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Classical to Quantum | The relativistic plane and its red complex number module structure | Wild Egg
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The relativistic plane, or timespace of a one-dimensional physics has a natural structure as a module over the red complex numbers, once we explicitly specify two signal worldlines corresponding to signals propagating right and left on the physical space. These signals could be interpreted in terms of light, or also sound, or in fact in a range of other ways. The red complex numbers C_r form an algebra, actually a *-algebra, with a distinguished notion of quadrance. Algebraically these are very close to the usual (blue) complex numbers, but more applicable to Special Relativity. The module structure that we here describe yields a simple way of coordinatizing the timespace, depending on the observer, who we assume is passing through the origin O which is the meet of the two distinguished signal lines, at time t=0, . We get yet more confirmation that the Lorentz transformations between the measurements made by homochronous observers are simply and directly just red complex number multiplications. This is part of the Classical to Quantum series meant for Members of the Wild Egg Channel and Patreon supporters. See the entire playlist at 
• Introduction to QM and the two slit e...
Transcript
0:00
hello everyone thanks for joining me on this adventure we're looking at the relativistic plane today from a red
0:08
complex number point of view so we're going to be giving coordinates to the plane but in a very flexible and
0:15
variable way so our coordinate system is not unique it's going to depend on our
0:20
choice of Observer and this will then allow us to connect the Lorent transformations in a
0:28
very pleasant way with red complex numbers as we've already seen but it will impose on the relativistic plane
0:35
what we call a red complex number of module structure so this is a quite a
0:42
different way of thinking about the geometry of the the plane the relativistic plane what we're doing is
0:48
we're starting from a basis of apine geometry but we're imposing something additional to it which uh is basically
0:57
two signal lines so the starting point is uh the empty plane the apine plane
1:04
like this but now with two additional lines in red which we call signal lines
1:10
okay and there's a signal line one and there's a signal line two and these two lines we're going to assume that they're
1:17
not parallel so they meet at a special Point called the origin denoted o now
1:23
associated with this we have a conjugation which sends a point a in the
1:28
plane to its conjugate a star so how is this defined this is defined in terms of
1:36
these red signal lines so what we do is we essentially create a parallelogram
1:42
through a with sides parallel to these given directions in this case here with these two green lines and this line here
1:50
which is parallel to Signal line two it meets signal line one at some point and
1:56
then we go an equal amount in the other Direction so we reflect a in the line
2:03
one in the direction of the line two to get a star so similarly if B is down
2:10
here we go in the direction of signal line two an equal amount to the opposite
2:17
side of signal line one to get B Star in this case C is actually on Signal line
2:24
one so it star is itself okay so this is the additional structure structure that
2:31
we're going to impose on the aine plane and we're going to think about these
2:36
lines as being possibly lines that light would travel or possibly lines that
2:43
sound would travel or perhaps some other means of communication in this
2:49
two-dimensional time space World which is modeling what's happening in a one-dimensional
2:55
physics Okay so uh maybe one of the first things
3:01
to to note is that once we have imposed just a single distinguished Point O then
3:10
that turns this apine space into a vector space so given the point O then let's
3:19
say we have points A and B we can then regard those points as being identified with vectors from o right so that point
3:27
a can also be identified with the vector OA and that means we can add the vector o a
3:33
and the vector o using a parallelogram law because we have parallel lines so we
3:38
can create this parallelogram and that gives us the point a plus b so that's a way of adding these two vectors once we
3:45
have a distinguished Point O so it's a vector space but it's more than just
3:52
that it also has a projective metrical structure we're going to talk about that
3:59
more so the ex existence of the two red signal lines imbus this story with
4:05
richer geometry essentially a metrical structure but not not quite but for us
4:12
today what's going to be really interesting is that this structure allows us to impose a module structure a
4:20
red complex number module structure so I have to remind you a little bit about
4:26
the red complex numbers they are a variant on the familiar regular complex
4:32
numbers where instead of having an i we have a j so we consider things of the form a * 1 + B * J or just for short a
