1. ## Symmetries in Physics

General Thoughts

I want to explore the idea of symmetry as it pertains to physics. I generally learn better when I try to explain things to people, particularly in writing.

We have a physics enthusiast group, but things have been rather dead these days on the group. So I decided to post here.

Unlike my computer architecture thread, where I knew everything I was presenting, and started at the very most basic level, and lost people's interest, I will be exploring a topic I am just learning, and will jump pretty quickly into the challenging material.

So symmetry is a fairly simple concept right? You kinda know it when you see it (or many people think so anyway). Mathematically speaking, one can have various symmetries, including rotational symmetry or reflective symmetry.

Noether's Theorem

In physics, a crucial set of symmetries lead to the conservation laws with which we are familiar. This phenomenon is due to Noether's First Theorem.

The informal statement of this Theorem is :
Originally Posted by wikipedia
To every differentiable symmetry generated by local actions, there corresponds a conserved current.
The hefty mathematical formulations are there in the wikipedia article. One of my goals is to be able to understand them, as well as general Gauge Theory and Variational Principles.

The summary of important results, however, is:
• The fact that the laws of physics are the same with respect to a translation of space coordinates leads to the conservation of linear momentum.
• The fact that the laws of physics are the same with respect to a rotation of space coordinates leads to the conservation of angular momentum.
• The fact that the laws of physics are the same with respect to a translation of the time coordinate leads to the conservation of energy.
• The fact that the laws of physics are the same with respect to what is known as the Gauge Transformation leads to the conservation of electrical charge.

Groups and Representations

Groups

Beyond Noether's Theorem, one very important mathematical concept is foundational, the notion of a mathematical Group.

A group is composed of a set, S, and a binary operator, *, such that:
1. The operation is closed (i.e. if a is in S, and b is in S, then a*b is in S).
2. There exist an element, e, in S, known as the "identity" such that e*a=a*e=a for all elements, a, of S.
3. For every element, a, of S, there exists an element a^-1 of a, such that a*a^-1=a^-1*a=e.
4. In all cases where a,b, and c are elements of S, a*(b*c)=(a*b)*c. This property is called "associativity."

Of course, our use of groups is to represent symmetries, by letting each element of a group be an operation under which something is the same, and letting composition of operations be the group operator.
Such symmetries are:
1. ...closed--because after each operation the desired object remains the same. The composition of two such operations(applying the operation on after another) will obviously keep the operation the same, and so the composition itself is part of the group.
2. ...has an identity--namely the operation of doing nothing.
3. ...has an inverse--doing the reverse operation.
4. ...is associative--composition is inherently associative.

Note something about the identity, that will give you a flavor of the type of thinking required to understand groups.

Suppose you have two elements, e and f, in S that satisfy the properties of an identity. Then we have e=e*f=f. So e = f. What this means in terms of symmetries is that the act of doing nothing EQUALS the act of doing something followed by the reverse operation, and all the other such things that are effectively equivalent. In the group of Rubik's Cube, turning a side 4 times 90 degrees, is the same as doing nothing. I think you get the idea.

Representations

Quite often we'd like to use a set of matrices with matrix multiplication as an operation to "represent" the symmetry group we are studying. This is especially useful in physics, since many of the equations of state can be turned into a linear system which in turn has matrix representations.

Representation theory is another thing I would like to understand more deeply.

There are some important classes of matrices to keep in mind.

I will denote the transpose of a matrix, A, with A^T. I will denote the inverse of a matrix, A, with A^-1. If A^-1=A^T (which means A*A^T=1), then A is called Orthogonal (because the column vectors of such a matrix are mutually orthogonal).

The group (you can verify it's a group) of all Orthogonal, nxn matrices is referred to as O(n). The group of all Orthogonal, nxn matrices with a determinant of 1 is referred to as SO(n).

Note that, generally speaking, orthogonal matrices have real value entries. But many things in physics use complex numbers. With this, the concept of a complex conjugate comes up.

I will denote the result of replacing all entries in A^T with their complex conjugate as A^(dagger). If A^-1=A^(dagger) (which means A*A^(dagger)=1), then A is called Unitary.

The group of all Unitary, nxn matrices is referred to as U(n). The group of all Unitary, nxn matrices, with determinant 1, is referred to as SU(n).

With this information, see if you can go from understanding this explanation of the standard model to this more advanced one.

Understanding the more advanced track of explanations at the official string theory web-site is another goal of mine.

Learning Goals

I'll stop there, but there is a lot more to this subject. I'll summarize my learning goals (so far) here:

1. Understand the derivations of Noether's Theorem, as well as general Gauge Theory and Variational Principles.
2. Understand Group Representation Theory.
3. Understand the more advanced track on the official string theory web-site.

I'm also a bit stuck up on the Action -> Reverse Action = Nothing bit. Energy as well as time is consumed in an action and reaction. Can't just neglect that. Not to mention quantum physics. But I'm no expert there. :P

3. Originally Posted by Fluffywolf
"Practical" is a very subjective adjective. If your aim is to understand string theory, this is indispensable.

