Classical to Quantum
Wild Egg Maths
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941 views Oct 3, 2023
We examine critically Einstein's ground breaking 1905 paper: "On the Electrodynamics of Moving Bodies" in detail in this series. In this video we try to understand Part I: Kinematical Part, the first section 1. Definition of Simultaneity, and raise some important questions about the meaning, and indeed to logical clarity, of this foundational piece of science.
We will be shortly advocating a different interpretation of Special Relativity, which in my view is more mathematically clear and elegant. It also touches base with earlier thinking of Poincare. But before we get to that, we must first try to understand Einstein's original thinking in this direction.
This is the fifth video of the series "Classical to Quantum" available to Members of the Wild Egg Maths channel and also to Patreon supporters. Here is the Playlist in the Wild Egg Maths channel for this series:
• Classical to Quantum
In this series, which is part of the Explore Research Level Maths program, we investigate how the relations between classical and quantum physics manifest themselves in pure mathematics.
Along the way we will learn about the history of Quantum Mechanics (QM), Einstein's Special Relativity, representation theory of Lie groups and Lie algebras, q-series, quantization, spherical harmonics, quantum groups, non-commutative geometry, C-* algebras, homogenous spaces, and touch base with the Dynamics on Graphs series (ADE graphs etc), the Q-series series, all at Wild Egg Maths, and the series on Diffusion Symmetry at Insights into Mathematics.
We'll also have a look at some of the main contenders for a decent physics theory of QM, including of course the Copenhagen school of Niels Bohr and Werner Heisenberg, Einstein's objections and Bell's famous inequality, and also Everett's Many Worlds theory, and de Broglie and Bohm/s pilot wave theory, along with other lesser known theories. Is QM really essentially incomprehensible? Can pure mathematics shed any light on the problem?
Some work of the author on the relation between quantization and dequantization, touching on geometric quantization, *-products and associated harmonic analysis topics in Lie theory will also figure.
0:02
hello everyone I'm Norman wallpaper
0:04
thanks for joining me on this adventure
0:05
we're looking at classical to quantum
0:08
mechanics
0:09
physics foundations of that subject and
0:13
we're carrying on with Albert Einstein's
0:16
incredibly important paper on the
0:18
electrodynamics of moving bodies today
0:21
what I want to do is I want to critique
0:24
or think a little bit about this first
0:27
section called the kinematical part
0:30
definition of simultaneity which I read
0:33
for you a little while ago and it only
0:36
has two pages okay and I want to examine
0:38
this with a view of you know what does
0:40
it mean what's he trying to say here and
0:44
are we comfortable with the Notions and
0:47
the way they are presented
0:49
so maybe I shouldn't have to reiterate
0:52
the great respect and admiration that I
0:55
share with pretty well everybody for
0:58
Albert Einstein and his uh courageous
1:00
and insightful and just remarkable uh
1:04
achievements in the direction of you
1:07
know both relativity and quantum
1:09
mechanics
1:10
but I also don't want to be afraid of
1:14
pointing out ambiguities or perhaps
1:17
logical difficulties or at least gaps
1:20
you know where they exist I think
1:22
there's something to be gained from that
1:24
if we have a healthy respect but also
1:28
not be unwilling to tackle uh
1:32
difficulties even if they're
1:33
historically kind of entrenched
1:35
so let's have a look at this all right
1:38
shall we
1:39
and uh we'll start from the beginning
1:40
and we're going to read it uh again I'm
1:42
just going to go over the the main
1:43
issues where I think there's some
1:45
questions arising
1:48
let us take a system of coordinates in
1:50
which the equations of Newtonian
1:52
mechanics hold good
1:55
so already here with this very initial
1:57
sentence there's all kinds of questions
2:00
or you know
2:02
issues that are being raised here and
2:04
Newtonian mechanics classically is
2:07
considered to be in a rather fixed
2:09
framework where we have a fixed notion
2:11
of space and time it might be useful in
2:16
this context to think of a
2:17
two-dimensional
2:19
space so maybe given with X and Y
2:22
coordinates and a Time coordinate T So a
2:25
Newtonian framework for that would be
2:27
basically a three-dimensional linear
2:30
space perhaps a euclidean space
2:33
involving also a metrical structure okay
