Quick Intro: Group Theory

A group is an abstract description of how a system can transform.

Now and again in higher-level Physics you might run into terms like this: SO(3), Lie Group, U(2), etc… These sound like strange and disconnected terms but they are all part of how we categorize systems into groups.

What is a group you might ask? A group is an abstract description of how that system can transform. Specifically, a group consists of a set of elements that create our group space. With this group there is an operator that allows us to combine any two elements into a third element. The operator is usually represented by a plus + symbol so that if A, B and C are group elements, a statement like A + B = C makes sense. You might notice that this looks a lot like simple algebra, and groups do form their own algebra in a sense, but they aren’t limited to the math operators that we learned in primary school.

Since a group is completely abstract, it can have different ‘representations’ that express the group. A cyclic group can be represented by the way we rotate an object, so that if you add two rotations, you might end up rotating the object 360 degrees and returning to the original position. Another example of a group representation would be how we can combine simple twists of string to form complex braids (the group in this case is called the braid group).

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One example of a group is the ways that we can flip a mattress. This helpful article explores group theory in that scenario: https://opinionator.blogs.nytimes.com/2010/05/02/group-think/

While there are infinite representations of groups, we try to reduce each group to it’s simplest form, known as the irreducible representation. This allows us to compare different systems in terms of their groups.

Breaking systems down into their fundamental groups allows us to predict transitions that appear in the group before we can even observe them. Group theory can also be used to understand symmetry, this perspective can not only be applied to the physical sciences, but the study of art.

As far as physics goes, here are a few common groups that come up a lot:

SO(3): The special orthogonal group in three dimensions. This group consists of simple rotations in 3 dimension space around a fixed point.

U(2): The unitary group of degree 2. This group consists of all 2×2 matrices that are unitary.

SU(2): The special unitary group of degree 2. Like the unitary group, but the determinant of each matrix is equal to 1.

There are a few ways of discussing different types of groups. Many groups are characterized by some kind of symmetry, which is defined by a parameter of the system that doesn’t change during a transformation. For example matrices in the special unitary group always have a determinant equal to 1, or the special orthogonal group doesn’t change the length of a given radius as it rotates.

A Lie group is any group that is defined by a continuous transformation. A particle might have discrete transformation between one energy state of another, but rotations can be made continuously.

There are so many different ways of looking at groups. Even if it doesn’t become your field of study, it can make for a very interesting way of looking at different systems. Here are a few sources where you can learn a little bit more about groups:

PDF/BOOK: Introduction to Group Theory for Physicists – Marina von Steinkirch
This book breaks gives us the mathematical formulation of group theory and looks at a number of groups that have special significance for Physics.
http://www.astro.sunysb.edu/steinkirch/books/group.pdf

VIDEO: Group Theory – Robert de Mello Koch
The first in a series of lectures that cover the study of group theory and representations. A great place to start if you have some experience with Quantum Mechanics.
https://www.youtube.com/watch?v=3wNPrSwbtQ8

The Principle of Least Action and Lagrangian Mechanics

The principle of least action is a idea that helped transition physics away from Newtonian methods and toward a more general description of physical systems.

Lately I’ve been studying methods of calculating path integrals, and while I would very much like to write about that, I think I should start with a more foundational topic. The principle of least action is a idea that helped transition physics away from Newtonian methods and toward a more general description of physical systems. It has been so important that it is still used across many different areas of physics today.

Principle of least action

To start, we should discuss what the principle of least action actually is. In the 1600s, Pierre de Fermat discovered that light always takes a path that takes the least amount of time to traverse. Knowing the initial point, and the final point of light, you would know what path it took by finding that path that minimized the time taken by the light. This would become known as Fermat’s principle or the principle of least time.

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The light ray moves slower when passing through a material. As per Fermat’s principle, the ray is deflected in such a way that minimizes the light takes to pass from beginning to end.

This was a novel idea, since older approaches of physics first considered the present state of a system and then apply laws to determine what further point it will transition to. In Fermat’s principle, you consider the endpoints of the system (beginning and end of the light path), and then you could derive the laws based on some kind of constraint (ie: minimizing the time that light travels).

Lagrangian Mechanics

As it turns out, light isn’t the only place you can apply this kind of approach. In Lagrangian mechanics, you could define the starting point of a system, and the end point of that system, then you’d try to minimize and maximize a particular quantity known as the action (S). The action is an abstract value that depends on the path the system takes; if find the path that minimizes or maximizes the action, that path corresponds to the real path that the system takes. This sounds pretty abstract but there is a well defined system for calculating the action for a path, and it comes from formula you can build for your system called the Lagrangian function or just the Lagrangian (L).

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 The Lagrangian is defined at each point in time. Integrating along the path of the system provides the action for that path.

So by deriving the Lagrangian function and applying the principle of least action, you can describe almost any physical system. This has been a very successful approach and was later adapted to a similar technique in what we call Hamiltonian mechanics.

There are some conceptual oddities of Lagrangian mechanics. Particularly, the idea of both the past and the future influencing the behaviour of the present can be a little discordant with our idea of causality. Regardless, it has been proven as an effective approach to understanding the behaviour of a large variety of scenarios in physics.

What does it mean?

The principle of least action is an extremely handy concept that lets us tackle some problems that would otherwise be extremely difficult in physics. A commonly used example is the problem of coupled oscillators. In this problem, a series of masses are connected by springs and strung between two walls. Deriving it using tradition newtonian methods can be extremely tedious for a large number of masses. On the flip-side using lagrangian mechanics can vastly simplify the problem. If you are interested in lagrangian methods for harmonic systems, this problem can illustrate can illustrate the power of lagrangian methods:

Coupled Harmonic Oscillators by Irina Yakimenko

https://people.ifm.liu.se/irina/teaching/sem6.pdf

Keep in mind that this approach can offer more than just mathematical power, it is conceptual change in how we look at interactions in the universe. Instead seeing the universe as of a series of sequential events, we can see a system of relationships across time and space that emerge from the optimization of a few constraints.

The principle of least action also provides interesting implications for our perspective of time. As mentioned earlier, since the lagrangian derives our equations of motion from past and future constraints, it violates our intuition about causality. Some have argued that it implies an eternalist universe where the past, present and future all exist simultaneously and are separated only by the dimension of time. This is largely philosophical and as such, is difficult to demonstrate with any certainty.

 

Here are a few resources for getting a handle on Lagrangian Mechanics and the principle of least action:

PDF: Introduction to Lagrangian and Hamiltonian Mechanics, Melanie Ganz

A short pdf that introduces the main concepts and equations of Lagrangian Mechanics

http://image.diku.dk/ganz/Lectures/Lagrange.pdf

BOOK: Mechanics: Volume 1, Landau and Lifshitz

The go-to book on classical mechanics. Uses Lagrangian methods from the beginning and covers many different methods applying them to mechanical situations.

https://www.amazon.ca/Mechanics-1-L-D-Landau/dp/0750628960

VIDEO: A Special Lecture: Principle of Least Action, Kenneth Young

A fantastic introduction to the material and great speaker.

https://www.youtube.com/watch?v=IhlSqwZBW1M