Ciro Santilli @cirosantilli 34

For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

As mentioned at buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.

The one parameter subgroup of a Lie group for a given element $M$ of its Lie algebra is a subgroup of $G$ given by:

Intuitively, $M$ is a direction, and $t$ is how far we move along a given direction. This intuition is especially vivid in for example in the case of the Lie algebra of $SO(3)$, the rotation group.

One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.

Physics (like all well done science) is the art of predicting the future by modelling the world with mathematics.

And predicting the future is the first step towards controlling it, i.e.: engineering.

Ciro Santilli doesn't know physics. He writes about it partly to start playing with some scientific content for: OurBigBook.com, partly because this stuff is just amazingly beautiful.

Ciro's main intellectual physics fetishes are to learn quantum electrodynamics (understanding the point of Lie groups being a subpart of that) and condensed matter physics.

Every science is Physics in disguise, but the number of objects in the real world is so large that we can't solve the real equations in practice.

Luckily, due to emergence, we can use uglier higher level approximations of the world to solve many problems, with the complex limits of applicability of those approximations.

Therefore, such higher level approximations are highly specialized, and given different names such as:

As of 2019, all known physics can be described by two theories:

Unifying those two into the theory of everything one of the major goals of modern physics.

Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) $GL(n)$, that has the same properties as the group.

Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).

Each such matrix then represents one specific element of the group.

This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.

Or more precisely, mapping each group element to a linear map over some vector field $V$ (which can be represented by a matrix infinite dimension), in a way that respects the group operations:

As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)

Motivation:

Bibliography:

Directly modelled by group.

For continuous symmetries, see: Lie group.

TODO find/create decent answer.

I think the best answer is something along:

A basic non-precise intuition is that a good model of reality is that electrons do not "interact with one another directly via the electromagnetic field".

A better model happens to be the quantum field theory view that the electromagnetic field interacts with the photon field but not directly with itself, and then the photon field interacts with parts of the electromagnetic field further away.

The more precise statement is that the photon field is a gauge field of the electromagnetic force under local U(1) symmetry, which is described by a Lie group. TODO understand.

This idea was first applied in general relativity, where Einstein understood that the "force of gravity" can be understood just in terms of symmetry and curvature of space. This was later applied o quantum electrodynamics and the entire Standard Model.

From Video "Lorenzo Sadun on the "Yang-Mills and Mass Gap" Millennium problem":

Bibliography: