The Peculiar Math That Could Underlie the Laws of Nature
21 Jul 2022 12:10:01 UTC
Topic 227854
(moderation:
There’s a suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions".
Peculiar Math
This is perhaps more relevant to the LHC forum, but they don't have Mike Hewson.
Jim1348 wrote: There’s a
)
There is a "And God said let there be light" math poster at my UUA church.
I have no idea if the equations make sense.
The fun part is a line drawn through some of the "mistakes" before it gets to the final "lighting" equation.
Tom M
A Proud member of the O.F.A. (Old Farts Association). Be well, do good work, and keep in touch.® (Garrison Keillor)
Jim1348 wrote: There’s a
)
Mike may know that discussing anything other than the LHC projects or VB and programming will just be ignored and this is the only place to discuss theoretical or experimental physics and calculus
Of course these days members that have been doing this as long as we have know that we just use our chosen browser to search for the answer to any questions.
It is just like talking to Tesla or Euclid about his 47th Problem and Pythagoras of Samos all the way up beyond Einstein and Stephan Hawking.......or Michio Kaku since he loves to sell books
That's quite an article and
)
That's quite an article and kudos to the researcher as it is always difficult if one studies things not of the current trend. As mentioned octonions are a number type in a similar vein to quaternions and complex numbers. Octonions. Sounds like beings from the planet Octonia. ;-)
Hmmmm. This will take some preparation. Or unpacking. What about I write a series of posts rather than one mega-post ?
Think of a space of N dimensions If N is small then we can visualise and calculate upon it without too much trouble. Conceptual problems arise if the space does not have features which are like our everyday three dimensional space that we are used to and have a lot of intuition about. A key feature, it turns out, is the definition of distance. I hope that you are familiar with the normal Euclidean 3D spatial definition of Pythagorus :
d = SQRT[(x2-x1)2 + (y2-y1)2 + (z2-z1)2]
..... where (x1, y1, z1) and (x2, y2, z2) are two endpoints with coordinates given in terms of an orthogonal set of linear axes. From this you can get the length of any vector. Just move the base or initial point of the vector to the origin of the coordinate system (0, 0, 0) and then the final point of the vector lies at some (x, y, z), and so d = SQRT[x2 + y2 + z2].
{ Also, for the moment, just remember that one can't always do such a thing as 'just move the vector'. There are 'curved' spaces where it doesn't make sense to do that and the lengths of things become dependent on a definition of distance that may vary from point to point. I'm thinking of General Relativity, and a total distance there is a summation of lots of 'little distances' along some curved path. More on that later, but in Euclidean space one can always 'parallel transport' a vector while maintaining it's length and which way it is pointing. }
This definition gives a positive number, or zero, for all distances between points in the space. The distance zero occurs if ( and only if ) both endpoints are in fact the same. If two points are different then we must have a non-zero positive number for the distance. In particular the triangle inequality is relevant which is the mathematical version of the statement : the shortest distance between two points is a straight line. If you pick another point not on that line then a triangle is formed and the distance b/w the original endpoints but going via that intervening point is at least as much as the straight line route. Or put another way we if we already have the shortest linear route between two towns along some straight road, say, then going via a third town will not decrease our travel distance. Only if that third town already lies on that shortest route will the distance be the same ( in which case we have collapsed the triangle to a single line ). We know this already, in fact the math was set up to give that property for us in the case of describing the world as we experience it.
