Classical to Quantum
Classical to Quantum | Red Complex Numbers and Special Relativity | Wild Egg maths
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The usual (blue) complex numbers get a lot of press. But the fascinating red complex numbers are arguably just as important in understanding our world. This is especially true once one gets on board with Einstein's Special Relativity, which dramatically changed 20th century physics by finally bringing to the fore Galileo's Invariance Principle. Red complex numbers have a somewhat richer story, as the role of null numbers, forming a "light cone" becomes crucial. These numbers are not invertible, so the red complex numbers don't form a field as do the blue ones.
However we will see that the parallels between these two structures are very tight. And it turns out that the algebra, and the geometry, of red complex numbers is exactly what is required to understanding Lorentz transformations and Special Relativity in a simple and elegant way (we will be explaining this in a few videos from now!) Of particular interest is that the usual rational parametrization of the unit circle in the regular blue complex numbers as a nifty analog here in the red situation.
This is all intimately tied to the fascinating three-fold symmetry of chromogeometry, which we will have to discuss at length in a different series. There is so much to be explored here for enthusiastic amateur mathematicians!
A big thanks to all my Patreon supporters and to Members of the Wild Egg maths channel.
0:00
hello everyone thanks for joining me on
0:02
this adventure towards special
0:03
relativity and quantum mechanics I'm
0:06
going to show you a much simpler and
0:09
more Elementary way of understanding
0:12
what's happening with the basic
0:14
equations of special relativity before I
0:17
do that though it's really good to have
0:21
a solid understanding of some algebraic
0:24
Frameworks upon which this special
0:27
relativity is built and that's the
0:31
theory of complex numbers actually red
0:34
complex numbers or hyperbolic versions
0:39
of the usual ukian complex numbers so
0:42
this is part of the general story of
0:44
Chromo geometry which has a three-fold
0:47
symmetry in planer geometry blue being
0:50
ukian and red and green
0:52
being relativistic kinds of geometries
0:55
that are closely connected but they're
0:57
sort of separate also in an important
0:59
way So today we're going to talk about
1:01
the red geometry and it's this geometry
1:05
and this algebra that is really the
1:07
underpinnings of special relativity and
1:10
if you understand this then the whole
1:12
story is dramatically
1:14
simplified and I'm going to present it
1:16
to you in a very parallel way to what I
1:19
did in my last video with the blue or
1:23
regular complex numbers so it's almost
1:25
sort of line by line uh similar except
1:29
crucially for the last slide on the
1:32
previous video that's a separate issue
1:35
that requires an entirely separate story
1:38
okay so we're talking about red complex
1:40
numbers so what is a red complex number
1:43
it's an ordered pair say Z equals TX and
1:46
that stands for T * 1 + x * J which
1:51
we'll also write as t + XJ and this is a
1:54
symbol that represents this 2x two
1:57
Matrix Matrix t x x T and here's our
2:01
symbol for red complex numbers so this
2:04
is a two-dimensional subalgebra of the
2:07
2x2 matrices just as the corresponding
2:10
Blue version was where there was a minus
2:13
sign down here okay so there's just a
2:15
little bit of a distinction so in the uh
2:18
previous version the the letter I was
2:21
representing the Matrix 0 1- 1 0 now the
2:24
letter J is representing the Matrix 0 1
2:27
1 0 okay the algebra is just taken from
2:32
the 2x two matrices so here's how we add
2:34
two such red complex numbers here's how
2:37
we scale or multiply 1 by Lambda and
2:40
here's how we multiply them so TX * r y
2:44
is T * r + x * y comma T * y + x *
2:50
R very similar to the usual story except
2:53
crucially there's a plus sign there
2:55
instead of a minus sign complex
2:58
conjugation is very similar if Z equal
3:01
TX then Z star is T minus X and then we
3:05
verify that z w star is the same as Z
3:07
Star * W star and we compute that Zed *
3:12
Z Star Zed is this thing here is T ^2
3:15
minus
3:16
x² and we call that the quadrants of Zed
3:20
okay t2 -
3:22
x2 and then the crucial property of
3:26
this quadrants is that it satisfies this
3:29
basic rule that the quadrants of zed w
3:32
