of light, where it has Planck momentum and Planck kinetic energy (or would have, but for gravitational radiation). The Minkowski diagram at top-left shows the "4D Lorentz rotation" of the moving frame of reference inhabited by the particle. The top-right projection shows time dilation approaching infinity at light speed, and the bottom-left projection shows length contraction in the moving frame of reference.
 * https://upload.wikimedia.org/wikipedia/commons/1/11/6D_Special_Relativity.gif
 * The Lorentz factor is the inverse-cosine of the phase angle (0 < φ < π/2), i.e. γ = 1/cos(φ), and the ratio of the particle's velocity to light speed is β = v/c = sin(φ). Time dilation and length contraction simplify to τᵩ = t∙cos(φ) and ʀᵩ = r∙cos(φ). The y-axis in the bottom-right projection represents the imaginary component of the particle's kinetic energy, while the x-axis represents the imaginary component of its potential energy mc2, in units of Planck energy (EP).


 * Thus, the particle's total energy Eᵩ is a function of √((t∙sin(φ))2 + (r∙sin(φ))2). At light speed, the particle's rest-mass energy and its momentum can be seen as inhabiting two imaginary spatial dimensions. A Planck mass at velocity v = c has total energy of √2∙EP, assuming no losses due to gravitational radiation. The particle's matter-wave now has a Compton wavelength λᵩ = unit Planck length, and is oscillating at Planck frequency. Rjowsey (talk) 04:23, 10 June 2015 (UTC)

Quantum wavefunctions in complex spacetime

 * Soak yourselves in this graphic for a while: https://upload.wikimedia.org/wikipedia/commons/3/34/SpinZero.gif — It's the 6D wavefunction of the Higgs boson, in supercooled vacuum. Top-left quadrant is the classic 4D Minkowski diagram; time is up the y-axis, space lies along the x-axis. The bottom-left projection pertains to momentum, aka kinetic energy; the top-right projection pertains to imaginary time and potential energy (kinda-sorta). The bottom-right quadrant shows the cross-product of the kinetic and potential energy sinusoidals. This bispinor represents ground-state spacetime, an elemental vibrating space/time string, aka "spin-foam". Then let's talk some more about time, and superposition, and probabilistic truthiness... :D Rjowsey (talk) 22:49, 5 June 2015 (UTC)


 * BTW, unity density is Planck density, and 0.25 density is an event horizon, like you'd find around a black hole. By way of contrast, have a looksee at the wavefunction for a spin-2 boson, which has phase offset of PI radians (180°), and whose bispinor, you'll notice, never oscillates into real time and real space at the same time. https://upload.wikimedia.org/wikipedia/commons/6/62/SpinTwo.gif — It's a CPT-inverted Higgs, and although purely imaginary, actually exists, because gravity. Rjowsey (talk) 04:54, 6 June 2015 (UTC)


 * The ground state of a canonical spin-1 boson's wavefunction, with phase offset of π/2 (↑-spin, R chirality), oscillates like so: https://upload.wikimedia.org/wikipedia/commons/e/eb/SpinOneUp.gif Its CPT-inverted (↓-spin, L-chiral) sibling, with a phase offset of 3π/2, oscillates like this: https://upload.wikimedia.org/wikipedia/commons/b/be/SpinOneDown.gif In 6D spacetime geometry, these "wavicles" manifest as massless photons propagating in null-time. At the next energy level, thus inhabiting an 8D or (3r+5i)-dimensional geometry, these bispinor wavefunctions manifest as gluons, (metaphorically) arranging themselves at the 8 vertices of a cuboctahedron, therefore congruent with String/E8 Theory. But that's way down the rabbit hole... ;-) Rjowsey (talk) 08:13, 6 June 2015 (UTC)


 * Finally, there's the spin-half fermions, e.g. the ground-state massless neutrino (my favourite invisible lepton). There are two L- and R-chiral fermions, plus two L/R-chiral anti-fermions, so the 4 fermionic wavefunctions having phase offsets of nπ/4 (where n = 1, 3, 5, or 7) oscillate like so:
 * https://upload.wikimedia.org/wikipedia/commons/1/10/SpinHalfUp.gif 
 * https://upload.wikimedia.org/wikipedia/commons/e/e7/SpinHalfUpAnti.gif 
 * https://upload.wikimedia.org/wikipedia/commons/9/98/SpinHalfDownAnti.gif 
 * https://upload.wikimedia.org/wikipedia/commons/4/46/SpinHalfDown.gif 