If you are planning on designing buildings, there is not much use for it (unless you plan to do it in space or at immense scales, I suppose).

Originally Posted by Fluffywolf
I'm also a bit stuck up on the Action -> Reverse Action = Nothing bit. Energy as well as time is consumed in an action and reaction. Can't just neglect that. Not to mention quantum physics. But I'm no expert there. :P
I'm not really sure what your saying or how it is related to symmetry in physics.

Newton's Third Law holds in the most "down to earth" cases. If there is no external force on a system, then yes, the sum of the internal forces is 0.

4. Just trying to see logic in it. Never took up physics very far. :P

So, if I am to understand the concept, is that the entirety of the universe equals nothing. As the universe is essentially one group, in which the sum of all force is 0?

Or did I just go off track. Maybe I should read the links you posted. :P

5. Maybe it is the word-choice, or maybe because my topic is extremely general, or because what you are asking is rather abstract, but I am having trouble understanding the question.

My first post included many things. The idea of symmetry is simple enough, I think. I was just exploring how symmetry shows up in physics.

One very powerful way is through Noether's Theorem--which basically says there is a conservation law for any symmetry of local action in physics. The ones I mentioned, the conservation of momentum, energy, and charge were all known well before Noether's Theorem.

Nowadays, theoretical physicists use symmetry as a way of trying to explain the other types of conservation laws seen: conservation of baryon number, conservation of lepton number, electron number, muon number, and sometimes Strangeness.

Any time you see a symmetry, you can associate a group to it. That's the second point I was trying to convey.

The final point is that groups can be represented by matrices, and this is what is done in The Standard Model and most theories beyond that, including the superstring theories.

6. All you need to know about symmetry with no equations whatsoever:

7. Please don't take this sarcastically. If you mixed a and b or even c+1 to infinity in one direction then shifted to the opposite direction mixing equally (equally=identical in every mixing pattern made previously) why do some things stay mixed and some do not. A rubix cube will unmix to it's original posture yet a bowl of flower and water will not. Granted the rubix cubes ingredients are bonded as one and the contents of the bowl were not at one time. Time and energy were consumed to create the rubix cubes structure before it was to be mixed at a later time. Is there a way to that mixed flower and water to be unmixed to their original singularities. A rubix cube can by taking it apart. It could rest in biological associativity and reflective symmetry manipulated/influenced by using the tool of guage transformation during while in the process of angular momentum. However it doesn't really jive with nature. Nature in this case taking the example of say translation of old english to modern english with the influence/manipulation of latin or greek.

8. Ah. I understand (I think). It's a theoretical measurement method to support the laws of physics as we know it by extrapolating all external factors. To better understand what the system itself does without the to and from relations to the environment.

If you tie a rock to a cord and the other end to a stick and wave it in circles at a steady speed. The rock should be moving in circles indefinatly, and the stick stay upright. However, due to external factors such as gravity. The momentum is dissappating. So to measure the system in its purest form, it focuses on removing all other influences, and understand the groups potential.

Heh, could theoretically create black holes that way. xD

Anyhow, clearly the formula's are quite a bit ahead of me so I'll leave it to the pro's. :P

9. Originally Posted by professor goodstain
Please don't take this sarcastically. If you mixed a and b or even c+1 to infinity in one direction then shifted to the opposite direction mixing equally (equally=identical in every mixing pattern made previously) why do some things stay mixed and some do not. A rubix cube will unmix to it's original posture yet a bowl of flower and water will not. Granted the rubix cubes ingredients are bonded as one and the contents of the bowl were not at one time. Time and energy were consumed to create the rubix cubes structure before it was to be mixed at a later time. Is there a way to that mixed flower and water to be unmixed to their original singularities. A rubix cube can by taking it apart. It could rest in biological associativity and reflective symmetry manipulated/influenced by using the tool of guage transformation during while in the process of angular momentum. However it doesn't really jive with nature. Nature in this case taking the example of say translation of old english to modern english with the influence/manipulation of latin or greek.
This is the 2nd Law of Thermodynamics, entropy increases with time. There seems to be an asymmetry in time, despite the fact that, at the level of individual particles, the laws of physics are time independent.

Explaining this asymmetry is one of the big open problems in physics.

Explaining other apparent asymmetries are also open problems in physics. For instance, why is there so much more matter, than anti-matter in the universe?

The notion of spontaneous symmetry breaking is often invoked for such mysteries.

I guess I could add that to the list of learning goals for this topic:
4) Really understand "broken symmetry"

10. Originally Posted by Fluffywolf
If you tie a rock to a cord and the other end to a stick and wave it in circles at a steady speed. The rock should be moving in circles indefinatly, and the stick stay upright. However, due to external factors such as gravity.
Gravity is a conservative force, so that is not the reason the momentum dissipates.

The reason is friction in its various forms (including drag).

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