equals the quadrants of Zed * the
3:34
quadrants of w and you can see uh maybe
3:37
it's really worth pointing out that the
3:38
quadrants in terms of the matrices is
3:40
just the determinant so this rule here
3:43
can also just be observed by being the
3:47
usual familiar product rule for
3:49
determinants in fact that's exactly the
3:51
same thing that happens also in the blue
3:53
story in terms of inverses if the
3:57
quadrants of Zed is non zero then the
4:00
inverse of Zed according to this
4:01
equation here is 1 / Q of Z * Z
4:07
Star now uh in terms of invertibility
4:10
there is a a difference here okay
4:13
because this quadrants can be zero in
4:18
more ways than in the Blue version in
4:21
the Blue version the quadrants was t2 +
4:23
x s which is only zero when both T and X
4:26
are zero but over here there's a lot
4:29
more Poss possibilities if T equals x
4:31
then you're going to get quadrant Z or
4:33
if T equals Min - x you're also going to
4:36
get quadrant Z so the non-invertible
4:39
elements the elements for which this is
4:42
not valid form a a cross actually in
4:45
this two dimensional um space so there's
4:48
more non-invertible elements that means
4:51
that this thing here is not a field in
4:53
the usual fashion and perhaps for that
4:56
reason this algebraic structure has seen
4:58
much much less uh development and
5:02
interest and understanding than the Blue
5:05
version but I think it's um it's good
5:08
for you to orient yourself the following
5:10
way that the notion of a field is not
5:12
really the crucial one here okay um this
5:16
thing here deserves to be an algebraic
5:19
structure on the same level as the other
5:21
complex numbers in many regards okay in
5:24
many
5:25
regards and uh it's the fact that we
5:28
still have an Inver ability formula like
5:31
this that makes a lot of things work but
5:36
of course if we dividing by something
5:38
then we as usual have to ensure that the
5:41
denominator is not zero okay and in this
5:44
case uh the corresponding statement is
5:46
we have to ensure that the denominator
5:48
doesn't have quadrants equal to zero
5:51
okay so it's this framework here that we
5:55
want to now investigate and see how many
5:57
of the things that we understood in the
6:00
blue geometry are holding also now for
6:02
this red geometry coming from these red
6:05
complex numbers so the first order of
6:07
business is perhaps to visualize these
6:09
things and we're going to do that in a
6:10
parallel way to what we did with the
6:12
blue complex numbers so we have a t axis
6:14
and an x-axis so there is the point one
6:17
and there is the point J now instead of
6:20
I it's j so a general Point say Zed TX
6:25
there's the T coordinate there and the x
6:28
coordinate is XJ
6:29
that's the red complex number t + XJ
6:34
what else do we see here well uh the um
6:37
complex conjugate of this thing here
6:39
there's Zed there's Z Star so that's T
6:42
minus X the X component has
6:45
negated um and crucially we see also
6:49
quadrants Z equals z so quadrants is
6:51
this thing here t2 - x squ so when the
6:54
quadrant is zero those are the
6:55
non-invertible elements we have these
6:57
two lines the line t = x and the line t
7:01
= - x so they're in red those are the
7:04
non-invertible elements that's a bigger
7:06
collection than just in the blue
7:08
geometry where only the origin was
7:12
non-invertible and what about the unit
7:14
circle in this context the place where
7:17
the quadrants of Zed is equal to 1 well
7:20
that's the equation t2 - X2 equal 1
7:22
which you recognize as a hyperbola this
7:24
case it's a hyperbola opening up and
7:27
down so there are two branches okay so
7:29
there's one branch here which is ASM
7:32
totic to these two lines okay so it goes
7:34
ASM totically up like that and
7:36
correspondingly down there like
7:38
this now on this diagram also we can see
7:43
the slope or velocity of Zed what is
7:46
that so if we look at the line through
7:48
the origin passing through Zed and here
7:52
is the line T equals 1 which is tangent
7:55
to the unit circle at the point one okay
7:59
then this line here meets that tangent
8:02
at the point 1 comma
8:05
V so one up V over is going to be
8:09
similar to t x okay so there's apine
8:13
connections here so the velocity is V of
8:18
Z is X over T exactly the same
8:20
definition as in the blue situation of
8:22
course for T not equal to
8:26
zero so that's now the picture that's
8:30
going to replace the very familiar
8:33
picture of the unit circle as an
