Symmetry, and Lie Algebra

Back by popular demand, this is the third instalment on my journey to learn Quantum Field Theory.

“I am waiting for another blog post.”

Ben Thamm, University of Edinburgh Student

Arguablly one of the most important concepts in the whole of modern physics is one of symmetry. Suppose we were to perform an experiment today, in Edinburgh, to measure the elementary charge. It’s easy to take for granted the fact that should this experiment be repeated at any time and anywhere, the results wouldn’t change (barring equipment error). Landau and Lifshitz’s Theoretical Physics starts by using the idea of symmetry to completely derive what is meant by mass, momentum, and energy. One can’t help but be in awe of the sheer elegance that lie amongst the mathematics.

Noether’s Theorem

Symmetry, to put is simply, is when a law of physics is invariant upon some transformation. This transformation can be any kind, a familiar one is Lorentz transformation. A more eloquent way to say a law is invariant under the Lorentz transformation, is to say it exhibits a symmetry of spacetime.

Let us look at the simplest case of Noether’s Theorem, namely that momentum is a conserved quantity.

Suppose we have a Langrangian of the following form:

\[\mathcal{L} \left( q^1, q^2 \dots q^N, \dot{q}^1, \dot{q}^2\dots \dot{q}^N\right)= \sum \frac{1}{2}(\dot{q}^i)^2- V(q_1\dots q_N)\]

where we have a collection of particles. The Euler-Lagrange equation tells us the following:

\[\dv{t}\pdv{\mathcal{L}}{\dot{q}^i} = \pdv{\mathcal{L}}{q^i}\]

Our momentum, takes the following form:

\[p_k = \pdv{\mathcal{L}}{\dot q^k}\]

Combining this with the fact that the spatial derivative of the Langrangian is 0, i.e.:

\[\pdv{\mathcal L}{q^k} = 0, k \in \{1\dots N\}\]

By virtue of the Euler-Lagrange equation, we can see that $p_k$ is conserved.

This was the simplest application of Noether’s Theorem. By applying this theorem to suitably chosen quantities/systems, we can demonstrate the conservation laws for various physical quantities!

Energy can be shown to be a conserved quantity by time translation, and angular momentum can be shown to be conserved by applying Noether’s Theorem to $SO(3)$ rotations. One most important take away is perhaps the following:

“Physics encapsulated in $\mathcal{L}$ should be time independent.”

As such,

\[\pdv{\mathcal{L}}{t}=0\]

which leads to the absolute mind-blowing result that our physics is the same throughout time, a very comforting thought to have indeed.

A rather elementary way of showing something is conserved, is to take the derivative of said quantity with respect to a chosen variable and demonstrating that the derivative is nil. This can be done to show the conservation of energy.

In effect, we want to write down an expression, such that when it’s hit with a time derivative, it becomes zero!

Dipping Toes in Group Theory

Another area that I encountered which I find to be quite fascinating is group theory. Never having really been formally introduced to the concept of Group theory, I thought I’d document my learning of it.

A rather familiar group for me personally would be the SO(3) group:

\[SO(3) :\left\{M \in \mathbb{R}^{3\times 3}| M^TM = \mathbb{1}_{3\times 3}, \det M = 1\right\}\]

where SO(3) contains all possible orthogonal rotations in 3D.

A Lie Group, as I understand, is a group that is also a manifold, and is composed in the following ways:

\[\mathcal{g} \times \mathcal{g} \rightarrow \mathcal{g}\]

\[(a,b) \rightarrow a\cdot b\]

In that regards, SO(3) is a three dimensional Lie Group. Each Lie Group, has an associated Lie Algebra, a vector space which has an antisymmetric product. The cross product \(\mathbb{R}^3, \cross\) would be an example. We can find the Lie Algebra rather simply for SO(2):

We know a finite SO(2) rotation to take the matrix form:

\[M = \mqty[\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha]\]

Suppose we want apply an infinitesimal nudge of rotation, in the counterclockwise direction. For small angles of $\varepsilon$, cosines can be approximated to 1, and sines can be approximated to said angle, i.e.:

\[ M \approx \mathbb{1} + \mqty[ 0 & -\varepsilon \\ \varepsilon & 0], 0<\varepsilon\ll 1\]

The Lie Algebra for SO(2) is said to be:

\[SO(2) = \left \{ \mqty[0 & -\varepsilon \\ \varepsilon & 0] \text{ for } \varepsilon \in \mathbb{R}\right\}\]

By following a similar process, we can write down the Lie Algebra of the SO(3) group.

\[SO(3) = \left \{ \mqty[0 & 0 & 0\\ 0 & 0 & -\epsilon \\ 0 & \epsilon & 0 ] + \mqty[0&0&-\delta\\0&0&0\\ \delta & 0 & 0] + \mqty[0 & -\rho & 0 \\ \rho & 0 & 0 \\ 0 & 0 & 0 ] \right \}\]

where the three matrices correspond a small nudge around $x$, $y$, and $z$ axes respectively. Said nudges are parameterised by $\epsilon$, $\delta$, and $\rho$.

This is the set of all antisymmetric matrices, and are the generators of SO(3), a topic of crucial importance in field theories.

A Hiatus… of Sorts

Unfortunately, with the start of a new semester, my brief escapade to a world beyond our intuition will need to be temporarily suspended. I am taking some interesting courses this year which I am very excited about. I hope to maintain a regular writing schedule of interesting physical phenomena on some of the materials I will soon be learning.


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