Classical to Quantum
Classical to Quantum | Special Relativity and green complex numbers | Wild Egg Maths
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Chromogeometry is a relatively recent discovery of mine which holds that there is a remarkable three fold symmetry in planar metrical geometries, between the usual blue Euclidean geometry, and two relativistic ones, called red, and green. Correspondingly there is a triple of complex number systems, the blue, red and green complex numbers.
While the latter two are not fields, they nevertheless share important properties with the more familiar everyday complex numbers all undergraduates learn about. The red and green complex numbers are particularly important for physics, as they underlie the geometry of Einstein's special relativity (SR).
In this video we are particularly interested in the linear algebra of transitions between the red and green geometries, corresponding to covariant and contravariant concepts in physics.
In many ways the blue and red geometries are parallel, and in other ways the red and green are parallel. However the green geometry, and associated algebra, is in some ways simpler than the other two. This provides us actually a powerful tool to study even Euclidean geometries.
In this video we introduce the green complex numbers, closely related to what physicists sometimes call null coordinates in SR.
A special thanks to Patreon supporters and Members of the Wild Egg maths channel.
0:00
hello everyone thanks for joining me on
0:02
this adventure towards quantum mechanics
0:05
we're looking at special relativity and
0:08
we've been thinking about blue complex
0:10
numbers and red complex numbers
0:12
I've told you those are intimately
0:14
connected now there's another version of
0:17
the red or relativistic Story coming
0:19
from Chromo geometry there's actually a
0:21
three full symmetry so there's a third
0:23
kind of geometry the green geometry and
0:26
an Associated Third Kind of complex
0:28
number the green complex numbers and
0:31
that's what we're going to talk about
0:32
today and the relationship between the
0:35
red and the green plays a key role so
0:38
there's a kind of a transformation to go
0:40
from red coordinates to Green
0:42
coordinates that's That's essential it's
0:45
really at the heart of what's happening
0:47
with special relativity on the
0:49
mathematical side
0:51
okay
0:52
the role in physics is that it's
0:55
connected with what physicists call null
0:57
coordinates in special relativity
1:00
okay so we're going to start just with a
1:03
pair of numbers t x so that's what a
1:06
green complex number is going to be for
1:08
us today and that stands for T times 1
1:11
plus x times K or just t plus x k for
1:15
short and we're going to think about
1:17
this as representing this two by two
1:19
matrix it's a diagonal matrix now with
1:22
entries T minus X here and t plus X here
1:26
C sub G will refer to this realm of
1:29
green complex numbers so this is another
1:31
sub algebra of the two by two matrices
1:35
and now we have three different
1:37
sub-algemas and pleasantly they're sort
1:39
of associated in a nice way with these
1:41
four basis vectors of the two by two
1:44
matrices the identity Matrix there this
1:46
Matrix which we could call i 0 1 minus
1:49
one zero this one J zero one one zero
1:52
and this one K now minus one zero zero
1:55
one so that's exactly what we need to
1:57
make this thing here expressible as
2:00
T times one plus x times k
2:04
now in other places I might use a
2:06
slightly different notation for example
2:08
you could also think of K as being one
2:10
minus one this will be convenient for us
2:12
with the applications to physics that we
2:15
have in mind
2:16
now algebraically K squared is the same
2:19
as J squared equals one so this sub
2:22
algebra is really isomorphic to the red
2:25
complex numbers span by one and J so we
2:28
have these three complex numbers the
2:29
span of these two and then the span of
2:31
these two and then the span of these two
2:33
okay and so that's a you know
2:37
so they're very natural almost canonical
2:40
structure that's sitting inside our two
2:42
by two Matrix world so we can adopt the
2:46
algebraic structure of the two by two
2:47
matrices and just restrict them to this
2:51
sub algebra of green complex numbers we
2:55
get that the sum is just the obvious
2:57
thing t x plus Ry is t plus r x plus y
3:00
the scalar multiplication is as expected
3:03
and the multiplication is exactly as it
3:06
is in the red story so TX times r y
3:09
equals t r plus x y
3:12
comma t y plus RX
3:16
and then just in parallel to what
3:19
happens in the red story
3:21
or in fact the blue story The Complex
3:23
conjugate of TX is defined to be T minus
3:26
X and then when we multiply complex
3:28
number say Z by Z Star TX times T minus
3:32
X then we get t squared minus x squared