8:35
ordinary Circle so we still say this is
8:38
a unit circle but it's the unit circle
8:40
in this
8:41
relativistic uh setting where
8:44
geometrically it looks like a hyperbola
8:47
to us so now we have a proposition which
8:50
tells us what the slope or the velocity
8:52
of a product is so the velocity of Z * W
8:57
is the velocity of Zed plus the velocity
9:00
of w / 1 plus the product V of z v of W
9:07
plus sign instead of a minus sign that
9:09
we had back in the blue situation and
9:12
we're going to introduce also a circle
9:13
sum notation for that so this is V of
9:16
Zed Circle sum subr V of w so it's the
9:21
red circle
9:23
sum almost the same as the blue circle
9:25
sum but not quite because of that plus
9:27
sign instead of a minus sign okay the
9:29
proof is very familiar if Z is TX and w
9:32
= r y then this expression here is x / t
9:36
y r 1 + x / t y r you multiply through
9:41
by TR R you get XR + YT and then the
9:45
bottom TR r + XY and that is the
9:47
velocity of ZW because the product of
9:50
these two things is T * r + x * y t * y
9:56
x * R and the ratio the second
9:59
coordinate divided by the first
10:01
coordinate is exactly what we're finding
10:04
here so those of you who have taken a
10:08
little bit of special relativity will
10:10
recognize this as a kind of a
10:12
relativistic velocity addition formula
10:15
of Einstein so very famous so what we're
10:18
saying here is that this particular
10:20
relation is just an elementary feature
10:24
of the algebra SL geometry of these red
10:27
complex numbers and has a direct
10:30
parallel in the blue situation which
10:32
we've talked about last
10:34
time okay so over and over again I
10:37
emphasize the importance of the rational
10:39
parameterization of the usual unit
10:41
circle so here also the rational
10:43
parameterization of this relativistic
10:45
unit circle is also crucially important
10:48
so the story follows very much uh the
10:51
blue situation so what I'm doing here is
10:53
exactly parallel to what we've discussed
10:56
when we were talking about blue complex
10:58
numbers so here's our unit circle
11:01
there's two branches and this applies to
11:04
both of them okay and here say is a a
11:07
point on this unit circle and we want to
11:11
sort of parameterize it there's the
11:12
parameterization so how do we get that
11:14
thing so what we do is we start with
11:17
this line tal 1 the tangent line to the
11:21
unit circle at the identity point1 and
11:23
we choose a point on it say w = 1 comma
11:27
H and and then what we did in the blue
11:30
case is we first of all we squared that
11:32
thing so if we take W and square it we
11:35
get 1 +
11:37
h2 comma 2 H and then what we did is to
11:41
this one here is we renormalized this
11:44
thing to make it of
11:47
quadrants one and what we do is we
11:50
divide this thing here by the quadrants
11:51
of the original W now so the quadrants
11:55
of the original W is 1^ 2 - h^ 2 or 1 -
11:59
h2 so that's what we have to do we have
12:01
to take this thing here and just divide
12:03
both sides by 1 - h2 that will ensure
12:06
that this thing here then has
12:09
quadrants uh one so it's on this
12:12
hyperbola now it's interesting to
12:14
observe that in the this relation here
12:16
if H is say between zero and one then
12:19
then these guys are positive and uh
12:21
we're sort of in in this range here as
12:23
exhibited here but if say h is bigger
12:26
like H is equal to you know five or
12:28
something we're going to have a negative
12:30
number here okay and that means that the
12:32
point that we're getting will actually
12:33
be on this Branch down here so this
12:35
Branch here is also covered this way and
12:38
while we have this thing here there's
12:40
the general point on the unit circle its
12:43
velocity is this quantity divided by
12:47
this quantity is 2 H over 1 + h2 and as
12:51
before we say that H is the half slope
12:54
of
12:55
Zed so in this example just to be a
12:58
little bit concrete
12:59
so um here is sort of like one and one
13:02
these are the the units on these two
13:05
scales so this would be say roughly H
13:08
equals a half so let's say that W is
13:10
equal to 1 comma half then you can check
13:12
that when you square it you're getting
13:15
um 5 quars 1 so the x coordinate
13:20
actually doubles so it happens to be one
13:21
so we get five quarters one and then the
13:24
quadrants of this thing is U quadrants
13:27
of w is 1 s minus a half squ so that's
13:31