The manifold of spacetime points

 * I tried to unpack your questions starting with "the manifold of spacetime points", as if I were tutoring a bunch of 16-year-olds. And that led me to the core concept in Einstein's interpretation of Poincaré's work on the Lorentz transformation, which Einstein conceptualised as the "mixing" of space and time, in special relativity and in GR. I've always felt uneasy about that notion, because I couldn't explain it simply, to kids, without complicated math and a bunch of hand-waving. Ditto for GR, where we find a 4D Minkowski "manifold", and a "metric", and x0 being rotated into xmumble. That same "space-time mixing" thing. And it works, perfectly well, right up to the event horizon of a black hole. Einstein realised, when Schwarzschild's solution predicted an impossible singularity inside a black hole, that his general relativity was "incomplete". Then Kaluza and Klein came along with a nifty 5D solution for GR, out of which fell Maxwell's equations, quite naturally. Einstein spent the next 30 years trying to find a way to fix his theory of relativity, and, reading his last few papers, he came very close. Meanwhile, Pauli, Dirac, Schrödinger, et al were burrowing into QM, using Hilbert's complex configuration/state space, which is very similar, mathematically, to the quaternion-based (3r+3i) space that Maxwell was (sorta-kinda) using behind his electromagnetics equations. Kids don't know anything about "manifolds", so they never ask that question. However, they intuitively understand what a "point of spacetime" is, so I begin there, by describing an infinitesimal volume of space, a Planck volume. It's utterly empty (vacuum), and absolutely cold (zero degrees K). It has Planck length (~10—43 metres) in x, y and z dimensions; you'd probably want to label those x1, x2, and x3, because you're thinking in a more abstract coordinate frame. Then you'd probably add a time dimension, labelled x0, as did I until a few years ago, when I found another way to understand special relativity. Geometrically. In Maxwell's 3r+3i space, the Lorentz rotation becomes a phase angle φ, on an imaginary plane, and β = v/c becomes sin(φ). The Lorentz gamma becomes cos(φ), because sin2φ + cos2φ = 1. There's no "mixing" of space into time. Space is what rulers measure, and time is what clocks measure. These fundamental dimensions don't get mixed up, so we don't actually need any kind of "manifold". Plus, that geometrical solution works perfectly all the way through a BH's event horizon, and keeps on working perfectly, all the way down to φ = 2π (= 0). The fundamental "manifold" is simply empty, flat, vacuum, with 5 dimensions. If we put some mass into this space, its phase angle increases as φ/2π = (Gm/rc2)2, which 16-year-olds can easily use to calculate what you'd think of as "curvature". In fact, they're able to derive that equation from first principles, from Newton's Law of gravity, once I show them how φ goes to π/2 at the Schwarzschild radius. They can understand that Einstein's "spacetime curvature" is actually an illusion at the "conformal boundary" of 5-dimensional space, because we can only observe its 3-dimensional surface. I explain to the kids that this is like a 3-dimensional hologram, because they want analogies, and examples, and simple animated graphics. Then, if we introduce some energy into this space, we can see that a harmonic oscillation happens, that there's an observable vibration in the fabric of spacetime, at some frequency f = E/h. If I were teaching this stuff to kids, I'd also mention that if we squeeze an entire Planck unit of energy EP (a Gigajoule) into this Planck volume of vacuum, we just made ourselves an infinitesimal black hole, because they love that idea! I'd also point out that the oscillation of this microscopic black hole is at Planck frequency ωP = EP/h (~1043 Hz), because that would make sense to them. The Dirac equation tells us that this fundamental harmonic oscillation has (at least) 2 orthogonal sinusoidal waveforms (functions of eiθ), one pertaining to kinetic energy, the other to potential energy. But these equations need 5 spatial dimensions and one of time, so I'd introduce Maxwell's ideas about imaginary spatial dimensions (which exist, but aren't exactly "real"). The analogy I'd use might be that of a sphere passing through a flat 2D surface, or I'd show them an animated tesseract, because that would help them visualise an imaginary volume "behind" or "inside" every point in space. Since now we have a simple 6-dimensional spacetime, they're able to understand those wavefunction animations, like ripples on a pond, like vibrations of a field. The unified one that Einstein was looking for. Rjowsey (talk) 04:20, 7 June 2015 (UTC)

Regarding time as an imaginary dimension:
 * In a 6D framework, time is what clocks measure. In Einstein's 4D Riemannian geometry, time is dimensioned as a distance (ct), and "spacetime" is a unitary object. Thus, a Lorentz rotation appears to mix time into space, to explain time dilation and length contraction. That's the way I learned to think about special and general relativity, and it's taken me many years to unravel that idea, and learn how to think in 5 spatial dimensions, evolving through time. In a (5+1)D geometry, distance is measured in metres, and time returns to being just time, measured in seconds. Time is what clocks measure; time is not a distance. The way I'd explain this to 16-year-olds would use the analogy of a world map, like the one on the wall in their classroom. Comparing the size of, say, Antarctica on this map to the continent at the bottom of a spherical globe, we can see that its size has become highly distorted when the mapmaker "projected" the surface of a sphere onto a flat piece of paper. This exact same distortion happens when we reduce the imaginary spatial dimensions of a 5D hypersphere into 3D Euclidean space, at the boundaries or edges we call an "event horizon". We see the distortion (at such conformal boundaries) as a warping of spacetime. This illusory effect happens at the "horizons" of our 3D hologram, so time appears to stop at the event horizon of a black hole, and conversely, time appears to be exponentially accelerated at the event horizon we call the big bang (according to the math at time-zero). The same distortion can be seen at a temperature near absolute zero, when time appears to stop, so light stops moving. This event horizon marks out a conformal boundary between our positive energy/forward time "3D universe" and a negative energy/reverse time "5D universe". Just like a hologram with blurry edges. That's the only way I can understand what the math is saying.


 * My main point is that 5D space is perfectly flat in every dimension, so simple Euclidean geometry rules. All that complicated partial differential calculus and those metrics and manifolds and tensors and Christoffel gammas and index acrobatics of Einstein-Minkowski's GR will (eventually) be replaced by a much simpler mathematical framework, based in pure geometry, just as Einstein's intuition told him was in there somewhere. He was right. Rjowsey (talk) 04:30, 8 June 2015 (UTC)