4:41
plus BJ and expressly we can think about that as being the Matrix a b b a so the
4:48
J here is represented by the The Matrix whose entries are zero 1 one Zer and
4:55
this Matrix has the property that when you square it you get the identity Matrix so that's the crucial property that J has j2 = 1 and so the
5:03
multiplication of two red complex numbers a plus BJ * C plus DJ is given
5:10
here it's AC plus b d because the J * J is 1 plus a d plus b c j the a * D and
5:22
the B * C we also have some familiar additional structures we have notion of
5:29
conjugation so that a + BJ star is a minus BJ notice I'm using star not bar
5:36
for conjugation and we have the quadrants of a complex number quadrants
5:42
of a plus b j is that number times its conjugate when you multiply that out you
5:49
get a 2 minus b2 because the complex conjugate of this is a minus BJ so
5:56
you're going to get it a^ 2 minus b^ 2 and the other uh J terms are going to
6:02
disappear so this is the quadrants of this complex number then the inverse of
6:10
it if it is invertible is given just as in ordinary complex numbers as the
6:15
complex conjugate divided by the quadrants so specifically it's a minus BJ over a^ 2 minus b2 and that of
6:23
course is provided that a2 minus b2 is not equal to zero another way of saying
6:29
what we're doing here is Zed inverse or 1 / Z if we multiply top and bottom by
6:35
Zed star then the Z * Z star is the quadrants of Zed so that's a simple way of seeing why this is the correct
6:41
formula for the inverse and here we have by now A
6:49
Familiar picture I hope this is the red complex number plane where the identity
6:56
one is vertical here and J is over here it's kind of a physics oriented picture so you think of this as like a time
7:03
space diagram uh quadrants of zero is representing these two lines here t = x
7:09
and t = Min - x so these are the non-invertible elements in this algebra
7:15
it's an algebra not a field because there's more than just one non-invertible element zero there's this
7:21
entire sort of cross of them the unit circle is replaced with this hyperbola
7:27
we can still call it a unit circle but it's determined by quadrants of Z equal 1 and it's this uh curve with two
7:35
branches asymptotic to our null directions the signal line directions in
7:43
terms of conjugation if there's a plus BJ then a minus BJ will be the
7:48
reflection in this vertical axis so this is not a field and maybe
7:55
that's one of the reasons why this structure has not gotten the press that it deserves and maybe physicists tend to
8:02
think that well it's nowhere near as important but that's really something that uh one should change one's
8:08
orientation about so the the property that a field only has one non-invertible
8:14
element is not really that crucial uh the fact that we have a quadrants here that we have very clear definition of
8:22
those non-invertible elements means that effectively we can do in this domain pretty well everything that we can in a
8:29
field we we yes we have to pay attention to when denominators are zero but we have to do that over fields as well so
8:35
this is a a really important algebraic um Arena and that's exactly what we need
8:42
in order to have this module structure on the relativistic
8:48
plane okay let me remind you of this term homones observers um with respect
8:55
to our relativistic plane so from now on the relativistic plane a two Rel okay
9:01
that's the two-dimensional time space plane with two distinguished signal
9:08
lines one and two and they meet the origin so if you have a point a which is
9:17
a non signal event so the points in this time space aine plane we can think of
9:24
them as being events in our one-dimensional physics world
9:30
so if we have such a point which is not a signal event so in other words it doesn't lie on either of the uh lines S1
9:39
or S2 then that induces a coordinatization of the plane okay how
9:45
well using basis vectors A and A star so here if here is a and it's not on the
9:52
signal line then a star is going to be different from it and a and a star when
9:58
regarded as vectors from the origin are going to form a basis for this two-dimensional space and that will then
10:06
be a coordinatization of the plane we'll be able to assign coordinates to points
10:11
in the plane using these two vectors as the basis vectors but for us even more important
10:19
is that this allows a red complex number
10:25
module action on this relativistic plane okay so how does this red module
10:33
structure come about so we're starting with a point a and we're assuming that's
10:38
not on the signal lines and so here is a star it's conjugate in our relativistic
10:46
plane now what we're going to do is for a red complex number Z equal a + BJ it's
10:57
a red complex number we're going to Define we're going to Define that
11:02
a do Z and we should think of that as the action of Zed on a okay so
11:11
specifically it's a DOT a plus BJ that's going to be the
11:17
point a * capital A plus b * capital a
11:23
star all right so this is a linear combination of a and a star are regarded
11:30
