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### Do the Wightman axioms uniquely fix the representation of the Poincaré group on the one-particle states given the representation on the fields?

Let $P := \mathrm{SL}(2,\mathbb{C})\ltimes \mathbb{R}^4$ be the universal cover of the connected component of the identity of the Poincaré group. Given a classical field $\phi : \mathbb{R}^{1,3}\to V$...
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### Spinors in 2+1 dimensions

I am trying to understand representations of the Poincare/Lorentz group, and in particular spinors, in 2+1 dimensions. I know some of the math, but I'm not sure about the physical interpretation of it ...
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### What is the matrix representation of the momentum operator (generator of translations) that is used in the commutators of the Poincaré Group?

So the commutators of the Poincareé group are given by \begin{eqnarray} [J_{i},P_{j}]=i\epsilon_{ijk}P_{k}, \quad [J_{i},J_{j}]=i\epsilon_{ijk}J_{k}, \quad [J_{i},K_{j}]=i\epsilon_{ijk}K_{k}, \quad [...
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### How to evaluate possible values of spin of two photon system?

Photon hasn't well defined quantity such as spin. Instead of it, it is characterized by helicity $h$. Let's assume state of two photons in CM frame (with $\mathbf k$ being the momentum of one of ...
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### How to arrive at the Dirac Equation from Poincare Algebra?

For the case of Galilean group, the time translation is given by the generator $H$. Hence, $$\mid\psi(t)\rangle\to \mid\psi(t+s)\rangle =e^{-iHs}\mid\psi(t)\rangle$$ Which immediately is the ...
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### The Wigner angle for two-particle state

Suppose we have the Wigner angle $\theta (\mathbf k, \Lambda)$, which is defined through the Lorentz group transformation $U(\Lambda)$ of one-particle state $|\mathbf k , \sigma\rangle$ ($\sigma$ ...
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### In what sense is BMS a symmetry? (What is kept invariant?)

Recently I've started to read about BMS (Bondi-Metzner-Sachs), and I've encountered several statements such as the following (from [1]). [I]t turned out that the asymptotic symmetry group at null ...
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### Poincare non-invariance in real world and field theory

This may be a very blunt question but I wonder why we always use Poincare invariant Lagrangians in field theory. After all, the entire world around us is by no means homogeneous, isotropic and so on. ...
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### Why are the particles called irreps of Poincare group? [duplicate]

Why are particle excitations called irreducible representation of the Poincare group? It will be very helpful if someone can illustrate with one concrete example of a particle. EDIT : But how does ...
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A global Poincare transformation on a scalar field induces $$\delta(a, \lambda)\phi(x) = [a^{\mu}+\lambda^{\mu\nu}x_{\nu}]\partial_{\mu}\phi(x). \tag{11.46}$$ In curved spacetime we replace $a^{\mu} ... 1answer 94 views ### Space-time translations and Propagator Let us assume to have the following scalar field theory $${\cal A}=\int d^4x\left[\frac{1}{2}(\partial\phi)^2-\frac{\lambda}{4}\phi^4\right]$$ where I used a quartic potential to fix the ideas. ... 1answer 68 views ### Does the spatial momentum of the ground state of a Poincare symmetric QFT vanish? Consider a flat space QFT, the Lagrangian (in general interacting) has Poincare symmetry, and$\lvert\Omega\rangle$is the ground state (or just merely no insertion at the far boundaries, from ... 0answers 156 views ### Finding Casimir operators for the Poincare group$ISO(1,2)$I was asked to write the generators for translations and Lorentz-transforms in 1+2 dimensions and then to find the Casimir operators. For the generators I can take the same ones as in 1+3 case $$P_\... 1answer 72 views ### Why is the Poincaré group non-abelian? Based on what I've learned, I gather the Poincaré group is the group of isometries of Minkowski spacetime and it is a non-abelian Lie group. Why is it non-abelian? Or perhaps rather, does the fact ... 2answers 136 views ### Scalar operators In Quantum Field Theory I am trying to learn Quantum Field Theory and I am stuck in a basic point. What is the definition of a scalar operator in QFT? That is, how does it transform under a Poincare transformation? Why do ... 1answer 155 views ### Misuse of \mathbf J^2 in classifying Poincare reps SO(1,3) has an infinite number of representations, classified by the Casimir invariant p^2. SO(3) also has an infinite number of representations, classified by the Casimir invariant \mathbf J^... 0answers 158 views ### Spin tensor from Noether theorem and spin tensor from Pauli-Lubanski vector Spin 3-vector directly from Noether theorem Let's have one of applications of Noether theorem: the invariance of action under Lorentz group transformations leads to conservation of tensor$$ \tag 1 ... 0answers 77 views ### Irreducible representation for the massless particle with helicity 2 and the Weyl tensor As it can be shown, the equations for the irrep with zero mass and helicity 2, -2 respectively can be given in a form $$\tag 1 \partial^{\dot {b}a}C_{abcd} = 0, \quad \partial^{\dot{b}a}C_{\dot{a}\... 1answer 97 views ### Verification of the Poincare Algebra The generators of the Poincare group P(1;3) are supposed to obey the following commutation relation to be verified:$$\left[ M^{\mu\nu}, P^{\rho} \right] = i \left(g^{\nu\rho} P^{\mu} - g^{\mu\rho} ... 1answer 152 views ### Notation for Translation Group Generators The generators of the translation group$T(4)$are given below:$P_0 \equiv -i \begin{pmatrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 &...
Let $F(x^\mu)$ is a scalar function; i.e. $F(x^\mu): \mathbb{R}^{1,3} \rightarrow \mathbb{R}$. How the Poincare Group $P(1,3)$ will act on it; i.e., by which formula I can calculate it for a specific ...