3/4 and so the The Zed you get by taking
13:34
this thing here and dividing by 34 that
13:37
gives us 5/3 comma
13:40
4/3 and that's the point that we're
13:42
getting here so five 5/3 this way and
13:45
4/3 this way notice there's a
13:48
Pythagorean triple floating around here
13:50
uh just like there is in the ordinary
13:52
blue situation but uh you know where the
13:55
hypotenuse is is in a different place
13:57
than uh ordinarily
13:59
and in this case here the velocity is 4
14:02
fths this over this is 4 fths there's a
14:05
velocity of 4 fths this would be a
14:07
velocity one now there's another way of
14:09
thinking about how to make this
14:11
parameterization instead of looking at
14:14
the point 1 H to start with we could
14:16
also look at the x axis through the
14:19
origin and just look at the point h i on
14:22
it and look at the point minus one on
14:26
the unit circle still and when when we
14:28
join minus1 to hi then we have a line
14:32
which is already
14:33
meeting the unit circle in one point so
14:36
it necessarily meets it in a second
14:38
point and that's another way of
14:40
obtaining this formula is just by
14:42
starting with this point and this point
14:44
and calculating where the second
14:46
intersection of this line is with the
14:49
the conic okay and um it's also worth
14:53
pointing out that just as in the blue
14:55
geometry this line here is parallel to
14:58
this line here okay they have the same
15:00
slope or velocity and furthermore
15:04
another feature that's also exactly the
15:07
same as in the blue situation is that if
15:10
you take this point here and the the
15:15
identity one and you take the midpoint
15:18
of that that's right that point there
15:20
that is also lying on here so that's
15:23
sort of another way of uh of thinking
15:26
about U about getting Zed you could uh
15:29
start with this line here and
15:30
essentially reflect this point in this
15:33
line it's actually a perpendicular
15:36
reflection you might ask what does
15:38
perpendicularity mean in this context
15:40
well there's an algebraic way of
15:41
thinking about that just in terms of the
15:43
sort of associated dot product but
15:45
there's also a geometrical way of
15:47
thinking about perpendicularity with
15:49
respect to a conic so um if you have
15:52
this conic here and you have a line like
15:54
this and you want to know what's the
15:56
perpendicular line okay what you do is
15:58
you think about where this um conic
16:01
meets the line okay maybe it doesn't
16:03
actually meet but let's say it suppose
16:04
it meets at that point there and you
16:06
look at the tangent at that line okay
16:09
this is a line through the center of the
16:11
of the conic okay so this is like a
16:13
radius okay and then if we take the
16:16
tangent at that that point where that
16:18
radius meets the conic that's going to
16:20
be the perpendicularity so that's
16:23
another way of saying what this thing
16:24
here is it's the line parallel to the
16:27
tangent uh to the conic at that point
16:29
there okay so it's quite a rich geometry
16:33
okay and it's exactly parallel to the
16:36
circle geometry that I showed you before
16:39
but there's no angles okay we're
16:40
avoiding
16:42
angles you might say well what about all
16:44
that beautiful Circle geometry that we
16:46
learned you know back in perhaps high
16:48
school or maybe just a fan of circle
16:52
geometry does that hold also for this
16:54
stuff for these kinds of circles yeah
16:57
basically yes yes yes so it's a a lovely
17:00
thing to do is to think about the
17:02
consequences of that uh of that
17:04
claim now because of that relation
17:07
quadrants of z w equals quadrants of Zed
17:09
quadrants of w it follows that our
17:12
unit circle is a group because if you
17:15
have two red complex numbers both of
17:18
quadrants one then their product will
17:21
also have quadrants one and the inverse
17:24
of one of them will just be the complex
17:27
conjugate because the coordinates Zed
17:29
which is in the denominator is one and
17:31
if Zed is on the unit circle then Zed
17:34
star is also on the unit circle so this
17:35
is a group okay and so naturally the
17:39
question is well what does the group
17:41
multiplication look like in the blue
17:44
situation you've probably leared and
17:46
I've hopefully convinced you that you
17:48
should think otherwise but you've
17:49
learned that you know um multiplication
17:52
of unit complex numbers is given by
17:55
adding angles right I told you you
