# From Symmetry Group to Physics Equations

To the extent that I know:

There are symmetry groups like the rotation groups SO(3), the Groups of Poincare Transformations,... If the physics of a system has a symmetry group G, then it can be described by a representation of G and the vector space acted on.

Correct me if I'm wrong.

If I'm not that wrong, I want to know a simplest example how we can interpret the physics of this system by studying properties of G's representation. (because I have been learning in the reverse way: first is the Hilbert Space of states, then the group of symmetry operators)

EDIT: I think the process of making a physics theory would be as following:

Corresponding to a specific "physics", there is particularly a Lie group (called G) of symmetry. Then we can build a framework by representing this Lie group as a group of linear transformations acting on a vector space V.

• Each element of V would be a state of physics.
• Each element of the Lie algebra (corresponding to G) would be an observable (this is what I want to know if it's true or false for sure)

Then we can apply quantum concepts as eigenstate, eigenvalue, distribution,...

Am I wrong? If I am wrong, how can I be fixed?

(I just read accidentally about representation theory last week and I'm kind of excited about the idea of promoting a theory from a somewhat simple (fundamental) object as a group of symmetry)

I have found a paper describing the way to construct quantum physics from symmetry group and representation theory:

http://www.math.columbia.edu/~woit/QM/qmbook.pdf

You are not wrong, the symmetries of a theory are essential to finding the right space of states. The space of states must carry a representation of all symmetries of the theory (though it might be the trivial one). For example, for a quantum system that is invariant under rotation (think of the hydrogen atom), the fact that we must represent the rotation group $\mathrm{SO}(3)$ on the space of solutions of the Schrödinger equation, since they are, as wavefunctions, the states, is naturally reflected in the fact that the solutions are (linear combinations of) the spherical harmonics $Y^l_m$, which are the basis vectors of all irreducible representations of $\mathrm{SO}(3)$, labeled by $l\in\mathbb{N}$. If $H_l$ denotes the representation of a certain $l$, the full space of states is $\bigoplus_{l\in\mathbb{N}}H_l$. So, you could have guessed the space of states by looking at the symmetry alone instead of solving the Schödinger equation! (I have neglected the radial and spin part in the above, but it gives the general idea, I think)

But a theory is (almost) always more than its symmetries. Many field theories have an action determining classical equations of motion and the quantum path integral, though not all. Quantum mechanics (almost) always has the Hamiltonian determining the time evolution, and the Quantomorphism symmetry (that post is not directly related, but Urs Schreiber tells a great story about how the passing from classical to quantum mechanics is intrinsically motivated by Lie theory, I think it might interest you) is not enough to fix it, it must be given.

The closest you get to determining the whole theory just by its symmetry are pure quantum gauge theories in low dimensions, where, in 2D, the topological structure of the spacetime together with the gauge group completely fixes the QFT and all observables (which aren't that many).

I'm not certain I have answered your precise question, so feel free to point it out if I missed the mark.

• It's sad. If we can have the theory from investigating the symmetry group it would be a beautiful way to build quantum physics. Really sad. Commented Jul 4, 2014 at 12:43
• @ACuriousMind: How do I match the idea of $SO(3)$ as the basis vectors of irreps of $SO(3)$ with its formal definition of some generators satisfying some commutation relation. What are the generators here, and how do you define the commutator in this case? More info about spherical harmonics as irreps of $SO(3)$ and some references will be appreciated.
– user7757
Commented Jul 4, 2014 at 19:32
• @ramanujan_dirac: You must be careful with the terminology! The generators with their commutation relation lie in the Lie algebra $\mathfrak{so}(3)$ of $\mathrm{SO}(3)$. The spherical harmonics can be seen to form irreps of $\mathrm{SO}(3)$ by observing that they transform under rotations just by composition, which gives that $H_l$ is a representation of the rotation group (since the harmonics form a vector space, and composition is linear). That they are irreps is then seen by just observing that they correspond exactly to the usual abstract irreps denoted by $|j,m\rangle$. Commented Jul 4, 2014 at 19:43
• @ramanujan_dirac: That $H_l$ is closed under rotation follows from the fact that $l(l+1)$ is the eigenvalue of the square of the angular momentum operator (which is the quadratic Casimir of $\mathfrak{so}(3)$) and that the rotations are generated by the angular momenta. Eigenspaces are invariant spaces, so $H_l$ is closed under rotation. Googling finds quite a lot on this subject, but I am afraid I have no particular source I could recommend. Commented Jul 4, 2014 at 19:52