In the two dimensional case all points lie on a plane, we may use two orthogonal linear axes for coordinate definition :
d = SQRT[(x2-x1)2 + (y2-y1)2]
... with the triangle inequality holding. If you extend to four dimensions, but want to stay Euclidean then
d = SQRT[(x2-x1)2 + (y2-y1)2 + (z2-z1)2 + (w2-w1)2]
for two points with coordinates annotated as (x, y, z, w). But it doesn't have to be that way. One can have some other definition of distance for a four dimensional space and indeed we have exactly that when applied to Special Relativity. If you're happy to work with units where the speed of light is deemed to be equal to one, then
d2 = (x2-x1)2 + (y2-y1)2 + (z2-z1)2 - (t2-t1)2
.. and you will notice that I didn't take the square root, because the right hand side of the above equation may be negative. That minus sign is the key point of difference comparing to Euclid/Pythagorus. If you have two 'events' in SR spacetime given in terms of coordinates (x, y, z, t), where t is a time coordinate, then the ( squared ) spacetime interval between those events may be negative, zero or positive. Yes, the distance definition implies that.
{ Also, for the moment, I am eliding the issue of coordinate systems vs reference frames. To be more exact : for a given reference frame, or observer, one can choose different coordinate systems with the proviso that such a choice doesn't alter the results for observable quantities ie. the physics remains the same. }
If d2 = 0 then the two events are joined by a light line or the path of a photon, and the fact that the distance is zero doesn't imply that the points must coincide. If d2 < 0 then we say it is a time-like interval and d2 > 0 it is a space-like interval. For instance one cannot 'travel' along a space-like interval because we would have to exceed light speed to do that. If two events are connected by a space-like interval then one event could not have influenced the other. We are used to time-like intervals in everyday life ie. light speed is the fastest possible, causes happen before effects, one can't travel backwards in time etc. So all this turns out to be quite useful and leads to mathematics which implies, say, E = mc2 and other stuff experimentally verified. This is called Lorentzian or Minkowskian spacetime after two of the guys that worked it out.
By now you should be feeling that your intuition is starting to slide. Do try to hang on. Thus far we have considered that coordinates are real numbers ie. the ones we are used to every day. It all comes down to whether you can always take a square root of a distance measure, the d2 as above, in it's various forms. d2 or it's square root or the expression for calculating it ( the right hand sides as above ) are also known as the metric, as in 'ruler' or measuring stick.
Let us now throw in complex numbers. These numbers are truly composite and to describe them requires using two real numbers as an ordered pair ie. (x, y) to represent a single complex number. You might at first think this would be just a 2D Euclidean space, but no it isn't. For one thing the value (0, 1) is unusual and has the deemed property that (0,1)2 = -1, which is another way of stating that we now have the square root of minus one. That changes everything. We will expand on that in the next post.
Please do throw some questions at me about the story thus far. ;-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Side Topic Of General
)
Side Topic Of General Relativity ( or why you could hate differential geometry ) For 'curved' spacetime, which is basically everything in the universe because there is always something rather than nothing and the something will have mass/energy and thus gravitate, the difficulty arises in describing spacetime without reference to a single set of linear axes that reach across all volumes. The key to this is Einstein's Equivalence Principle which implies that any sufficiently small volume in spacetime must be 'flat' and not curved ie. Minkowskian. One deduces properties in the large by aggregating properties of the small.
{ In what follows I imagine that one could view spacetime from without, a God's Eye view. I say 'traverse' when I mean 'measure along'. We aren't concerned here with how actual physical masses move in spacetime, or what separated observers will see and experience, they are separate issues. Here we are trying to mathematically cope with a system that has curvature. }
So if the separation of two events is not too large ( in space and/or time ) we can view the spacetime between those two events being represented as a linear map. So there are little axes/arrows that point in the principle directions in space and time, small signposts telling you what are the directions of greatest increase for each particular coordinate. One set of signposts for each event. Generally signposts for different events don't align. However for close points in spacetime these signposts can be as well aligned as you like, or put another way : to some degree of tolerance they will be identical provided you don't stray too much. If you think they are not well enough aligned then just zoom-in some more to a smaller spacetime volume, pick two points, and repeat as required. GR assumes that this must always apply. Nothing weird happens to spacetime in the small.
