Speed Lectures on Modern Physics
Hello. I will now start a series of lectures taking up the challenge
of explaining in a very short time the fundamental theories of
modern physics : special relativity, classical physics and quantum
physics, without any prerequisites.
By making it short, I don't mean doing any superficial work of
popularization, that does not really explain the theories, but that
in the name of simplification and understanding, actually eliminates
any effective understanding, to just roughly describe some
No. What I will give here is the full, accurate, mathematical
expression of these theories. And I will make them much more
intuitive and faster to learn than you can find anywhere else.
So that just a short time of these lectures, such as one hour of a
new lecture per day for a few days (with the rest of the day
re-playing it and caring to understand it), will let you learn
essentially as much as by attending the lectures currently provided
by any educational institution for a few weeks or even months. Well
okay it won't be complete in the same way, I won't explain all the
same things and give you the tools to solve the same problems, but
it will still be as good, by better explaining things in other ways.
How is that possible ? Because as I explained in the first video,
these institutions never really cared to find the best possible ways
to teach physics. It isn't so hard to do the best when others,
aren't really trying to compete. But anyway, you will see.
Ok, let's start now by a lecture on geometry.
Why introduce geometry to start a series of lectures on physics.
Because geometry is the first theory of physics. It is a theory of
physical space. And for modern physics, space geometry is somehow
just a physical substance among others, in interaction with others,
and similar to others. It is the physical substance that we most
easily and naturally understand. Its right mathematical expression
is very similar to those of other theories of physics, so that its
study is the key to understand other theories, both in a purely
mathematical sense, and by its way of providing a visual intuition
that can help to understand other physics theories too.
The beginning of this lecture will perhaps seem slow and obvious,
but be careful, it will then skyrocket and go very fast. You can
find a slower and more detailed approach, in my web site
(settheory.net). (2 min 30 sec)
So, we start with the geometry that we are familiar with, that is
the Euclidean geometry. We seem to live in a 3-dimensional Euclidean
space. Still we are also used to do plane geometry : the
2-dimensional Euclidean geometry, easier to draw and to visualize:
the geometry of a plane, or of a paper. But why not also
consider 1-dimensional geometry. That is, the geometry of a line. It
is even simpler.
(show : space, a paper, a wire)
So in fact we have several geometries. But what is a geometry ? A
geometry is a mathematical theory. Any mathematical theory is the
description of a system of objects. In particular, each geometry
describes a system of points. And the same geometry, the same
description, can refer as well to several systems of points, that
are of the same kind.
For example, the same Euclidean plane geometry can describe the
system of points formed by this sheet of paper, or it can as well
describe the other system of points formed by that sheet of paper.
Because these two systems are of the same kind.
From now on we shall call "space" a system of points described by
any geometry, that is not necessarily 3-dimensional, unless
Now, how does a geometry describe a space. Like any mathematical
theory, it has a vocabulary. This is the list of names (or symbols)
available in the language of this geometry. These names may have
different forms of use, to describe different kinds of things : they may
describe combinations of points, or differents kinds of figures, giving
them qualifications by which some figures will be distinguished from
some other figures.
Examples in the language of Euclidean
geometry : distance, straight line, square, curve, angle.
But an interesting fact with most geometries as opposed to other
mathematical theories, is that their language has clear limits, as
there are different figures that cannot be distinguished from each
other but remain perfectly similar, no matter the length of the
story we may try to tell about them.
For example in Euclidean plane geometry, everything that can be said
about this square can as well be said about that other square.
(5 min) A description of a figure by
the language of geometry cannot specify where this figure is located
inside the considered space. And if we have different spaces of the
same kind, described by the same geometry, a description of a figure
cannot say in which space it is. For any figure in this plane, it is
possible to draw a figure in that other plane, that will look the
Because if we could not do it, it would mean that these two spaces
don't have the same geometry. Because one of them would have the
quality that such a figure can be drawn in it, while the other would
not have this quality.
Now as both figures in a different space are similar, it means that
there is a correspondence between the spaces, we say an isomorphism,
that maps every point of the one to a unique point of the other and
conversely, that maps one figure to the other, and preserves all
structures of geometry, that is, any quality that is attributed to any figure.
