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1

A spinor has two components that describes its transformation under a lorentz transformation, classified by two indices $\Psi_{a\dot{b}}$, the first transforms under the left part of the lorentz group $SO(1,3)\approx SU(2)_L \otimes SU(2)_R$ (in more detail $\Psi_{a\dot{b}}\rightarrow L(\Lambda)_a^{a'} R(\Lambda)_\dot{b}^{\dot{b}'}\Psi_{a'\dot{b}'}$ where ...

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I'm not very strong in group theory, so could someone please explain in simple terms what it means for the left handed parts to transform as triplets and right handed parts to transform as singlets? How would you go about writing down such terms in a Lagrangian? Groups are abstract. They have elements that can be "multiplied" and they have other ...

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The unitary representation of Galileian group already includes a representation of Weyl-Heisenberg group. The boost generator $K$ and the generator of translations $P$ satisfy $[K,P]= imI$ where $m$ is the mass of the system. Therefore, the subgroup whose generators are $m^{-1}K, P, I$ is the wanted unitary representation of Weyl-Heisenberg group. As a ...

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This notation is typical of the terrible habit of high energy theorists to label irreps by their dimension, and some educated guess is required to figure out what is $50$ and what is $50^*$. I will label representations of su(5) by their highest weight (or Dynkin labels), i.e. by the 4-dimensional vector of non-negative integers ...

4

The confusion here arises because we are not fully analogous to non-relativistic QM here. Given a (quantum or classical) field $\phi$, we usually specify whether it is a "scalar", "spinor", "tensor", whatever field. This refers to a finite-dimensional representation $\rho_\text{fin}$ of the Lorentz group the field transforms in as an element:  \phi ...

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The definition suggested by joshphysics and clarified by Qmechanic already exists in the literature under then name of representation operator. This is discussed in, e.g., Sternberg's Group Theory and Physics, as well as the somewhat more elementary text An Introduction to Tensors and Group Theory for Physicists by Jeevanjee.

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The 4-vector $A^{\mu}$ is an operator which acts on a Hilbert space with states $|a\rangle$. These things are called tensor operators - see chapter 4 of Howard Georgi's book Lie Algebras in Particle Physics. So, they have a matrix representation $\langle a|A^{\mu}|b\rangle$ which I'll write as $A^{\mu a}_{\ \ \ b}=\langle a|A^{\mu}|b\rangle$ to emphasize the ...

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