Regarding the metric, particles, fields and gravity:
 * The hyperspace I'm describing is flat in 5 spatial dimensions, two of which are (mathematically) imaginary, so they are unmeasurable by any real ruler; they're non-observable, hidden, invisible. They're not compactified, nor rolled up. Time is just imaginary time, so it, not ict. We inhabit 3D space, moving through time, and we observe event horizons at the conformal boundaries, which are demarcation horizons between our 3D reality and 5D hyperspace. There's also a 7D hyperspace, where we find the quarks and gluons, inhabiting a cuboctahedral geometry (see E8 Theory for details). Then there's a 9D hyperspace, and so on, apparently ad infinitum. Gravity is the complex conjugate of time in n-dimensional hyperspace. When a gravitational "field" couples to mass, it imparts negative momentum, so that gravitating masses attract other masses, just like positive charges impart negative momentum to negative charges via the electrostatic "field". Alternatively, we can conceptualise this energy as being transferred via "virtual particles", e.g. gravitons and photons. These concepts are equivalent, just as "particles" and "waves" are equivalent concepts. The 6D metric you seek probably looks more like √(Δx12 + Δx22 + Δx32 ± Δx42 ± Δx52), but I'm focussed on unification and QM at present, and such metrics aren't particularly relevant to 6D geometrical algebra. I'm giving you details of this hyperspatial math, e.g. those wavefunction animations for the Higgs, leptons, photons, anti-leptons and the graviton. I don't have a "theory"; I have a mathematical framework, based on Maxwell's mathematics and the Planck Units, which appears to be a thoroughly robust foundation for the main currently-accepted theories of physics, viz. electromagnetism, special and general relativity, and quantum mechanics. It's also congruent with Loop Quantum Gravity, String Theory, the Holographic Hypothesis, E8 Theory, etc. Rjowsey (talk) 06:42, 8 June 2015 (UTC)

James Clerk Maxwell
As a physics history geek (and one-time cognitive psychologist), I'm deeply interested in how certain concepts, styles of thinking, and mathematical techniques/tools have, over time, particularly the past century, brought us to an impasse, the so-called "crisis in physics".

In this "Complex Spacetime" topic, I intend to trace this history in some depth. Beginning with the little-known fact that Maxwell's ideas about spacetime were instinctively complex, i.e. his geometry had two invisible imaginary dimensions to accommodate the E (electric) and H (magnetic) potentials, plus another for gravity (the complex conjugate of time), all three being orthogonal to real Euclidean space. In his mind, and in his math, these comprised 6 distinct and different spatial dimensions. Thus, he used a "heretical form" of quaternion math, stating emphatically that tensors and vectors were inadequate for encapsulating his 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 very fabric of space itself.

His quaternion notation was eliminated from the Treatise at the insistence of his publisher, over his strenuous objections, because very few people could understood the math. His 20 electromagnetic equations were then "simplified" by Oliver Heaviside, who dispensed with Maxwell's potentials, stating them to be "mystical" and "metaphysical". Richard Feynman restored the electromagnetic potentials to physics some 50 years later, stating them to be essential to Quantum Electrodynamics (QED) theory.

Minkowski was a mathematician, not a physicist, so he pressured Einstein to eliminate the imaginary unit from General Relativity. Thereby, for the past 100 years, while Quantum Mechanics dived deeply into complexity, Special and General Relativity remained stuck with a 4-dimensional approximation to complex spacetime, the "Minkowski space". Which works extremely well (although the math is hideously complicated), right up to the event horizon of black holes, where it goes to infinity, and results in physically impossible singularities at the center. It's also broken at the instant of the "big bang", the beginning of spacetime itself, time-zero.

It will be shown that 21st century unified physics must be based on a 6D complex spacetime, the mathematics of which seamlessly merge Relativity with Quantum Mechanics, at all scales. From such a complex geometry (plus phasors), electromagnetism naturally falls out, the Planck units fall out, special and general relativity fall out, and quantum spin falls out, including the spin-0 and spin-2 boson. Rjowsey (talk) 00:43, 9 June 2015 (UTC)

Mostly units?
Why is this page mostly about units? We already have articles on SI units, Planck units, natural units, etc...

Also, dimensional analysis of any physical quantity is trivial, it follows from the defining equation of the quantity, and generalizations to more/less space and/or time dimensions is also trivial. The real and imaginary parts of a complex-valued quantity (e.g. wave function) have the same units/dimensions, because both are real numbers (we're not including the unitless factor of i in the definition of "imaginary part" right?), and the modulus of the complex-valued quantity makes no sense otherwise.


 * Correct, the imaginary unit is dimensionless. Rjowsey (talk) 23:54, 8 June 2015 (UTC)

I'm not nominating for deletion, but fail to see the use of this article so far. If it had mathematical context, say ℂ3 with defined metrics and other things I don't know, followed by possible physical applications, all with reliable secondary sources, then it would be an interesting and worthwhile article. M&and;Ŝc2ħεИτlk 15:34, 8 June 2015 (UTC)


 * Early days yet. There's lots more work to do on this topic. Rjowsey (talk) 23:54, 8 June 2015 (UTC)


 * Yes I know, just saying it would be ideal to cut out the tables of units, link to the main units articles where detailed tables should be, so there is more room for the content of this article. M&and;Ŝc2ħεИτlk 10:45, 9 June 2015 (UTC)


 * Very good point. The tables are there for total math transparency, especially for the 16-year-olds. It's clutter, but there's no other tabulation on WP for an L/T dimensional reduction of the fundamental units. I'll certainly think some more about how to get rid of them. Any suggestions welcomed. Cheers. Rjowsey (talk) 10:53, 9 June 2015 (UTC)

I had a look at the "selected papers". These are solid, but there is a huge gap between these papers treating the AdS/CFT correspondence between superstring/M-theory in 10-11 spacetime dimensions (with a given background metric and curled up dimensions) and a conformal field theory on a flat boundary of that spacetime, and the 3 complex spacetime dimensions of Maxwell. I can see this article evolve into something nice (it's nicely written), but it doesn't tell a consistent story as it stands. It does not explain where the "imaginary" dimensions come from. True, it's probably convenient to work with the complex plane or higher dimensional analogues in conformal field theory. But this does not make spacetime complex. References are needed addressing the reader intermediate between the sixteen-year-old and the mature M-theory expert.