But still there is a deep meaning in saying that these 2 spaces,
with the exact same properties, are still different spaces.
It is that there is not just one isomorphism between them, by which they
can be identified, but usually even an infinity of them. Indeed, if we
choose a figure here, then we can find a similar figure there, but this
does not say which corresponding figure there we should pick up because
there are many other figures which are similar as well. So we do not have
any way to describe a specific choice of isomorphism to pick up, through which
the identification should be made, rather than any other possible isomorphism.
So, we do not have a better reason to confuse this figure with that rather than
that other figure. But doing both confusions would be wrong because similar
figures there must be considered distinct and not confused. So we cannot
accept either the legitimacy of any confusion (specific correspondence)
between similar figures inside similar but independent spaces.
And so the different spaces should be recognized as essentially
distinct from each other too.
But if 2 isomorphisms are different, how can we understand their
difference ? This difference can be represented by composing one of
these isomorphisms by the inverse of the other, that is, as an
isomorphism from one of these spaces into itself. An isomorphism
from a space to itself, is called an automorphism. In an Euclidean
plane or 3-dimensional space there are 2 important kinds of
automorphisms: the translations, and the rotations. And there can be
other ones depending on the precisely chosen geometry, that will be
presented when needed. (7:30)
Now after these generalities that go for any geometry, let us start studying
specific geometries, starting with a simple but still an important one :
The geometry of the line.
The geometry of the line is usually not explicitly mentioned in
school, even where a space of that kind is actually there behind what is done,
as the habbit is to conventionally assume that an arbitrary choice was made of
2 points in this line, to be called 0 and 1, to identify the line with the set of real numbers.
This habbit of expressing the geometry of the line as
identified to the set of real numbers, is defended by the argument
that once identified the points to real numbers, we can do all the
work we need on them, by means of the usual operations
among real numbers: the operations of addition, subtraction,
multiplication and division. So we can work in this geometry using
But there is something pitiful in this approach to the geometry of
the line : the choices of positions of 0 and 1 constitute
supplementary structures that do not belong to
the intended geometry of the line : the symbols 0 and 1 are names that distinguish
points that were not naturally distinguished in that geometry. Instead, adding them
to the language changes the geometry into another theory
that is more rigid, and even so rigid that it does not admit any
automorphism anymore, except of course the identity function, that
is the function that lets every element fixed. In this
formalism, every point is identified to a number; and as every real
number can be completely specified by its decimal expansion, no real
number can ever have all the same properties as another real number.
Instead, the geometry of the line admits similarities between objects. Thus, for
a formalization of the geometry of the line to be exactly faithful to its
intended meaning, it should directly see as similar
the objects intended to be similar, and thus should not involve any choices of 0
and 1. But what are these similarities ?
Let us describe the automorphisms of the line. We said, there is no
more automorphisms once two points are fixed, labeled by the numbers
0 and 1. However, we can give these labels to any 2 different
points, and any such choice is similar to any other choice. So,
any combination of 2 different points is similar to any other such combination.
This means that if we have 2 lines, we arbitrarily choose 2
different points to be labeled 0 and 1 in one line, and another 2
different points in the other line, then there exists a unique
isomorphism between both lines that sends the 0 here to the 0 there,
and the 1 here to the 1 there. And then, as isomorphisms preserve everything,
they preserve the attribution of a real number to a point that is determined by a
choice of 0 and 1. So, every point is
sent to the point that is represented by the same real number,
for the representations determined by these choices in each
line. (10 min)
So, what is this representation of points of a line by real numbers
? Since any geometric line can be identified to the set of real
numbers as soon as 0 and 1 are chosen, we can just say that the set
of real numbers is a geometric line together with a choice of a 0
and a 1. Then, for any other line with a choice of 2 points to be
labeled 0 and 1, the identification of points to real numbers is
defined as the one given by the only isomorphism between lines that
sends the 0 to the 0 and the 1 to the 1.
But what does the use of real numbers have to do with the
automorphisms of the line that is not identified with them ?
Interestingly, the same operations usually assumed to operate
between real numbers, are also the ones that can describe these
automorphisms, but for this, they need to be reinterpreted to be
used in that different framework.