2:36
but in such a space every point is
2:40
uniquely associated with a triple of
2:42
numbers X Y and Z which are the X Y and
2:45
and T coordinates of that of that
2:47
position so that is I guess our first
2:51
guess as to what he's what he's saying
2:53
here that we're going to take such a
2:54
system of coordinates but there is a
2:56
question there is a question of whether
2:58
he's introducing a model for
3:01
our world saying well here's a
3:03
mathematical setup in which we're going
3:05
to investigate some some Notions from
3:07
physics or whether he's saying let's
3:09
take the space of the world that we are
3:12
living in and impose on it
3:15
such a coordinate system and examine the
3:19
space that we're in it in the in the
3:21
light or in in the in the the mental
3:24
context of such a Newtonian coordinate
3:27
system
3:30
so if a material point is at rest
3:32
relatively to this system of coordinates
3:35
its position can be defined relatively
3:36
there too by the employment of rigid
3:38
standards of measurement and the methods
3:39
of euclidean geometry and can be
3:41
expressed in Cartesian coordinates so he
3:43
is assuming a a euclidean structure
3:47
um but in the next paragraph we see that
3:52
this euclidean structure doesn't really
3:54
extend in his mind to time so if we wish
3:57
to describe the motion of material point
3:59
we give the values of its coordinate as
4:00
functions of the time now crucially now
4:03
we must bear carefully in mind that a
4:04
mathematical description of this kind
4:05
has no physical meaning unless we are
4:07
quite clear as to what we understand by
4:09
time
4:11
now strictly speaking Newton might
4:15
object to this that Newton might say
4:17
well we are just assuming a
4:19
three-dimensional structure in which one
4:21
of the coordinates is time so there's no
4:23
there's no question about what time is
4:26
it's just the third coordinate in this
4:27
two-dimensional
4:29
space together with a one-dimensional
4:31
time okay so
4:34
um
4:34
but I think what this makes clear is
4:37
that Einstein is thinking about space as
4:40
having a system of coordinates so we
4:41
have a system of coordinates on space
4:43
but not necessarily
4:46
also for time so whether this is
4:49
actually really a newtonian's system or
4:52
not is is a question
4:55
okay
4:56
okay so but I I think it's quite
4:59
reasonable to say you know we need to
5:01
investigate what we understand by time
5:03
you know even if even if we are working
5:05
in some kind of mathematical framework
5:07
we I mean when we want to apply it to
5:10
the physical world we have to make some
5:12
kind of Correspondence we have to say
5:13
well what in the physical
5:16
actual World actually corresponds to
5:18
this mathematical idealization of a Time
5:20
axis
5:21
so we have to take into account that all
5:24
our judgments in which time plays a part
5:25
are always judgments of simultaneous
5:27
events
5:29
and in a Newtonian system you might say
5:31
that corresponds to the idea that if you
5:34
specify a t coordinate in this
5:36
three-dimensional space time that we're
5:39
considering here then that defines a
5:41
plane okay and then that plane then
5:43
consists of all the points whose time
5:45
coordinate is that given one
5:48
so yes and those are all considered
5:50
simultaneous
5:52
um but Newton would say yeah but we
5:54
don't need to say anything extra about
5:56
that that's not really saying anything
5:57
more than than saying well when we
6:00
specify an x-coordinate we're talking
6:01
about also a plane in a different kind
6:04
of Direction which consists of all the
6:06
the points in the space time whose x
6:10
coordinate is the given one
6:14
okay so then he talks about simultaneous
6:16
events uh for example the time of
6:18
something is really a statement about it
6:20
corresponding to uh you know um position
6:24
on your watch or your clock and we're
6:27
noting a simultaneity between an event
6:29
and and the position of uh the clock
6:32
hands
6:34
so it might appear possible to overcome
6:36
all the difficulties attending the
6:37
definition of time by substituting the
6:39
position of the small hand of my watch
6:41
for time okay
6:44
um and in fact such a definition is
6:46
satisfactory when we are concerned with
6:48
defining a Time exclusively for the
6:49
place where the watch is located where
6:52
this notion of simultaneity is
6:55
well it's maybe sort of self-evident
6:57
especially if we're thinking about
6:59
making observations with light which
7:02
goes so fast that we can't register how
7:05
fast it's taken for the light to come
7:07
