Tuesday, December 5, 2017

Advanced Berenst#in Bears Theory - Complex Euclidean Space

I wrote a blog post several years ago about the Berenst#in Bears that, somehow, against all odds, managed to get picked up on the internet.  It's been posted on Reddit and other outlets several times, and accounts for about 95% of all traffic to my blog.

The post is of course intended to be silly, but part of the silliness is that hidden inside this ridiculous proposition of universes merging over children book names is a very mathematically complicated thesis about the nature of space and time.

I don't think I did a very good job explaining what that thesis is, because I often see my blog being referenced as claiming either time travel, the many-worlds hypothesis, or something else about quantum mechanics are responsible for us finding ourselves in this piteous state of being in the wrong universe with the wrong cartoon bears.

In this post, my goal is to outline what exactly it is that I was "claiming," so that if the story gets picked up again (unlikely now that more major outlets have picked it up), maybe people can actually at least summarize what I'm saying correctly.

TL;DR the theory is about changing the geometry of spacetime by allowing all four dimensions to be general complex numbers.

Let me start with what the theory is not.


Firstly, and probably most importantly, this is not a real scientific theory.  It's barely a scientific hypothesis.  At the moment, it doesn't lead to any predictions.  Until there are concrete predictions and experimental evidence, it's just principled imagination.  But it is very principled imagination.

Secondly, it's not a theory about the names of cartoon bears.  If you twist it hard enough (as I did) you could use it to explain that, but that isn't what it's about, really.

Thirdly, it's not a theory of quantum mechanics.  There have been similar ideas used in quantum mechanics, but the theory isn't that quantum mechanics causes universe splits/mergers.

Fourthly, it's not a theory about time travel, though I have written extensively about time travel elsewhere.

The theory is actually about geometry and algebra.

Usually when describing space and time we use a set of four numbers.  Three give the location in space, and the fourth gives the time when something occurs.  When we describe objects in the universe in terms of both their position and time, we are describing objects in spacetime.

These numbers are what are called "real" numbers, "real" here meaning that they describe things that we encounter on an everyday basis.  The "real" numbers includes the natural numbers (counting numbers): 0, 1, 2, 3, etc... and the negative numbers: -1, -2, -3, etc... The reals also includes fractions: 4/7, 1123/456, etc... The reals also include irrational numbers, like $\sqrt{2}$, which can't be written as fractions of whole numbers, and transcendent numbers, like $e$ and $\pi$, which can't be written with nonrepeating decimal expansions.  The best way to think of the reals is to imagine a ruler; the whole numbers are just the thick lines, but the reals are the lines and all of the spots between the lines.

The reals have a lot of great properties, like commutativity, distribution, closure, existence of multiplicative inverse, and some others you learned in 7th grade and forgot.  But they also have one bad problem, which is that they are not algebraically closed.  This means that I can write a polynomial using only real numbers, but the equation will not have any real solution.  As a simplest example, consider
$$x^2 + 1 = 0.$$
That equation only uses real numbers, but there is no real number that solves it.  There is no real number which, squared, gives -1.  This is something undesirable, mathematically.

If you paid attention in 11th grade math, then you maybe remember that there is a solution to the above equation, the famous number $i = \sqrt{-1}$.  The number $i$ is called $i$ because it is an imaginary number.  Notice the distinction: real numbers, vs imaginary numbers; one refers to normal things in the real world like lengths of chairs, while the other is impossible and doesn't exist in the real world.  As in, there is nothing with a length of $\sqrt{-1}$, and you can't have $\sqrt{-1}$ liters of water.

If you have $i$, then you can make more imaginary numbers.  Numbers like $2i$.  Or $456.32i$.  Or $i \sin(7)$.  Lots of them.  In fact, there as many imaginary numbers as real numbers, because for any real number $x$, just make the imaginary number $ix$.

These imaginary numbers actually turned out to be incredibly useful.  Mathematicians initially regarded them with great skepticism, as back then Euclid defined mathematics, and numbers were largely defined in terms of ratios of real things to each other.  For instance, mathematician and author Charles Dodgson (who wrote Alice and Wonderland and Hunting of the Snark) was famously opposed to them.  But using these imaginary numbers in their calculations proved to actually be able to return useful results -- results that they otherwise could not get with the usual methods.  So they gradually were accepted by mathematicians, and today we learn about them in high school trigonometry.

