Scruffy's math program for kids

Lemme see, SO3 and SU2 are algebraic groups, their names come from the "group catalog" (universal classification of groups, which I already mentioned is about 20 years old - even though, physicists have been using these names for over 100 years).

SO3 stands for "special orthogonal group in 3 dimensions", it's special because all the determinants are 1. Basically it's the group of rotations in 3 dimensions, so the matrices (rotation operators) might look something like this:

| Cos t, Sin t, 0 |
| - Sin t, Cos t, 0 |
| 0, 0, 1|

where t is an angle, usually written as theta.

We apply this to a 3d vector (x,y,z) like this:

rotated result = MV

where M is the rotation operator (matrix), and V is our vector. In this example we're rotating in the XY plane and Z is left alone (hence the 1 at the bottom right of the matrix).

However we can write our 3d vector as a 2x2 Pauli vector, like this:

| z, x + yi |
| x - yi, z |

and apply the SU2 rotations to it (SU2 is also a Lie group, it's the special unitary group in 2 dimensions, unitary meaning that the inverse equals the Hermitian conjugate). In this case, our vector transforms in a different way - we have to use a double sided transformation with half the angle. Like this:

V' = U V U°

where U° is the inverse (Hermitian conjugate) of U, and U looks like this:

| e^(-it/2), 0 |
| 0, e^(it/2) |

In this case the angle t (theta) is halved, and each side of the double sided transformation performs half the rotation.

This construction is also how we get spinors from vectors, the SU2 matrices are "spin operators". But you'll notice that

U V U° = (-U) V (-U)°

so there are two SU2 matrices that perform the exact same rotation, and this is why we say that SU2 is a "double" cover of SO3.

So this is how we understand 1/2-spin particles in relation to 1-spin particles. For a 1/2-spin particles the SU2 transformation is 1-sided, whereas for a full spin particles it's 2-sided.

Very logical, right? Simple, easy, nothing more than matrix math. But we're using groups, and geometric rotations, and we're using them to understand particle physics.

This is a great example of Scruffy's math program for kids. We already used the exact same math to understand the quantum spin states that underlie quantum computing, and now we're using it to understand particle physics, and it's nothing more than rotation and matrix multiplication.

Similarly, we want to introduce kids to "connections" before they graduate high school. Probably 12th grade, would be a great time for it. When we do rotations of a point on a sphere, we want to show the kids the tangent plane associated with each point. Then we can introduce them to "tangent bundles" and show them the Levi-Civita connection for parallel transport
 
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Using the "Pi" formula ,one can't get an exact answer. But ,by the very nature of things , there HAS to be an EXACT answer.
Well that depends on how you define "exact".

This formula (just one example) exactly defines 𝜋

1737046090619.png



I suspect you really want to define "exact" as being computable in finite time but some things are not. But we often don't need a numeric representation at all, we can just use the symbol 𝜋 to represent it and still do mathematics with it easily. The symbol 𝜋 can be used to compactly represent the algorithm on the RHS.

In pure mathematics the physicalized values of things are seldom important to mathematicians.
 
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This mathematician explains this far better than scruffy who simply wants to impress you, his motto is "why use one word when twenty five will do".

 
Well that depends on how you define "exact".

This formula (just one example) exactly defines 𝜋

View attachment 1066274


I suspect you really want to define "exact" as being computable in finite time but some things are not. But we often don't need a numeric representation at all, we can just use the symbol 𝜋 to represent it and still do mathematics with it easily. The symbol 𝜋 can be used to compactly represent the algorithm on the RHS.

In pure mathematics the physicalized values of things are seldom important to mathematicians.
And that is the trouble with Math. The answer to my question has an EXACT answer. Perhaps the only way to find it is to measure the diameter and the circumference manually.
 
And that is the trouble with Math. The answer to my question has an EXACT answer. Perhaps the only way to find it is to measure the diameter and the circumference manually.
The term "exact" isn't absolute in the physical sciences though. You can measure things of course but there are always uncertainties, measuring devices have limited precision, need calibrating and so on, so there is no "exact" in the physical realm.

The number 𝜋 appears within pure mathematics without any relation to geometry, mathematically the ratio is no longer the "definition" because pure mathematics does not rely on physical measurements to define fundamental things like this.