{ Which is, by the way, one problem when you ( try to ) include Quantum Mechanics, and things like spacetime separations may become not well defined at some small scale. Sharp jumps, gaps, loops or infinities et al. }
The distances then become those of Special Relativity and easier to understand. To find the distance b/w two more distant events one chooses a ( curved ) path to traverse along, and for that path choose a set of consecutive points that are close enough to use a flat metric in between. The distance of the total curve then becomes the sum of all the lengths of the itty-bitty steps. This is made precise by the use of calculus ie. the method of limits of sums of displacements. As per SR light paths play a special role.
Another way of thinking about the distance b/w two close events ( call them A and B ) is to imagine another event ( call it C ) somewhere in between those two. That third event can be represented within two ( linear ) maps : one defined for each of the first two points. As you go from A to B via C then one could gradually transform from a representation in one map to that of the other. You may know this already from maps of the Earth's surface that look 2D but are representing a surface curved in 3D. For instance the traditional Mercator projection of the entire globe makes Greenland look enormous in area. But you could do a Mercator projection centered on Greenland which would much more sensibly estimate it's true area but somewhere else will be quite distorted. Having the same point ( or small region ) drawn on two maps helps you translate b/w two metrics/maps used.
All this is actually horribly complicated, and the correct math is derived by a pretty painful analysis, theorems and whatnot, to yield expressions that can be just as hard to solve as to set up. But it is all correct in that for the circumstances in which predictions are possible and results measurable, they agree to high precision. The glorious example is the orbit of Mercury, closest planet to our Sun. The spacetime isn't really very curved compared to many other places in the Universe, but curved enough to produce a discrepancy in the point of closest approach to the Sun ( when compared to the predictions, say, of Newtonian based orbital mechanics ).
The non-gravitational force laws assume that the underlying space upon which particles move is fixed. No mucking about with varying curvatures of spacetime. In GR the masses/energies distort space and time, and matter/light move according to the curvature induced. Nuff said.
Cheers, Mike
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Let's Do Complex Numbers ( or
)
Let's Do Complex Numbers ( or why you could dislike them ) Complex numbers underlie many areas of engineering and physics. They are essential for quantum mechanics. The 'complex' bit, I think, is that there are several main representations of complex numbers, of which I will outline four here. It is a simple construction of geometry which connects them.
(1) The ordered pair representation You know what an ordered pair is. It is two numbers with a first one and a second one. The usual notation for this is (a, b). This is not set notation where the order doesn't matter, here the order does matter and generally (a, b) is not the same as (b, a). For complex numbers both elements of a pair are the real numbers which you are familiar with in everyday life. The first in a pair is called the real part, and the second in the pair the imaginary part.
The thing that makes ordered pairs of real numbers represent complex numbers is the operations to be performed and the special status of (0, 1) in the scheme of things. It turns out that (0, 1) can function as the square root of minus one, that is
(0, 1)2 = (0, 1) * (0, 1) = (-1, 0)
The operations are as follows :
ADDITION : (a, b) + (c, d) = (a + c, b + d)
SUBTRACTION : (a, b) - (c, d) = (a - c, b - d)
.... these seem straightforward as you just add/subtract the corresponding entries.
MULTIPLICATION : (a, b) * (c, d) = (ac - bd, ad + bc)
.... is not really obvious, and then there's
DIVISION : (a, b)/(c, d) = ((ac + bd)/(c2+d2), (bc - ad)/(c2+d2))
.... which seems outright bizarre. Note this loses definition entirely if c = d = 0 ie. (0, 0) is the zero element and you can't divide by it.
NB With these expressions as above the + symbol is overloaded ie. it means 'add the two complex numbers either side of me' or 'add the two real numbers either side of me' depending on where it is. Likewise for the minus symbol - and /.
