What is energy? What is physics?

In your high-school physics classes, you may have studied the dynamics of various physical systems, but you may have also heard that physics is the "general science", that is universally applicable. One may wonder if physics can be formulated as some general framework that can handle any type of system without more specific assumptions about such system, and what interesting conclusions can be drawn about physics in such a general setting.

Fundamentally, we seek to make predictions about some observables. You may imagine that all the observable information about a system is given by some abstract "phase space" of "state vectors" such that each observable is some function of the state vector. For example if the phase space is parameterized by position and momentum, then the position and momentum observables would be projection functions and various other observables can be written as compositions thereof of functions with these projections e.g. $E=\frac{p^2}{2m}+U(x,p)$.

The question of why position and momentum are sufficient to specify our phase space in so many real-life situations is a rather advanced one -- here's a paper that explains that this general "first-derivative sufficiency" in physics arises as a special case of something to do with Fisher information [1] -- but I don't understand Fisher information.

The Lie theory of dynamics

One may say that we are particularly interested in the time-evolution of such observables, $dA/dt$. If we can find a differential equation for this, then we can find the value of $A$ given some initial condition. But more generally we may not have an initial condition as such and maybe the problem we're trying to solve isn't even a differential equation as such but we just want to speak more abstractly about the dynamics of the system.

This should all be screaming "Lie groups" to you. 

One may imagine that time evolution is the flow under a certain vector field. 

More precisely, suppose the state of a system is given by some $\psi=(x, p)$, which represents the position and momentum of a particle at time $t$. Then the "position at time $t$" is an observable given by the function $X(\psi)=x$, the projection onto the $x$ co-ordinate -- but we may also think about the observable "position at time $t+\Delta t$", which is some function $\Phi^{\Delta t}X$ that projects onto a different co-ordinate system.

This co-ordinate transformation on the phase space is a type of canonical transformation -- we will soon define this generally, but it is important to realize that motion/time-evolution is a type of canonical transformation. 

You may imagine that such canonical transformations form a Lie group, and as is standard the Lie algebra to this Lie group consists of vector fields (flows) on the phase space. 

Noether and symplectic geometry

The question, then, is what sort of transformations count as "canonical transformations", or equivalently, what sort of vector fields are we interested in. What kind of "co-ordinate transformations" are acceptable?

We want to be as general as possible, so we do not wish to restrict what sort of paths are predicted on the phase space -- rather, we want to restrict the relationships between the predicted paths, i.e. how a distribution evolves under time evolution. We won't go into details of why this is true (because I do not fully understand it yet -- apparently it is called Liouville's theorem and has to do with "conservation of information", which has to do with probabilities inferred from some symmetries, see a priori probability), but we expect the vector fields to be solenoidal, i.e. have zero divergence. Analogous to how irrotational vector fields are precisely those that are gradients of functions, solenoidal vector fields are precisely those that are the symplectic gradients of functions (see reddit for a proof):

$$\nabla\cdot \vec{v}=0\leftrightarrow \exists g, \vec{v}=\frac{\partial g}{\partial y}\hat{x}-\frac{\partial g}{\partial x}\hat{y}$$

This "symplectic gradient" always points along the contours of $g$ -- thus, these flows conserve the quantity $g$. More generally, the change in some observable $f$ over the flow that is the symplectic gradient of $g$ is given by $\vec{v}\cdot\nabla g$, and is called the Poisson bracket of the two observables:

$$\{f,g\}=\frac{\partial g}{\partial y}\frac{\partial f}{\partial x}-\frac{\partial g}{\partial x}\frac{\partial f}{\partial y}$$

It should be intuitively clear that this Poisson bracket corresponds to the Lie bracket of their symplectic gradients -- if the vector fields commute, they must be constant under the flow of the other, etc.

The Hamiltonian and conservation of energy

The evolution of a classical Newtonian state is given by 

$$\dot{x}=p/m$$

$$\dot{p}=F$$

We wish to find the observable $H$ such that this evolution is represented by the symplectic gradient of $H$, i.e. so that:

$$p/m=\{x, H\} = \frac{\partial H}{\partial p}\frac{\partial x}{\partial x}-\frac{\partial H}{\partial x}\frac{\partial x}{\partial p}=\frac{\partial H}{\partial p}$$

$$F = \{p, H\}=-\frac{\partial H}{\partial x}$$

Integrating, the quantity we want is

$$H=\frac{1}{2m}p^2-\int F dx$$

This quantity is called the "energy" of the system -- often we call $p^2/2m$ the "kinetic energy" and $U=-\int F dx$ the "potential energy". In particular, it is immediately clear that energy is conserved. In general, the physics of a system can be completely specified by some "Hamiltonian" $H$, this rather generalized definition of energy, whose symplectic gradient represents time translations.

The Killing form; factorising non-Abelian Lie groups

It could be fun to try and define a "dot product" on a Lie algebra.

