# Questions tagged [group-theory]

Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. Groups are used in physics to describe symmetry operations of physical systems.

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### Has this group something to do with the cone of light?

Consider the group $V=(-1,1)$ with addition $+_{rel}:V\times V\to V$ defined as: $$v+_{rel}w=\frac{v+w}{1+vw}$$ This group is analogous to the relativistic velocities where the speed of light equals ...
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### Emergence of rotational symmetry on 2D square lattice

On page 74 of David Tong's Statistical Field Theory lecture notes, it is said that $(\partial_1\phi)^2 + (\partial_2\phi)^2$ respects both $D_8$ (that includes discrete four-dimensional rotation ...
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### An invariant of the gauge group $G$ that is totally symmetric with three indices in the adjoint representation

In Ch.19 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.680 the property of a quantity $$\mathcal{A}^{abc}=\mathrm{tr}\left[t^a\{t^b,t^c\}\right]\tag{19.132}$$ ...
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### Representation of the Lorentz group using matrices of $SL(2,\mathbb{C})$

There is a correspondence between the Lorentz group and the group $SL(2,\mathbb{C})$. To each Lorentz transformation $\Lambda$ we can associate two matrices $\pm A(\Lambda) \in SL(2,\mathbb{C})$ such ...
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### Symmetry relation of Wigner-Eckart

I saw a symmetry relation following from the the Wigner-Eckart Theorem looking like this $$(\xi j|| T_L || \xi'j') = (-1)^{j-j'} (\xi' j'|| T_L || \xi j)^*$$ I know that it must come somehow under ...
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### Quantum representation of a system of identical particles

I'm studying mathematics and I began a course in quantum statistics, in which I got to the discussion related to indistinguishibility of particles. My professor's notes are not very clear and ...
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### Representation and Lie algebra of $SO(3)$

Studyng the book Group Theory in Physics of Wu-Ki Tung, I have read: "... every representation of the [$SO(3)$] group is automatically a representation of the corresponding Lie algebra, (...) a ...
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### Winding number in 4D & $SU(2)$ group

In the book Quantum field theory by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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### Difference between $\tilde{\textrm{Diff}}_+(S^1)$ and ${\textrm{Diff}}_+(S^1)$

In this paper, where Liouville theory is being studied on a strip, after equation 2.3 it is mentioned that the conformal transformations of the strip are given by the same chiral and anti-chiral ...
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### What is a rotation group and how do we get its unitary representation?

The rotation group is ${\rm SO(3)}$. It is the group of $3\times 3$ orthogonal matrices $\{g(\theta)\}$ with unit determinant. So these are already defined in terms of $3\times 3$ matrices. But we use ...
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### Why do particle physicists often use $\otimes$ instead of $\times$ to denote the direct product of groups? [closed]
For example, sometimes one sees $SU(3)\otimes SU(2)\otimes U(1)$ instead of $SU(3)\times SU(2)\times U(1)$. My understanding is the the product here is just the usual direct product (aka Cartesian ...
I am trying to follow the Particle Data Group's instruction (PDF link) to combine SU(N) multiplets. On page 3, they show an example calculation of SU(3)'s $\textbf 8\otimes \textbf 8$. I understand ...