To describe this in details, let us split the group of all
automorphisms into different components, corresponding to a split of
the double choice of positions of 0 and 1, as a combination of
Of course an obvious split would be to say that on the one hand we
choose the 0, that we shall also call the origin, on the other hand
we choose the 1, so these are both structures. But the second one is
similar to the first (and we must request both chosen points to differ).
Interestingly, we can replace the choice of the 1 by a combination of
two other independent structures.
And these are the two most important other
kinds of possible additional structures to that geometry:
2 structures : orientation and unit of distance, can be conceived
independently of the origin, and they are also independent of
They can replace the 1, or, are an equivalent expression of this choice, in
the sense that for a given origin, they are definable from it
(as we said) and also they can define it. Namely, the 1 can be defined as
the point on the side of 0 specified by the orientation, at the distance from it given
by the unit of distance. But no such definitions can be expressed
if the origin is not specified.
distance between points 0 and 1, to be taken as unit of
- The orientation, that is the
distinction between left and right, for saying that the 1 is located on the right side
of the 0. So it is a choice among 2 possibilities.
As these structures are outside the geometry of the line, they are usually
not preserved by its automorphisms. Instead, automorphisms can be measured
by their effects on these structures. In this way, automorphisms can be analyzed
as composites of 3 transformations, where each transformation can only disturb a different structure:
|Name||Formula : x becomes||Modified structure|
|Translation by a||x+a||Origin
|Dilation by a positive number k||kx||Distance unit|
|Symmetry w.r.t. the origin||-x||Orientation|
Let us represent graphically these operations of addition and
multiplication: first for addition
- Keeping fixed the 0 and the orientation, the change of
distance unit is operated by dilations centered on 0,
numerically expressed as the multiplication by a positive
- Keeping fixed the 0 and the distance unit, the possible
switch of orientation is operated by the symmetry with respect
to 0, numerically expressed by the negation (the change of
- Keeping fixed the orientation and the distance unit, the 0 can
be moved to any other point by a translation, numerically
expressed as the addition with a fixed number.
0 a b a+b
There are 2 ways to interpret this figure.
One way is to say that the horizontal lines are copies of each
other, or at least they are lines with the same unit of length and
the same orientation, so that we have here an isomorphism between
these 2 copies, mapping the (0 a) pair to the (0 a) pair there. And
there we have another isomorphism between for these structures but
not for the 0, sending this pair (0 a) to the pair (b a+b). So, we
obtain a+b by applying on a the translation that maps 0 to b.
So we see a as moved by the translation that maps 0 to b.
But as the addition is commutative, a+b=b+a, another view is to say
that b is moved by the translation of the below line that maps 0 to
a. And this translation is somehow symbolically represented by this
oriented pair 0 a : having fixed this pair, we can process this
translation on any point b below by applying to it the same movement
of translation, that sends 0 to a.
Then for multiplication: we have a similar figure but where the
lines of correspondence, instead of being parallel, are required to
have a common intersection with the line that connects the 1's.
The principle is exactly the same, except that the roles of the 0's,
that could be moved, are now played by the 1's, while the 0's now
remain always fixed. So, dilations are now allowed and balanced with
the required preservation of one more point. But we can also use
another graphical representation of the same process, where both
copies are no more parallel but intersect at 0, making its
preservation automatic, so that now the correspondences between both
lines appear parallel again.
Ok. Now as we made some observations about the geometry of the line,
and before going further, let us look again at the properties of
what we just presented.
Each dilation, that preserves the 0 point, can be described as the
dilation centered on this point that is the preserved structure, and
with a ratio described by a real number. This description does not
involve any specific length nor orientation.
On the other hand, each translation, which preserves both structures
of length unit and orientation, can itself be described by... a
length and an orientation, but no center. Just like the structures
that it preserves. So it resembles what it preserves, except that,
here the preserved structures, the length unit and the orientation,
are not directly involved. Separate choices of length and
orientation are involved instead. But still these new choices can be
described using the first ones, by a nonzero real number: that is
the ratio between these lengths, with a negative sign if the
orientations are opposite. So, any choice of length and orientation
provides the language to describe any other choice using a number.
This is not the case for a choice of a center: once chosen a point
as center, we cannot describe where another point is, by just a
number. We need to specify a length and orientation instead.