You know, you might've already realised that the cross product is a Lie bracket of sorts -- you know, given its antisymmetry and the whole $a^\mu b^\nu - a^\nu b^\mu$ representation of the wedge product and all that. It's a short exercise to verify that the Lie algebra $\mathfrak{so}(3)$ of $SO(3)$ is the algebra of skew-symmetric matrices, and with the Lie bracket $XY-YX$ is isomorphic to $\mathbb{R}^3$ with the cross product.

Well, the dot product on $\mathbb{R}^3$ has an interesting connection to $SO(3)$ -- it is precisely the form that is invariant under the action of $SO(3)$. Well, but that's $SO(3)$ acting on $\mathbb{R}^3$ -- what is that action in the notation of $\mathfrak{so}(3)$? As it turns out (and you can work this out), it is precisely the adjoint map $\mathrm{Ad}_gX:=gXg^{-1}$ which corresponds to this "rotating $X$ by $g$". It's not really that unexpected, if you ask me -- conjugation is always the natural way to transform matrices in linear algebra when vectors are multiplied on the left.

So the "dot product" is an $\mathrm{Ad}$-invariant bilinear form. In fact, adding a symmetricity requirement allows us to just bother with norms (as a symmetric inner product can be determined from the norm, through the cosine rule). Conjugation basically allows you to determine the "contours" of this norm or inner product. The question is: can we determine the bilinear form -- up to scaling -- just from "$\mathrm{Ad}$-invariant symmetric bilinear form" alone?

This is equivalent to asking "is the orbit of some non-zero $X$ under conjugation by $G$ equal to $\mathfrak{g}$?" (so that the norm of that $X$ would suffice to determine all norms -- do you see why?) Well, this is equivalent to asking "is $X$ contained in some non-trivial ideal?" (prove that these are equivalent!), and this is equivalent to asking "does $\mathfrak{g}$ have any non-trivial ideals?" (do you see why?)

A Lie algebra without nontrivial ideals is called a simple Lie algebra. Our demonstration above shows that a simple Lie algebra has a unique $\mathrm{Ad}$-invariant symmetric bilinear form, determined by the value of $\langle X, X\rangle$ for some non-zero $X$.

Even before we actually derive what this form must look like, we can derive one important consequence of automorphism invariance: $\langle X, [X, Y]\rangle = 0$ (prove it!), i.e. the tangent to an automorphism curve is perpendicular to the position vector at every point. The understanding of the group as acting as a "rotation group" on its Lie algebra in the adjoint representation really makes sense!

Someone tell me if they know how one may "derive" the trace-form formula from this characterisation rather than pulling it out of the blue and then proving it is the unique $\mathrm{Ad}$-invariant symmetric bilinear form. Here's something I started to write:

Here's an idea for the base length (i.e. to define the scaling): $X$ has length 1 iff the length of $[X,Y]$ equals the length of $Y$ for all $Y$ perpendicular to $X$ -- equivalently: $\forall V\in\mathfrak{g}, |[X,[X,V]]|=|[X,V]|$. We need to check that this condition is well-defined, i.e. that:

1. Given an $X$, $|[X,[X,U]]|=|[X,U]|$ for some $U$ not a multiple of $X$ implies that $|[X,[X,V]]|=|[X,V]|$ for all $V$.
2. $X$ satisfying $|[X,[X,V]]|=|[X,V]|$ implies that all conjugates $gXg^{-1}$ of it satisfy it too. This is trivial from considering $V=gV'g^{-1}$ (since the identity is true for all $V$).

Is the first one even true outside $\mathfrak{so}(3)$ -- for all simple Lie algebras?

One may come up with the idea of defining a form $\langle X, Y\rangle = \mathrm{tr}[X,[Y,\cdot]]$ (example of some weak motivation -- the vector triple product $x\times(x\times v)$ has as eigenvectors the vectors $v$ perpendicular to $x$ and the eigenvalues depend on the length of $x$) and check that this is indeed an $\mathrm{Ad}$-invariant symmetric bilinear form, and is thus unique up to scaling for simple Lie algebras. This form is called the Killing form.

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Factorisation of Lie groups

We have seen the classification of connected Abelian Lie groups: they are products of circles and lines. We wonder if such a classification is possible for more general Lie groups.

The natural way to "factorise" groups by taking quotients over normal subgroups -- we wonder if this means that all Lie groups can be written as direct products of simple Lie groups (groups that don't have a nontrivial connected normal subgroup -- can you see why "connected" matters?). Well, not really -- the quotients need not be subgroups at all, after all. Instead, the "factorisation" takes the form of what is known as a group extension. A group for which it is a direct product is called a reductive Lie group -- and its Lie algebra is the direct sum of simple Lie algebras, or a reductive Lie algebra.

It is more conventional in the literature to define a simple Lie algebra excluding the one-dimensional/abelian case. In this definition, direct sums of simple Lie algebras are semisimple Lie algebras, and reductive Lie algebras are direct sums of semisimple and abelian Lie algebras.

TBC: Cartan's criterion, solvability, nilpotency