Not just imaginary numbers come up as solutions to equations.  If you remember the quadratic formula (and of course you do!), if you have an equation of the form
$$0 = ax^2 + bx + c,$$
where $a, b$, and $c$ are coefficients, then the solution is
$$x = \frac{-b \pm  \sqrt{b^2-4ac}}/{2a}$.
There are two solutions, formally, but if $b^2 < 4ac$, then there is no real solution, since the term in the square root will be negative, and the square root of a negative is an imaginary number.  That imaginary number is then added to a real number.  For instance, for
$$0 = x^2 - 2x + 5/4$$
the solutions are
$$x = 1+i, 1-i.$$
These combinations of real and imaginary numbers are called complex numbers.  A complex number is two numbers in one: a real part and the imaginary part.

The complex numbers include the whole numbers (just write them as $1+0i$, $2+0i$, ...), the
rationals (write as $5/4 + 0i$, ...), and the reals (like $\pi + 0i$, ...), but also all of the imaginary numbers ($0 + 1i$, $0 + \pi i$, ...), and any combination of a real and imaginary number ($5/4 + \pi i$, ...)

As it happens, the complex numbers actually share all of those nice properties the real numbers have: such as commutitivty, distribution, closure.  But they also have the one nice property that the reals didn't have.  The complex numbers are algebraically closed.  Every polynomial equation written entirely in terms of complex numbers has a solution that is a complex number.  This fact was so important to mathematicians back in the day that they actually call it the Fundamental Theorem of Algebra.

More so, it can be proved that the complex numbers are the unique set of numbers that have all of these properties and preserve the properties of real numbers.

So complex numbers are very important, mathematically.

Complex numbers are also very, very important physically.  There are a number of key physical results that are expressed most elegantly using complex numbers instead of real numbers.  When you're solving Newton's Law and drawing Free Body Diagrams, like you do in intro physics classes, then complx numbers don't really come up.  But if you're resolving phasors in electronics, or working with oscillatory systems in classical mechanics, or describing electromagnetic waves in E&M, then your life will be made much simpler by describing these in terms of complex numbers.

They are very useful, but when we get into the world of quantum mechanics, complex numbers cease being merely useful and become indispensable.  The Schrodinger Equation (the backbone of quantum mechanics) is formulated in terms of complex numbers, and the solutions to it will be complex-valued functions.  You literally can't do quantum mechanics without complex numbers.

As the quatum theories become more complicated and fundamental -- going from the "classical" quantum mechanics to Dirac's relativistic quantum mechanics to field theories to nonabelian field theories -- you find you need more and more complicated algebraic superstructures.  At some point the complex numbers become insufficient and now you need Clifford algebras, then Gell-Mann numbers, then anticommuting numbers, and more.  These numbers have a lot of other "nice" algebraic properties, but they still aren't as nice as the complex numbers.

As it happens, complex numbers are the only algebraically closed, commutative field (with characteristic zero).  That makes them very special.  A lot more special than the reals, and a lot more special than larger or more bizarre algebraic sets.

But while we use these numbers in quantum mechanics and field theory, we don't really use them in a lot of real-world engineering applications; and when we do use them for real-world applications, we take the real part of our solutions at the end and drop the imaginary part.  The answers we eventually implement in the real world of engineering of course have to be in terms of real numbers, because real numbers are what actually describe the world around us.

Or do they?

That question is really the point of what I'm saying.  The complex numbers are a very special, very nice algebraic structure that get used constantly throughout physics and mathematics, and yet are relegated to a sub-status beneath the reals because we don't observe them corresponding directly to anything we measure in the universe.

But what if the complex numbers do correspond to ontologically real classical quantities?

In particular, what if complex numbers corresponded to distances and lengths and times?  What if those four numbers that describe where a thing is in space and time were considered to be complex numbers, and if we formulated relativity and the rest of our physical theories in terms of spacetime variables that are complex-valued?

What would the implications of that be?

I don't really know, actually.

Now, yeah, this idea sounds crazy, if for no other reason than because it was originally used in regards to anthropomorphic cartoon bears from other universes.  But a much milder form of this idea was originally proposed by Einstein himself, and has remained in use in quantum field theory in the form of a Wick rotation.

The original proposal by Einstein didn't propose making all four spacetime variables complex, but just to make the one time variable purely imaginary.  The reason for this is the way distances are measured in special relativity.  I've explained this property of special and general relativity in several other posts (like this post on time dilation in Interstellar, or this one on time travel), but the result is that, in general relativity, the spacetime distance (properly called an "interval") between two points in spacetime is given by
$$\Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2  - \Delta t^2,$$
which looks a lot like Pythagorus’s famous theorem in four dimensions, except for that pesky "-" sign there on the t.  To solve this, Einstein introduced a new variable which I will call $$w = it$$.  Then
$$\Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 + \Delta w^2,$$
and we have regained the famous result for distances in Euclidean space.