 * I agree. I began the discussion with deep dimensional analysis, because L/T analysis of the Planck Units demonstrates unequivocally that 2 extra spatial dimensions exist, which a priori must be orthogonal to 3D spatial dimensions, therefore they're mathematically imaginary. Thus, "imaginary" spatial dimensions arise axiomatically from the mathematics of a 5D hyperspace; the "imaginary time" dimension comes from Einstein [1905]. By definition, a spacetime with imaginary dimensions is complex. The idea of imaginary "higher" dimensions is certainly not a new one. I vaguely recall first hearing about the concept (of 6D hyperspace) some 40-odd years back, from a postgrad tutor who was reading an (unpublished) manuscript of Pauli's, while doing lit review for his thesis, something to do with spinning black holes (under our Head of Dept, Prof Roy Kerr, who's world famous in NZ). Pauli's hyperspace ansatz was apparently an extension, or generalisation, of Kaluza-Klein theory, but (I think) using imaginary dimensions instead of compactified or cylindrical dimensions. Or maybe that was my tutor's idea, or possibly even Prof Kerr's (can't recall exactly). The notion of imaginary spatial dimensions was originally David Hilbert's, I think, but it never went mainstream. I understand that Gödel tried to convince Einstein to use imaginary dimensions in his unified field math, but the old fella wouldn't have a bar of it. Not "ponderable" enough, I guess. Rjowsey (talk) 23:54, 8 June 2015 (UTC)


 * This,
 * ...because L/T analysis of the Planck Units demonstrates unequivocally that 2 extra spatial dimensions exist, which a priori must be orthogonal to 3D spatial dimensions, therefore they're mathematically imaginary.
 * illustrates my concerns. It is, to me, an ad hoc argument with loose connections to the conclusion (imaginary spacetime). Note that none of the "selected papers" talk about imaginary spacetime. They talk about real dimensions. They also talk about compactified dimensions, something you tossed off as belonging to the dustbin on Q's talk page;


 * First, forget compactified dimensions — that's an very obsolete notion that belongs in the dustbin of history.
 * Your assurances here on the talk page that things unequivocally follows is not a good substitute for rigor and references. It is not my intention to be overly negative. I do think the topic deserves an article, but, again, as it stands it doesn't tell a consistent story.


 * It's not an ad hoc argument, it's just math. Those 2 extra spatial dimensions have to be imaginary, mathematically. They're Wick-rotated from the 3D Euclidean x,y,z axes. All those 5 spatial dimensions have to be orthogonal, to each other. Maxwell realised that, which is why he "converted" to quaternions. Damn, I'm trying to put 20+ years of reading physics history into a few "selected papers" and "further readings", to hook into with the literature. You asked. Give me time... :-) Rjowsey (talk) 10:18, 9 June 2015 (UTC)


 * I'm just getting warmed up! Total newb to WP. Thanks for your help, btw. Appreciated. Rjowsey (talk) 10:18, 9 June 2015 (UTC)


 * You probably get the dimensions of units right. This has no "obvious" implication for the dimension of spacetime. If so, it would have been acknowledged hundreds of years ago. (As you write in the article, Maxwell's ideas weren't acknowledged.) Again, your assurances that it is "just math" wont pass the test. You need solid references from this century - or at least, the past century. YohanN7 (talk) 11:00, 9 June 2015 (UTC)


 * Yep. you're right. I'm working on it. Any help would be gratefully accepted. Cheers! Rjowsey (talk) 11:18, 9 June 2015 (UTC)


 * And oh, write a lead YohanN7 (talk) 09:52, 9 June 2015 (UTC)


 * Oh yes, I'm on it! See current intro. So much more to do. The abstract is always the hardest part, after the conclusion. :D Rjowsey (talk) 10:18, 9 June 2015 (UTC)

The major part deals with units. Expressing what strikes the mind in units of length and time, one finds that the "highest power of length" that is needed (why not more? e.g. time rate of change of power takes you out of the matrix) is $L^{5}$. Hence we need six-dimensional spacetime ($L^{0}$ through $L^{5}$). But we only see three dimensions. Ergo, the rest are "imaginary". The Maxwell equations can be formulated in these three complex dimensions. Is this the general idea? A more recent reference than Maxwell is probably needed.

A citation for the following would also be appreciated:
 * The space/time matrix represents a complex 6-dimensional Hilbert space, with internal symmetries corresponding to the SU(3) × SU(2) × U(1) unitary group, consistent with the Standard Model.

YohanN7 (talk) 22:14, 8 June 2015 (UTC)


 * Yes, that's the general idea. Maxwell used quaternion math because his understanding of the EM (and gravitational) fields was that they inhabited 3 imaginary "extra dimensions" in space. I'll dig around in Feynman's QED, try to find you a more recent reference. And yes (the intuition is strong with this one), there must be more than 2 imaginary spatial dimensions, e.g. 7 spatial dimensions are needed for wavefunctions with 3 phase angles, such as quarks and gluons. Ad infinitum, I suspect. Regarding the assertion that this "matrix represents a complex 6-dimensional Hilbert space", which part of this needs a citation? I agree it definitely needs further elaboration. Rjowsey (talk) 23:54, 8 June 2015 (UTC)
 * On reflection, it'd probably be better to pull that reference to Hilbert state space for the time being, since (while it's blindingly obvious to me) it needs further elucidation and/or citations. Tip o' the hat. Rjowsey (talk) 00:52, 9 June 2015 (UTC)


 * The $SU(3) × SU(2) × U(1)$ group does not act directly on the spacetime manifold (whatever its dimension) as far as I know. It is the Lorentz group that does (or the diffeomorphism group in GR). YohanN7 (talk) 11:08, 9 June 2015 (UTC)


 * SU(3) × SU(2) × U(1) symmetry is applicable to complex 6D spacetime. Rjowsey (talk) 21:53, 9 June 2015 (UTC)


 * Quantum Electrodynamics is an abelian gauge theory with the symmetry group U(1). The mathematics of the bispinor inhabits a complex 5D space having 3 real and 2 imaginary spatial dimensions (which exist, but aren't REAL). Spacetime is complex, any way you look at it! Oh wait, unless you're Minkowski, and it's 1910... Rjowsey (talk) 23:47, 9 June 2015 (UTC)

Intro (Lead)

 * The Minkowski space of special and general relativity (GR) is a "pseudo-real" vector space, since while the time dimension of special relativity is explicitly imaginary, the "time dimension" x0 of GR is not.

