Questions tagged [poincare-symmetry]

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Commutator of the Pauli-Lubanski vector operator and the generator of translations $P^\alpha$

I'm trying to obtain the commutation relation between the Pauli-Lubanski vector operator and the generators of the Lorentz Group: $$[W^\mu,P_\sigma]=[\frac{1}{2}\epsilon^{\mu\nu\lambda\rho} P_\nu M_\...
3
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2answers
42 views

Casimir operators for Poincare algebra

I have seen at various places the comment that the operator $P_\mu P^\mu$ is a Casimir operator of Lorentz algebra and thus it satisfies a on-shell condition like $P_\mu P^\mu=m^2$. Given the Poincare ...
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1answer
41 views

Proof that the commutator of angular momentum and 4-momentum is 0

I have this commutator $[P^2,J_{\mu\nu}]$ that I'm supposed to prove is zero. If we expand it (given that $[P_{\alpha}, J_{\mu\nu}] = i(g_{\mu\alpha}P_{\nu} - g_{\nu\alpha}P_{\mu})$ and $[P_{\alpha}, ...
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0answers
28 views

How to derive the general expression for eigenvalue for the square of Pauli Lubanski operator?

After some trials, I managed to get the correct eingenvalue $(\frac{-3}{4}m^2)$ for $W^2$, where $W$ is the Pauli Lubanski pseudo vector. The expression for each $J^{\mu \nu}$ is a sum of a 4x4 ...
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1answer
42 views

Application of a Poincaré group element to a scalar function

Let $f(x)$ be a scalar function and let's say that we want to know how it transforms when it's subjected to a translation (by a vector $a^{\mu}$), rotation and a Lorentz boost. Thus we can write an ...
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20 views

Intuition for 2 commutation relations related to the Poincare group

For some commutation relations related to the Poincare group, there are intuitive way of understanding it. For example, $[J_i,H]=0$ can be understood as the conservation of angular momentum along i ...
6
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1answer
166 views

General relativity as a gauge theory of the Poincaré algebra

Let the Poincaré algebra be given without any factors of i as $[P_\mu,P_\nu]=0$, $[M_{\rho \sigma},P_\mu]=\eta_{\sigma\mu}P_\rho-\eta_{\rho\mu}P_\sigma$, $[M_{\mu\nu},M_{\rho\sigma}]=\eta_{\nu\rho}...
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2answers
79 views

What is the problem with a generalized kinetic term as $K^{\mu\nu}(x)\partial_\mu\phi\partial_\nu\phi$?

For field theory in flat spacetime, the most general kinetic term that I can think of for a field is $$K^{\mu\nu}(x)\partial_\mu\phi\partial_\nu\phi$$ where $K^{\mu\nu}(x)$ is an arbitrary second rank ...
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0answers
45 views

Why do we use affine groups in gauge theory? What is the purpose?

When we study General Relativity in the frame of gauge theory, what's the importance of affine group?
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1answer
40 views

Poincaré Group element of pure spacetime translation

If we make a spacetime translation of the coordinates of a event $ x^{\mu}$ such that $x' ^{\mu} = x ^{\mu} + a^{\mu}$, the element $\eta _{\mu \nu} x'^\mu x' ^\nu $. Must be invariant : \begin{...
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27 views

Can local rotations lead to a gauge theory?

I was reading about the relation between electromagnetism and the complex Klein-Gordon field. The KG field had a global $U(1)$ symmetry and upon demanding that even a local phase transformation must ...
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1answer
87 views

What is the value of $W_\mu W^\mu$ for massless particles?

What is the value of the quantity $W_\mu W^\mu$ for massless particles where $W^\mu$ is called Pauli-Lubanski vector defined as $W^\mu=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}P_\nu J_{\alpha\beta}$. ...
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1answer
51 views

Generators of $su(2) \oplus su(2)$ in the $(A, B)$ representation

Let us call the generators of $su(2)$ in the spin $A$ or spin $B$ representation $J^A_i$ and $J^B_i$ respectively. What are the generators of $su(2) \oplus su(2)$ in the $(A, B)$ representation ? ...
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95 views

Doubt in Weinberg's book on QFT

In chapter 3 of his book on QFT (volume 1), while discussing the symmetries of the S-matrix, Weinberg makes the following statement For any proper orthochronous Lorentz transformation $x\rightarrow ...
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3answers
244 views

The role of Lorentz tranformations

My questions concern the role of Lorentz transformations in Special Relativity and General Relativity, as described in the following fragment of the series of GR lectures: https://www.youtube.com/...
4
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3answers
109 views

As $SL(2,\mathbb{C})$ is a double cover of the Lorentz group, is $SL(2,\mathbb{Z})$ a discrete subgroup of the Lorentz group?