3:35
times 1 which we'll just write as t
3:36
squared minus x squared and that's the
3:38
quadrants of Zed and then the quadrant
3:42
satisfies the usual fundamental property
3:45
that we know and love quadrants of z w
3:47
equals quadrants of Z times quadrants of
3:49
w and it allows us to define the notion
3:52
of inverse Z inverse equals one over the
3:54
quadrant of Z times Z Star provided that
3:56
the quadrants of Z is not equal to zero
4:01
and
4:03
geometrically well here is the familiar
4:08
looking story but now with basis vectors
4:11
1 and K
4:13
okay so
4:15
um
4:16
in terms of this basis Vector the
4:19
quadrants is t squared minus x squared
4:21
so then we have quadrants of Z equals
4:22
zero and the quadrants of Z equals one
4:24
this hyperbola with two branches as
4:27
before
4:29
but you know so it looks like it's
4:32
exactly the same as complex numbers in
4:35
the red story but it's it's not
4:37
crucially so
4:39
um
4:40
there's something different about this
4:42
green geometry which actually in some
4:45
ways makes it even easier to compute
4:47
with than the red one even though
4:49
algebraically they're pretty well sort
4:51
of isomorphic but nevertheless this
4:54
three-fold symmetry then becomes a a
4:57
large kind of motivating understanding
5:01
and in fact somewhere else I'll have to
5:03
talk about my student Kevin lamonta's a
5:07
remarkable work on super Catalan numbers
5:09
and integral formulas integrating over
5:14
over spheres in in in quite a general
5:17
fashion
5:19
um over over finite Fields as well which
5:22
really exploits this Chromo geometry
5:25
story of we have a blue geometry we have
5:28
a red geometry we have a green geometry
5:30
we can go from the blue to the red just
5:32
by a change of sign and we can go from
5:34
the red to the green by this
5:36
transformation that we're going to talk
5:38
about now and how probably the the green
5:41
story is in some ways simpler
5:43
arithmetically than the other two okay
5:46
so let's talk about change of basis for
5:49
green complex numbers there is an
5:51
alternate basis which appears very
5:53
naturally in this particular context
5:55
which simplifies the arithmetic
5:58
dramatically so here's the identity
6:00
Matrix one here is the K Matrix minus
6:04
one zero zero one and together they span
6:07
the diagonal matrices but the diagonal
6:10
matrices also have another very natural
6:12
basis just the matrices with a single
6:16
one in either the top left quarter or
6:20
the bottom right corner let's call them
6:22
N1 and N2 so this is an alternate basis
6:26
for this sub algebra that means we can
6:29
look at how the arithmetical operations
6:31
look like in this basis as well as in
6:35
this basis so the relationship between
6:37
the two is pretty simple to see the
6:40
identity Matrix is just the sum of N1
6:42
plus N2 while K is minus N1 plus N2 and
6:46
conversely if we wanted to express these
6:48
ones in terms of those ones well then N1
6:51
is this one minus this one over two well
6:55
N2 is this one plus this one over two
6:59
so we have these sort of two different
7:01
kind of Transformations they're inverses
7:03
to each other so this one here we can
7:05
express it in this sort of interesting
7:08
Matrix form but the elements of this
7:11
Vector here are not numbers they are
7:13
themselves vectors okay so this is a
7:16
vector whose components are the vector
7:19
is one
7:20
and K and here's another Vector whose
7:24
components are the vectors N1 and N2
7:27
and this statement here is just saying
7:29
that this column Vector 1K is
7:33
obtained by multiplying by The Matrix
7:35
one one minus 1 1 on the left of N1 and
7:39
- so we get N1 plus N2 for 1 and minus
7:43
N1 plus N2 for K and then inverting
7:48
we just bring this to the other side and
7:50
invert it the inverse of this is this
7:53
Matrix here the determinant is one minus
7:55
minus one or two so there's a one-half
7:57
and then you need to change those and
8:00
multiply these diagonal Elements by
8:02
minus one getting one minus one one one
8:05
so then N1 M2 equals one half one minus
8:08
one one times one k this is a matrix way
8:11
of writing those things
8:13
okay so we have these two bases now and
8:17
that means that correspondingly we have
8:19
actually two ways of thinking or looking
8:21
at the story here so here's the story
8:24
that we started with with basis elements
8:26
one and K in the usual sort of
8:29
horizontal vertical and horizontal
8:31
positions and in that story N1 and N2
8:35
are over here so N1 is the um one minus
8:40
K over two so it's halfway between 1 and
8:43
minus K and N2 is 1 plus K over 2 so
8:46
it's halfway between 1 and K
8:49
but we can take this picture and just
8:51
rotate it and just rotate it
8:55
um
8:55
a an eighth of a turn okay so then these
9:00
black axes go to here so we get one here
9:03
and K here and then N1 is here and N2 is