I have a digital volt meter here, it will display measured voltages with four digit precision like this 5.0033 but is that exact? If I were to use a more expensive device with higher precision that same voltage might appear as 5.003286

This astonishing equation is named Euler's Identity, it effectively defines 𝜋 without any reference to geometry, the physical world.

1737128472563.png


So don't think of 𝜋 as being defined by geometry but rather as geometry being defined in terms of 𝜋
 
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The term "exact" isn't absolute in the physical sciences though. You can measure things of course but there are always uncertainties, measuring devices have limited precision, need calibrating and so on, so there is no "exact" in the physical realm.

The number 𝜋 appears within pure mathematics without any relation to geometry, mathematically the ratio is no longer the "definition" because pure mathematics does not rely on physical measurements to define fundamental things like this.

I have a digital volt meter here, it will display measured voltages with four digit precision like this 5.0033 but is that exact? If I were to use a more expensive device with higher precision that same voltage might appear as 5.003286

This astonishing equation is named Euler's Identity, it effectively defines 𝜋 without any reference to geometry, the physical world.

View attachment 1066613

So don't think of 𝜋 as being defined by geometry but rather as geometry being defined in terms of 𝜋
Somewhere in the misty haze of Mathematical relationships there is an Absolute value to this question. Just as an ounce weighs and ounce ,the EXACT number of Diameters to form a circumference has an answer. In the REAL Physical World. Perhaps our math ain't up to it yet.
 
Somewhere in the misty haze of Mathematical relationships there is an Absolute value to this question. Just as an ounce weighs and ounce ,the EXACT number of Diameters to form a circumference has an answer. In the REAL Physical World. Perhaps our math ain't up to it yet.
Still not sure what you're after.

Pi has been expanded to a million digits, you can see the results here:


Pi is a "transcendental" number, it is not solvable in a finite sequence. Simple things like √2 are in the same category.
 
Somewhere in the misty haze of Mathematical relationships there is an Absolute value to this question. Just as an ounce weighs and ounce ,the EXACT number of Diameters to form a circumference has an answer. In the REAL Physical World. Perhaps our math ain't up to it yet.
If by "exact" you mean a finite number of decimal digits then I agree, there is no "exact" representation of either irrational or transcendental numbers.
 
I believe the "Ether" theory was debunked.
lol - well, depending on your point of view I suppose. Quantum field theory references fields which "could" be construed as a form of ether, except there are lots of them, not just one.

String theory is a real challenge. The most promising variant of it is called E8xE8, where E8 is the largest and most complicated "exceptional Lie group" having 248 dimensions. These are the dimensions of the smallest irreducible representations:

1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, .....

Someone (Jeffrey Adams) has actually calculated these matrices. The minimal construction is the 16 dimensional special orthogonal group SO(16), but there are also 128 new generators that transform as Weyl-Majorana spinors.

Anything this complicated is bound to describe a lot of stuff, but there are actually some experiments that confirm predictions. Wheeler reported that cobalt-niobium under certain conditions show 2 of the 8 E8 peaks predicted by the theory.

This is what the E8 root system looks like, projected down into 3 dimensions.

1737236057387.jpeg


This is the 2 dimensional version.
1737236171214.jpeg


E8 goes back to Wilhelm Killing in 1890, but Elie Cartan is the guy who put it on the map.

Here is a summary of the Lie algebras:

 
I believe the "Ether" theory was debunked.
It was eventually abandoned because observations became explicable with relativity and the aether plays no role in that theory.

The whole aether thing was due to the very important question as to what are electromagnetic waves propagated "in". Waves are fluctuations in stuff like pressure, in some medium like water waves or sound waves.

But light? what is there for these weaves to propagate "in"? so they hypothesized and later believed in, the aether.

Maxwell's field equations predict the speed of light but they do not define what that speed is relative to, it's just a speed.

The thinking was that that speed was relative to the aether and so everything made sense if this "aether" existed but as you likely know, efforts to demonstrate its existence all failed.

There is one thing to note though, although the aether was abandoned the problem of some kind of absolute reference remains in physics, even both relativity theories seem to rely on some kind of absolute reference frame - it's a puzzle.
 
It was eventually abandoned because observations became explicable with relativity and the aether plays no role in that theory.

The whole aether thing was due to the very important question as to what are electromagnetic waves propagated "in". Waves are fluctuations in stuff like pressure, in some medium like water waves or sound waves.

