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How are symmetries precisely defined?

In basic physics courses it is usual to see arguments on symmetry to derive some equations. This, however, is done in a kind of sloppy way: "we are calculating the electric field on a semicircle wire on the top half plane on the origin. Since it is symmetric, the horizontal components of the field cancel and we are left with the vertical component only".

Arguments like that are seem a lot. Now I'm seeing Susskinds Theoretical Minimum courses and he defines a symmetry like that: "a symmetry is a change of coordinates that lefts the Lagrangian unchanged". So if the lagrangian of a system is invariant under a change of coordinates, that change is a symmetry.

I've also heard talking about groups to talk about symmetries in physics. I've studied some group theory until now, but I can't see how groups can relate to this notion of symmetry Susskind talks about, nor the sloppy version of the basic courses.

So, how all those ideas fit together? How symmetry is precisely defined for a physicist?

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What is a physical theory/model?

A given physical theory is typically mathematically modeled by some set $\mathscr O$ of mathematical objects, and some rules that tell us how these objects correspond to a physical system and allow us to predict what will happen to that system.

For example, many systems in classical mechanics can be described by a pair $(\mathcal C, L)$ where $\mathcal C$ is the configuration space of the system (often a manifold), and $L$ is a function of paths on that configuration space. This model is then accompanied by rules like "the elements of $\mathcal C$ correspond to the possible positions of the system" and "given an initial configuration of the system and it's initial velocity, the Euler-Lagrange equations for $L$ determine the configuration and velocity of the system for later times."

What is a symmetry?

If we think of physics as being a collection of such models, we can define a symmetry of a system as a transformation on the set $\mathscr O$ of objects in the model such that the transformed set $\mathscr O'$ of objects yields the same physics. Note, I'm deliberately using the somewhat vague phrase "yields the same physics" here because what that phrase means depends on the context. In short:

A symmetry is transformation of a model that doesn't change the physics it predicts.

For example, for the model $(\mathcal C, L)$ above, one symmetry would be a transformation that maps the Lagrangian $L$ to a new Lagrangian $L'$ on the same configuration space such that the set of solutions to the Euler-Lagrange equations for $L$ equals the set of solutions to the Euler-Lagrange equations for $L'$. Even in this case, it is interesting to note that $L$ need not be invariant under the transformation for this to be the case. In fact, one can show that it is sufficient for $L'$ to differ from $L$ by a total time derivative. This brings up an important point;

A symmetry does not necessarily need to be an invariance of a given mathematical object. There exist symmetries of physical systems that change the mathematical objects that describe the system but that nonetheless leave the physics unchanged.

Another example to emphasize this point is that in classical electrodynamics, one can make describe the model in terms of potentials $\Phi, \mathbf A$ instead of in terms of the fields $\mathbf E$ and $\mathbf B$. In this case, any gauge transformation of the potentials will lead to the same physics because it won't change the fields. So if we were to model the system with potentials, then we see that there exist transformations of the objects in the model that change them but that nonetheless lead to the same physics.

How do groups relate to all of this?

Often times, the transformations of a model that one considers form actions of groups. A group action is a kind of mathematical object that associates a transformation on a given set with each element of the group in such a way that the group structure is preserved.

Take, for example, the system $(\mathcal C, L)$ from above. Suppose that $\mathcal C$ is the configuration space of a particle moving in a central force potential, and $L$ is the appropriate Lagrangian. One can define an action $\phi$ of the group of $G= \mathrm{SO}(3)$ of the set of rotations $R$ one the space of admissible paths $\mathbf x(t)$ in configuration space as follows: \begin{align} (\phi(R)\mathbf x)(t) = R\mathbf x(t). \end{align} Then one can show that the Lagrangian $L$ of the system is invariant under this group action. Therefore, the new Lagrangian yields the same equations of motion and therefore the same physical predictions.

Often times the objects describing a given model involve a vector space. For example, the state space of a quantum system is a special kind of vector space called a Hilbert space. In such cases, it is often useful to consider a certain kind of group action called a group representation. This leads one to study an enormous and beautiful subject called the representation theory of groups.

Are groups the end of the story?

Definitely not. It is possible for symmetries to be generated by other kinds of mathematical objects. A common example is that of symmetries that are generated by representations of a certain kind of mathematical object called a Lie algebra. In this case, as in the case of groups, one can then study the representation theory of Lie algebras which is, itself, also an huge, rich field of mathematics.

Even this isn't the end of the story. There are all sorts of models that admit symmetries generated by more exotic sorts of objects like in the context of supersymmetry where one considers objects called graded Lie algebras.

Most of the mathematics of this stuff falls, generally, under the name of representation theory.

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Symmetry is present when something $x$ doesn't change under some transformation $T$:

$$T(x)=x$$

In an infinite cylinder, there is radial symmetry because if you move at constant height and radius, you see the same figure.