9:07
here we might call these axes say Z1 and
9:10
Z2 just to give them some names okay as
9:12
opposed to the T and X axes
9:16
so from this point of view we're
9:17
thinking of N1 and N2 as sort of the
9:20
fundamental basis vectors and expressing
9:23
everything in terms of linear
9:25
combinations of N1 and N2 where over
9:27
here we think about 1 and K as being the
9:30
the basis and we're expressing
9:31
everything in terms of multiples of 1
9:33
and K so we have this transformation
9:37
the advantage of this thing here is that
9:39
the multiplication is much simpler in
9:43
in these coordinates than it is in these
9:46
coordinates so I want to go rather
9:47
carefully through the change of
9:49
coordinates here because it's a little
9:51
bit subtle we're touching base with what
9:53
physicists refer to as covariant and
9:56
contravariant vectors or tensors
9:59
this is a very simple example but
10:01
there's sort of like two different ways
10:03
that we can think about a transform you
10:06
think about transforming the basis
10:08
vectors themselves or we can think about
10:11
transforming the coordinates
10:13
that are required to express an
10:15
arbitrary Vector in terms of those basis
10:17
vectors it's kind of a complementary
10:19
aspect or almost like a duality between
10:22
these two different ways of looking at
10:24
things and this particular change of
10:27
coordinate is right at the heart of
10:30
special relativity as I'm going to
10:31
explain to you
10:33
reasonably shortly okay so we really
10:36
want to understand exactly the
10:38
arithmetic involved here even though
10:40
it's very very Elementary
10:43
okay so we started with a green complex
10:45
number Z say TX and I remind you that
10:48
was T times 1 plus x times K we saw that
10:51
we could rewrite things because one is n
10:54
1 plus N2 and K is minus N1 plus N2 so
10:57
we can rewrite this thing as
10:59
T minus x times N1 plus t plus x times
11:03
N2
11:05
so that's a linear combination of N1 and
11:07
N2 say alpha 1 N1 plus Alpha 2 and 2.
11:13
so these are the new coordinates Alpha
11:15
One Alpha two are the new coordinates
11:17
with respect to the basis N1 N2 and I
11:20
should point out that you know this
11:21
thing here this this thing that we're
11:23
talking about as a matrix okay it's a
11:25
diagonal matrix whose entries are just
11:28
alpha 1 on the top and Alpha 2 on the
11:32
bottom
11:33
okay that's all
11:36
this is telling us how to go from the
11:40
coordinates in the 1K basis to the
11:43
coordinates in the N1 and two bases so
11:46
to summarize what we've just done the
11:48
Alpha One Alpha 2
11:51
are obtained from the TX by multiplying
11:54
by this same Matrix that we've seen
11:56
before but now
11:57
on the right
11:59
so TX times 1 1 minus 1 1. so T minus X
12:03
that's giving us the Alpha One and t
12:06
plus X is giving us the alpha 2.
12:08
and if we want to invert that whether we
12:11
multiply by the inverse of this Matrix
12:12
which we've already seen before but we
12:14
multiply it on the right so we get TX
12:17
equals Alpha One Alpha two times here is
12:19
that inverse Matrix one half one minus
12:21
one one
12:24
foreign
12:25
so to give you an example here is the
12:28
story again now but now reoriented so
12:30
that our null basis vectors are in the
12:33
sort of central position so we have N1
12:36
in here and N2 here
12:38
there's one
12:39
there is K
12:41
so here is some
12:43
um
12:44
green complex number say N1 plus 2n2 so
12:49
in the N1 N2 coordinates it's one in
12:52
this direction and two over in this
12:55
direction so because we're using square
12:57
brackets here to distinguish the round
12:59
brackets here this would be the one
13:01
comma two
13:03
now if we wanted to rewrite it in terms
13:06
of 1 and K well geometrically that means
13:10
you have to re-express it you have to
13:11
tilt your head and you have to write
13:12
this Z now as a combination of this
13:14
basis vector and this basis Vector okay
13:16
so it turns out to be three halves of
13:19
one plus one half times K or three
13:24
halves comma one-half and how could we
13:27
get that okay well so this would this is
13:30
like the um the T okay and that's like
13:33
the X so there's T and X we're starting
13:35
off with one two and we multiply by this
13:38
thing here so one two
13:41
um one times one that's three and that's
13:43
divided by two so that's three halves
13:44
and one two uh minus one plus two that's
13:47
one divided by two so that's a half okay
13:49
so just going from here to Here by
13:51
multiplying by that Matrix on the right
13:55
and in terms of the quadrants okay the
13:59
quadrants well in
14:01
the 1K coordinates you take these
14:05
coefficients three halves and one half
14:06
you take three halves squared minus one
14:08
half squared which is um
14:11
uh two but another way of doing it is in