(2) The real and imaginary part representation where a and b are real numbers :
(a, b) as above is written as a + bi or a + ib
... here i is the square root of minus one with it's traditional symbol, although others are used eg. j commonly used in engineering. The + sign here does not mean addition but rather means 'pair up the two numbers either side of me'. The above operations are the same of course but with an adjustment of symbols :
ADDITION : (a + bi) + (c + di) = (a + c) + (b + d)i
SUBTRACTION : (a + bi) - (c + di) = (a - c) + (b - d)i
MULTIPLICATION : (a + bi) * (c + di) = a * (c + di) + bi * (c + di) = ac + adi + bci + bd i2
= (ac - bd) + (ad +bc)i
... where I have expanded and evaluated as if in ordinary algebra and replaced i2 by -1 when I see it.
DIVISION : (a + bi)/(c + di) = [(a + bi)(c - di)]/[(c + di)(c - di)]
{ NB (c - di)/(c - di) = 1 ie. I can multiply by 1 and not change the value of something. }
= [(ac - bdi2) + (bc - ad)i]/(c2 + cdi - cdi - d2i2)
= [(ac + bd) + (bc - ad)i]/(c2 + d2)
= (ac + bd)/(c2+d2) + i*(bc - ad)/(c2+d2)
.... some rigamarole there to get a real number in the denominators. Again, you can't divide by 0 + 0i
When I can get some decent diagrams I will present a third way of looking at complex numbers.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
(3) The Geometric
)
(3) The Planar Orthogonal Geometric Representation Because each complex number has two parts - real & imaginary - then you can plot them on a plane with two orthogonal axes ( the real axis and the imaginary axis ). To distinguish that from other 2D planes we call this whole plane onto which complex numbers can be plotted the Argand plane. Each point on the plane corresponds to a complex number and vice-versa. The values a and b then represent the displacements along the respective axes, or the coordinates of the point/number (a, b).
.... that is pretty much that. You can see that we might draw a vector from the origin at (0, 0) or 0 + 0i to the point. You can add complex numbers just like adding vectors - nose to tail - as many of them as you like and the resultant vector will be the total/final vector/number/point. Subtraction is achieved by adding the reversed second vector to the first. So what about multiplication and division then ? To make better sense of that then read on ....
(4) The Planar Polar Geometric Representation Let's stay on the Argand plane but use a different coordinate system, as follows :
..... where the general point I've denoted by z with components x and y. Each point on the plane still has a vector from the origin to it but now is measured as r, the distance from the origin and theta the angle the vector makes with the positive real axis ( anti-clockwise is positive by convention ). I could write :
x = r * cos(theta)
y = r * sin(theta)
or transforming the other way :
r = (x2 + y2)1/2
theta = Tan-1(y/x)
.. here Tan-1 is a clever function that gives theta = 90o ( or PI/2 ) if x = 0 and y > 0, and also gives theta = 270o ( or 3*PI/2 ) if x = 0 and y < 0. The point (0, 0) is a rather subtle one, as in polar form r = 0 and theta is undefined due to 0/0 being undefined. If you like the vector from the origin to the origin, a null vector, hence has no direction. Theta is often called the argument of a real number and r the magnitude. Now a clever chap ( Euler ? ) discovered that the natural exponential function ( based on the real line ) had an extension to the Argand plane such that :
z = x + i y = r * cos(theta) + i * r * sin( theta) = r * e(i*theta)
{ the most famous example is ei*PI = -1 or ei*PI + 1 = 0 which is another way of expressing (-1, 0) + (1, 0) = (0, 0). A step to the left and then a step back to the right. }
which seems a bit arcane, until you realise that for two complex numbers R1e(i*theta1) and R2e(i*theta2) their product is :
R1e(i*theta1) * R2e(i*theta2) = R1 * R2 * e(i*theta1) * e(i*theta2) = (R1* R2) * e[i*(theta1+theta2)]
or, thinking vector-wise, you scale the length of one vector by the other's length and rotate it by the argument. For division
R1e(i*theta1) / R2e(i*theta2) = (R1 / R2 ) * e(i*theta1) / e(i*theta2) = (R1/ R2) * e[i*(theta1-theta2)]
{ .... yep, R2 cannot be zero }
It certainly is not immediately apparent why these are terrific results. But they are. We have multiplication and division of vectors. This leads to all manner of helpful concepts and relationships.
