Questions tagged [poincare-symmetry]

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Poincare invariant Lagrangian?

I only see it mentioned that we want Lorentz invariant Lagrangians in quantum field theory, but I would expect that we additionally also need translational invariance, i.e. Poincare invariance. After ...
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Extending Wigner's Classification with Gauge Symmetry

In Wigner's Classification, as far as I understand, one uses unitary irreps. of the Poincare group for treating an elementary particle, since then mass $m$ and helicity $h$ emerge naturally as ...
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Is every unitary representation of the Poincare group a direct sum of Wigner's irreducible representations?

Is Wigner's classification of unitary irreducible representations of the Poincare group [1] sufficient for constructing all unitary representations of the Poincare group by taking direct sums? The ...
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Correlation function of 4-currents on a general QFT

Given $V^\mu(x)$ a 4-current of a general unitary, Poincaré invariant QFT, I need to show that the correlation function: $$iC_{\mu\nu}(x-y) = \langle {\tilde{0}}|T\left[V_\mu(x)V_{\nu}^\dagger(y)\...
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Does the location of the Hilbert space of momentum eigenstates in QFT change under time translations and boosts?

I have two questions concerning Wigner's transformation law for irreps of the Poincare group: \begin{equation} U[\Lambda,\vec{a}]\vert p,\sigma\rangle=e^{ip\cdot a}D_{\sigma'\sigma}[\Lambda;p]\vert \...
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Spin as Poincaré invariant label

I was thinking about how we construct unitary representations for the Poincaré group in the case of massive particles. We move to a frame where the particle is at rest, and here the little group that ...
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Space-Time Symmetries and Scaling

Typically, when a course examines symmetries of space-time and their consequences (e.g. symmetries of Lagrangians and conserved quantities), either the Lorentz group or the Poincaré group are ...
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Proof for any Propagator = Klein-Gordon Green's Function times Spin-Projector

I have seen a sketch for this some time ago, but I'm unable to find it again. Consider a wave function $\phi^{\mu_1\ldots\mu_n}$ who satisfies the eigenvalue equations of the two Casimir operators of ...
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Wigner's classification

Could someone offer a very clear explanation of Wigner's classification of particles as infinite-dimensional unitary representations of the Poincare group?
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Spin generator of massless fields

How can I derive an expression for the spin generator $S^{\mu\nu}$ of massless fields (preferably fermions) and show, in this way, that the $S^{0i},\ i=1,2,3$ components of the spin generator vanish??
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Representation of Poincaré group and quantum field

How can we understand the quantum field $$\phi(x) = \int \frac{d^3p}{(2\pi)^3\sqrt{2E}} ( a_p e^{-ipx}+ a^\dagger_p e^{ipx}) ,$$ where $ a_p$ and $a^\dagger_p$ are creation annihilation operators, ...
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Method of Induced Representations for inhomogeneous $SO^+(p,q)$ group

In Weinberg Chapter $2$, he classifies all the irreducible unitary representations of the inhomogeneous Lorentz group $SO^+(1,3)$ via the method of induced representations. Can a similar analysis be ...
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Commutators in Poincare algebra

Consider the method of induced representations for the Poincare algebra, i.e. given a field $\phi$ (which need not be a scalar field despite its notation), we have the commutator $$[J^{\mu\nu},\phi(0)]...
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Needs of unitary representation for QFT

Weinberg book states the following (pg. 231): "There is no problem in working with non-unitary representations, because the objects we are now concerned with are fields, not wave functions, and ...
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Are there finite dimensional representations of the Poincaré algebra with non-nilpotent momentum generators?

The matrix representations of the Poincaré algebra that I am familiar with, have a nilpotent set of momentum generators: $p^\mu p^\nu = 0$. I am wondering whether it is possible to have finite ...
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Is spectrum of Hamiltonian all you need?

This should be well-known, but I don't seem to know it... Quantum mechanics is defined by a Hamiltonian, and a Hamiltonian (as any Hermitian operator) is determined by its spectrum. Hence, it seems as ...
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Relativistic many-particle dynamics as a field theory subsector

In more than 2 dimensions, the so-called "no-interaction theorem" of Leutwyler (see this article for a proof) states that naive attempts at constructing $n$-particle classical relativistic ...
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What is the relationship between the Galilean group and the Poincaré group?