This is blatantly incorrect. The "imaginary" time referred to is a reference to a choice of coordinates that is made to hide from sight the fact that the metric is indefinite. Such a choice is highly impractical in GR. See MTW proclaiming the "death of $ict$". That is all there is to it.


 * Not quite. Read Minkowski_space. Rjowsey (talk) 02:18, 11 June 2015 (UTC)


 * Exactly so. The imaginary time coordinates makes the indefinite quadratic form appear as a positive definite form and Lorentz transformations as ordinary rotations. They aren't. YohanN7 (talk) 09:48, 11 June 2015 (UTC)

You implicitly ascribe to the wave function amplitude physical reality. The physical reality of it is highly debatable, and it is more commonly than not believed to be non-measurable. Physicists aren't happy at all to ascribe physical reality to non-measurable quantities. (Exceptions include the Dirac sea, but such ideas have not survived.) YohanN7 (talk) 11:31, 9 June 2015 (UTC)


 * I don't understand which part of my statement is "blatantly incorrect". Einstein's 1905 special relativity explicitly defined the time dimension as imaginary, viz. √-1ct. Then in 1915, he defined the time dimension as x0. The imaginary unit has disappeared, thanks to Minkowski. Hence, 100 years of confusion about what's real and what's imaginary, plus a whole bunch of bewildered bickering about "choice of coordinates", "correct interpretation", etc. Complex 6D spacetime coordinates don't make any sense in 3D Euclidean space, nor in 4D Minkowski space. Only the real parts that can be detected and measured make any sense: the observables. The universe contains gazillions of tiny N-dimensional objects (where N = 5,7,9,...) that we conceptualise as quantum wavefunctions, or particles, or wavicles — what kind of 4D coordinate system and metric should we use to describe vast numbers of those little things? What coordinate system should we use inside a black hole? At the instant of the big bang, time-zero? Are quantum particles "real things", or are they infinitesimal half-imaginary/half-real ephemera flickering in and out of reality? What kind of 4D spacetime manifold or metric can possibly make sense when the Dirac equation shows us that a quantum of 6D space exists in both real and imaginary dimensions: https://upload.wikimedia.org/wikipedia/commons/3/34/SpinZero.gif Rjowsey (talk) 21:15, 9 June 2015 (UTC)


 * To a hypothetical 5D creature, an electron is a solid real thing, inhabiting a volume; it's a perfectly ordinary spherical ball. To a 3D creature, it's an infinitesimal point, with some kind of weird "quantum spin" going on, and "entanglement" happening, not to mention "zitterbewegung". To a 4D creature, a tesseract is just a cube. To a 3D creature, a tesseract doesn't really make much sense. To a 2D creature, a sphere is some kind of bizarre circular-shaped thing, which seems to grow from a point, then shrink, then disappear into some kind of imaginary dimension, which can't be real... Rjowsey (talk) 22:04, 9 June 2015 (UTC)

Status of article
User:Rjowsey has been adding links to this article, suggesting that it should be regarded as satisfying WP criteria and is ready for scrutiny. I have consequently drawn broader attention to this at WT:PHY. I feel that the article, as it stands, would fail several criteria, and should at best be considered a draft. —Quondum 14:10, 10 June 2015 (UTC)


 * A message-box kept appearing in the article header reminding me that this article is an "orphan". I commented it out, then removed it (it was distracting), but it kept returning. Pesky little bot! So I followed its advice, and put a few "see also" links into articles which seemed pertinent to some of my article's content, which I note you've reverted. Grace Hopper said: "It's easier to ask forgiveness than it is to get permission", so please accept my humble apologies. Rjowsey (talk) 22:39, 10 June 2015 (UTC)


 * Its author is a very obvious crank. See above. YohanN7 (talk) 18:23, 10 June 2015 (UTC)


 * YohanN7, I infer that you consider me "whimsically eccentric", which I'll cheerfully confess to (on occasions). If there's some part of my historical narrative, mathematics, physics, hyperspatial geometry, whatever, that you don't quite understand, I welcome further questions and/or constructive criticism of the content. Rjowsey (talk) 22:39, 10 June 2015 (UTC)


 * The mindset of the author is certainly important. This is your religion, and you are clearly not interested in having the correctness of your claims scrutinized. You have managed to make the article giving the impression that all, from Maxwell and Einstein through Hawking to Witten support that spacetime is obviously complex. None of the articles I have read makes any mention of complex spacetime (you keep adding more, so I haven't read all). This is akin to fraud.