But light? what is there for these weaves to propagate "in"? so they hypothesized and later believed in, the aether.

Maxwell's field equations predict the speed of light but they do not define what that speed is relative to, it's just a speed.

The thinking was that that speed was relative to the aether and so everything made sense if this "aether" existed but as you likely know, efforts to demonstrate its existence all failed.

There is one thing to note though, although the aether was abandoned the problem of some kind of absolute reference remains in physics, even both relativity theories seem to rely on some kind of absolute reference frame - it's a puzzle.
There is one thing that bothers me. And computers refuse to answer. If 2 Astroids are traveling at 3/4 light speeg ,going right at each other , their relative speed ,to each other ,is faster than C. Yet computers will not recognize this simple fact.
 
There is one thing that bothers me. And computers refuse to answer. If 2 Astroids are traveling at 3/4 light speeg ,going right at each other , their relative speed ,to each other ,is faster than C.

No, actually it isn't. Velocities don't add up by simple summation, you have to use a "Lorentz transform", iow special relativity.

Velocities only add by simple summation in the limit as v => 0. (Which covers most of the cases we encounter in our daily lives). Once you get up above 1/4 c or so the errors become significant.

There is an "apparent" exception to c called "phase velocity", it's not a real exception, it only seems that way to us.

There is another type of exception that's perhaps more real but it has nothing to do with the actual velocity, only the information "carried by" the velocity. It's a form of "completion phenomenon" that happens in neural networks and many-dimensional quantum systems, and it depends on pre-existing information compartments.


Yet computers will not recognize this simple fact.

Here's a video that describes the relativistic calculation of mutual velocity. If you have two asteroids A and B, there are 3 velocities involved: A as seen by B, B as seen by A, and A and B as seen by an inertial third party (you or me).

 
There is one thing that bothers me. And computers refuse to answer.

If 2 Astroids are traveling at 3/4 light speeg ,going right at each other
Where is the observer that's measuring their speed? how is it being measured?
their relative speed ,to each other ,is faster than C. Yet computers will not recognize this simple fact.
The "relative speed" depends on the situation of the observer who's making the measurements.
 
The observer is ON one of the astroids.
Simplest case, is moving in a straight line, and not accelerating (so, constant velocity). This is called an "inertial" reference frame.

In such a reference frame, distance and angles confirm to a "local Cartesian coordinate system", meaning distance can be measured by a yardstick and time by a clock.

The fundamental principle of physics is that the laws of physics are the same from any reference frame. So if as you say the two asteroids are moving directly at each other, we should be able to exchange reference frames, in other words A looks to B exactly as B looks to A. This is what the Lorentz transformation does, it leaves the physical observables invariant when we swap reference frames.

The weirdo thing about our universe is, the geometry isn't flat, it's hyperbolic. In inertial reference frames this structure is called "Minkowski space". To understand it, you can define an "event" as occurring at a point in space and time. So, in a flat Euclidean geometry the distance between two events would be given by

D = t^2 + x^2 + y^2 + z^2

where t is the time difference and x, y, and z are the space differences.

But Minkowski space doesn't work like that. It works like this:

D = t^2 - x^2 - y^2 - z^2

It is an example of a "non-Euclidean" geometry.

So, using this, you can figure out the Lorentz transformation for any two inertial events.

First, in keeping with the fundamental principle of physics, the speed of light is the same (constant) from any reference frame. Therefore, you can calculate the Lorentz factor as follows:

L = 1 / sqrt (1 - v^2 / c^2)

where L is something like the "amount of compression" as the velocity increases. When v is near 0, the compression factor is 1, which means you see something like the actual velocity. But when v is near c, the compression factor grows without bound, meaning that what you see is always "a fraction of" the speed of light, and you'll never quite hit the actual speed of light (cause if you did you'd be dividing by 0, making the compression infinite).

You can derive this equation from the "Lorentz group", or alternatively using the Minkowski metric. Because of the reciprocity of reference frames, it only describes a situation where one of the reference frames can be translated (or rotated, or sheared) to the origin. And left fixed there, during the measurement of the spacetime events.

If you're accelerating, or equivalently if your spacetime is curved, you have a more complicated situation and you have to use general relativity instead of special relativity. In general relativity, the assumption of yardsticks and clocks breaks down (because of the curvature), and you have to use a "metric tensor" to measure distances and angles.
 

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