as vectors in this
11:36
plane well there's some immediately sort of obvious properties that if we have
11:42
two points A and B in our apine plane said is a red complex number and say
11:47
Lambda is an element in our field so we're working over a field the generic
11:52
one for us it's just the rational numbers but I do want you to think that
11:58
there's a possibility of doing this over other fields as well okay specifically
12:03
over finite Fields so what are the linear properties
12:08
well if we take the sum of two vectors A plus b and act on it by Zed then we get
12:14
the sum of a do Z Plus B do Z so it's linear in a that's because a and this
12:22
conjugation action are depending linearly on a if we multiply a by a scalar Lambda in
12:31
our fields we we stretch it we dilate it by a factor of two or by a factor of
12:36
minus 1/3 and then act on Zed well it's linear again so the factor of Lambda
12:43
comes out front that's Lambda of a do Z and that's also in fact a do Lambda Z
12:49
because we could also multiply the red complex number by lbda that's just stretching it
12:56
in the red complex plane and then there's also a linearity with respect to the complex number addition
13:04
a. Z + W is a. Z Plus
13:10
A.W and now module associativity for a a
13:16
point in our relativistic plane and Zed and W two complex numbers if we take a
13:22
and we act on it by Zed and then we act on that by W it's the same as acting on
13:30
a by the product of Z * W okay so let's go over proof of this
13:37
just to sort of see how the algebra works so if Zed is say A Plus B J and W
13:45
is C+ DJ then we take a point a and we act on it by Zed by definition that's
13:53
little a * capital A Plus little B * a star that that's the effect of Zed
14:00
acting on a and we now have to act on that by W well acting on W what does
14:05
that do well it's because of the linearity that's little a * a acted on
14:12
by W plus little B A star acted on by W
14:17
so w is C plus DJ how does it act on something like this well it multiplies it by C and also adds D times its
14:26
conjugate and similarly W acting on this thing multiplies by C plus d * the
14:34
conjugate so here we just get a c * capital A here we get a d * a star here
14:41
we get b c * a star and when we take the conjugate of this okay B is just an
14:48
element in our field so its conjugate is just B and the conjugate of a star is a
14:54
itself so we get plus BDA and then when we combine these things in terms of A's we have AC plus
15:01
BD and in terms of a Stars we have a d plus BC we see that's just
15:06
a acted on by ZW because ZW the product
15:11
of these two things is AC plus BD plus a d plus b c j so that's what we're doing
15:17
here we're multiplying this thing here by the first one and this thing here by the second
15:23
one okay so we have this important associativity of the module
15:29
structure okay so now we can introduce a coordinates on our relativistic plane
15:34
we're
15:57
coordinatization precisely when this notation is happening when the a
16:04
coordinate of B is Zed so there's a capital B with a subscript capital A
16:11
equals Zed and we we say this in either one of two ways we say that the a coordinate of B is Zed or that the
16:19
coordinate of B with respect to a is Zed all right so what we're doing here is
16:24
we're saying okay pick a point a okay we're going to use that Point as sort of
16:29
the basis of a coordinate system which is dependent on this point a and that
16:35
means that any other point can be obtained by multiplying this fixed point a by a suitable red complex number and
16:43
we're going to call that the coordinate of B with respect to a that number Z
16:49
that complex number Zed okay it's going to be the coordinate of B with respect to
16:55
a so now what we're doing is we're saying okay we're going to coordina tize this time space not with a pair of
17:02
coordinates time comma space but actually with a single coordinate which
17:08
is a red complex number and of course this red complex
17:14
number that is assigned to B depends on a so sort of from the point of view of
17:20
a this is what we have to multiply a by to get
17:26
B okay so now we have ort of two facts uh if B sub a equals Zed so if the
17:34
coordinate of B with respect to a is Zed then a subb equals Zed inverse in other
17:40
words the coordinate of a with respect to b is z inverse why is that well B sub
17:48
a equals Zed that's the same as b equal a do Z by definition right that's the
17:55
definition right here B sub a equal Z that means that Bal a do
18:04
Z all right but then if we take B do Z inverse okay well there's B it's a do Z
18:12
so if we dot it with zed inverse then by the associativity of the module structure this is the same as a do Z * Z
18:20
inverse and that's a do1 and it's easy to check that's another property but I
18:25
didn't say it explicitly that a DOT one is just a so that's telling us that the
18:32
coordinate of a with respect to b is z inverse you need to multiply B by Zed
18:37
inverse to get a so if you need to multiply a by Zed to get B then you have
18:44
to multiply B by Zed inverse to get back to