So there ! That's complex numbers. I'll have a lie down for a while and think of how to present to you quaternions, the next step up in complexity, while you ponder thus far. Questions ?
Cheers, Mike.
( edit ) As a quick example : what is the unity vector on the Argand plane, meaning what vector/number when multiplying another leaves that second vector unchanged ? Is the unity vector unique ?
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
And I always thought
)
And I always thought that Socrates died in 399 BCE
Our Australian version of a Greek philosopher
It is almost 3:30am here so too late for derivative or partial derivative
goodnight
LOL ;-) Goodnight to you
)
LOL ;-)
Goodnight to you also.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Some More About Complex
)
Some More About Complex Numbers The answer to the unity question was the real number one, 1 + 0i, or (1, 0). If you multiply it by any vector/number/point it will leave it unchanged. The unity is unique but it's representation may not be, as for traveling in a circle one returns to the same position then the argument of e(i* theta) may be plus or minus any multiple of 2 * PI ( the number of radians in a circle ).
Now, look upon the Argand plane. The origin is our chosen centre of view, with the real axis going off positively to the right and the imaginary axis going off positively upwards. Put you fist of your right hand on the origin, then stick the thumb out : the way your fingers are curled is toward the direction ( counter-clockwise ) of the increase in angle for the polar representation. The direction of the thumb, out of the Argand plane towards you, is a way of representing any rotation as a vector. The length of the vector increases with greater rotation, is negative or into/below the plane if the direction is clockwise/negative, and is by construction orthogonal to the plane within which the rotation occurs.
Did you know that the complex number i, or 0 + i, or (0, 1) can represent a rotation by PI/2 in the clockwise direction ? It ought to as
i = 1 * i = 1 * [0 + i*1] = 1 * [cos(PI/2) + i * sin(PI/2)] = 1 * ei*(PI/2)
... thus in the polar representation it is a vector of length one and angle argument PI/2 in the anticlockwise direction from the real axis.
If I multiply some vector/complex number p by i then I will get say p', and the two vectors should be at right angles, or orthogonal. Hence letting p = a + bi
p' = i * p = i * ( a + bi ) = ai + i2 * b = -b + ai
If one takes the plain ( or Euclidean ) dot or inner product of the original and rotated vectors one gets
p . p' = (a + bi) . (-b + ai) = - a * b + ab * i2 = -ab - ab = - 2ab
this is not zero, as one would expect orthogonal vectors to be ! For complex numbers/vectors the inner product is defined as one of them times the complex conjugate of the other. A complex conjugate of a complex number is obtained by multiplying the imaginary part by minus one. So the complex conjugate of (-b + ai ) is (-b - ai). Now take the complex inner product :
(a + bi) . (-b - ai) = - ab - ab * i2 = -ab + ab = 0
and that zero means the vectors at right angles. Such is the rule for the complex plane, and complex spaces of higher dimension in general.
Speaking Of Higher Dimensions Shall we dip a toe into quaternions ? You may know that for purely real numbers a higher dimensional space is called Rn, where n is the number of independent axis directions, and that means that a 'right angle' is between each and every pair of axes in the space. If n is more than 3 then one has a hard time visualising this construct. So it is better to rely on rules assumed to be true for low dimension spaces and keep those rules when generalising to the case of n > 3.
So how does one generalise in terms of complex numbers? Well for many reasons, William Rowan Hamilton and others ( eg. Gauss ) found that going to four dimensions was a good ploy. Just how good we shall see. What is the generalisation of an ordered pair then ? An ordered quadruple of course ! Four components per number/vector. Hamilton chose to keep one real number in first position of the quadruple and make the other three imaginary. So
q = (a, b*i, c*j, d*k) = a + bi + cj + dk
is the general quaternion. If i is just like it was for the complex numbers, what are j and k ? How to describe them is our next task.