That is the original way that Einstein dealt with the issue of the spacetime metric.  An equally valid alternative is to use different space variables,
$$X = ix, Y = iy, Z = iz,$$
and then the interval can be defined as
$$\Delta s^2 = \Delta t^2 + \Delta X^2 + \Delta Y^2 + \Delta Z^2,$$
which is also a different but perfectly valid definition.  The difference between these two definitions is that one gives
$$\Delta s^2 =  - \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 ,$$
and the other
$$\Delta s^2 = \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2,$$
which are referred to as two different sign signature conventions.  The first has the signature (-+++) and the second signature (+---), but as long as you are consistent with this convention, then they define the same physics.

In the modern study of special and general relativity, we don't use imaginary time or imaginary space to define the spacetime interval, and instead use a more complicated system involving tensors.  These tensors turn out to be vital in general relativity where the fundamental equation for gravity, the Einstein Equation, has to be expressed in terms of tensors.

But in special relativity, actually, it turns out you don't really need all the mechanics of tensors.  Actually, you can do everything with imaginary time and your answers will work out fine.

More interestingly, still, famous particle physicist and textbook author J.J. Sakurai, in his "Advanced Quantum Mechanics," goes into relativistic quantum mechanics and field theory and QED and still doesn't introduce tensors.  Sakurai was able to reproduce all of the physics of Dirac's theory without the need for a signed metric tensor, by using imaginary time components instead.

As Sakurai editorially states in the text, if you're working in special relativity, tensors are an unnecessary complication that only muddle the mathematics; you only need to use them if you plan on merging quantum mechanics with general relativity.

And while you most certainly do need tensors for general relativity... you don't need them for the sign convention.  You can use proper metrics of a "proper" Reimann geometry (as opposed to the pseudo-metrics of our pseudo-Reimann geometry) and stick to an imaginary time component.

So while we have dropped imaginary time in relativity in favor of more complicated mathematical tools, imaginary time still works just as well as it did when Einstein first described it.

In quantum field theory, especially if you are using Feynmann's path integral approach, you get to a point where you introduce a "fictitious rotation" of the time axis so that time becomes imaginary.  This puts you in complex Euclidean space.  Once in this space, results that did not use to work earlier, or were unsolvable earlier, can now be solved.

My professor at the time remarked that this complex Euclidean spacetime is, in a sense the "truly" covariant theory.

In physics, we use the word "covariant" a lot, and it has several formal definitions, but the more intuitive definition you get after using it a lot is that a theory is "covratiant" when it properly and correctly includes time and time-components into its description of physical quantities in spacetime.  Or another way to say: when your theory treats time as just another dimension without special attention.  Any theory that is covariant can have space and time combined into a single geometric entity like a vector or a tensor, where the time components become just one of the components.

Complex Euclidean spacetime is then the "most covariant", because in complex Euclidean spacetime, time is really treated just like space.

But while it's more elegant it isn't fully there, because time isn't being treated just like space.  Time is being treated as imaginary, and space as real.  Why that asymmetry?  Since the asymmetry breaks "true covariance", then lets remove it, and treat them exactly the same.

Making both time and space either both real or both imaginary restores the symmetry between time and space, but ruins the spacetime interval.

Another way to restore symmetry is to allow for complex space and time variables.  Four complex directions with a Euclidean metric.

This is really sort of obvious in some regards.  Complex numbers are, as I said, that special algebraically complete field.  They are enormously useful in making predictions, and keep popping up in our more complicated theories at lower levels.

We actually frequently use complex spatial dimensions in calculations.  I'm speaking really just to physicists at the moment, but when doing contour integrals over a spatial variable, you are implicitly assuming that there is an imaginary component to space.  You complexified the real variable.  You do a calculation that only works with complex numbers.  But then, once you finish the calculation, you sort of "forget" that you did, and pretend it's still just real.

But why forget?  Why pretend?  You had to complexify it to get your answer; why de-complexify?

Instead, let's maintain that all four spacetime dimensions are complex numbers, not real numbers that only have complex components when convenient.

But what does any of this mean, really?  What are the implications?  Sure, it's elegant in a certain sense, but what does it predict, what does it explain, what new physics can we get from it, and how can we test if its true?

And I actually have no idea, really.  I'm embarrassed to say it, but I do not have the mathematical background to analyze it.  I can handle calculus with twenty(-thousand) real variables, and can handle calculus with a single complex variable, but with four complex variables I really start to lose track of what is where and when.