What is the relationship between the Galilean group and the Poincaré group? Are they siblings within the Lie group? Or does the Poincaré group contain the Galilean group as a subgroup? I'm not so much ...
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Why should the infinite-dimensional representation of Poincare group induced by the unitary representation of little group be unitary?

In Weinberg's Quantum Field Theory (Vol. I, pages 64-67) it is stated that a unitary representation of little group induces a unitary representation of the Poincare group. But I don't understand how ...
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Sign error in representation of angular momentum on fields

Consider a field $\phi$ that belongs to a representation of the Poincaré algebra. We assume that under translations $\phi\mapsto\phi'$, where $\phi'(x')=\phi(x)$. This fixes the action of the momentum ...
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Intuitive understanding of single-particle states in the interacting QFTs

I am trying to understand to which realistic objects one can apply the definition of single-particle states as irreducible representations of the Poincare group. Let me start with a couple of ...
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Wigner classification of particles vs generic Hamiltonian spectrum

Wigner tells us we should associate infinite dimensional unitary irreps of the Poincaré group with particle states. His classification using eigenvectors of the spacetime generators $P^\mu$ and the ...
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What combination of Poincare generators can be used to transform to a uniformly accelerating frame?

Is there a combination of Poincare generators which can be used to transform from an inertial frame in Minkowski space to a uniformly accelerating frame? i.e Are there some co-efficients (probably ...
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The specific derivation of Poincare algebra

In my note it says for Poincare transform $\delta x^\mu=a^\mu+\omega^\mu_\nu x^\nu$: to derive the commutator relation of algebra, we consider 2 consecutive transforms $$\delta_2\delta_1 x^\mu=(\...
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What's the difference between the spin matrix and the generator of the Poincaré algebra?

Searching for articles and research papers on the Poincaré gauge theory, I found https://arxiv.org/abs/gr-qc/0302040 a good three lectures of Blagojevich on the subject. So straight away on the second ...
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Representations of the Poincaré group

I am currently trying to understand the representations of the conformal group. I am following the script by J. D. Qualls. At page 29, the author finds the effect of $L_{\mu\nu}$ by "studying the ...
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The Cartan sub algebra and Killing form of the Poincaré algebra

Doing some studies on Group theory, I worked Frederic Schuller's lectures on youtube where he classifies all semisimple Lie algebras by the Dynkin's diagrams; I should say it was interesting. Trying ...
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Lorentz Transformation of massive Spin-1 fields

First, let me give a summary of the relevant background: In the QFT book by Schwartz, in Chapter 8.2.2, we derive a Lagrangian for a massive Spin-1 field: $$ \mathcal{L}=\frac{1}{2} A_{\mu} \square A_{...
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Little Group of massive particles in moving frame

My understanding of the little group till now was, that we take some standard-momentum and define the little Group as the subgroup of the Poincaré group that leaves this standard-momentum invariant. E....
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Equivalent representations of Poincare group

This question is regarding the paperA wavelet transform for the Poincaré group We have the Poincare group action as $$U(a, \Lambda)\psi(x, t)=\Psi(\Lambda^{-1}(x-a))$$ Then the author defines another ...
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In what meaningful way can we talk about the generators of the super-Poincaré algebra being "spinors" or "4-vectors" because of their indices?

It is often presented without much justification that the generators of the super-Poincaré algebra carry indices that imply they are elements of a representation space of the Poincaré algebra. In ...
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Coleman–Mandula theorem and its assumptions on QFT

The description of the Coleman–Mandula theorem on Wikipedia starts with the following assumptions: Every quantum field theory satisfying the assumptions, Below any mass M, there is only a finite ...
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Translation of generators when derive representations of Poincare group on fields

As stated in section 4.2.1 of Di Francesco's book on conformal field theory. In order to find out representation of Poincare group on fields, we can start by studying the subgroup that leaves the ...
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Spin in Lorentz group and Poincare group

