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1answer
52 views

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 ...
4
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1answer
115 views

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 [...
3
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1answer
111 views

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 ...
2
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0answers
102 views

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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0answers
12 views

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 $|\mathbf k , \sigma\rangle$ ($\sigma$ ...
0
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1answer
26 views

How to get explicit value of Wigner angle for massless one-particle state transformation?

The one-particle massless state $|\mathbf p , \sigma\rangle$ is transformed under the Lorentz group $U(\Lambda) \equiv U(\Lambda , 0)$ as $$ U(\Lambda)|\mathbf p, \sigma \rangle = \sqrt{\frac{(\Lambda ...
6
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2answers
92 views

Poincare representations for interacting field theory

I was going through Rudolf Haag's memoir http://link.springer.com/article/10.1140%2Fepjh%2Fe2010-10032-4 and came across these lines: '..in quantum field theory (or for any system of interacting ...
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1answer
51 views

Homework-lile questions about Poincare transformation [closed]

Here is a page from a paper which I am currently reading. This page mainly talk about Poincare symmetry. Now I can not understand how is Eq.(3.32) is derived. Also Eq.(3.28) looks odd to me. Why ...
8
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1answer
191 views

Spin state after boost

I am working through Weinberg's QFT book, and in problem 1 in chapter 2 I ran into copious amounts of algebra, so I am trying to "cheat" a little by using some assumptions, but am unsure of their ...
1
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1answer
49 views

Translation Transformation on the Invariant Interval (Spacetime) [closed]

So we know that the invariant interval in a two-dimensional spacetime in special relativity is given by $$ s = -c^2t^2 + x^2 = -c^{'2}t^{'2} + x^{'2}$$ So this scalar should hold true in all frames....
0
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1answer
53 views

Coleman Mandula theorem and translations

I don't know what Coleman Mandula theorem is, however if I were forced to say something about it, I will say it is a statement that suggests that internal and spatial symmetries have no unique ...
2
votes
1answer
67 views

Explicit form of the translation operator generators in the Poincare group?

Let $P_0$ be the generator for temporal translation and $P_1, P_2, P_3$ be for spatial translations. Let $p_μ$ be the momentum operator in the $x_μ=x^μ$ direction. I watched a lecture where the guy ...
0
votes
1answer
50 views

Confusion with Thomas precession

Suppose an inertial frame $S^\prime$ is moving with a relative velocity $\textbf{v}=v\hat{n}$ w.r.t another intertial frame S with their axes parallel and $\hat{n}$ is an arbitrary direction. In that ...
0
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0answers
68 views

Are there (interesting) Poincare-invariant QFTs with non-invariant Lagrangian densities?

In all QFTs I know, the Lagrangian density is completely invariant under the Poincare group, $$ \mathcal L \to \mathcal L. $$ On the other hand, the action would be invariant even if the Lagrangian ...
3
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2answers
183 views

Confusion with Weinberg's QFT book, volume 1, chapter 3: time translation and Heisenberg picture

Sorry if this is a naive question, but I am new to QFT. In the treatment of scattering in section 3.1 of The quantum theory of fields, vol.1, Weinberg first presented the general transformation rule ...
6
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1answer
185 views

Transformation of photons under Lorentz transformation

This question is a continuation of one of my earlier post. In this post,I asked about the transformation of photon fields under rotation. Here I generalize the question to Lorentz transformation, and ...
0
votes
1answer
68 views

What is the group transformation property of photons under rotation?

Both the photon and the W boson are spin-1 particles. Under rotation W boson must transform under the 3-dimensional representation of SU(2). However, the photon has two degrees of freedom (or helicity ...
1
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1answer
78 views

Poincare group representation and complete set

In Weinberg's book of Qft, chapter 2 of volume 1, he uses the eigenstates of the four-momentum to construct the unitary irreducible representations of the Poincare group. My question is, since $P^\mu,...
1
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3answers
180 views

Symmetry at quantum level in quantum field theory

In nonrelativistic quantum mechanics, a symmetry is a transformation on states in the Hilbert space which keeps the Hamiltonian invariant and this implies that the generator of the transformation must ...
4
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1answer
139 views

Why are one-particle states called representations of Poincaré group?

The one-particle states in the Hilbert space of a quantized relativistic field theory are said to form representations of the Poincaré group. Why is that? I mean, popular texts in QFT do not ...
2
votes
2answers
127 views

Does every Hilbert Space carry a representation of Poincare group?