18:52
a and second proposition and this is very important
18:57
for for physicist this is going to provide a a much simplified way of doing
19:03
algebra with Loren Transformations so it becomes not trivial but it becomes very simple just
19:11
basic algebra using complex numbers of this red variety so what is this
19:17
proposition is that if we have three points say a b and c and if the
19:24
coefficient of B with respect to a is Zed and if the coefficient of C with
19:30
respect to b equals W then the coefficient of C with respect
19:36
to a is the product Z * W the product of these two red complex
19:43
numbers and another way of saying that is C Suba over here is Zed which is B Suba
19:52
time W which is C subb and if we put it in the other order it's C subb time B
20:01
sub a this form of it I think is easy to remember C sub a equal C subb * B sub
20:11
a well let's see why this is true so B sub a equals Zed what does
20:19
that mean okay that means that b equal a
20:24
do Z you have to multiply a by Zed in order to get B and this one here C sub
20:31
Bal W that means that c equal B do W you have to multiply B by W to get C all
20:40
right hence C which is b. W is equal to a do Z dot w because B is
20:52
a do Z so we're replacing the B here with a do Z and then we use the module
20:58
associativity that we've already had this is a do z w so that's telling us
21:08
that the coefficient of C with respect to a is zed w because you have to
21:14
multiply a by ZW in order to get C so
21:20
that's indeed the conclusion that we're trying to get
21:27
to let's have a look at an example all right let's have a look at an example so
21:33
here is relativistic plane there's the origin here are two signal lines signal
21:41
line one in red signal line two in red here's a point a here's its conjugate a
21:48
star here's a point B here's its conjugate B star note that we're not
21:54
fussing around about you know forward light cones here right now we're being very flexible as to where these points
22:01
are they're not restricted other than they're not allowed to lie on the signal
22:06
lines and there's another Point C up here and it's conjugate C star so let's
22:14
see if we can figure out the various relationships between these
22:19
things so first of all B sub a the
22:25
coordinate of B with respect ECT to a so let's let's scroll up here and say okay
22:31
how do we do that what is the coefficient of B with respect to a well
22:38
that's basically asking how do you express b as a linear combination of
22:44
these two basis vectors determined by a basis Vector o a and o a star okay so to
22:51
get to B uh you have to sort of go negative
22:58
a down here and then 2 a star okay so in terms of these two basis vectors B is
23:06
equal to minus1 comma 2 and that's just another
23:12
way of saying that if we take this point a and we multiply it by -1 + 2 J -1 + 2
23:21
J then we're going to get minus a + 2 a star minus a plus 2 a star which is B
23:27
okay that's what's happening down here the coordinate of B with respect to
23:34
a is min -1 + 2 J so suppose we were
23:39
interested in inverting that and describing a as a combination of B and B Star okay
23:52
well that's in red module language that's trying to determine a subb the
24:01
coefficient of a with respect to B but we've seen that that's just the inverse
24:07
of B sub a so we just take this red complex number minus one plus 2J and
24:15
look at its inverse how do you form the inverse you take the complex conjugate
24:20
up in the numerators that's minus 1 - 2 J and you have to divide by the quadrants of this thing which is what
24:27
well you take take the square of the first coefficient so that's Min - 1^ s and you subtract the square of the
24:35
J so squared so you subtract 2^ squar so
24:40
Al together we have -1 - 2 J over Min -3 for a total of 1/3 + 2/3 J so let's see
24:50
what that's saying back up here that's saying that if we want to express a as a
24:57
the combination of the vector B and B Star we should take 13 of B so imagine this Vector going
25:06
from o to B and 1/3 of it which would be about down to there right and then what
25:13
should we do uh we should add 2/3 of J so then we have to add 2/3
25:21
of this Vector from o to B Star so that be about up to there right so we're
25:28
going 1/3 down here and then 2/3 o star
25:34
so that's the effect of multiplying B by
25:40
the red complex number 1/3 plus 23 J you multiply B by this you get 1/3 B plus
25:47
2/3 B Star and that is indeed uh corresponding with what seems to be
25:54
happening up here okay what's next how about the coefficient of C with respect to a
26:02
well let's do it geometrically first C is up here so with respect to a so we
26:10
have to go two in the a direction and one in the a star Direction so that's to say that we have to multiply this
26:16
capital A by the red complex number 2 + J okay multiplying by 2+ J we get 2 A
26:24
plus a star which is C all right so there's 2 + J so if we want
26:32
to invert that and get a sub c in other words the coordinate of a with respect
26:41
to C we have to take 2+ J inverse it's the complex conjugate divided by the
