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Restoring Poincaré symmetries in Hamiltonian lattice field theories

I can imagine how the continuum limit of a non-relativistic quantum field theory discretized on a spatial lattice restores the Galilean symmetries of the original theory. But how does this work for ...
mavzolej's user avatar
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Poincaré algebra and supersymmetric spaces

If i understand correctly, a supersymmetry algebra should contain as a subalgebra the Poincaré algebra, however for a supersymmetry algebra the corresponding supersymmetric (Minkowski) space has ...
Tomás's user avatar
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3 answers
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Physical interpretation of reducible but indecomposable reps of the Poincaré group?

As I understand it, there are representations (reps) of the Poincaré group that are reducible but still indecomposable (i.e., cannot be expressed as a direct sum of two subreps). This would be ...
WillG's user avatar
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1 vote
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Why every projective irreducible representation of $(ISO(2,1)^{\uparrow}=SO(2,1)^{\uparrow}\ltimes \mathbb{R}^3)$ is equivalent?

Why every projective irreducible representation of the connected poincare group $(ISO(2,1)^{\uparrow}=SO(2,1)^{\uparrow}\ltimes \mathbb{R}^3)$ is equivalent to a projective irreducible representation ...
Mahtab's user avatar
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The establishment of Out-of-Time-Ordered Correlators (OTOCs) from Lyapunov Exponents (LEs)?

the OTOC in quantum system is $$F(t) = \langle \hat{W}^\dagger(t) \hat{V}^\dagger(0) \hat{W}(t) \hat{V}(0) \rangle_{\beta} $$ the Lyapunov exponent is $$\lambda = \lim_{{t \to \infty}} \lim_{{d(0) \to ...
Sara's user avatar
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Why $\sigma$ wouldn't change under standard Lorentz transformation in Weinberg QFT V1 Ch2

Weinberg defines one particle via state vector $\Psi_{p,\sigma}$ that is eigenvector of four-momentum and a label $\sigma$ to denote all other degrees of freedom. This is physically understandable, ...
Ting-Kai Hsu's user avatar
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1 answer
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Is there a systematic way to construct the parity and charge conjugation operator for any Poincaré irreducible representation?

I am currently taking an undergraduate introductory QFT course. However, the proceeding will be about classical field theory, the results of which I assume will carry over mutatis mutandis into ...
Silly Goose's user avatar
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2 answers
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Confusion with Weinberg's QFT book, Volume I, Equation 2.5.3 (one-particle states as irreps of Poincare group)

I am reading Sec. 2.5 of Weinberg's Quantum Theory of Fields, Volume I. There he talks about the classification of relativistic one-particle states according to their transformation under the Poincare ...
Solidification's user avatar
1 vote
0 answers
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Poincaré group conservation laws: 10 of 7? [duplicate]

According to the Wikipedia page about the Poincaré group, we get 10 conservation laws using Noethers theorem. 10 generators (in four spacetime dimensions) associated with the Poincaré symmetry, by ...
Riemann's user avatar
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Projective representation in Weinberg's QFT book

Weinberg's QFT vol 1 has excellent discussion on symmetries and projective representations. Because physical states are represented by rays in the Hilbert space, symmetries are realized as projective ...
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Why representations? [duplicate]

I've been studying Talagrand's What is a Quantum Field Theory? lately and I have some questions regarding the scheme he presents. Essentially the state of affairs as of where I am in the book is that ...
Lourenco Entrudo's user avatar
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1 answer
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Poincaré invariance and uniqueness of vacuum state

I'm trying to understand the Poincaré invariance of the vacuum state in Minkowski spacetime, how it implies the uniqueness of the vacuum state, and why there's no unique vacuum state in general ...
Samuel Jaramillo's user avatar
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1 answer
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Unitary representation of physics groups in Wu-Ki Tung book

I am reading the Wu-Ki Tung book "Group theory in physics" and I'm trying to put the various pieces (chapter) together to understand how he gets the unitary irreducible representations of ...
Andrea's user avatar
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6 answers
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What exactly is a quantum field?

As I understand it, a relativistic quantum field is an operator-valued function of spacetime that transforms under some finite dimensional irreducible representation of the Lorentz* group: \begin{...
QFTheorist's user avatar
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1 answer
126 views

In what sense are fields representations of the Poincare group?

As far as I know, a representation is a homomorphism from the group to a vector space $V$ which preserves the group multiplication, i.e., if $(\pi,V)$ is a representation of the group $G$, then ...
QFTheorist's user avatar
4 votes
1 answer
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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}{(...
Gleeson's user avatar
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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 ...
Jack Euler's user avatar
1 vote
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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. ...
Kuro_'s user avatar
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0 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 ...
qavidfostertollace's user avatar
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1 answer
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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 ...
David Shaw's user avatar
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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. ...
Eren's user avatar
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3 votes
0 answers
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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 ...
Flo's user avatar
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1 answer
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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 ...
Panopticon's user avatar
1 vote
1 answer
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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 ...
korni1990's user avatar
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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 ...
Tanmoy Pati's user avatar
4 votes
0 answers
128 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
180 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
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5 votes
1 answer
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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}^{+...
SolubleFish's user avatar
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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 ...
Wintermute's user avatar
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1 answer
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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} \...
user132849's user avatar
2 votes
1 answer
227 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
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1 answer
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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 $E(-J_1 - K_2)$. ...
Gleeson's user avatar
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5 votes
3 answers
371 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
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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-...
user1379857's user avatar
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2 votes
2 answers
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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 ...
Ryder Rude's user avatar
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2 votes
0 answers
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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_{\...
Ayz's user avatar
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4 votes
1 answer
264 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
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1 answer
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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 ...
Simplyorange's user avatar
2 votes
0 answers
68 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 ...
Evangeline A. K. McDowell's user avatar
5 votes
0 answers
104 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
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0 answers
61 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
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1 vote
1 answer
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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 ...
Jungwoon Song's user avatar
2 votes
1 answer
465 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
272 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
214 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
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2 votes
0 answers
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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 ...
Bastam Tajik's user avatar
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3 votes
1 answer
274 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
123 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
60 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
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1 vote
0 answers
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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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