 * If the article is to stay, it must clearly state that this is a minority opinion (i.e. fringe ) among researchers, and that there is certainly no deductive trail that unequivocally leads to the conclusion that spacetime is complex. You must also supply references that are relevant to the content of the article. Supplying references to E8 theory is just muddling the waters. Are we supposed to be impressed? YohanN7 (talk) 05:20, 11 June 2015 (UTC)


 * Re: "it must clearly state that this is a minority opinion (i.e. fringe) among researchers". That's a good point, actually. I guess I could address that by pointing out that complex spacetime manifolds have been mooted for over 100 years, by many respected and gifted physicists, in the context of both SR and GR, but that relativity math has been dominated by partial differential calculus over a pseudo-Reimannian (4D Minkowski) manifold, so complex spacetime never gained any mainstream traction. As to whether it's "fringe", that sounds more like someone's been drinking the cool-aid, off with the fairies, so I'd prefer to think it was "eccentric" perhaps, even "whimsical". Maybe I should annotate this historical/mathematical physics narrative as a "minority opinion"...? What do you think? How would you narrate the story of how numerous and various theoreticians over the past century have assiduously explored the relationship of relativity to complex spacetime manifolds, with little success? TIA. Rjowsey (talk) 09:19, 11 June 2015 (UTC)


 * I agree with Rjowsey, that considering spacetime over the complex (complete field) rather than the real field is mainstream, not fringe. Have a look at these recent articles on the importance of \(\mathbb{C}\) vs. \(\mathbb{R}\) (for the base field) when it comes to QM prediction results (DOI: 10.1038/s41586-021-04160-4; DOI: 10.1063/PT.6.1.20220120a) and complex-time (kime) for the longitudinal dimension(s) (DOI: 10.1515/9783110697827; DOI: 10.1016/j.padiff.2022.100280), which may also be cited in the article. 192.54.94.255 (talk) 14:17, 23 June 2022 (UTC)
 * In this Talk:Imaginary_time talk page, there are additional details addressing the question "Does the existence of both real time and imaginary time mean that time is complex and therefore two-dimensional?". Iwaterpolo (talk) 15:44, 29 January 2024 (UTC)


 * It may be more constructive to focus comments here on the merits of the article and how to handle it, than on the contributing editor(s). Something like being fringe does not disqualify it as a standalone article.  The first thing to focus on is whether the topic is notable.  The lead should make it clear what the topic is, but this remains unclear to me.  The title suggests something along the lines of a complex manifold, whereas the intent may be a real manifold in which some dimensions are treated as being imaginary (whatever that might mean mathematically).  In any event, the topic should be considered as a topic in its own right in a notable source.  —Quondum 20:14, 10 June 2015 (UTC)


 * The fringe page states "an idea that is not broadly supported by scholarship in its field must not be given undue weight in an article about a mainstream idea", with which I am fully in agreement. As regards notability, which states "Wikipedia articles cover notable topics—those that have gained sufficiently significant attention by the world at large and over a period of time", I concur completely. The issue of unification (GR+QM) has definitely gained a great deal of attention in the (physics) world at large, over a lengthy period of time, and the problem of complexification (of GR) is considered "highly notable" by many scholars in physics and related fields. You say "The lead should make it clear what the topic is, but this remains unclear to me." — point taken; I'm working on clarifying this intro, such that it's precise, concise and succinct. Please be patient. I'm new to writing in WP, and there's a fairly steep learning curve for us "seniors". Any assistance is greatly appreciated, e.g. your link to complex manifold may prove quite useful. Many thanks. Rjowsey (talk) 22:39, 10 June 2015 (UTC)


 * Quondum Tip o' the hat for that "complex manifold" pointer. As soon as I looked at the lead, I recognised that idea (I've forgotten a whole lot more math than I remember!). Added link to intro. Hopefully, it won't deter the "average lay-person" from reading further. :D Rjowsey (talk) 23:56, 10 June 2015 (UTC)


 * Question: if the Whitney embedding theorem tells us that every smooth n-dimensional manifold Cn can be embedded as a smooth submanifold of R2n, could this mean that higher-dimensional complex manifolds only increase their dimensionality in pairs, like 5D —> 7D —> 9D —> ... (3 + 2n)D ? Because that's what I'm seeing in my hyperspatial math. Equations for quantum spin, and electromagnetic propagation, need 5 (3r+2i) dimensions of space, plus time, because 2 phasors are needed, one for the position-momentum sinusoid, the other for time-energy. Those describing quarks and gluons appear to need 7 dimensions of space (because cuboctahedral geometry, so 4 phasors). The Z and W bosons will likely need 6 phase angles, so 9 spatial dimensions, and to fully describe the Higgs boson, I suspect we'll need 11 (= 3r+8i). See E8 Theory. Rjowsey (talk) 02:07, 11 June 2015 (UTC)

(I'm hopping in from WT:PHY.) I recall that the idea of complex spacetime, reducing to mixed signature spacetime, does have a history in the older literature, and it may warrant an article. I remember it being four-complex dimensional though, not three. The complex manifold page is a good place to start. The appearance of physical constants in the article, and their relation to two extra spacetime dimensions, seems odd. Also, the relation to E8 Theory is, I believe, specious. E8 does have rank 8, but that's the dimension of maximal Cartan subalgebras, which I don't think has to do with spacetime dimensions. Dilaton (talk) 03:16, 11 June 2015 (UTC)


 * Yep, true dat. Point taken. My imagination took wings, I think, glimpsing some possible connection between 11D String/M-Theory with Lisi's E8 (they don't play nicely together, at all). Yes, various complex spacetimes pop up quite regularly through physics history, but they never get any mainstream traction. Rjowsey (talk) 03:26, 11 June 2015 (UTC)


 * PS: regarding "the appearance of physical constants", I'm writing the topic up that way, because I've found it the best way to teach youngsters how to use the complex math framework I'm trying to describe. The "Complex Spacetime" title is tentative, and I'm certainly open to suggestions, if you can think of something better. :D Rjowsey (talk) 03:39, 11 June 2015 (UTC)


 * OK, thanks for injecting some enthusiasm into staid Wikipedia. I hope you understand, though, especially if things go poorly for your edits, that flights of fancy are usually unwelcome here, and edits need to be backed by good secondary sources. For the matrix of physical constant fractions, that strikes me more as related to dimensional analysis. Dilaton (talk) 04:18, 11 June 2015 (UTC)


