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9
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2answers
1k views

Irreducible Representations Of Lorentz Group

In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as ...
3
votes
1answer
179 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} ...
5
votes
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 ...
0
votes
0answers
11 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
votes
1answer
23 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 ...
0
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1answer
48 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
votes
1answer
184 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 ...
6
votes
2answers
85 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 ...
1
vote
1answer
47 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 ...
0
votes
1answer
49 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
56 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
49 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
votes
0answers
64 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
votes
2answers
176 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 ...
0
votes
1answer
67 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 ...
0
votes
2answers
91 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 ...
6
votes
1answer
154 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 ...
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.
3
votes
1answer
483 views

Unitary Lorentz transformation on quantized Dirac spinor

I am stuck again on page 59 of Peskin and Schroeder. In particular, I do not know how they get equation (3.110). Let me first give some background in the way that I understand it (but I might be ...
1
vote
3answers
170 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 ...
1
vote
1answer
68 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 ...
4
votes
1answer
133 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
121 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
votes
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
votes
1answer
28 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 ...
3
votes
1answer
340 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
vote
0answers
57 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 ...
0
votes
1answer
90 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
65 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
votes
0answers
146 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 ...
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 ...
2
votes
2answers
389 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 ...
3
votes
2answers
133 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
votes
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 ...
1
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0answers
153 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
votes
0answers
75 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 ...
2
votes
1answer
150 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 ...
1
vote
1answer
96 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} ...
0
votes
1answer
59 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 ...
0
votes
1answer
109 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
306 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
votes
0answers
89 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 ...
2
votes
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
vote
1answer
58 views

Poincaré symmetry and linearized gravity

When working with linearized gravity, is Poincaré symetry assumed to be the symmetry of space-time?
10
votes
2answers
251 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
votes
1answer
337 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?
1
vote
0answers
147 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 ...
3
votes
0answers
214 views

Reducing massive representation of the Poincare group to the massless one

I want to ask about the connection for massive and massless representation of the Poincare group. Sorry for the awkwardness. First I must to represent the formalism for both of cases. Massive ...
1
vote
1answer
404 views

Trying to rhyme Peskin and Schroeder with Weinberg

This is a follow up question of this one. In the Vol 1, Weinberg derives how a unitary operator $U(\Lambda)$ acts on one-particle states, which is given by equation (2.5.2): \begin{equation} ...