26:46
quadrants which is 2^ 2 - 1^ 2 we get 2/3 -3 J 2/3 minus 1/3 J so that's telling
26:57
us that if we want to express a in terms
27:03
of C we have to multiply that by 2/3 minus
27:09
1/3 J so if we take 2/3 of this Vector
27:14
here that would be about there okay and then we have to
27:20
subtract 1/3 of the vector from o to C star so 13 of that is maybe about there
27:29
and we have to take negative of that so yes that seems to work 2/3 OC is roughly
27:35
there and then we um add minus 13 of C
27:42
star that gets us to a see so we're doing these calculations
27:48
just in the world of red complex numbers okay we're not using matrices we've
27:55
stepped back from the Lorent trans as matrices and uh you know it's it's just
28:01
it's so much simpler okay now let's illustrate the important um way of um of
28:09
comparing uh what happens when you have um three observers okay so suppose that
28:16
we're interested in C subb in other words the the coordinate of C with
28:22
respect to B well one way of expressing that it's c sub a time a
28:31
subb right so if we're interested in C subb how do you express
28:39
c as in the coordinate system of B whose
28:45
basis vectors are OB and OB star we want to express C that way okay so just using
28:51
the algebra C sub a is 2 plus J there it is
28:57
a subb we've calculated here it's 1/3 + 2/3 J we do the multiplication so 2 *
29:05
1/3 is 2/3 and the 1 * 2/3 is another 2/3 so that's 4/3 and the coefficient of
29:12
J is well there's a 2 * 2/3 that's 4/3 and there's a 1 * 1/3 that's 1/3 for a
29:19
total of 5/3 so we get 4/3 + 5/3 J that's the coefficient of C
29:28
with respect to B 4/3 plus 5/3 J so let's have a look
29:34
there okay so 4/3 5/3 so to get to C we have to take
29:42
4/3 of B so that's in this direction here 4/3 maybe around there okay and
29:49
then we have to go from here up to to C so how do we do that we're going to go
29:54
5/3 of this that's almost twice it's about the same
30:00
thing right so we go 4/3 B that's down here plus 5/3 B Star that's that that's
30:09
telling us that if we multiply this point here by the red complex
30:15
number 4/3 + 5/3 J we're going to get
30:22
C great and if we wanted to get B
30:28
in terms of C so B Sub C we would have to take this 4/3 plus 5/3 J and invert
30:34
it in the world of right complex numbers there's the complex conjugate there's the quadrants and the result is - 4/3 +
30:42
5/3 J let's just check that so - 4/3 + 5/3 J that's B in terms of C okay so to
30:52
get to B we have to go minus 4/3 of this so that's uh - 4/3 that would be uh down
31:00
down somewhere around here okay and then we have to go plus 5/3 of C star so
31:07
there's C star o to C star we have to take 5/3 which is almost twice that so that's going from down here off to be
31:15
yeah that sounds about reasonable great so our conclusions
31:21
first of all there are as many coordinate systems now we're talking about hochron coordinate systems which
31:28
um depend on having the fixed Point O as
31:34
sort of our center of our observations so all of these observers have their
31:39
world lines going through o and have agreed that that's going to be the zero
31:46
time for them so there's as many such coordinate systems on our relativistic
31:51
plane as there are non-s signal events so there's lots of different coordinate frames okay and and that's the whole
31:58
point of Relativity there's not just one frame but now with respect to a given Observer the coordinates of all events
32:05
or observers can be identified with red complex numbers so once we have an observer a then every other event or
32:12
Observer can be identified by multiplying that a by a suitable unique
32:18
red complex number something in the form a plus BJ okay so that point there is
32:25
with respect to a 3 + 5 J this point over here might be Min - 4 + 7 J Etc
32:36
every point is coordin tized by a red complex number and
32:43
finally and finally the Lorent Transformations just map to Red complex
32:50
number multiplications via this module structure right to go from one
32:56
coordinate system to another coordinate system that's exactly this associativity
33:01
that we are are talking about in this proposition here it's exactly this law here that
33:09
allows us to go from observing uh something in in one coordinate system to
33:17
another coordinate system so if we know um say the uh the coordinates of C with
33:24
respect to B and we know the coordinates of B with respect to a then we can figure out the coordinates of C with
33:30
respect to a just by taking this this product this is basically the Lorent Transformations this is the embodiment
33:37
of the the previous lecture when I said that Loren Transformations were just red complex number multiplications there's
33:44
the red complex number multiplication explicitly now we have a nice sort of language to talk about going from one of
33:51
these coordinates to another uh very smoothly and it's all um
33:58
just uh algebra which is at some level quite familiar it's just like complex
34:03
number arithmetic except that we're in the red World rather than in the blue
34:10
world