# Tag Info

Accepted

### Lie algebra in simple terms

It's an enormous subject, but briefly: Lie groups are smooth groups. Technically, Lie groups are sets that are both a smooth manifold, like a sphere for instance, and also have a group structure (...
• 2,950
Accepted

### Why are relativistic quantum field theories so much more restrictive than non-relativistic ones?

One of the reasons relativistic theories are so restrictive is because of the rigidity of the the symmetry group. Indeed, the (homogeneous part) of the same is simple, as opposed to that of non-...

### What does it mean for particles to "be" the irreducible unitary representations of the Poincare group?

Irreducible representations of the Poincare group are the smallest subspaces that are closed under the action of the Poincare group, which includes boosts, rotations, and translations. The point is ...
• 95.5k

### Why do all fields in a QFT transform like *irreducible* representations of some group?

Gell-Mann's totalitarian principle provides one possible answer. If a physical system is invariant under a symmetry group $G$ then everything not forbidden by $G$-symmetry is compulsory! This means ...
• 172k
Accepted

• 172k

### The anticommutator of $SU(N)$ generators

Indeed, the anticommutator $$S^{AB}\equiv \{T^A,T^B\}$$ is not in the Lie algebra, but, rather, in the universal enveloping algebra (formed by sums of products of generators); and, as you ...
• 48.9k

### Are all representations of a finite group unitary?

Every representation $(D,V)$ of a finite group $G$ is equivalent to a unitary representation. It is often termed as Weyl's unitary trick. This works by simply redefining your inner product by ...
• 3,192
Accepted

### $su(1,1) \cong su(2)$?

You may indeed identify the generators in the way you did. However, the Lie algebras and Lie groups are different because – as quickly said by Qmechanic – you must use different reality conditions for ...
• 174k

### Comprehensive book on group theory for physicists?

There is a new book called Physics From Symmetry which is written specifically for physicists and includes a long, very illustrative introduction to group theory. I especially liked that here concepts ...
### Why $SU(3)$ and not $U(3)$?
Suppose that $\text{U}(3)$ was the gauge group. We can decompose this as $$\text{U}(3)=\text{U}(1)\times\text{SU}(3),$$ which implies that in addition to the $\text{SU}(3)$ that has eight generators ...