8
votes
Accepted
Basic Facts about Lie Algebras
Yes, OP is right: P&S do not distinguish between the abstract Lie group and its defining representation, which is usually the fundamental representation.
- 170k
4
votes
Basic Facts about Lie Algebras
You learnt that in your group theory course, that sounds right. But you must consider the topology. Lie groups are manifolds, so they have tangent spaces (this is an additional structure to being a ...
- 141
3
votes
Accepted
Find commutator $[P_\mu,K_\nu]$ in conformal group
It is best to work with generators rather than group elements. The generators have a representation in real space, that is $P_\mu = -i\partial_\mu, D = -i x^{\mu} \partial_\mu, M_{\mu\nu} = i(x^{\mu}\...
- 116
3
votes
Confusion in group theory at the example of the Lie algebra $su(2)$
Yes, the $T_i$ are supposed to be a basis of the algebra as a vector space. Therefore, choosing n-by-n matrices $\rho(T_i)$ that represent each of the $T_i$ defines a linear map $\rho : \mathfrak{su}(...
- 107k
3
votes
A derivation of the canonical commutation relations (CCR) written by Dirac?
It is not a derivation per se, more like a suggestive argument. Dirac is assuming that the usual Poisson bracket on the algebra of functions can be replaced by a bracket $\{\cdot,\cdot\}$ on the ...
- 170k
2
votes
Lie bracket in general relativity
Note that the Lie bracket of the basis vectors vanishes. In a coordinate chart $x$, the partial derivative operators $\left\{\frac{\partial}{\partial x^i}\right\}$ constitute a basis for the tangent ...
- 51.8k
2
votes
Importance of raising/lowering/ladder operators
That is just the thing about quantum mechanics. It is based on the fundamental principle that interactions involve the exchange of quanta. As a result of this principle each particle field taking part ...
- 12k
2
votes
Can the $SU(3)$ gauge field be put in geometric algebra terms?
The 6D geometric algebra bivectors span
$$spin(6)$$
that is isomorphic to Pati-Salam's
$$SU(4)$$ which in turn contains
$$SU(3)_{color} * U(1)_{B-L}$$
- 3,078
2
votes
How did the two copies of the Witt algebra become two copies of the Virasoro algebra in the CFT?
$L_n$ seemed to have nothing to do with $\ell_n$.
There exists a specific relation between these two. The symmetry of the action is reflected on the correlation functions (which are the 'observables' ...
- 2,721
1
vote
Importance of raising/lowering/ladder operators
The construction of raising and lowering operator is very natural when there is an underlying Lie algebra. Thus yes there's some systematic way of constructing them. In fact, if the Lie algebra is ...
- 38.8k
1
vote
Confusion in group theory at the example of the Lie algebra $su(2)$
...and wouldn't then linear combinations of them yield new representation?
Not necessarily. Suppose you have a representation in matrix form like:
$$
[M_i, M_j] = i\epsilon_{ijk}M_k
$$
A linear ...
- 7,589
1
vote
Accepted
Car moving on a ball in space
I'll assume in the following that the mass of the car $m$ is negligible compared to the mass of the ball $M$, so during the motion, the center of the ball remains fixed. However, in order for the ...
- 1,647
1
vote
Accepted
Coefficient of effective chiral Lagrangian of $\pi\pi$ scattering
Importantly, observe what you might know, namely the method in the madness of the Lie algebra elements involved,
$$L=\frac{F^{2}}{4} \rm{Tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U)= \frac{F^{2}}{4} \...
- 48.1k
1
vote
Accepted
Lie bracket in general relativity
At the same time, the curvature of spacetime has to do with the Lie bracket
It does not. If you have two vector fields $V$ and $W$, you can define new vector field $X\equiv[V,W]$ such that for every ...
- 5,553
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