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Questions tagged [poincare-symmetry]

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Massive chiral fermions

The main question is: why does nobody care about massive chiral fermions? It is well-known that in QFT (in the axiomatic framework of Garding & Wightman) the quantum field transform according (...
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Understanding the idea behind the super-Poincaré algebra

On the Super-poincaré algebra wiki page (https://en.m.wikipedia.org/wiki/Super-Poincaré_algebra), it says: "If Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry ...
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Is the antisymmetrisation of $a^{\mu}b^{\nu}\epsilon_{\mu\nu}= a^{[\mu}b^{\nu]}\epsilon_{\mu\nu}$ with antisymmetric tensor $\epsilon$ mandatory?

When in tensor algebra the product of 2 vectors with a antisymmetric tensor appear, is antisymmetrisation compulsory ? Given an antisymmetric tensor $\epsilon_{\mu\nu}$, is $$a^{\mu}b^{\nu}\epsilon_{...
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How is deSitter group tranformations different from poincare group transformations

In QFT, we have studies Poincare group of massive and massless particles. Is the deSitter group also useful to study such things? What exactly is the main role of this group in QFT? I just know the ...
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Check that the Poincaré's transformations form a group structure

How can I answer to this question ? I know that this is a Lorentz transformation + a translation but I don't know how to start. What's the difference between group/group structure ?
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How is Inönü-wigner contraction done?

I have read that little group for the massive particles is $SO(3)$ and for the massless particles is $E(2)$ in 4 dimensions. How does one take zero mass limits for the representations and show that it ...
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Poincare transformations and “three kinds of infinitesimal variations”

I'm currently reading these$^1$ lec. notes as an introduction to relativistic QFT. In chapter two (pp.57-61) he introduces the concept of field variations along with some formulas for the different ...
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Poincaré invariance of linearly polarised plane wave

I am reading a book that just quotes the Lie group generators and the discrete subgroups that leave a linearly polarised plane wave unchanged. And I have no idea how to derive them. Context The ...
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Doubts on representations of poincare group and QFT

I am studying Poincare group and encountered the term massless representations of the Poincare group. I know Poincare group is studied by the studying the little group of various momenta, massless and ...
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3answers
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Lorentz transformation of vector field

Under a Lorentz transformation, a vector field transforms as: $A'_{\mu}(x')=\Lambda^{\nu}_{\mu}A_{\nu}(x)$ My question is, why is the Lorentz transformed vector field evaluated at $x'=\Lambda x$, ...
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1answer
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Poincare group and free theories

How exactly is the Poincare group related to the free relativistic theories in quantum field theory? I know Poincare group is the Lorentz group along with translations but don't see any connected why ...
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Variation of vector field under Lorentz transformation and gauge transformation

In a paper I am reading, it is stated that under a Lorentz transformation, the coordinates transform as $x^{\mu} \to \Lambda^{\mu}_{\nu}x^{\nu}$, and so the change in the (vector) field at the same ...
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Derivatives in Poincare' gauge theory

I have been reading the lectures: http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_Supergravity.pdf about Poincare' gauge theory. The Poincare' group is considered as semidirect ...
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How to write the Poincare transformation for an arbitary path in Minkowski space?

So lets Say for arguments sake we have some vector $V^{a}$ and we drag it along some path $\gamma_{1}$ in Minkowski space $R^{3,1}$. For a straight path (represented by a vector $\Delta\overrightarrow{...
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Poincaré and Galilei group - notation

On this slide it just says that $\mathcal{P}$ and $\mathcal{G}$ are the Poincoré and Galilei groups, but I do not understand what they are made of. What does $\mathbb{R}^{1,3}$ mean? Why does $\...
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Doubt in Weinberg's book on Quantum Field Theory

In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as: \begin{equation} U(1+\omega,\epsilon)=1+\dfrac{1}{2}...
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What sort of particles corresponds to the $(1,1/2)$ representation of the Lorentz group?