 * Cheers. Regarding "flights of fancy are usually unwelcome here", I understand perfectly. I certainly won't indulge in any speculative (nor "religious", nor "fraudulent" (LOL)) content on my topic page, but user talk is apparently a good deal more, shall we say, free-flowing. But I'll be more careful in future. Tip o' the hat. You're a gentleman and a scholar, sir. Rjowsey (talk) 08:43, 11 June 2015 (UTC)


 * From Quondum's talk page:
 * Because I live, eat, sleep and breathe imaginary space/time dimensions, they've come to feel very "real" to me.
 * Metaphorically speaking, this is your religion. Then your Einstein reference (early in lead) makes absolutely no reference to imaginary time. The statement is thus fraudulent and is not a simple mistake on your part as you have made clear on this talk page. The same lack of connection between the contents of the article and what the references actually say exists for the other references as well. You are, purposefully or not, misleading the reader as far as I can tell. YohanN7 (talk) 09:34, 11 June 2015 (UTC)


 * I agree, a better WP reference to Einstein's $$\scriptstyle{ct\sqrt{-1}}$$ imaginary time would be history of special relativity. Thanks (metaphorically, whimsically, religiously, and fraudulently). LOL. Rjowsey (talk) 10:34, 11 June 2015 (UTC)


 * Sorry, no, WP does not count for a reference. Besides, Menyhért Palágyi's imaginary time predates Einsteins SR and there is no connection between the two. You added whimsically to the list yourself. Good. YohanN7 (talk) 11:05, 11 June 2015 (UTC)


 * Re: "WP does not count for a reference." You're kidding, right?! Rjowsey (talk) 11:33, 11 June 2015 (UTC)
 * No. YohanN7 (talk) 11:38, 11 June 2015 (UTC)
 * OK, that √-1ct reference is in Einstein's 1905 electrodynamics paper. You want to dig out the exact ref/cite? That's be helpful... Rjowsey (talk) 12:11, 11 June 2015 (UTC)
 * I can't find it in the English translation. But even if there, it still does not make time imaginary. I see the same problem with the other references, the most important one being Wesson. I do not have that book and can only read chapter 1 (Google search). It is clear that Wesson is not talking about complex dimensions since he is talking about a signature of the metric. (Bilinear forms over the complex vector spaces do not have signatures). This means that there is not a single reference to your article that supports it. Don't you agree that this is problematic? YohanN7 (talk) 12:29, 11 June 2015 (UTC)
 * I don't see why you're spending so much energy on this. The article was created, as you are learning, without a single reference about the topic, and it seems that nearly every reference relates to an essentially incompatible theory and the claim it is attached to is typically not supported by the reference.  It contains numerous vaguely associated and broadly ranging claims, and the author has deflected (e.g by claiming to agree, and saying that he's new and working at it) but not addressed the issues raised.  Concepts raised as contradicting concepts in talk pages have paradoxically been incorporated too, e.g. the idea of a complex manifold.  Many of the assertions are a synthesis of what have been said on various talk pages.  IMO, this is nothing less than self-publication on WP, and fully deserving of being called "not even wrong".  I had suggested that this be developed in a sandbox, but as that suggestion was ignored, it should be considered as a candidate for WP:AfD.  This author has, from the start, also shown a familiarity with editing WP that belies the claims of being a total newb, which might have other implications.  —Quondum 13:30, 11 June 2015 (UTC)

My posts above were not meant to rush writing the article, indeed it usually takes a few days just to get it into a starting shape. Even so, we still have the tables of units.

On another matter, the the author of this article seems to misunderstand spin (in particular "total spin"), making basic errors even an average graduate in physics (like me) would not... see his user and talk pages.

Moreover, perhaps user:Siddhant Singhji would like to chime in on his/her views of this article, when possible. M&and;Ŝc2ħεИτlk 14:27, 11 June 2015 (UTC)


 * Having seen this exchange, I think it is clear that this article does nog belong here per elementary wp:NOR. Let's get rid of it. - DVdm (talk) 19:58, 11 June 2015 (UTC)

Thanks all. It's been fun! Learned a lot. I'm done with this, and outta here. :D Rjowsey (talk) 20:27, 11 June 2015 (UTC)

More false references
Excerpt from Einstein's A GENERALIZATION OF THE RELATIVISTIC THEORY OF GRAVITATION:
 * The above field is described by a tensor $g_{ik}$ with complex components. These components shall satisfy a condition of symmetry which constitutes the natural generalization of the condition of symmetry of the metric field of the theory of gravitation to the complex domain, which we call "Hermitian symmetry":
 * (1) $g_{ik} = \overline{g_{ki}}|undefined$
 * The components are continuous functions of the four real coordinates $x_{l}, ..., x_{4}$.