These properties concerned each of both line geometries that were
almost rigidified by choosing structures that only left one possible
dimension of movement for the automorphisms. It looks different for
the geometry without structure where both dimensions of
automorphisms remain available, where a dilation has a center, and a
translation has a length and orientation, which were not among
structures and could not be specified by numbers.
Now do you think these phenomena are accidental and specific to the
geometry of the line, or do you think they are more generally valid
for other geometries, because of some logical necessities ?
In fact, much of these properties are not specific to the line, but
come from some universal logical necessities. Let us explain this,
by coming back to the basic and general definitions. Let us write
An automorphism of a space
is a transformation of this space to itself
that preserves all basic structures of its geometry
And thus also all others definable from them
such as the notion of "automorphism" itself
but not always specific automorphisms among them
What it preserves the notion of automorphism understood as the
distinction between which transformations are automorphisms and
which are not. So if we take an automorphism and we transform it by
another, then the transformation we obtain is also an automorphism,
that is similar to the previous one, but maybe different from it.
But how does an automorphism transform another one ?
More generally, if we have 2 spaces, each has its own group of
automorphism, but they are similar, so there are isomorphisms
between them. So how does an isomorphism between them sends an
automorphism of the one into an automorphism of the other ?
For example let us take 2 planes. In one plane, choose an
automorphism, for example a rotation, as shown by this drawing : the
rotation sending this figure into that figure. Then take a second
plane, similar to the first one, so there exists some isomorphisms
between them. Now what is the image of this automorphism of this
plane when transported into the other plane by that isomorphism
between both planes ? It is a transformation of that plane, also a
rotation, that we can draw in this way.
So the transformation we get there is indeed an automorphism, and
this can be explained in 2 ways. One way is of course that the
isomorphism preserves every structure, so that if a transformation
here has a property of being an automorphism, then its copy there
has the same property. Another way is to say that the transformation
we obtain there can be described as a composite of 3
Since all 3 of them are isomorphisms, all structures are preserved
at each step, so they are also preserved after all steps, so the
resulting transformation is an automorphism too.
Now let us ask another question. Take 2 spaces, choose an
automorphism inside each space, we will be interested to take
similar ones but it does not matter here, and then choose an
isomorphism between both spaces. Now the question is, at which
condition does that automorphism coincide with the image of this one
by the correspondence ? In the case of rotations, they may be the
same if they have the same center and angle, or they may be
different, if they have a different center, or if the orientation is
Now to visualize the condition, consider any figure or just any
point in one space, and its image in the other. So the
condition is that for every 2 points that correspond to each other
by the isomorphism, their images by the respective automorphism in
each space also always correspond to each other by the isomorphism.
We can understand this in terms of the expression we gave of the
copy of an automorphism transported by an isomorphism, as a
composite of 3 transformation. But we can also express it as the
equality between both composites of the isomorphism with an
automorphism, that happen in a different order.(20 min)
Now taking the particular case when both spaces are the same, the
condition for an automorphism to preserve another one, in other
words, for the latter to be invariant by the former, is simply
expressed saying that their composite is independent of the order
between them, that is the choice of which one we apply before the
other. This property is called saying that both automorphisms
So generally for any two bijective transformations f and g of a
space, or bijections of a space to itself, we say that f and g
commute to mean fog = gof, which is both equivalent to say that f
preserves g and that g preserves f.
This has a very interesting consequence: that any automorphism of
any space, in fact generally any bijection of any set into itself,
preserves itself : fof is obviously independent of the order.
Ok, you may find it not very interesting because this case is easy
but exceptional. But what is more interesting is that the same
result also applies to any two transformations of a space that may
now be different but only required to belong to a same group of
transformations that in itself forms a space with 1 dimension, with
a shape that may be that of a line or a loop, but that is connected,
I mean it is in one piece, it is not several lines.
Now if we represent its dimension in the form of time, this groups
appears as a transformation of our space, that depends on time, and
the evolution of this transformation along time has the property of
always following the same movement. For example : in a plane, we
have the 1-dimensional group of all rotations around any given
point. They are parametrized by the angle of the rotation, that is
proportional to time, so that the progress of this rotation in time
is always given by keeping everywhere the same rotational movement.