Every irreducible massive unitary representation of the Poincaré group is specified by a mass and a non-negative half integer spin. Every massless irreducible unitary representation of the Poincaré ...
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Gravity from algebra

Can someone provide me a reference that describes the gauging of the Poincare algebra to obtain Einstein's relativity? "It is well known that Einstein’s formulation of gravity can be obtained by ...
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GR as a gauge theory: there's a Lorentz-valued spin connection, but what about a translation-valued connection?

Given an internal symmetry group, we gauge it by promoting the exterior derivative to its covariant version: $$ D = d+A, $$ where $A=A^a T_a$ is a Lie algebra valued one-form known as the connection ...
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Fractal discrete networks approximations and Poincaré invariance?

In the link below one may find interesting paper by Sabine Hossenfelder, about finite networks symmetry due to Poincaré group. According to this thesis locally finite networks cannot be Poincaré ...
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Physical meaning of the Casimir operators of Poincarè algebra

If one considers the algebra $su(2)$, it is well known that the Casimir Operator is $$ C=L_1^2+L_2^2+L_3^2. $$ It corresponds to the total angular momentum and correctly is a conserved quantity. ...
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What about the (1, 1/2), or (3/2, 1/2) representations of the Lorentz group?

All irreducible finite dimensional complex representations of the Lorentz group can be specified by two positive half-integers, i.e. $(j_1, j_2)$. The $(0,0)$ representation is the trivial scalar ...
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Why do we want supersymmetry transformations to form a group?

I am getting introduced into supersymmetry reading Ryder "Quantum Field Theory". I have taken an introductory course on QFT last semester so I am far from being an expert. Sorry if this is a silly ...
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1answer
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Is there a type of supersymmetry where supercharges have spin 3/2?

Thinking of supersymmetry operators $Q$, they mix fields with a certain spin with fields with spin $1/2$ higher or lower. Thinking of open bosonic strings from string theory, the different modes are ...
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Why do we require that all fields are scalars under spacetime translations?

In his book A Modern Introduction to Quantum Field Theory, Michele Maggiore affirms that We require that all fields, independently of their transformation properties under the Lorentz group, behave ...
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More than one Poincare-invariant vacuum in case of spontaneous symmetry breaking?

It is said in axiomatic quantum field theory approach that Poincare-invariant vacuum is unique. However, in case of spontaneous symmetry breaking - such as Mexican hat example - it seems that more ...
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Wigner Classification via the orbit structure of the Lorentz group

I have been reading and writing a lot about the Wigner Classification of irreducible unitaries of (the universal cover of) the Poincaré group lately, both from a physicist's and a mathematician's ...
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Does Haag's theorem say covariant transformation of interacting field is not possible?

In https://www.physicsforums.com/threads/haags-theorem-perturbation-existence-and-qft.177865/#post-1384425 #2 post by meopemuk (Eugene) say that Haag's theorem says: $$U(\Lambda)\Phi(x) U^{-1}(\...
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Can cut-off regularisation cause a Poincaré anomaly?

Momentum cut-off regularisation leads to non-covariant results, i.e., it breaks the Poincaré covariance of the theory. Is there any guarantee that Poincaré covariance is always restored when we remove ...
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What is the mathematical formulation of the universality of physics in spacetime?

Consider a general spacetime manifold $\mathcal{M}$ of a given dimension (usually $D = 4$). I call two physical constraints that should be imposed on any reasonable classical theory of physics : ...
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Is there any feature which distinguishes the Hamiltonian in the Poincare algebra?

The Poincare algebra is defined as \begin{align*} i[J^{\mu\nu},J^{\rho\sigma}]&=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}+\eta^{\sigma\nu}J^{\rho\mu}-\eta^{\sigma\mu}J^{\rho\nu}\\ ...
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Is local Lorentz + diffeomorphism invariance equivalent to full local Poincaré invariance?

Consider classical General Relativity without the torsion field (the affine connection is already assumed to be symmetric from the start). It is well known that this theory is independent of the ...
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Why aren't infinite-dimensional representations of the Poincaré group classified by *two* half-integers?

It is known that to specify a finite-dimensional irreducible representation of the Lorentz group, one needs to specify two half-integers, $(j_1,j_2)$. For instance, the left-handed and right-handed ...
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Is the Rindler vacuum invariant under Poincare symmetries?