from the watch to the eye or from the
7:09
eye to The observed event
7:15
but it is no longer satisfactory when we
7:17
have to connect end time series of
7:18
events occurring at different places or
7:21
what comes to the same thing to evaluate
7:22
the times of events occurring at places
7:24
remote from The Watch
7:26
so I think he really is thinking about
7:28
having coordinates on the space like on
7:31
the two-dimensional space say in the
7:32
example I talked about earlier and then
7:35
thinking about well how do we actually
7:36
establish a notion a consistent notion
7:39
of time from this
7:43
now
7:45
okay in this next paragraph we might of
7:47
course content ourselves with time
7:48
values determined by an observer
7:50
stationed together with the watch at the
7:52
origin of the coordinates
7:54
and coordinating the corresponding
7:55
positions of the hands with light
7:57
signals given out by every event to be
7:59
timed and reaching him through empty
8:01
space
8:06
so
8:08
okay so he started positing an observer
8:10
in quotes perhaps together with this
8:13
watch
8:14
you know and strictly speaking probably
8:16
we should put watch in quotes too right
8:18
because you know what is a watch exactly
8:21
I mean are we assuming that we already
8:23
have a sophisticated physical structure
8:27
like this let me remind you that
8:29
historically clocks have been hugely
8:31
problematic things to construct
8:33
accurately you know the notion of of a
8:36
clock or an accurate clock
8:38
um that actually registers you know not
8:41
events like days and years but actually
8:43
you know minutes and seconds even uh
8:46
that's that's relatively recent it's
8:47
only a couple of centuries old okay the
8:50
technology inside this Seiko watch is
8:53
remarkable and and difficult to
8:55
understand and appreciate okay so we
8:58
should not think about this thing here
9:00
as some kind of you know Elementary
9:03
god-given physical aspect of the world
9:05
this is actually a highly uh structured
9:10
complicated device and I think we should
9:14
be suitably nervous about
9:17
pinning our understanding of fundamental
9:21
physical reality on a prior uh a prior
9:26
you know abundance of of such technical
9:29
devices
9:30
doesn't ring true to me
9:33
okay anyway
9:35
um so what what's he mean here well he's
9:38
saying good we might be able to get away
9:39
with just having a single Observer
9:42
um stationed with a watch at the origin
9:44
and and looking at things with light
9:47
signals
9:49
um given out by every event to be timed
9:51
and reaching him through empty space so
9:53
if something event is happening there
9:54
then there's a light signal that that
9:55
coming from that event and reaches him
9:57
through empty space
9:59
but this coordination has the
10:01
disadvantage that it's not independent
10:02
of the standpoint of the Observer with
10:04
the watch or clock as we know from
10:05
experience
10:07
I'm not entirely sure exactly what he's
10:10
saying here I mean there is the issue of
10:11
if I get a light signal from some remote
10:14
event okay maybe I can register that but
10:17
how do I know where and when that light
10:20
signal started
10:22
all right I mean I don't have to go over
10:24
there and and look at the uh the
10:27
coordinates of that of that point and or
10:29
or get somebody else to register when
10:31
that um that light signal sort of
10:34
emerged
10:39
so we arrive at a much more practical
10:41
determination along the following line
10:42
of thought
10:44
okay so he's now
10:46
maybe getting around some of the
10:48
ambiguity here by trying to pin down
10:50
more clearly what he means by sort of
10:52
time in in this uh semi-newtonian setup
10:58
Okay so
11:00
have a look at this line if at the point
11:02
a of space there is a clock an observer
11:05
at a can determine the time values of
11:07
events in the immediate proximity of a
11:09
by finding the positions of the hands
11:11
which are simultaneous with these events
11:18
okay
11:19
um I do have the flag that you know he's
11:22
talking about a point a of space
11:24
and I suppose by that he means a point a
11:27
in this Newtonian mathematical model of
11:31
space
11:33
if Galileo was around Galilee would say
11:36
yeah but he's can't be talking about
11:37
real space because in real space the
11:41
notion of a point is hugely problematic
11:45
why well because it depends on your your
11:48
your motion right if I go like this and
11:50
I say well consider this point here this
11:51
is point a and this here is point B
11:54
well that only makes sense if you are
11:57
sort of viewing it in you know in a
