Questions tagged [poincare-symmetry]

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Infinite dimensional representation of generators of the Lorentz group

I was reading Schwichtenberg's "Physics from Symmetry 2nd Edition". In Section 3.7.11, there is the discussion on the infinite dimensional representations. If we consider a transformation of ...
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1answer
60 views

Why the little group of a massless spin-1 particle is ISO(2) rather than SO(2)?

As the title suggests, after going through a lot of Wikipedia, and references of books, I have learned that: The little group of a massive spin-1 particle is SO(3), while the little group of a ...
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47 views

Commutator of tetrad and Lorentz generators

While reading "Four lectures on Poincaré gauge field theory" (available at RG) the authors present a relationship between a tetrad $e^i_{\;\gamma}$ (with Latin indices coordinates, Greek ...
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Commutator of gauge Poincaré group

I'm reading along with Hehl, von der Heyde, and Kerlick's General relativity with spin and torsion on gauging the Poincaré group to include spin into GR. They introduce the covariant derivative $$D_\...
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40 views

Why does the Hamiltonian density transform as a scalar field and not as the zeroth component of a vector field?

Page 191 of Weinberg Vol.1. Break the Hamiltonian operator up into two parts $H=H_0+V$, where $H_0$ is the free Hamiltonian and $V$ is the interaction. Write the interaction operator $V(t)$ as a local ...
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179 views

Which particles do not fit into Wigner's picture?

In his accepted and highly upvoted answer to Why particles are thought as irreducible representation in plain English? @Valter Moretti finishes his ADDENDUM with "Finally not all particles fit ...
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153 views

Is there a formal distinction between Lorentz boosts and the others types of transformations of the Poincaré group?

The doubt arises as space translations can be associated to homogeneity of space, time translations to homogeneity of time and classical rotations to isotropy of space. These properties of space leads ...
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1answer
48 views

How many elementary particles are predicted by Wigner's classification?

My understanding is that when it comes to the correspondence between representation theory and particle physics, every irreducible representation of the Poincare group has a corresponding fundamental ...
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120 views

How is it possible for quantum fields in different irreps of the Poincaré group to interact?

In QFT, elementary particles correspond to irreps of the universal cover of the Poincaré group, and the full quantum field is then the direct sum of fields living in different irreps. So the unitary $...
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138 views

Inönü-Wigner contraction of Poincaré $\oplus$ $\mathfrak{u}$(1)

Metric = (-+++), complex $i$'s are ignored. Using the following decompositions of the Poincaré generators, I can write the Poincaré algebra as I can get the Galilei algebra using the following ...
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Masses at finite density

I am reading this paper, where they make the statement: Second, and less importantly, given our high-energy upbringing, we might be tempted to refer to an excitation’s zero-momentum gap as its “...
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1answer
60 views

Commutativity and Associativity of Poincare Transformations

Commutativity and Associativity of Poincare Transformations: For commutativity I showed that $2$ successive transformations does not commute with the same transformations reversed. $$(Λ_2 Λ_1, Λ_2 ...
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Representation of the Poincaré group by means of exponential

Let $(\Lambda,a)$ be a Poincaré coordinate transformation. Let $U$ be an unitary representation of the Poincaré group on some vector space. Is it always possible to express $U(\Lambda,a)$ in the ...
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What is CPT, really?

The naive statement for the "CPT theorem" one usually finds in the literature is "relativistic theories should be CPT invariant". It is clear that this statement is not true as written, e.g. ...
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63 views

Does $\partial^\mu X^\nu = \partial_\mu X_\nu$ (Neumann Boundary Conditions)?

Problem I am trying to prove that the Neumann boundary condition , $\partial_\sigma X_\mu=0$ , implies that no momentum flows out of the end of an open string. I'm told that the associated conserved ...
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586 views

Why does having a representation of the Poincaré algebra imply conservation of energy, momentum and angular momentum?

Considering that Poincaré algebra is given by the following relations $$i[J^{\mu\nu},J^{\rho\sigma}]=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}-\eta^{\sigma\mu}J^{\rho\nu}+\eta^{\sigma\...
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37 views

Finite Poincare transformation in momentum space?

In position space, a finite Poincare transformation changes the coordinate four-vector $x^\mu$ as $$x^\mu\to x'^\mu=\Lambda^\mu{}_\nu x^\nu+a^\mu$$ where $\Lambda^\mu{}_\nu$ is a Lorentz ...
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67 views

Relationship between the Klein-Gordon equation and Poincaré invariance

Derivations of the Klein-Gordon equation such as the one given by Phoenix in here, are based on studying the wave equation of the wave function of a relativistic particle. In this case, the Klein-...
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112 views

Does Fabri-Picasso theorem imply non-conservation of charge?

The vacuum expectation value of the square of the Noether charge operator $$\langle 0|\hat{Q}^2|0\rangle=\int_{\rm all space} d^3\textbf{x}\langle0|\hat{j}_0(0)\hat{Q}|0\rangle$$ diverges in case of a ...
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83 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_\...
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55 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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53 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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37 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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43 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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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 ...
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240 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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82 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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46 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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46 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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31 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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114 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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52 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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107 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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291 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/...
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151 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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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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146 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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150 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
74 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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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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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
129 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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490 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
56 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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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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77 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
74 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
93 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
169 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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228 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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