Questions tagged [poincare-symmetry]
The poincare-symmetry tag has no usage guidance.
198
questions
2
votes
1answer
45 views
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
votes
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 ...
1
vote
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}, ...
0
votes
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 ...
1
vote
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 ...
1
vote
0answers
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
votes
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}...
0
votes
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 ...
1
vote
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?
0
votes
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{...
1
vote
0answers
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 ...
1
vote
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}$.
...
0
votes
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 ?
...
0
votes
0answers
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 ...
4
votes
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
votes
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 ...
1
vote
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]=\...
0
votes
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 ($\Lambda,a)$ ...
1
vote
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:$$W^2=-\frac{1}{2}M_{\mu\nu}M^{\mu\nu}P_{\alpha}P^{\alpha}+M^{\mu\nu}M_{\...
0
votes
1answer
69 views
Where to learn about Poincaré Group properties?
I am studying my first QFT course, and there seems to be a lot that I was not taught in previous courses.
In my first assignment, I have to prove several properties about the Poincare group, but I ...
2
votes
2answers
113 views
Why physics should be the same in all inertial frame? [closed]
One of the postulates of special relativity is that physics should be the same in all inertial frame.
Suppose we have two observers $A$ and $B$ suppose that $A$ is accelerated.
Now suppose that we ...
1
vote
0answers
36 views
Do gauge fields not transform like functions of the coordinates under translations?
By "transform like a function of the coordinates," I mean that under an infinitesimal translation $x^\mu \to x^\mu + \epsilon^\mu$, to first order in $\epsilon^\mu$ the function $f(t,\mathbf x)$ ...
4
votes
1answer
91 views
Transformation of the derivative of the scalar field in Ramond's book about QFT
In the book by Pierre Ramond about quantum field theory, he explores in chapter 1.4 (p.13) the behavior of fields under Poincaré transformations. He starts by explaining that infinitesimal ...
1
vote
0answers
229 views
Significance of the Little group
My current understanding of the Little group is that it is the symmetry of a given state in the Fock space. This means that given a massive or massless particle in n dimensions, I can tell the number ...
1
vote
2answers
51 views
Lorentz boost tensor notation confusion
I have been given this$$
\delta X^{\mu}=\omega_{\mu \nu}\left(M^{\mu \mu}\right)_{\sigma}^{\rho} X^{\sigma}
$$
and I think it should be equal to this but I'm confused if I'm doing it correctly
$$
\...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
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) ...
1
vote
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 ...
0
votes
0answers
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 (...
2
votes
0answers
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 ...
0
votes
2answers
52 views
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_{...
2
votes
0answers
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 ...
0
votes
0answers
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 ?
3
votes
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 ...
5
votes
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 ...
2
votes
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 ...
3
votes
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 ...
2
votes
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$, ...
1
vote
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 ...
0
votes
0answers
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 ...
2
votes
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 ...
0
votes
0answers
73 views
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{...
1
vote
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 $\...
2
votes
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}...
3
votes
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 ...
2
votes
0answers
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 ...
7
votes
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,
$$
where $A=A^a T_a$ is a Lie algebra valued one-form known as the connection ...
1
vote
1answer
223 views
Physical meaning of the Casimir operators of Poincarè algebra
If one considers the algebra $su(2)$, it is well known that the Casimir Operator is
$$
C=L_1^2+L_2^2+L_3^2.
$$
It corresponds to the total angular momentum and correctly is a conserved quantity.
...