More stupidly, the major problem that keeps confronting me is that I've run out of symbols to use for all of the variables when I use separate variables for the real and imaginary parts.  I want to use arabic letters for some of them, but LaTeX will not let me use arabic letters as math-like symbols (if you know how, please respond in the comments).

I've attempted some scribbles on paper, but they don't really give me a lot and I'm not sure where to take it.

At the moment, the only "prediction" I can work out of the theory is my idea of spacetime then dividing into "hexadectants" -- 16 separate regions, each of which has its own "real number" physics.  But I use prediction in scare-quotes because it is not at all a scientific prediction, since it is impossible to test if the universe is broken into hexadectants.

Why hexadectants?  Why 16?

Earlier, I talked about metric signature, and said there were two: (-+++) and (+---).  Those both give us the same special relativity, provided we define certain quantities consistently in each.  The difference between them is whether we make the time have a negative (-+++) or positive (+---) sign in the Pythagorean equation.

Notice, that singles out time as the "first" variable.  It makes time special.  That's asymmetric.  If we treat time just like every other variable, then it may very well be that the second variable, the one we think of as space, could be a time variable.  There could be a special relativity using a signature of (+-++) and another one using (-+--).  These both also give the same special relativity, but now in this physics, what we call "time" is a spatial variable and what we call a spatial variable is time.

Carrying on, there are two more pairs:  (--+-), (++-+), (---+), (+++-)

All of these different signatures describes a hexadectant with special relativity identical to ours.  There are 8 of them.

In addition to these "parallel" hexadectants, there some weirder places with two spatial and two time dimensions:
(++--), (--++), (+-+-), (-+-+), (+--+), (-++-)

There are six of those.

Lastly, there are four hexadectants that are true, true covariant regions, in that time works exactly like space:
(++++), (----)

These last two have been described in pretty great detail in Greg Egan's Orthogonal series.

That makes sixteen total.  These different hexadectants aren't a natural division of complex spacetime, but come about by a choice of whether you consider the real or imaginary part of one of the variables to be a dimension.  In our "hexadectant" we take the real of one and the imaginary part of three (or vice-versa), but that's an arbitrary choice, so it's worth considering all the different choices.

(Also, choosing one specific direction in space and calling it the "x" direction and another the "y" is arbitrary -- I haven't taken rotation into account here.)

But there are, obviously, a lot of unanswered questions.

What does it mean, in terms of physical observables, to have extension into the imaginary axis?  Why have we never observed objects moving along the imaginary axis before?  Why aren't standard forces complex?  Why can't we move into the imaginary axis, or push things along the imaginary axis?

Are you even able to describe a consistent physical theory like this?  What happens to cross terms in the metric?  What happens to quantum mechanics?  What happens to field theory?  What happens to normal classical mechanics?

At the moment, I can't even describe an object moving in one direction in a Newtonian framework with this theory -- but maybe these concepts only become relevant or observable at extreme conditions that I don't even know what they are because I haven't figured it out yet.

So, anyway, that's the "advanced" Berenst#in Bears theory.  It's not really about quantum mechanics, and isn't even a little bit about time travel.  It's really about complex numbers and geometry.  It's about symmetrizing spacetime to be optimally covariant and algebraically nice.  It's about strange extra degrees of freedom of movement that we aren't aware of and the weird consequences of having them.

TL;DR the theory is about changing the geometry of spacetime by allowing all dimensions to be general complex numbers.

It's not a scientific theory, even though it's framed in scientific language.  But, if it could be worked out by someone smarter than me (someone who knows multivariate complex analysis), then it could possibly lead to predictions that could make it a scientific theory.  Or... could lead to nothing but dead-ends.  Who knows?  Not me, because I haven't done the math yet.

I hope you enjoyed this excessively long essay on imaginary physics.  I'm sorry that I lost probably lost you halfway through, but I'll try to include some links to general-level articles throughout.  A quick wiki-check of any unfamiliar term is probably enough to get the gist of what I'm saying, if you aren't already familiar.

2 comments:

Reece Lockhart said...

Hello Reece, This is strange,..but hey..Anyhoo, my name is Reece as well..with a C like you. Not going to take up much of your time, but just for pondering sake, if you want you can check my YouTube stuff for any similarities other than our names in this rabbit hole. I make my vids for me to remember my wierd life and my son.. Feel free to contact me at SharkticonGnaw68@Gmail.com my lil vids are under Sharkticon Gnaw on youtube. Sorry, I'm still learning so I don't know how to link. Take care, Reece. ☺

Reece said...

Are these your videos? On blogger you link with HTML tags, but other platforms have different standards.