More generally, when we quantize fields in the Rindler space and obtain the Fock space of Rindler particles - does that carry a unitary representation of the Poincare symmetries? It should not, ...
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Casimir Operators and the Poincare Group

Following along in QFT (Kaku) he introduces the Casimir Operators (Momentum squared and Pauli-Lubanski) and claims that the eigenvalues of the operators characterize the irreducible representations of ...
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Does $[P_j,B_k]=i(Mc^2)\delta_{jk}$ imply particle number conservation?

From reading Weinberg's Quantum Theory of Fields, Vol. 1, I learnt that for the Galilean group $[P_j,B_k]=i(Mc^2)\delta_{jk}$, and for the Poincare group $[P_j,B_k]=iH\delta_{jk}$ where $P_j$ and $B_k$...
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Definition of particles in Schwartz, Quantum field theory and the standard model

I would like to check if my understanding of the definition of a particle is right. I also have some questions on things I still don't understand. On pages 109-111 of Schwartz, Quantum field theory ...
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Can Poincare representations be embedded in non-standard Lorentz representations?

My impression for how Poincare and Lorentz representations are linked in $3+1$ dimensions is: Assuming positive mass for simplicity, irreducible representations of the Poincare group are indexed by ...
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Is there any non-trivial sense in which the spacetime symmetries in the SM gauge QFT are made local?

I know there is a trivial sense in which they are made local by tensoring them with the internal local gauge symmetries through a direct product as per Coleman-Mandula theorem, but it bothers me a bit ...
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Massive axial field interacting with massive fermions: number of independent components

Assume a model of massive fermion $f$ interacting with an axial boson $A_{\mu}$: $$ \mathcal{L} = -\frac{1}{4}A_{\mu\nu}A^{\mu\nu}+\bar{f}i\gamma_{\mu}\partial^{\mu}f+\bar{f}\gamma_{5}\gamma_{\mu}f A^{...
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How are the two independent states of polarization of photon related to the two helicity states?

(1) In the canonical quantization of the free electromagnetic field, the Coulomb gauge condition $$A^0=0,~~ \nabla\cdot\textbf{A}=0\tag{1}$$ implies that the polarization vector $\epsilon^\mu$ ...
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SUSY: No-Go theorems?

According to the Coleman-Mandula theorem, there is no trivial unification or the Poincaré symmetry and the internal gauge symmetries. However, if we give up the commutator bounds, and we allow for ...
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Poincaré group and BMS symmetry

How we can derive BMS symmetry by Poincaré group?
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1answer
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Lorentz Transformations And Space-time origins

When we restrict our attention to homogeneous Lorentz Transformations, we restrict two observers to have the same space time origin. Yet they could be moving w.r.t each other with a constant velocity. ...
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Difference between dynamical and geometric phase

How can one differentiate between dynamical and geometric phase produced in the time dependent Schrodinger evolution of a quantum system? How we can describe or define these two terms separately?
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What's the Lie group generated only by dilation and Poincaré symmetry?

Given space $\mathbb{R}^{1,d-1}$($d\ge3$), the total conformal group is $SO(d,2)$ generated by $1$-dilation, $d$-translation, $d$-special conformal, $d(d-1)/2$-Lorentz transformation. But we know ...
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Particle definition [duplicate]

I came across this statement in the book "Quantum Field theory and the Standard Model". "A particle can be defined as a set of states that mix only among themselves under Poincare transformations." ...
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Topology of the Poincaré group $\mathcal{P}(1,3)$

This is planned as a Q&A session, hopefully it serves people who seek a mathematical foundation to (relatively) known results in standard textbooks on QM or QFT. Question: What is the topology of ...
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1answer
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Conservation of momentum in scattering process

On page 60 of Quantum Field theory and the standard model from Schwartz, he talks about scattering process with the $S$ matrix. He says: "Since the S-matrix should vanish unless the initial and ...
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Implications of Poincare symmetry: spin and mass

Is it correct for me to say that the symmetry of a quantum system with respect to the Poincare group leads to the concept of mass and spin? The postulates of the special theory of relativity demand ...