The group $SL(2,\mathbb{C})$, the group of $2 \times 2$ complex matrices with determinent $1$, is a double cover of the Lorentz group. (These transformations can be understood as Mobius ...
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2answers
65 views

Commutation of position four-vector with spacetime derivatives

I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134 $$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\...
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2answers
101 views

Finding properties of Poincare Transformation

I have started studying the Poincare group for the first time, in preparation for my first QFT course, and I wish to be able to solve the following problem: A Poincare transformation ($\Lambda,a)$ ...
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1answer
113 views

The conmutator of the square of Pauli-Lubanski vector and the generators of Poincare group

I'm working on trying to solve the following problem: Using the following expressión for the square of Pauli-Lubanski vector:$$W^2=-\frac{1}{2}M_{\mu\nu}M^{\mu\nu}P_{\alpha}P^{\alpha}+M^{\mu\nu}M_{\...
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1answer
69 views

Where to learn about Poincaré Group properties?

I am studying my first QFT course, and there seems to be a lot that I was not taught in previous courses. In my first assignment, I have to prove several properties about the Poincare group, but I ...
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2answers
113 views

Why physics should be the same in all inertial frame? [closed]

One of the postulates of special relativity is that physics should be the same in all inertial frame. Suppose we have two observers $A$ and $B$ suppose that $A$ is accelerated. Now suppose that we ...
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0answers
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Do gauge fields not transform like functions of the coordinates under translations?

By "transform like a function of the coordinates," I mean that under an infinitesimal translation $x^\mu \to x^\mu + \epsilon^\mu$, to first order in $\epsilon^\mu$ the function $f(t,\mathbf x)$ ...
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1answer
91 views

Transformation of the derivative of the scalar field in Ramond's book about QFT

In the book by Pierre Ramond about quantum field theory, he explores in chapter 1.4 (p.13) the behavior of fields under Poincaré transformations. He starts by explaining that infinitesimal ...
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229 views

Significance of the Little group

My current understanding of the Little group is that it is the symmetry of a given state in the Fock space. This means that given a massive or massless particle in n dimensions, I can tell the number ...
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2answers
51 views

Lorentz boost tensor notation confusion

I have been given this$$ \delta X^{\mu}=\omega_{\mu \nu}\left(M^{\mu \mu}\right)_{\sigma}^{\rho} X^{\sigma} $$ and I think it should be equal to this but I'm confused if I'm doing it correctly $$ \...
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68 views

Generators of the Poincaré group

I am specifically interested in constructing the generators of an Poincaré group for a 2+1 dimensional Euclidean field theory. But I am pretty new to the subject, so I would like to ask some basic ...
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70 views

Representation of Poincaré group

Let's consider the most general Lorentz transformation: $x'^{\mu} = \Lambda^{\mu}_{\ \ \nu} x^{\nu} + a^{\mu}$. These transformations form the Poincaré group. The generators of translations of this ...
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1answer
65 views

Doubt in the Poincaré algebra and one-particle states

I am studyng the algebra of Poincaré group and the definition of one particle states using the Weinberg book "Quantum theory of Fields" (vol. 1), but I'm having a hard time understanding part of the ...
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1answer
79 views

Reference request for Lie algebras

My future adviser just published a beautiful paper, https://arxiv.org/abs/1904.08304, and I am looking for some references/textbooks to look into the following concepts: Lie algebra (central) ...
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1answer
133 views

How is this derivation of a field transformation, in Weinberg's QFT book, performed?

I am reading Weinberg's book Quantum theory of fields. Could you explain to me the following things? Vol.1, page 60 (transcribed from this image): To first order in $\omega$ and $\epsilon$, we ...
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136 views

Massive chiral fermions

The main question is: why does nobody care about massive chiral fermions? It is well-known that in QFT (in the axiomatic framework of Garding & Wightman) the quantum field transform according (...
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79 views

Understanding the idea behind the super-Poincaré algebra

On the Super-poincaré algebra wiki page (https://en.m.wikipedia.org/wiki/Super-Poincaré_algebra), it says: "If Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry ...
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2answers
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Is the antisymmetrisation of $a^{\mu}b^{\nu}\epsilon_{\mu\nu}= a^{[\mu}b^{\nu]}\epsilon_{\mu\nu}$ with antisymmetric tensor $\epsilon$ mandatory?