17:58
should let let go of that okay so here
18:00
also there's no angles involved okay
18:03
there is a nice geometrical way of
18:05
thinking about the product it's the same
18:08
lorm product as before okay so let me
18:12
explain this through an example so here
18:14
we have two points U1 and U2 on our unit
18:17
circle and to be specific let's let U1
18:21
have half slope 1/2 so there's the point
18:25
1/2 there's the line from minus one
18:28
through that point on the x-axis and it
18:30
meets the conic at the other point
18:33
U1 and uh it's actually in terms of
18:37
parameters it's 5/3 4/3 we've seen that
18:39
point already so U2 here's another one
18:42
it's associated with the half slope Min
18:44
three Hales which is over here okay so
18:48
if we look at the line from minus one
18:50
through this thing here then it's going
18:51
to meet the conic at a second Point down
18:54
here there's U2 and you you just use the
18:56
formula you know the 1 + h^ 2 over 1 -
19:00
h^ 2 comma 2 H over 1 - h^ 2 um to to
19:05
get - 13 fths comma 12 fths so - 13 fths
19:10
down here and 12 fths in that direction
19:14
and again you see a Pythagorean
19:17
triple in that in that uh relation there
19:21
okay what's the product U1 * U2 well you
19:24
just do the product use the red complex
19:27
number formula and you get U on U2 is -
19:29
17
19:30
15 8 15 which is where it's right
19:35
here okay so um here's how you can think
19:38
about that uh the lmer Meer product is
19:42
the following that what you do is you
19:43
just take the line joining U1 and U2 and
19:47
you consider a parallel line through the
19:52
the uh the point one the identity of our
19:54
group okay and where that line meets the
19:57
conic in the second point because
19:58
there's one point on it the second point
20:00
will meet it at a unique point and that
20:02
is the product U1 and
20:04
U2 and uh so there's another Pythagorean
20:08
triple 8 15 17 and that should also have
20:11
a half slope right what's the half slope
20:14
of that well that's given by taking
20:16
these half slopes and taking the red
20:18
circle sum so that's H1 plus H2 over 1 +
20:21
H1 H2 okay plug in this and this so if
20:25
we have um H1 plus H2 uh2 plus minus 3es
20:31
that's minus one so it's minus one in
20:33
the numerator and 1 + H1 H2 uh the
20:37
product is - 3/4 1 + - 3/4 is 1/4 so -1
20:42
over 1/4 that's -4 so the actual half
20:45
slope is over here -1 -2 - 3 -4 over
20:49
here somewhere okay and um the line from
20:52
minus one to that point there does go
20:56
through U1 U2
20:59
so it's still a
21:00
group okay and uh you know I think it's
21:04
it's crucial that physicists see this
21:06
and you know there's all kinds of
21:08
exercises and and investigation that can
21:11
be done basically you can take any of
21:13
those sort of dozens or perhaps hundreds
21:15
of exercises that relate a complex
21:18
number geometry and the the unit circle
21:21
and and the group Theory aspect of that
21:23
and you can translate you can find
21:26
analoges of those directly in this red
21:30
story okay um there's all kinds of
21:33
questions that are that are that are
21:34
raised here um questions in terms of you
21:36
know forier analysis can you do for
21:38
analysis on these things and what does
21:40
it look like these are all very
21:41
interesting questions that I hope to
21:44
address at some point um but right now
21:47
uh we've gotten a good solid
21:49
introduction to Red complex numbers but
21:53
crucially I haven't talked about the
21:56
analog of the so of momentum story where
21:59
we have energy and momentum and
22:01
Associated two vector and and the you
22:04
know various relations um around eals
22:08
mc2 so um we're going to talk about
22:10
that later um that's also a lot of it is
22:14
sort of inherent also in in the geometry
22:16
uh of these red complex numbers but in
22:19
in my next video what we need to do is
22:22
we need to look at the other
22:24
relativistic story I told you there were
22:26
three planer geometries in Chromo
22:29
geometry right the blue one the red one
22:31
there's also a Green version so it turns
22:33
out that the Green version while it's
22:35
pretty similar to The Red version it's
22:38
also noticeably different and gives you
22:40
an important different perspective which
22:42
turns out to be crucial for physics so
22:45
we're going to have to have a discussion
22:46
about the green complex numbers and also
22:49
how they uh connect with special
22:54
relativity