Questions tagged [poincare-symmetry]

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Representation of The Poincare 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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Weyl Spinor Representation and Single Particle States

I'm trying to study representation theory for quantum field theory. Let me first summarize my current state of (hopefully correct, please correct me if I'm wrong about something) knowledge: Single ...
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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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Generators of the Poincare algebra

I am currently reading the book "Conformal Field Theory" by Francesco, Mathieu and Senechal. In chapter 4.2.1, in order to determine the form of the generators in the conformal group, they ...
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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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Lorentz or restricted Lorentz group?

We say (or we observe empirically) that the laws of physics are Lorentz invariant, i.e their form does not change under transformations of the Lorentz group. The weak interactions are not invariant ...
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168 views

Proof of Poincaré algebra with Poisson bracket

I'm trying to prove that the generators of Poincaré group in Poisson bracket close the well-known Poincaré algebra. In particular, I don't know how to prove that $\{P_\mu,P_\nu\}=0$. Let's take an ...
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Are particles in curved spacetime still classified by irreducible representations of the Poincare group?

For QFT in Minkowski space, the usual story is that particles lie in irreps of the Poincare group. Wigner's classification labels particles by their momentum and by their transformation properties ...
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Poisson-Bracket representation of the Poincaré group and symmetries of dynamical systems

In canonical formalism we know that a symmetry for the dynamical system can be expressed by $\{H,f\}=0$, where $H$ is the hamiltonian of the system and $f$ is the smooth function associated to the ...
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Do 2d CFTs define healthy 4d QFTs?

When doing 2d CFTs we typically complexify coordinates and formally consider $\mathbb C^2$, with the understanding that, in the end, we are to restrict to the real slice $\bar z=z^*$. If we do not ...
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Invariance of inner product under Poincare transformation

The Poincare transformation reads, $$x\rightarrow x^\prime=\Lambda x +a $$ The scalar product is preserved under Lorentz transformation. However I do not see how it is preserved under the more general ...
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Does potential energy break Poincare invariance down to invariance under Lorentz boosts?

Landau and Lifshitz (Mechanics, Vol. 1) derive the form of the Lagrangian for a free particle by requiring invariance of action under Galilean transformation and assuming homogeneity and isotropy of ...
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Relations between the spin of representations of Lorentz group and Poincare group

It is known that Finite dimensional irreducible representations of Lorentz group can be indexed by two half integers $(s_1,s_2)$ and the sum $s_1+s_2$ is called the spin. Infinite dimensional unitary ...
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What does it mean for particles to “be” the irreducible unitary representations of the Poincare group?

I am studying QFT. My question is as the title says. I have read Weinberg and Schwartz about this topic and I am still confused. I do understand the meanings of the words "Poincaré group", &...
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Mass as a coupling and mass as a Casimir operator

In Poincare group, we consider mass as a Casimir of the group. Hence it is a constant in various frames (I do not mean old fashion Lorentz transformation). But, in the quantum field theory mass is the ...
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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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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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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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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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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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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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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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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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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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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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