We know all infinite dimensional Hilbert Spaces are unitarily equivalent. It should follow therefore that if I have an unitary representation of say Lorentz or Poincare group on one infinite ...
0
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0answers
39 views

Galileons and the brane origins of their Galilean invariance

I've been reading through a paper on Galileons by K. Hinterbichler et al. in which they discuss the brane origin of their Galilean invariance (starting on page 9) http://arxiv.org/abs/1008.1305 The ...
0
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1answer
29 views

How proof the Lorentz algebra using the Poincaré algebra? [closed]

Show that $$[J_{i},J_{j}]=i\varepsilon_{ijk}J_{k},\quad [K_{i},K_{j}]=-i\varepsilon_{ijk}J_{k}, \quad [J_{i},K_{j}]=i\varepsilon_{ijk}K_{j},$$ using $$[M_{\mu\nu},M_{\rho\sigma}]=ig_{\nu\rho}M_{\...
3
votes
1answer
391 views

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 ...
2
votes
1answer
78 views

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. ...
1
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0answers
58 views

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 ...
3
votes
1answer
196 views

Local translations in curved spacetime

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} ...
0
votes
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. ...
2
votes
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 ...
2
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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_\...
0
votes
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 ...
3
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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 ...
4
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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^...
1
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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 ...
2
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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}\...
1
vote
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} ...
2
votes
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 &...
0
votes
1answer
61 views

Action of the Poincare Group on a Scalar Function

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 ...
2
votes
2answers
399 views

Derivation of the full generator of the Lorentz transformations

Let us study the subgroup of the Poincare group that leaves the point $x=0$ invariant, that is the Lorentz group. The action of an infinitesimal Lorentz transformation on a field $\Phi(0)$ is $L_{\mu \...
0
votes
2answers
95 views

Poincare Symmetry of Nambu-Goto action

How do I show invariance under the Poincare transformations of the action for a relativistic string, $$S=-\frac{1}{2 \pi \alpha'} \int{\text{d}^2 \zeta}\sqrt{-\det(\partial_{\alpha}X^{\mu}\partial_{\...
0
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1answer
166 views

Equality of masses of particle and antiparticle

Usually we say that equality of masses of particle and antiparticle follows from CPT-theorem. But do we need it for showing this equality? The first method to show that is following. The equation ...
3
votes
1answer
332 views

S-operator lorentz invariance

How to show that $\hat {S}$-operator must be lorentz-invariant operator? $$ |\Psi (t)\rangle = \hat {S} | \Psi (0) \rangle , \quad \hat {S} = \hat {T}e^{-i\int \hat {H}_{I}d^{4}x}. $$ I have read ...
2
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0answers
91 views

One question about Weinberg's derivation of arbitrary spin fields expressions

In his book "QFT" (vol. 1) Weinberg writes the expression for an arbitrary spin massive field: $$ \hat {\Psi}_{a}(x) = \sum_{\sigma = -s}^{s} \int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2 \epsilon_{\...
2
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0answers
82 views

General covariance and global Poincaré algebras

Reading an article (page 7) I read this: Just as ordinary general covariance may be regarded as the local gauge symmetry corresponding to the global Poincare algebra and local gauge invariance ...
1
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1answer
61 views

Poincaré symmetry and linearized gravity

When working with linearized gravity, is Poincaré symetry assumed to be the symmetry of space-time?
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2answers
255 views

From representations to field theories

The one-particle states as well as the fields in quantum field theory are regarded as representations of Poincare group, e.g. scalar, spinor, and vector representations. Is there any systematical ...
8
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1answer
353 views

Representations of the Poincare group

Which type of states carry the irreducible unitary representations of the Poincare group? Multi-particle states or Single-particle states?
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0answers
154 views

Two pairs of projection operators of the Dirac equation

The Dirac equation may be interpreted as the action of projection operator $\frac{1 - \Delta}{2}\Psi = 0$, where $$ \Delta = \begin{pmatrix} 0 & \Delta_{b \dot {a}} \\ \Delta^{\dot {b}a} & 0 ...
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1answer
100 views

Five-component field

Recently I was reading about 5-component field $(\varphi , \psi_{\mu})$, for which $$ \hat {p}^{\mu} \varphi = mc\psi^{\mu}, \quad \hat {p}_{\mu}\psi^{\mu} = mc\varphi . $$ This field refers to the ...
9
votes
1answer
1k views

Generators of Poincare Groups

How can I determine the generators of the Poincare Group, $P(1,3)$ explicitly? Here $P(1,3)$ means a matrix Lie group.