I am currently learning representations of Lorentz group and Poincare group by Harald's Introduction to Supersymmetry Chap.1. I have 2 questions about the definition of spin. Provided that finite ...
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Systematically constructing a Lagrangian with given Poincare representations for particles

In one approach to constructing field theories, we begin with some desired particle content (e.g. a massive spin-1 particle), and then we construct a corresponding classical Lagrangian (e.g. the Proca ...
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Commutation relation of four vectors [closed]

I was trying to prove that: $$[P_\mu, J_{\rho \sigma}] = i(\eta_{\mu \sigma} P_\rho - \eta_{\mu \rho} P_\sigma) $$ $\textbf{Attempt}$ $$\begin{align} [P_\mu, J_{\rho \sigma}] = [P_\mu, x_\rho P_\sigma ...
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Pauli-Lubanski square operator

The Pauli-Lubanski operator is given by: $$ W_\mu = -\frac{1}{2} \epsilon_{\mu \nu \rho \sigma} J^{\nu \rho}P^{\sigma}$$ Thus the square operator is given by: $$ W_\mu W^\mu = W^2 = -\frac{1}{4} \...
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What is the role of the mass Casimir invariant in Galilean and what it's actual role in special relativity? [closed]

What is the role of the (mass) Casimir invariant of the algebra of relativistic symmetries in Galilean and what it's actual role in special relativity?
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Does every Hilbert space contain the generators of the Poincaré algebra?

I'll pose my question in the form of a statement, and you can tell me if there's anything wrong with what I say: Every Hilbert space which describes a system that obeys the "fundamental ...
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No-interaction theorem in classical relativistic mechanics

In classical relativistic Hamiltonian mechanics there is a so-called "no-interaction theorem" (see, for example, this article for a proof). Roughly, it states that if we have an $N$-body ...
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Poincaré group and the Bargmann theorem

In Valter Moretti's book (Spectral Theory and Quantum Mechanics), page 578, it is said that the Poincaré group is semisimple, but Wikipedia says otherwise. Moretti mentions it in order to ensure that ...
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Wightman axiom 2: what kind of representation?

Both in Wikipedia and on page 98 of Streater, Wightman, PCT, Spin and Statistics and all that, the second axiom postulates that a field must transform according to a representation of the Poincaré ...
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How does parity act on relativistic one-particle states?

Please allow me to set the context based on my understanding before I present the question. In quantum field theory, one-particle states are the basis states of the infinite-dimensional unitary ...
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Square of Pauli-Lubanski operator

I am following Ashok Das QFT book (pg. 152-153) on the calculation of the Pauli Lubanski operator. The Pauli Lubanski vector operator is defined as $$W^\mu=\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}P_\nu ...
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Little-group and spinor helicity

I'm struggling a bit to understand the little group in the context of massless momenta and the spinor-helicity formalism. I'll clarify notation and my understanding through a brief recap, and put the ...
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How to define the components of the Poincare group?

I know that the Poincare group/inhomogeneous Lorentz group can be defined as: $$ x^\mu = (t,-x) \\ t \rightarrow t^\prime = \gamma x + \delta t + b^0 \\ x \rightarrow x^\prime = \alpha x + \beta t + b^...
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Casimir of $SO(3)$, $SO(2)$, $IO(1,3)$, $T(4)$

It is known that $SO(3)$, a semisimple group of rank 1, has one Casimir $J^2$, and one can use this information to classify its irreps with the eigenvalues of $J^2$ and $J_3$: $(j,m)$. Now, only $j$ ...
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Question for ${\cal N}=1$ supersymmetry representations

Please see this lecture note: https://arxiv.org/abs/1011.1491. In section "2.2.5 Massless supermultiplet" the author defines a Casimir and says it is zero. How can we confirm it? We take the ...
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On Poincare group’s Casimir operators

We’ve defined Casimir operator for a group as an operator which commutes with all generators of that group. For the Poincare group we’ve found two Casimir operators: $p_\mu p^\mu$ and $W_\mu W^\mu$ ...
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Symmetry and Coordinate invariance

How are spacetime symmetries different from simple general coordinate invariance? Physical laws should be coordinate independent. Are Poincare invariances not simply changing coordinates?
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