Questions tagged [poincare-symmetry]
The poincare-symmetry tag has no usage guidance.
310
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Question about proof of Weinberg-Witten theorem
In proving the Weinberg-Witten theorem, there is a step where one needs to show
\begin{align*}
\lim_{k' \to k}\langle k, \sigma | J^{\mu} |k', \sigma \rangle &= \frac{q k^{\mu}}{k^0}\frac{1}{(...
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What's the definition of spin for a particle in $d$-dimensional Minkowski spacetime?
Consider a relativistic quantum theory in d-dimensional flat spacetime. Neglecting possible internal symmetries, a particle is defined as a system whose Hilbert space furnishes the support of an ...
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Coordinate Transformation rule of QFT field Vs. Differential geometry
I am taking my first QFT course and this has bugged me for an entire semester. Also while browsing, it looks to me that really no one in this website or any textbooks have given consistent answers.
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Why do we need to consider the full Poincare group to get unitary representations?
I am trying to study and understand QFT from the perspective of symmetries. I was referred to this super helpful answer by @ACuriousMind : https://physics.stackexchange.com/a/174908/50583. I still ...
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Some question about the irreducible representation of Poincare group
I am writing a note about the Poincare group and I am trying to explain that argument that one-particle state transforms under irreducible unitary representations of the Poincare group. However, there ...
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Does the generator of boosts depend on the Hamiltonian?
I feel like the generator of boosts on the Fock space depends on the Hamiltonian. I have the following argument for this :
We take the classical field theory on the phase space, and take the initial ...
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How do Poincare group act on Classical field, Quantum field operator, Field configuration states, Fock space states?
I will try to make each of my statement as clear as possible, if any of the statements are wrong prior to my question, please point them out:)
For simplicity, we work in free QFT with scalar field.
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What is the single-particle Hilbert space in the Fock space of QFT?
In Quantum field theory, the fields are operator-valued functions of spacetime. So for a scalar (spin $0$) field $$\psi: \mathbb{R}^{3,1} \rightarrow O(F),$$ where $O(F)$ is the space of operators on ...
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Deriving the gauge group from the little group
Arguments from the "little group" are used to show that the internal degrees of freedom of a massive particle transform under $SO(3)$, while the internal degrees of freedom of massless ...
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Poincare invariance of the vacuum
When quantizing the free scalar field, we define positive frequency modes according to
$$\frac{\partial}{\partial t} \phi_{\omega}=-i \omega \phi_{\omega}. $$
In the mode expansion we then separate ...
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General element of the Poincare group
Consider the generators of the Poincare group are $M^{\mu\nu}$ and $P^\mu$. The Lorentz group elements are $g(\omega)\in SO(1,3)=e^{-\frac{i}{2}\omega_{\mu\nu}M^{\mu\nu}}$ and the translation group ...
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Coadjoint orbit method (geometric quantization) for second quantisation of free relativistic particles
The quantum theory for relativistic particles can be obtained by constructing the (projective) unitary irreducible representations associated to the integral coadjoint orbits of the Poincare group.
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Doubt in Poincare Algebra
So, I have been reading Lecture notes on "Supersymmetry and Extra Dimensions" (PDF), taken by Flip Tanedo (notes of the course of SUSY and Extra Dimension taken by Professor Quevedo, ...
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Lorentz generators on Fock space
Consider a free massive relativistic scalar field in $d+1$ dimensions. Its Hilbert space can be taken to be the bosonic Fock space on $\mathfrak h = L^2(\mathbb R^d)$:
$$\mathcal F = \bigoplus_{n=0}^{+...
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Why must Lie superalgebras always contain both $Q$ and $\bar{Q}$?
In four dimensions Lie superalgebras naturally arise by relaxing the presupposition in the Coleman-Mandula theorem that the symmetry is not classified by a Lie algebra. It is then typically stated ...
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Poincaré group representation generator commutators
I am currently trying to understand how to derive the commutation relations for the generators of the Poincaré group. What I am reading instructs to start with:
$$U( \Lambda, a) = e^{\frac{i}{2} \...
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Do generators of translations transform as *covariant* vectors under a homogeneous Lorentz transformation?
Using the composition law of Poincaré transformations, it is easy to see (cf. e.g. Ref. 1 this answer) that under a Lorentz transformation
$$\underbrace{U(\Lambda,0)P^\mu U(\Lambda,0)^{-1}}_{P'^{\mu}}=...
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Computation involving Pauli-Lubanski vector
I am trying to check that the "1" component of the Pauli-Lubanski vector for a massless particle with $P^{\mu} = (E, 0, 0, E)^{\mu}$ is $E(-J_1 + K_2)$, but I keep getting $E(-J_1 - K_2)$.
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Why the Pauli-Lubansky and momentum operator build an irreducible representation of Poincarè group?
