Questions tagged [poincare-symmetry]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
1 answer
86 views

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}{(...
Gleeson's user avatar
  • 321
3 votes
1 answer
77 views

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 ...
Jack Euler's user avatar
1 vote
0 answers
79 views

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. ...
Kuro_'s user avatar
  • 11
1 vote
0 answers
52 views

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 ...
qavidfostertollace's user avatar
2 votes
1 answer
151 views

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 ...
David Shaw's user avatar
0 votes
0 answers
67 views

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 ...
Ryder Rude's user avatar
  • 6,308
0 votes
0 answers
80 views

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. ...
Eren's user avatar
  • 21
3 votes
0 answers
81 views

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 ...
Flo's user avatar
  • 31
1 vote
1 answer
81 views

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 ...
Panopticon's user avatar
1 vote
1 answer
146 views

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 ...
korni1990's user avatar
  • 329
0 votes
0 answers
39 views

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 ...
Tanmoy Pati's user avatar
4 votes
0 answers
104 views

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. ...
Mike Kiss's user avatar
3 votes
1 answer
167 views

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, ...
Alex's user avatar
  • 85
5 votes
1 answer
199 views

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}^{+...
SolubleFish's user avatar
  • 5,443
1 vote
0 answers
61 views

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 ...
Wintermute's user avatar
2 votes
1 answer
200 views

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} \...
user132849's user avatar
2 votes
1 answer
157 views

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}}=...
Mr. Feynman's user avatar
  • 1,679
0 votes
1 answer
63 views

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 $E(-J_1 - K_2)$. ...
Gleeson's user avatar
  • 321
5 votes
3 answers
317 views

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 ...
Filippo's user avatar
  • 353
1 vote
0 answers
51 views

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-...
user1379857's user avatar
  • 11.4k
2 votes
2 answers
123 views

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 ...
Ryder Rude's user avatar
  • 6,308
2 votes
0 answers
101 views

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_{\...
Ayz's user avatar
  • 41
4 votes
1 answer
200 views

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 $\mathcal{P}$ ...
Silas's user avatar
  • 502
2 votes
1 answer
92 views

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 ...
Simplyorange's user avatar
2 votes
0 answers
64 views

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 ...
Everiana's user avatar
  • 1,598
5 votes
0 answers
85 views

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 ...
Jahn Dorian's user avatar
2 votes
0 answers
58 views

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 ...
ungerade's user avatar
  • 1,344
1 vote
1 answer
46 views

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 ...
Jungwoon Song's user avatar
2 votes
1 answer
331 views

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 ...
Michael Levy's user avatar
3 votes
3 answers
216 views

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 ...
Michael Levy's user avatar
2 votes
1 answer
178 views

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, \...
Vesemir's user avatar
  • 23
2 votes
0 answers
98 views

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 ...
Bastam Tajik's user avatar
  • 1,242
2 votes
1 answer
234 views

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 ...
Megahyttel's user avatar
2 votes
2 answers
110 views

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 ...
2000mg Haigo 's user avatar
3 votes
0 answers
59 views

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 ...
Luke's user avatar
  • 2,210
1 vote
0 answers
53 views

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)\...
lambda's user avatar
  • 21
0 votes
2 answers
75 views

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 \...
Luke's user avatar
  • 2,210
0 votes
1 answer
82 views

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 ...
Marcosko's user avatar
  • 350
0 votes
0 answers
84 views

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 ...
Uroc327's user avatar
  • 153
2 votes
0 answers
112 views

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?
Jasper's user avatar
  • 21
0 votes
1 answer
84 views

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??
schris38's user avatar
  • 3,799
2 votes
0 answers
221 views

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, ...
Display name's user avatar
2 votes
0 answers
34 views

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 ...
Ishan Deo's user avatar
  • 1,558
1 vote
1 answer
341 views

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)]...
user984949's user avatar
0 votes
2 answers
285 views

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 ...
Andrea's user avatar
  • 373
3 votes
1 answer
72 views

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 ...
Stijn's user avatar
  • 128
3 votes
1 answer
171 views

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 ...
William Nelson's user avatar
6 votes
1 answer
279 views

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 ...
Aleksandr Artemev's user avatar
1 vote
1 answer
181 views

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 ...
Real Pattern's user avatar
4 votes
0 answers
157 views

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 ...
Kong Yeu's user avatar

1
2 3 4 5
7