14:14
terms of the alpha coordinates the
14:17
quadrants is even simpler it's just the
14:19
product of alpha one times Alpha two
14:22
right because we're talking about the
14:23
determinant of this Matrix so it's just
14:25
Alpha One times Alpha two so in these
14:27
coordinates it's one times two
14:32
so please make sure that you understand
14:34
this example a very simple
14:37
perhaps you know one of the most simple
14:40
kinds of change of basis stories that we
14:43
can discuss just a two-dimensional
14:46
situation with very simple
14:47
transformation matrices but of crucial
14:51
importance for a special relativity
14:54
so let's look at this transform which is
14:56
operating at two different levels at the
14:59
level of changing basis vectors and also
15:02
changing coordinate
15:04
vectors really so the physicists refer
15:07
to this distinction as covariant and
15:10
contravariant so there's a green complex
15:13
number Z equals TX in the usual initial
15:16
coordinates involving 1 and K and here's
15:19
another way of expressing what that is
15:21
if we allow this sort of curious usage
15:25
of a basis Vector where the basis
15:29
vectors themselves are in a vector so in
15:31
my linear algebra course I do this
15:32
occasionally so you may have seen this
15:35
before it's an interesting kind of thing
15:38
to do but this is like a regular Matrix
15:42
multiplication TX 1K you get T times 1
15:46
plus x times K that's giving us that
15:48
combination
15:50
now the same
15:52
number complex number can also be
15:54
written in terms of the N1 and N2
15:56
coordinates say with coefficients or
15:59
coordinates Alpha One and Alpha 2. in
16:01
that case we write it with square
16:02
brackets just to distinguish it between
16:04
you know this one here okay
16:06
and what that is is taking this Vector
16:11
Alpha One Alpha two of numbers and
16:13
multiplying it by this column Vector
16:14
again of basis vectors it's just another
16:17
way of saying it so
16:20
the there's a relationship between this
16:24
column vector and this column Vector
16:25
okay given by some transformation and
16:28
there's also a corresponding
16:29
relationship between this coordinate
16:32
vector and this coordinate Vector given
16:33
by another transformation and these two
16:36
Transformations are are very closely
16:38
connected and we can sort of see that
16:40
here let's introduce say l to be this
16:43
two by two Matrix that we've seen before
16:45
1 1 minus one one and its inverse is one
16:48
half one minus one one
16:50
so what we've seen is that
16:53
if you take this Alpha One Alpha 2 and
16:56
you take the TX and you ask what's the
16:58
relation the answer is this one here is
17:00
obtained from this one here the TX by
17:03
multiplying on the right by L right
17:07
and
17:11
on the the basis Vector side this one k
17:14
okay is obtained from this one by
17:18
multiplying by L on the left
17:21
so there's corresponding inverse
17:23
statements perhaps I should write those
17:26
down just for completeness so the
17:28
inverse statements would be t x equals
17:32
Alpha One
17:33
Alpha 2 times
17:36
l
17:38
inverse
17:40
and inverting this one
17:43
N1 N2
17:45
equals inverting
17:49
we have to multiply by L inverse on the
17:52
on the other side so we put an L inverse
17:54
on this side
17:55
times 1 k
17:59
so the role of the L is involved in
18:02
going from coordinates
18:05
here to coordinates here but also from
18:09
basis vectors here to basis vectors
18:12
there
18:13
and if you do both of them at once you
18:15
see that they cancel out if you want to
18:17
go from here to here for example if you
18:19
want to replace if you take TX and you
18:21
multiply Inside by L then you're going
18:23
to get Alpha One Alpha 2. but to get
18:25
something that's the same as what you
18:27
started you need to multiply by L times
18:29
L inverse so if you put an L times L
18:31
inverse then the TX times L is this one
18:35
here
18:35
while the L inverse times 1K
18:39
over here is N1 N2
18:42
right so that's why this is equal to
18:45
this because you go from this one to
18:47
this one by sticking in l and an L
18:49
inverse in there and doing those two
18:52
Transformations together
18:55
Okay so we've touched base with this
18:58
third important geometry and algebra the
19:01
green complex numbers for which the
19:04
multiplication ultimately is a little
19:06
bit simpler because in this basis the
19:09
multiplication is just a point wise
19:11
multiplication right so so
19:14
arithmetically
19:16
this is a much simpler kind of Realm to
19:18
deal with so that's why physicists like
19:21
this because it's a nice transformation
19:23
that after you've done the
19:24
transformation the algebra simplifies
19:26
dramatically
19:28
now we're in a really good position to
19:30
understand what special relativity is
19:32
really all about