# group theory

1. ### Other Textbooks for tensors and group theory

Hello, I am an undergraduate who has taken basic linear algebra and ODE. As for physics, I have taken an online edX quantum mechanics course. I am looking at studying some of the necessary math and physics needed for QFT and particle physics. It looks like I need tensors and group theory...
2. ### A Selection rules using Group Theory: many body

Hello, I am newish in group theory so sorry if anything in the following is not entirely correct. In general, one can anticipate if a matrix element <i|O|j> is zero or not by seeing if O|j> shares any irreducible representation with |i>. I know how to reduce to IRs the former product but I...

19. ### I About Arnold's ODE Book Notation

In Arnold's book, ordinary differential equations 3rd. WHY Arnold say Tg:M→M instead of Tg:G→S(M) for transformations Tfg=Tf Tg, Tg^-1=(Tg)^-1. Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a...
20. ### I Tensor representation of the Lorentz Group

I've been trying to understand representations of the Lorentz group. So as far as I understand, when an object is in an (m,n) representation, then it has two indices (let's say the object is $\phi^{ij}$), where one index $i$ transforms as $\exp(i(\theta_k-i\beta_k)A_k)$ and the other index...
21. ### Left invariant vector field under a gauge transformation

1. Homework Statement For a left invariant vector field γ(t) = exp(tv). For a gauge transformation t -> t(xμ). Intuitively, what happens to the LIVF in the latter case? Is it just displaced to a different point in spacetime or something else? 2. Homework Equations 3. The Attempt at a...
22. ### I Minimum requisite to generalize Proca action

Hello guys, In 90% of the papers I've read about diferent ways to achieve generalizations of the Proca action I've found there's a common condition that has to be satisfied, i.e: The number of degrees of freedom allowed to be propagated by the theory has to be three at most (two if the fields...
23. ### I Breaking down SU(N) representation into smaller groups

Hi all I have a shallow understanding of group theory but until now it was sufficient. I'm trying to generalize a problem, it's a Lagrangian with SU(N) symmetry but I changed some basic quantity that makes calculations hard by using a general SU(N) representation basis. Hopefully the details of...
24. ### I What is difference between transformations and automorphisms

Could you please help me to understand what is the difference between notions of «transformation» and «automorphism» (maybe it is more correct to talk about «inner automorphism»), if any? It looks like those two terms are used interchangeably. By «transformation» I mean mapping from some set...
25. ### Are these homomorphisms?

1. Homework Statement Are these functions homomorphisms, determine the kernel and image, and identify the quotient group up to isomorphism? C^∗ is the group of non-zero complex numbers under multiplication, and C is the group of all complex numbers under addition. 2. Homework Equations φ1 ...
26. ### A How to calculate the degeneracy of an energy band?

Could somebody write the guide for calculate the degeneracy of energy band by group theory? For instance, the valence band of Si and Ge in Gamma point. Thanks a lot!
27. ### I What are the groups for NxNxN puzzle cubes called?

The group of moves for the 3x3x3 puzzle cube is the Rubik’s Cube group: https://en.wikipedia.org/wiki/Rubik%27s_Cube_group. What are the groups of moves for NxNxN puzzle cubes called in general? Is there even a standardized term? I've been trying to find literature on the groups for the...
28. A

### Dic12 element orders

1. Homework Statement The dicyclic group of order 12 is generated by 2 generators x and y such that: $y^2 = x^3, x^6 = e, y^{-1}xy =x^{-1}$ where the element of Dic 12 can be written in the form $x^{k}y^{l}, 0 \leq x < 6, y = 0,1$. Write the product between two group elements in the form...
29. A

### Contractions of the Euclidean Group ISO(3) = E(3)

1. Homework Statement Consider the contractions of the 3D Euclidean symmetry while preserving the SO(2) subgroup. In the physics point of view, explain the resulting symmetries G(2) (Galilean symmetry group) and H(3) (Heisenberg-Weyl group for quantum mechanics) and give their Lie algebras...
30. ### I Proof that Galilean & Lorentz Ts form a group

The Galilean transformations are simple. x'=x-vt y'=y z'=z t'=t. Then why is there so much jargon and complication involved in proving that Galilean transformations satisfy the four group properties (Closure, Associative, Identity, Inverse)? Why talk of 10 generators? Why talk of rotation as...