I see that I'd need to use \sin(x) = 1/2i (e^{ix}-e^{-ix}), but it seems like the factor \exp(i\vec{p}\cdot \vec{x}) is factored out, which can't be true. They way I see it is
\exp(-ip\cdot x)-\exp(ip\cdot x)=\exp(-iEt+i\vec{p}\cdot\vec{x})-\exp(iEt-i\vec{p}\cdot\vec{x}), in which it is not...
Hi everyone,
I'm going through some lecture notes on Quantum Field Theory and I came across a derivation of an explicit form of the Pauli Jordan Green's function for the Klein-Gordon field.
The equations used in my lecture notes are equivalent to the ones in...
Hi everyone,
In one of the assignments in a course on classical field theory I'm given the action
S = \int d^4 x \mathcal{L}
where
\mathcal{L} = -\frac{1}{16\pi} F_{\mu \nu} F^{\mu \nu} - A_{\mu}j^{\mu}.
I'm now supposed to construct the canonical momenta \pi_\mu = \frac{\delta...
Hello everyone,
I'm going through some lecture notes and there are some things I don't understand about the whole derivation of the angular momentum multiplet.
It's said that the skew-symmetric 3x3 matrices J_i are the infinitesimal generators of the rotation group SO(3). Later, however...
Hi everyone,
I'm trying to find a general rule that expresses the product of two rotation matrices as a new matrix.
I'm adopting the topological model of the rotation group, so any rotation which is specified by an angle \phi and an axis \hat{n} is written R(\hat{n}\phi)= R(\vec{\phi})...
Hey guys,
How come a representation \rho of a group G is always equivalent to a unitary representation of G on some (inner product) space V ?
Can anyone provide a good source (book, preferably) which states a proof?
Thanks
Thanks for your quick answer.
Does that, then, imply that the Casimir operator is actually \sum_i \pi(J_i)^2?
Do the commutation relations satisfied by the Lie group generators carry over to the operators under the representation map? I can see that
[\pi(J_i),\pi(J_j)] = \pi(J_i)\pi(J_j) -...
Hi,
I'm confused by a sentence in a set of lecture notes I have on quantum mechanics. In it, it is assumed there is some representation \pi of SO(3) on a Hilbert space. This representation is assumed to be irreducible and unitary.
It is then said that the operators J_i, which are said to...
Hello everyone,
Say I have two irreducible representations \rho and \pi of a group G on vector spaces V and W. Then I construct a tensor product representation
\rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right)
by
\left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v...
Hi everyone,
I'm having some trouble with the concept of the direct sum and product of representations.
Say I have two representations \rho_1 , \rho_2 of a group G on vector spaces V_1, V_2 respectively. Then I know their direct sum and their product are defined as
\rho_1 \oplus \rho_2 : G...