We know that particle states in QFT are identified with irreducible representation of Poincarè group, in particular they can be identified using Pauli-Lubansky and (squared) Momentum operator (wich ...
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What are the discrete representations of the Poincare group, analogous to the $(j_1, j_2)$ representations of the Lorentz group?
The finite dimensional representations of the Lorentz group are given by two half-integers $(j_1, j_2)$. One can break up the Lorentz Lie algebra into two $\mathfrak{so}(3)$ Lie algebras, and the half-...
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Is the defining property, of the Quantum theory Hilbert space, the Heisenberg algebra or the Poincare algebra?
In non relativistic quantum mechanics, we state the Heisenberg algebra $[X, P]=i\hbar$ as one of the postulates. The rotation and translation algebra is discussed later, after we've already defined ...
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Pauli-Lubasnki pseudovector on tensor field
We had the infinitesimal Poincaré transformations in the lecture with the Pauli-Lubanski vector:
$$W^{\mu}=\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}P_{\nu}M_{\rho\sigma}$$
with:
$$P_{\nu}=-\partial_{\...
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What is the difference between Poincare symmetry and coordinate independence in field theory?
I know that there several questions that deal with this question but I‘ve found no satisfactory answer. In QFT we want that a scalar field is invariant under Poincare transformations $\mathcal{P}$ ...
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Inonu-Wigner contraction in Weinberg Volume I
In volume I of Weinberg quantum theory of fields, on page 61, Weinberg derived the commutation relations of the generators $H,P_i, J_i,K_i$ of the Poincare algebra, then he tried to take the ...
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Breaking conformal symmetry to Poincare symmetry
If I have a free massless scalar field $\phi$ in (3+1)D Minkowski spacetime, the vacuum state is not only Poincare-invariant, but actually also conformally invariant. Is there a way to break the ...
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Weinberg–Witten theorem in other dimensions
The Weinberg–Witten theorem was proven in $3+1$ spacetime dimensions. Has anyone extended this to arbitrary dimensions? In other words, the theorem in its form as given in the above link, with the ...
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Poincaré with spontaneously broken translations
What is the physical interpretation of Poincaré symmetries with spontaneously broken spatial and temporal translations?
Is there an interesting low-energy effective model for it and what are its ...
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Is Newton’s third law of motion formed from Poincare symmetries?
So I know that Newton's third law states that every action has an equal reaction, making a symmetry. But just like how Poincare symmetries form conservation laws, do any Poincare symmetries form ...
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In the Poincaré group, what are explicit representations of translations, boosts, and rotations?
Context
In [1], cowlicks asks, ``How can the Gallilean transformations form a group?''. In [1] Selene Routely explains that the Galilean transformations form a group of dimension 10. Routely explains ...
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What we talk about when we talk about Lorentz transformations?
Context
In [1], cowlicks asks the question, ``How can the Gallilean transformations form a group?'' It is clear what a group is. Borrowing liberally from [2],
"A group is a set $G$ together with ...
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A question about the Weinberg QFT Vol.1 Section2.4
I'm self-studying Weinberg QFT, and I'm confused about the connection between the momentum operator and the generator of translations
In Section 2.4, Weinberg shows the Lie algebra of Poincare Group,
\...
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Why is spacetime in the adjoint representation? [closed]
Given the fermionic dimensions, it seems like naively speaking that real spacetime dimensions transform in the adjoint representation of the fermionic dimensions' generators which is a 3+1 dimensional ...
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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:
$$iC_{\mu\nu}(x-y) = \langle {\tilde{0}}|T\left[V_\mu(x)V_{\nu}^\dagger(y)\...
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Does the location of the Hilbert space of momentum eigenstates in QFT change under time translations and boosts?
I have two questions concerning Wigner's transformation law for irreps of the Poincare group:
\begin{equation}
U[\Lambda,\vec{a}]\vert p,\sigma\rangle=e^{ip\cdot a}D_{\sigma'\sigma}[\Lambda;p]\vert \...
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Spin as Poincaré invariant label
I was thinking about how we construct unitary representations for the Poincaré group in the case of massive particles. We move to a frame where the particle is at rest, and here the little group that ...
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Space-Time Symmetries and Scaling
Typically, when a course examines symmetries of space-time and their consequences (e.g. symmetries of Lagrangians and conserved quantities), either the Lorentz group or the Poincaré group are ...
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Wigner's classification
Could someone offer a very clear explanation of Wigner's classification of particles as infinite-dimensional unitary representations of the Poincare group?
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Spin generator of massless fields
How can I derive an expression for the spin generator $S^{\mu\nu}$ of massless fields (preferably fermions) and show, in this way, that the $S^{0i},\ i=1,2,3$ components of the spin generator vanish??
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Representation of Poincaré group and quantum field
How can we understand the quantum field $$\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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