In the Lagrangian case, if you change coordinates, the Lagrangian doesn't change. $L(x') =L(x)$

In group theory, group elements will represent some kind of transformations. This will have some associated symmetry.

For example:

  • $GL(n,\mathbb{R})$ (group of all real matrices) preserves points to be in $\mathbb{R}$.
  • $SL(n,\mathbb{R})$ (group of all real matrices with $\det=1$) preserves volumes. Recall that we can define a volume as determiant of vectors.

  • $O(n)$ (rotation group) preserves distances (dot product with an euclidean metric).

  • And many more...

And note that there are many symmetries not at all related with Physics, like 2125922464947725402112000 symmetries of Rubik's cube, which is described by the Rubik's cube group.

As you dive in Physics, you'll learn many more symmetries: diffeomorphisms, gauge fixing, CPT in QFT, Noether's theorem...

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Fixed points $T(x)=x$ is a bit too strong a condition for symmetry, unless you mean $x$ is a set invariant under $T$. (E.g., the $n$-dihedral group's natural action on regular $n$-gon in the plane transforms the points in the $n$-gon to other points in the $n$-gon; points outside the $n$-gon on the plane are not left invariant in the same way...) –  Alex Nelson Feb 11 at 16:21
    
@AlexNelson I mean if $x$ is some property (like the trace or whatever), $T(x)=x$, i.e it remeains unchanged. –  jinawee Feb 11 at 16:32
    
Ah, good, my error :) –  Alex Nelson Feb 11 at 16:36
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Symmetries indeed have a broad and powerful impact in physics, and I will only be able to scratch the surface of the subject in this answer, but I will try to give you a glimpse of the subject.

In the most simple framework, you mention an electrostatic problem. In such a problem, the key factor is the geometric symmetries which apply to the charged particles. For example, if the charged volume is symmetric with respect to the plane $z=0$, then the whole physical system admits this symmetry. As a consequence, the eletric field respects the same symmetry. So for a point $r$ located on this plane, $E(r)$ must be equal to its symmetric with respect to such plan, which imposes that its $z$ component be 0.

So here we see an example where a geometric property, respected by the causes, is said to have to be respected by the effects, and gives us a clue as to the properties of such effects.

The more general formulation is indeed Susskind's formulation, but you have to consider "Lagrangian" in his mouth, as meaning "the fundamental equation which the system obeys". So what he really means is that if a symmetry leaves the equations that drive a system unchanged, then such physical system is said to respect such symmetry. And there are usually very deep conclusions to be drawn from this mere fact (think about central forces for example).

The second definition above indeed is the same as the previous simple case: All I wrote about the charges and the electric field is contained in Susskind's definition, if you replace "coordinates" with "space coordinates" and "Lagrangian" with "Maxwell's equations", which are only a simpler formulation of the Lagrangian in a specific context.

So really, what you hear in basic physics courses is the proper definition, but indeed expressed in a somewhat sloppy way, and applied in a restricted context.

Group theory is closely linked to system symmetries because all the symmetry operations which leave a physical system unchanged form a group: you can check for yourself that the composition of two such symmetries leave the equations of the system unchanged, that the identity transformation leaves the equations of the system unchanged, and that for each transformation that leaves the equations of the system unchanged, its inverse leaves them unchanged as well. So you are naturally dealing with group algebra. It plays a crucial role in condensed matter for instance, because the symmetries of a crystal determine the symmetries of the potential in which the electrons move, and thus the symmetries which leave the Hamiltonian unchanged (equivalent to talking about the Lagrangian or about the equations of motion in a simpler setting). Since the energy levels of the electrons are the eigenvalues of the Hamiltonian, this has consequences on the structure or such energy levels: you can deduce degeneracy maximal degeneracy degree of energy levels in the crystal from simple symmetry considerations, and levels whose degeneracies are lifted if such or such symmetry is broken by an applied external constraint.

I would like to mention translational symmetry, which is so simple one sometimes forgets it - but translational symmetry for all space vectors (in other words homogeneity of space) allows you to show that momentum is conserved, i.e. that a particle moves with a constant velocity if mass is constant. A more restricted translational symmetry is found in crystals, where translational symmetry only holds for vectors of the underlying lattice, which leads to non less powerful conclusions with Bloch's theorem and its fundamental applications in transport theories and many other areas of condensed matter physics.

Finally I would like to emphasize that when Susskind says "coordinates", he does not mean "space coordinates" only. Time symmetry is another important symmetry. More generally, any coordinate, in the sense of any generalized coordinate as a function of which the Lagrangian or Hamiltonian may be written, may be the object of symmetry operations.

As a conclusion I would recommend the reading of the first volume of the course of theoretical physics, by Landau and Lifschitz. You will find beautiful insights based on symmetry in chapters I.1 to I.9

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