Another example is a movement of translation, that keeps the same
direction and speed along time. Another example is the group of
dilations of a line with a fixed center.
So, these transformation groups are such that if we take a small
transformation among them, that is one happening in a small time and
that is just a little bit different from the identity function, then
all others, happening in larger amounts of time, are essentially
obtained by composing this small transformation with itself a large
number of times. For example a rotation with angle 50 degrees, is
obtained from the small rotation of 1 degree around the same center,
by iterating it 50 times.
Now in this situation, we can notice that all these transformations
commute with each other.
Because, for any choice of a small transformation in this group, as
small as we wish, and for any 2 numbers n and p, composing the same
transformation with itself, or in other words, repeatedly applying
it n times and then p times, gives the same result as applying it p
times and then n times, because n+p=p+n as natural numbers.
So, since any 2 transformations in the group can be approximated as
precisely as we wish by different numbers of iterations of a common
sufficiently small one, and since these approximations of these 2
transformations exactly commute, we deduce that any 2
transformations in the group also exactly commute.
For example, 2 rotations around the same point of a plane, exactly
commute: they preserve each other...
Now we have an explanation for the phenomenon that we described
earlier: that when we have a space whose group of automorphisms is
1-dimensional and connected, then all these automorphisms preserve
each other. So, they are invariant, which means that every one of
these automorphism can be specified in the language of the geometry
of that space using a number. It does not require to break that poor
symmetry of the space, it does not require to choose anything in the
space among similar other things.
Because for an automorphism to be similar to another one, it means
it would be transportable into it, by a third automorphism with
which it wouldn't commute.
Which is excluded by our assumption that the automorphism group is
1-dimensional and connected.
Notice that if it was just 1-dimensional but not connected, it
For example, take the geometry of the line together with a unit of
distance but no orientation. Its automorphism group is in 2 pieces:
one line is the group of translations, and the other line is the set
of all symmetries with respect to any center. To specify an
translation, we need not only a length, expressible by a number
using the distance unit, but also an orientation, which is not
expressible here. And to specify a symmetry, we need to choose its
center, which is a point, similar to any other point.
Let us now complete the explanations of the above observations.
We took a space, that was the oriented line, whose automorphism
group was 2-dimensional. We rigidified its geometry by introducing
an additional structure, whose value was picked up inside a given
1-dimensional set of similar objects : in one case it was a point,
in the other case it was a length.
When it was a point, the other points remained similar to each
other, as they were moved to each other by the remaining dimension
of automorphisms with left the chosen point fixed.
And since we explained that these automorphisms could anyway be
expressed numerically, it is logically impossible to translate this
numerical choice as the choice of another point or length, whose
similarity with other objects of that kind makes it impossible to
measure in a numerical form.
But when the choice was that of a length, the other lengths were no
more similar to each other, as they could be expressed as a
numerical multiple of the given length. The automorphisms, that
preserved the given length could no more move the other lengths into
each other, but they preserve every other length too. Therefore, the
property for an automorphism to preserve one given length, is a
property that does not depend on which is the length that we ask it
to preserve. These automorphisms are the translations, and this
notion can also be defined in the initial geometry without any
choice of unit of length.
Now in this initial geometry, we have this definition of the
distinction of the translations among other automorphisms, and this
set of translations is here 1-dimensional. These translations in the
line happen to be similar to each other. This isn't logically
necessary in general, and we will see later another geometry with a
definable subgroup of automorphisms that are not similar to each
other, but must be defined by numbers.
Now if the translations are similar to each other, then the choice
of a translation constitutes an additional structure, that once
added to the language, rigidifies the geometry. From the 2
dimensions of the group of automorphisms, one dimension is removed.
We have only one dimension left. But which dimension is this ? It
anyway contains the dimension of translations. Why ? Because since
the group of translations is 1-dimensional and connected, the chosen
one inside is necessarily invariant by all others, which thus remain
automorphisms of the rigidified geometry.
This is why the translations of the line are essentially the same
kind of geometrical object as those that these translations
preserve. That is, a length together with an orientation.
Now let sum up the different concepts that we just described, how
they can be expressed by the basic operations, and how can these
operations be graphically represented.