11:59
frame with me in which we're both kind
12:02
of stationary
12:04
or where we both appear to be stationary
12:06
crucially okay but in fact maybe we in
12:09
fact certainly we are moving
12:11
dramatically through through the world
12:13
because of the rotation of the earth and
12:15
the uh the orbiting of the Earth around
12:17
the Sun which which Galilei would have
12:19
been well aware of okay so Galilee would
12:21
have said yeah but it just looks like
12:23
this is defining a point it's not really
12:24
it would be like me saying hey guys
12:27
consider this point uh a here
12:30
see the end of my finger that's the
12:31
point a
12:33
and let's introduce the second point
12:35
this is the point B
12:36
right you might think well wait a minute
12:38
you're not defining points in space
12:41
because those things are moving
12:43
well that's exactly the same problem
12:44
that we have even in a more
12:46
simple-minded fashion by just thinking
12:48
that you know
12:49
we can define a point in space
12:51
independent of the obvious motion of all
12:54
these these systems in which we are
12:56
parts of
12:58
while we're getting philosophical I also
13:01
have to say a little bit of you know at
13:02
some point we're gonna have to talk
13:03
about this notion of an observer and
13:06
especially with quantum mechanics this
13:07
becomes a crucial uh you know
13:11
crucial topic which which we'll have to
13:13
say much much but even at this point
13:16
here we should probably flag okay that
13:19
there's there's this implicit assumption
13:22
here that
13:24
we can make sense of having an observer
13:27
you know that the Observer is somehow
13:29
independent of these things and and can
13:31
objectively
13:33
make
13:35
measurements and and deductions and and
13:37
apply theories and so on
13:40
that's a huge assumption okay it very
13:43
well might be that an observer say it
13:46
like this one here is actually you know
13:48
a biological creature
13:50
which is it's self-controlled by laws of
13:55
biology and chemistry which ultimately
13:57
depend on laws of physics
13:59
and that you know any notion of
14:03
self-determination and and Free Will
14:05
that this Observer thinks that they have
14:08
is simply a consequence of their uh
14:10
thought processes not being sufficiently
14:12
uh
14:13
foreign scale and clear okay so
14:19
um it might actually be that you know
14:21
this Observer is just a part of the the
14:24
phenomena just as much as you know
14:25
spaceships or or clocks or or balls
14:29
being thrown up in the air
14:30
okay so that's a topic we'll have to
14:34
return to very interesting questions of
14:36
Free Will which I'll also
14:39
hugely important in talking about
14:41
quantum mechanics but let's just flag
14:43
that there's an issue there
14:45
okay so let's give Einstein the benefit
14:48
of the doubt here and and suppose you
14:50
know he's he's thinking perhaps more of
14:52
a mathematical model for a space rather
14:55
than space itself so we have this
14:56
mathematical model in which
14:59
space coordinates are given in the
15:01
Newtonian fashion but time coordinates
15:04
are not okay and then what he's he's
15:08
saying is that if we have two points A
15:10
and B in this space okay that makes
15:13
sense perhaps in that context then an
15:16
observer near a can determine a sort of
15:20
a local time by looking at things close
15:22
to a and having a watch and
15:24
um an observer at B could also do the
15:27
same thing assuming that the clock at a
15:30
and the clock at B are in all respects
15:33
resembling each other
15:36
now there there's another huge
15:37
assumption which
15:40
plays a big role in in special
15:42
relativity the this assumption that we
15:44
can create frames reference frames
15:47
perhaps far apart with essentially
15:49
identical clocks
15:51
so maybe I can get uh you know my Swiss
15:54
or Japanese Factory to churn out lots of
15:56
identical watches
15:58
but if I want to you know send copies of
16:00
these watches to Neptune or to uh Alpha
16:04
Centauri or or to Andromeda you don't
16:06
have to accelerate them I may have to do
16:08
a lot of acceleration and decelerations
16:10
and it may take a lot of time and are
16:13
the clocks at the result of that still
16:16
the same in all respects as the ones
16:18
that we started out with that's a very
16:20
big assumption probably realistically
16:23
unwarranted
16:25
but in fact what he's proposing here is
16:28
in fact even though he's not saying it
16:30
is a way of actually getting around that
16:32
problem so
16:34
then we get this this crucial little
16:36
section here we have so far defined only
16:38
in a time and a b time we have not
16:40
defined a common time for a and b for
16:43
the latter cannot be defined at all
16:44
unless we establish by definition that
16:47
the time required by light to travel
16:49
from A to B time in quotes equals the
16:51
time again in quotes it requires to
16:53
travel from B to a
16:57
now
16:59
he's right to put those times in quotes
17:02
and we should
17:03
flag that that use of time is different
17:06
from the the a time story and the B
17:09
times story separately
17:10
to posit that there's also a notion of
17:13
of time uh from an event over here in a
17:17
to a time over here in B is already and
17:20
some kind of additional assumption
17:22
what does that mean
17:27
okay so he's trying to get at that now
17:29
by by saying well let array of light
17:31
start at the a Time T sub a from a
17:33
towards B and let it at the B time B
17:35
reflected so it's reflecting light
17:38
uh off of B from a and recording when it
17:41
gets back so it starts at time T sub uh
17:45
a t sub a and that gets back at T sub a
17:48
prime okay
17:50
and he's saying then we can use this
17:54
reflection property of light
17:56
to synchronize the two clocks to to
17:59
guarantee that they have sort of a
18:00
common Time by this equation here which
18:04
I've pointed out is is really equivalent
18:06
to saying that the the the B time
18:09
coordinate of the light hitting it and
18:11
being reflected is the average of the
18:15
initial and final times red at a
18:19
so that's a definition of when these two
18:22
clocks at A and B are synchronized
18:25
and I think this is a very important
18:26
idea that that we can use light signals
18:30
to somehow you know
18:32
um
18:33
make sure that we're talking about
18:35
something that's sort of consistent
18:37
see he addresses and he does it in a way
18:41
that's not that comfortable for a
18:43
mathematician okay so he says we assume
18:46
that this definition of synchronism is
18:48
free from contradictions and possible
18:50
for any number of points and that the
18:52
following relations are universally
18:54
valid and then he says a few things in
18:57
terms of sort of a symmetry of this
18:59
notion of synchronization and a kind of
19:00
a transitivity
19:03
at this point as a mathematician I would
19:06
prefer to say well look you're working
19:07
in a mathematical model okay so you're
19:11
actually free to make these definitions
19:14
in the mathematical model and then you
19:17
could actually
19:18
aspire to
19:22
verifying that these things are true in
19:25
other words you you make the
19:26
corresponding definitions in this
19:27
mathematical model and then you you
19:29
check you compute you you create
19:32
theorems that say that this notion of of
19:35
synchronization it's like a relation is
19:37
actually satisfying this reflexive and
19:39
transitive property
19:41
that's the kind of thing that a
19:42
mathematician would want to do
19:46
okay but he's not a mathematician fine
19:48
but anyway so that's just something to
19:50
think about and then in the last part of
19:51
it that's with the help of certain
19:53
imaginary physical experiments we have
19:55
settled what it is to be understood by
19:56
synchronous stationary clocks located at
19:58
different places okay and have evidently
20:00
obtained a definition of simultaneous or
20:03
synchronous and of time
20:06
which Newton would maybe say well yeah
20:08
okay so that's just another way of
20:10
establishing what I would just have
20:12
assumed automatically that we have this
20:15
linear system ultimately which is
20:17
three-dimensional which includes time as
20:19
as a coordinate but you're you're able
20:22
to to get at at the time coordinates
20:25
sort of from uh one local position by
20:30
by using the the path of light so then
20:33
he says further in agreement with
20:35
experience we further assume the
20:37
quantity 2 a b over T prime a minus t a
20:41
equals c to be a universal constant the
20:44
velocity of light in empty space
20:48
okay this is very big
20:51
um big sense here so this is essentially
20:53
the the second postulate of special
20:56
relativity that
20:57
um
20:58
that this velocity of light in in empty
21:01
space is um is a well-defined um concept
21:08
now
21:09
you see
21:11
I'm nervous here because what he's doing
21:14
he's introducing light okay which is
21:16
very much a physical thing that we don't
21:18
have a good mathematical theory for
21:20
maybe somewhat better nowadays than his
21:23
time but you know light was a
21:25
complicated thing he had this
21:26
electromagnetic aspect they thought it
21:27
was waves uh they thought it required an
21:30
ether like waves typically do some kind
21:32
of underlying medium in which to
21:34
propagate
21:35
um
21:36
okay but
21:38
how do we how do we build that into this
21:40
mathematical model that we're we started
21:42
out with right we
21:43
it wasn't saying well let's let's
21:45
consider the physical world he was
21:47
saying let's consider a mathematical
21:48
model of the physical world so now if he
21:51
really wants to introduce light into
21:52
this
21:54
into this mathematical model he would be
21:56
better perhaps to say well let's assume
21:58
that you know certain lines in in this
22:00
in this affine space or euclidean space
22:03
that we've posited uh you have certain
22:05
properties and certain past certain
22:07
parallelism or certain slope condition
22:08
that's Universal okay so that it's more
22:11
of a something that's built into the
22:14
mathematical side but what he's doing
22:16
here is he's now going back to the real
22:18
world saying that you know actually in
22:20
the real world according to Michaelson
22:22
and morally and and uh you know
22:24
um electromagnetism uh Theory at that
22:27
time that the velocity of light appears
22:29
to be the same in empty space so he's
22:31
making some concept about the world
22:33
but
22:35
how's that okay as soon as he leaves the
22:37
Newtonian framework then Galileo
22:41
steps up and says wait a minute okay
22:43
wait a minute
22:45
if you're working in the in the real
22:46
world okay not some some idealized
22:49
mathematical model then in the real
22:51
world the notion of velocity I'm sorry
22:52
is not a well-defined one at least not
22:55
unless you spend a lot of time thinking
22:57
about it there's because of a velocity
22:59
of of anything it doesn't make sense
23:01
right in a Galilean system if I ask a
23:04
question
23:05
what is the velocity of this eraser that
23:07
is a nonsensical question because it
23:09
obviously depends on
23:12
The Observer
23:14
and if we want to compare the velocities
23:16
obtained by different observers say of
23:19
this eraser then that means we have to
23:21
have a prior way of synchronizing the
23:24
the unit systems in those two observers
23:26
you know I mean this Klingon warship
23:28
might not be using meters and seconds
23:30
and using something entirely different
23:33
okay so to say that we're getting the
23:35
same velocity of light
23:37
requires a prior consistency of units
23:41
across
23:42
um the board and then we're exactly in
23:44
that same kind of position that he was
23:45
worrying about before about how to
23:47
establish the the similarity or the
23:51
synchronization of of clocks not to
23:53
speak of the synchronization of lengths
23:55
which are implicit here in this a b
23:57
which is supposed to be a a distance
24:00
they're actually probably a better
24:01
displacement
24:03
um from A to B
24:05
so you know there's lots of brilliant
24:09
stuff here he's he's thinking about
24:11
things in in a novel way
24:14
um he's uh moving away from an entirely
24:19
a Newtonian point of view even though
24:21
he's sort of adopting it but where he's
24:23
actually sitting and uh what the real
24:26
meanings of a lot of these words that
24:29
he's using is is not at all really very
24:31
clear and finally it is essential to
24:34
have time defined by means of stationary
24:35
clocks in the stationary system and the
24:37
time now defined being appropriate to
24:39
the stationary system we call it the
24:40
time of the stationary system okay so um
24:43
there's a distinguished system in which
24:45
things are fixed which is the time of
24:48
the stationary system
24:50
um
24:51
but overall I honestly if if I was an
24:55
editor of
24:57
you know a mathematics Journal say in
24:59
1905 and somebody had submitted this to
25:02
me um completely honest okay I don't
25:05
know if I would accept it
25:07
um I would say you know there's some
25:09
interesting ideas here but you have to
25:10
expand considerably on on what you're
25:13
saying and you have to clarify much much
25:16
more clearly the meaning of the terms
25:18
that you're using and whether you're
25:20
actually talking about the physics of
25:21
space or some mathematical model of it
25:24
and if you are talking about a
25:25
mathematical model of it then you need
25:28
to spell out more clearly what exactly
25:30
that is and what the underlying geometry
25:32
that you're assuming is
25:34
okay
25:37
but it's still brilliant stuff for sure
25:40
right but um
25:42
there's questions here and these are
25:45
important questions to think about