(my emphasis)

As you can see, there is no reference to any $3r + 3i 3D$ complex spacetime of Maxwell. The behavior of Rjowsey is becoming unacceptable. YohanN7 (talk) 12:15, 12 June 2015 (UTC)


 * I doubt that Maxwell had any conception of spacetime having more than four real dimensions, even if he may have associated spacetime with a space that would be considered one-dimensional over the quaternions (I haven't looked at the detail). I'm guessing that Maxwell may have considered functions (fields) on the real four dimensional spacetime to take on values in higher-dimensional vector spaces (we do this all the time, whether in the form of tensors, multivectors or whatever).  So let's not associate $3r + 3i 3D$ spacetime (with its six real dimensions) with Maxwell's name.  —Quondum 13:33, 12 June 2015 (UTC)


 * Maxwell didn't, I have his 2-volume EM treatise (Dover) and can't find any complex dimensions. All his work is mostly 19th century style real algebra and calculus, can't even find complex analysis used anywhere. M&and;Ŝc2ħεИτlk 14:17, 12 June 2015 (UTC)


 * Maxwell is now not mentioned except in connection with his equations.
 * Can anyone get to this reference?
 * Since other references have proved to not support what has been written in the article, this one too needs to be checked. YohanN7 (talk) 14:27, 12 June 2015 (UTC)
 * Since other references have proved to not support what has been written in the article, this one too needs to be checked. YohanN7 (talk) 14:27, 12 June 2015 (UTC)


 * Found the pdf just by googling his name and the title of the paper (can't post the link, it seems disallowed by WP). Reading it now... M&and;Ŝc2ħεИτlk 14:33, 12 June 2015 (UTC)
 * In that paper, the Geometry is Riemannian and 4d complex, the line element (squared) i.e. ds2 is complex-valued, with the real part corresponding to gravity and the imaginary part for electromagnetism. To quote page 76:
 * "(Posulate) 2. We postulate for the physical world a complex four-dimensional Riemannian geometry where for evident physical reasons the coordinates are real."
 * The author uses a mixture of the real coordinates x, y, z, t and 3d vector calculus, as well as 4d complex-valued tensors. M&and;Ŝc2ħεИτlk 14:40, 12 June 2015 (UTC)


 * Thanks. I found the paper now. YohanN7 (talk) 14:55, 12 June 2015 (UTC)


 * To a non-expert reading this article and some articles it links to, Soh's appears to be the first explicitly complex spacetime in the article and the preceding paragraphs are just historic background. Is that right? If so, it might be as well to omit them and leap straight to saying that in 1932, Soh published a paper tht explicitly made use of complex space-time - except that that does lead us straight to the next problem, that Soh's paper is WP:PRIMARY and we have no secondary sources indicating its quality or influence. NebY (talk) 15:19, 12 June 2015 (UTC)


 * In Soh's setting spacetime is still real as it seems. I haven't read it though, just located his postulate #2. But just as in quantum mechanics, one can have complex valued functions (like the metric) of real spacetime. This applies to the Einstein paper too.
 * Reference #7 genuinely refers to complex spacetime, and another reference (gone now for some reason) treats EM in complex spacetime. YohanN7 (talk) 15:33, 12 June 2015 (UTC)
 * But that paper doesn't seem to be published. YohanN7 (talk) 15:42, 12 June 2015 (UTC)

Irrelevant and questionable quotes
I removed Feynman's quote about Euler's formula, because so what if he thinks it's "the most remarkable formula in mathematics" or whatever... Lots of people do (myself included). The equation itself was totally pointless since it is the basics of complex number algebra and analysis, and this article starts from complex numbers so people should already have some familiarity with the formula.

At the time of writing, what's the point of this quote by Einstien??


 * "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[5] "

All Einstein seems to be saying is that he has not accomplished his unified theory of gravity + EM. It does not suggest that complex spacetime is a possible thing to consider, because he is not saying that. It should be deleted as completely irrelevant along with the reference. Anything about Einstein using complex spacetime should be in the history of this article.

As for the other quote...


 * "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[8]"

I can't find it in the cited PDF reference, even tried pasting the first few words "I really do not yet" and "Schro" (for Schrödinger) in the "find" tool of the PDF reader and nothing came up. Still, this quote, even with the bracketed "[complex]" seems to suggest the equations are complex, and nothing is implied about spacetime coordinates themselves being complex. This quote also should be deleted as irrelevant, but possibly keep the paper in some "further reading" or "external links" section. M&and;Ŝc2ħεИτlk 11:20, 14 June 2015 (UTC)


 * This is all probably to keep the reader and other editors confused. YohanN7 (talk) 15:26, 14 June 2015 (UTC)


 * I went ahead and deleted them. Much of the stuff about the KK theory, string theory, and unification uses real spacetime, so everything in the history section on real coordinates needs to be reduced and anything to do with complex spacetime should be expanded on. M&and;Ŝc2ħεИτlk 23:00, 14 June 2015 (UTC)

Riemann–Silberstein vector
The use of the complex Riemann–Silberstein vector has been perpetuated by Ezra T. Newman in the 1973 article cited and again in 2004 "Maxwell fields and shear-free null geodesic congruences", Classical and Quantum Gravity 21(13). This use of complex spacetime is a long-running thread. — Rgdboer (talk) 23:53, 7 September 2017 (UTC)


 * What does the Riemann–Silberstein vector have to do with complex spacetime? YohanN7 (talk) 07:24, 8 September 2017 (UTC)

Newman (1973) first sentence: "complex Minkowski space", third sentence "complex Poincare group". He uses R–S vector W to represent a source-free vacuum EM field in complex spacetime. — Rgdboer (talk) 01:27, 9 September 2017 (UTC)


 * I see. Newman is cited in the article. I have, of course, no objections whatsoever if you want to beef out on the content of his paper (or any other paper on the present subject). But this isn't the reason you post here I suspect? YohanN7 (talk) 09:57, 9 September 2017 (UTC)

Yes, there’s more. Citing Newman (1973) and leaving the impression that the history of the idea started there is wrong. That is why the R_S vector is important: it shows prehistory and sources. The complex spacetime is also implicit in use of M(2,C) with Cartan's spinors in three dimensions. In 1938 Cartan neglected to cite previous study of that algebra though it had been studied as biquaternions. So an accurate report on complex spacetime would acknowledge the antecedent algebra on C4 and note that it had been appropriated as spacetime, for instance, by Ludwik Silberstein in his 1914 textbook. — Rgdboer (talk) 23:48, 12 September 2017 (UTC)