{ Hint : each 'imaginary' axis unit vector squares to -1, and all four are in mutually orthogonal directions }
Cheers, Mike.
( edit ) Quaternions are not in Cn, the space of four complex coordinates. That is quite a different space entirely.
( edit ) If i was along the imaginary axis in complex numbers and thus an imaginary number, what do we call j and k ? Do we call j an other imaginary number, with k an other-other imaginary number ? ;_0
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Aside : Hewson, We Have A
)
Aside : Hewson, We Have A Problem ( Or Why You Should Avoid Gimbal Lock ) Remember when we talked of the origin of the complex plane while using the polar representation of complex numbers ? At the origin the argument or angle that it makes with the real axis is undefined. I said it was due to zero divided by zero not being defined, which is quite true, but there is more to it than that.
The issue is generally called degeneracy of coordinates, in this case all angles lead to the same point if the radius from the origin is zero. Another word often used is singularity. It's important to note that this depends on the choice of coordinate system, so for example the same point in rectangular coordinates ( real part and imaginary parts ) has no such problem. In this latter case the origin point has no specially identifiable property, other than we had to chose a point to call the origin. So why does this matter ?
Go to either the North or the South pole on Earth. As it so happens these points are chosen because they are special physically : if you join the two points by a line then you have the axis of rotation of the Earth. That certainly matters eg. the Coriolis effects. But suppose the Earth did not spin - a globe not rotating - and for that matter ignore magnetism too. You could still use the same latitude and longitude system as is the favorite for such round objects. But what is the longitude at either the North or South poles ? Degeneracy again, where one could traverse straight across the pole and suddenly have your longitude jump by 1800 or PI. In a navigation system reliant on such a coordinate system choice these are special situations and require some 'corner case' rules, as it were, to resolve. Crossing over any other non-polar points on Earth doesn't have that problem.
In some 3D systems where we are describing the directions within the entirety of space we could, and sometimes do, use coordinate systems that have angles as parameters. Often used are the three angles that a vector makes with each of three axes. Usually they are called yaw, pitch and roll. It's a matter of choice but there is some degeneracy lurking here too.
Enter the gimbal This device has many uses. The one I'm thinking of is for navigation where one prefers to have constant knowledge about direction without having to always 'look out a window' and check. The classic case mentioned here is from the aerospace industry and all manner of guidance. It often has a gyroscope ( a spinning mass ) in it's centre whose axis of rotation remains the same regardless of the movement of the craft that is wrapped around it. Conservation of angular momentum applies, at least in principle, if the device is freely moving and has no friction/torque etc to change that. In the various motions of the craft you have that constant direction with respect to the outside universe and can thus read some angles of the craft with respect to the gyroscope, and so deduce the crafts' orientation with respect to the outer universe. A three axis gimbal looks something like this :
Trouble arises when, say, two of the three rings come to lie the same plane ( because of craft motions ) and we have a circumstance called 'gimbal lock', as then those rings might subsequently rotate around the same axis together. The two rings now behave as one ( that's the 'lock' ). Two gimbal rings in a 3D space .... one inevitably loses orientation information about one 3D axis. It's a singularity/degeneracy problem. One has to 'reset' the gadget to get out of that or even resort to other trickery in advance to avoid it. This situation nearly happened on Apollo 11 ( and Apollo 13 but for other reasons ) but the system designed to prevent it actually stopped the device from working. LOL not ! They had to literally look out the window to re-align it. Alternatively, as one of the astronauts ( Collins in the Command Module ) pointed out at the time, if you had access to a fourth gimbal ring .....
So why the long winded explanation, eh ? It turns out that quaternions can solve this problem basically because they are four numbers or one more than the number of dimensions of the space it might be describing vectors within. Yep, one can use quaternions to specify directions in 3D space and they don't gimbal lock ! There is never ( important ) degeneracy when used for this purpose. It is now the favored approach to dealing with directions and changes thereof in modern video games. More on that later.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal