Ciro Santilli @cirosantilli 34

How atomic clock works? answer by Ciro Santilli on Physics Stack Exchange.

In simple terms, if you believe in the Schrödinger equation and its modern probabilistic interpretation as described in the Schrödinger picture, then at first it seem that there is no strict causality to the outcome of experiments.

People have then tried to recover that by assuming that there is some inner sate beyond the Schrödinger equation, but these ideas are refuted by Bell test experiments, unless we give up the principle of locality, which feels more important, especially in special relativity, where faster-than-light implies time travel, which breaks causality even more dramatically.

The de Broglie-Bohm theory is a deterministic but non-local formulation of quantum mechanics.

Mechanics before quantum mechanics and special relativity.

It appears that Maxwell's equations can be derived directly from Coulomb's law + special relativity:This idea is suggested by the charged particle moving at the same speed of electrons thought experiment, which indicates that magnetism is just a consenquence of special relativity.

Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

Experiments not explained: those that quantum electrodynamics explains like:See also: Dirac equation vs quantum electrodynamics.

The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.

In special relativity, it is impossible to travel faster than light.

One argument of why, is that if you could travel faster than light, then you could send a message to a point in Spacetime that is spacelike-separated from the present. But then since the target is spacelike separated, there exists a inertial frame of reference in which that event happens before the present, which would be hard to make sense of.

Even worse, it would be possible to travel back in time:

A 4D gradient with some small special relativity specifics added in (the light of speed and sign change for the time).

A law of physics is Galilean invariant if the same formula works both when you are standing still on land, or when you are on a boat moving at constant velocity.

For example, if we were describing the movement of a point particle, the exact same formulas that predict the evolution of $x_{land}(t)$ must also predict $x_{boat}(t)$, even though of course both of those $x(t)$ will have different values.

It would be extremely unsatisfactory if the formulas of the laws of physics did not obey Galilean invariance. Especially if you remember that Earth is travelling extremelly fast relative to the Sun. If there was no such invariance, that would mean for example that the laws of physics would be different in other planets that are moving at different speeds. That would be a strong sign that our laws of physics are not complete.

The consequence/cause of that is that you cannot know if you are moving at a constant speed or not.

Lorentz invariance generalizes Galilean invariance to also account for special relativity, in which a more complicated invariant that also takes into account different times observed in different inertial frames of reference is also taken into account. But the fundamental desire for the Lorentz invariance of the laws of physics remains the same.

Unifies both special relativity and gravity.

Not compatible with the Standard Model, and the 2020 unification attempts are called theory of everything.

One of the main motivations for it was likely having forces not be instantaneous, but rather mediated by field to maintain the principle of locality, just like electromagnetism did earlier.

This single experimental observation/idea is the basis for all of special relativity.

Special relativity is the direct result of people bending their backs to accommodate for this really weird fact.

When we particles particles, the action is obtained by integrating the Lagrangian over time:

In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.

Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.

E.g. for a 2-dimensional field $f(x,y,t)$:

Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with $x\in {\mathbb{R}}^2$:

And in the context of special relativity, people condense that even further by adding $t$ to the spacetime Four-vector as well, so you don't even need to write that separate pesky $t$.

The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.

A little suspicious that it bears the name of Lorentz, who is famous for special relativity, isn't it? See: Maxwell's equations require special relativity.

The equation that allows us to calculate stuff in special relativity!

Take two observers with identical rules and stopwatch, and aligned axes, but one is on a car moving at towards the $+x$ direction at speed $v$.

TODO image.

When both observe an event, if we denote:It is of course arbitrary who is standing and who is moving, we will just use the term "standing" for the one without primes.

Then the coordinates of the event observed by the observer on the car are:where:

Note that if $\frac{v}{c}$ tends towards zero, then this reduces to the usual Galilean transformations which our intuition expects:

This explains why we don't observe special relativity in our daily lives: macroscopic objects move too slowly compared to light, and $\frac{v}{c}$ is almost zero.

The following aspects of Maxwell's equations make no sense without special relativity:

When charged particle though experiment are seen from the point of view of special relativity, it becomes clear that magnetism is just a direct side effect of charges being viewed in special relativity. One is philosophically reminded of how spin is the consequence of quantum mechanics + special relativity.

Bibliography:

This one might actually be understandable! It is what Richard Feynman starts to explain at: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).

The difficulty is then proving that the total probability remains at 1, and maybe causality is hard too.

The path integral formulation can be seen as a generalization of the double-slit experiment to infinitely many slits.

Feynman first stared working it out for non-relativistic quantum mechanics, with the relativistic goal in mind, and only later on he attained the relativistic goal.

TODO why intuitively did he take that approach? Likely is makes it easier to add special relativity.

This approach more directly suggests the idea that quantum particles take all possible paths.

Full set of all possible special relativity symmetries:

In simple and concrete terms. Suppose you observe N particles following different trajectories in Spacetime.

There are two observers traveling at constant speed relative to each other, and so they see different trajectories for those particles:Note that the first two types of transformation are exactly the non-relativistic Galilean transformations.

The Poincare group is the set of all matrices such that such a relationship like this exists between two frames of reference.