When in tensor algebra the product of 2 vectors with a antisymmetric tensor appear, is antisymmetrisation compulsory ? Given an antisymmetric tensor $\epsilon_{\mu\nu}$, is $$a^{\mu}b^{\nu}\epsilon_{...
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33 views

How is deSitter group tranformations different from poincare group transformations

In QFT, we have studies Poincare group of massive and massless particles. Is the deSitter group also useful to study such things? What exactly is the main role of this group in QFT? I just know the ...
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46 views

Check that the Poincaré's transformations form a group structure

How can I answer to this question ? I know that this is a Lorentz transformation + a translation but I don't know how to start. What's the difference between group/group structure ?
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0answers
58 views

How is Inönü-wigner contraction done?

I have read that little group for the massive particles is $SO(3)$ and for the massless particles is $E(2)$ in 4 dimensions. How does one take zero mass limits for the representations and show that it ...
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2answers
344 views

Poincare transformations and “three kinds of infinitesimal variations”

I'm currently reading these$^1$ lec. notes as an introduction to relativistic QFT. In chapter two (pp.57-61) he introduces the concept of field variations along with some formulas for the different ...
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0answers
56 views

Poincaré invariance of linearly polarised plane wave

I am reading a book that just quotes the Lie group generators and the discrete subgroups that leave a linearly polarised plane wave unchanged. And I have no idea how to derive them. Context The ...
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1answer
120 views

Doubts on representations of poincare group and QFT

I am studying Poincare group and encountered the term massless representations of the Poincare group. I know Poincare group is studied by the studying the little group of various momenta, massless and ...
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3answers
379 views

Lorentz transformation of vector field

Under a Lorentz transformation, a vector field transforms as: $A'_{\mu}(x')=\Lambda^{\nu}_{\mu}A_{\nu}(x)$ My question is, why is the Lorentz transformed vector field evaluated at $x'=\Lambda x$, ...
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1answer
91 views

Poincare group and free theories

How exactly is the Poincare group related to the free relativistic theories in quantum field theory? I know Poincare group is the Lorentz group along with translations but don't see any connected why ...
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99 views

Variation of vector field under Lorentz transformation and gauge transformation

In a paper I am reading, it is stated that under a Lorentz transformation, the coordinates transform as $x^{\mu} \to \Lambda^{\mu}_{\nu}x^{\nu}$, and so the change in the (vector) field at the same ...
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0answers
84 views

Derivatives in Poincare' gauge theory

I have been reading the lectures: http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_Supergravity.pdf about Poincare' gauge theory. The Poincare' group is considered as semidirect ...
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How to write the Poincare transformation for an arbitary path in Minkowski space?

So lets Say for arguments sake we have some vector $V^{a}$ and we drag it along some path $\gamma_{1}$ in Minkowski space $R^{3,1}$. For a straight path (represented by a vector $\Delta\overrightarrow{...
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1answer
99 views

Poincaré and Galilei group - notation

On this slide it just says that $\mathcal{P}$ and $\mathcal{G}$ are the Poincoré and Galilei groups, but I do not understand what they are made of. What does $\mathbb{R}^{1,3}$ mean? Why does $\...
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1answer
134 views

Doubt in Weinberg's book on Quantum Field Theory

In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as: \begin{equation} U(1+\omega,\epsilon)=1+\dfrac{1}{2}...
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1answer
146 views

What sort of particles corresponds to the $(1,1/2)$ representation of the Lorentz group?

Every irreducible massive unitary representation of the Poincaré group is specified by a mass and a non-negative half integer spin. Every massless irreducible unitary representation of the Poincaré ...
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103 views

Gravity from algebra

Can someone provide me a reference that describes the gauging of the Poincare algebra to obtain Einstein's relativity? "It is well known that Einstein’s formulation of gravity can be obtained by ...
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2answers
205 views

GR as a gauge theory: there's a Lorentz-valued spin connection, but what about a translation-valued connection?

Given an internal symmetry group, we gauge it by promoting the exterior derivative to its covariant version: $$ D = d+A, $$ where $A=A^a T_a$ is a Lie algebra valued one-form known as the connection ...
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1answer
223 views

Physical meaning of the Casimir operators of Poincarè algebra

If one considers the algebra $su(2)$, it is well known that the Casimir Operator is $$ C=L_1^2+L_2^2+L_3^2. $$ It corresponds to the total angular momentum and correctly is a conserved quantity. ...