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Questions tagged [lorentz-symmetry]

Lorentz symmetry is a fundamental symmetry of [tag:special-relativity] describing the invariance of physics with respect to changes of orientation and boosts of inertial reference frames. These symmetry transformations are called transformations.

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What is the Lorentz group composition of two electrons?

We know that the wavefunction of an electron transforms as Dirac spinor $(1/2,0)⊕(0,1/2)$ under the Lorentz group $SO(3,1) \sim SU(2)\times SU(2).$ Which representations can we form with two ...
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Question about special relativity in Carroll lecture note

In the lecture of General relativity by Carroll, page 7 is written that: Notice the distinction between this situation and that in the Newtonian world; here, it is impossible to say (in a ...
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Space (Lorentz-Fitzgerald) contraction

Lorentz transform can be written, according to wikipedia, as: $$ \begin{align} ct' &= \gamma \left( c t - \beta x \right) \\ x' &= \gamma \left( x - \beta ct \right) \\ y' &= y \...
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Could be possible to build a 4-vector in special relativity whose spatial component was the electric field E?

Hi everyone and sorry for my English. I would like to know if I can build a legitimate 4-vector as $E^\alpha=(E^0,\mathbf{E})$. I'd like you to check if my way is correct. 1- We already know that $\...
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Coordinate transformations

In Einstein's "The Meaning of Relativity" I don't understand the relation between $b_{\nu\alpha}$ in equation (3a) and $\lambda$ that pops up in equation (2b). I understand the fact that there's a ...
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What is the reasoning behind 1 step in Einstein's derivation of the Lorentz Transformation

In Einstein's book "Relativity" there is a wonderful derivation of the Lorentz transformation, requiring no more than high school algebra (pp. 117 - 121). It is quite clear but I do not understand ...
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About $(0,1/2)$ representations

While studying representations of Lorentz group, we get the generators to be $J_{i}$ - rotations and $K_{i}$ - boosts. We define $N_{i}^+$ and $N_{i}^-$ operators and these operators obey the same ...
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Confusion with covariance of the Dirac Equation

I'm studying the covariance of the Dirac equation: I start with the Dirac equation for observer A: $$(i\hbar\gamma^\mu\partial_\mu-mc)\psi(x)=0$$ and since $$\psi=S^{-1}(\Lambda)\psi'(x'),$$ I get: $$...
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Lorentz transformation of electric and magnetic field vs. 4-vector

I have a very general question about Lorentz transformations of electric and magnetic fields vs. 4-vectors . It arised from my previous post. I will describe the difficulty I encountered. ...
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Generators in Field Theory and Derivatives

Let's consider a representation of the multiplicative group $(0,\infty)$ on Minkowski space $\mathbb{R}^4$ by dilations. \begin{align} \rho:(0,\infty)&\rightarrow\text{GL}(\mathbb{R}^4)&\\ a ...
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Gauge invariant and Lorentz invariant in Weinberg's QFT textbook (eq. 8.1.5)

In Weinberg's QFT textbook, using a gauge transformation $$A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu}\epsilon(x)\tag{8.1.3},$$ it has: $$\delta I_{M} = \int d^4 x \frac{\delta I_{M}}{\delta A_{...
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Why do we differentiate a 4 vector with respect to proper time to obtain 4-velocity?

The coordinates of an event in spacetime are given by the 4-vector $(ct, \mathbf{r})$, where $\mathbf{r}$ is the spacial coordinates of the event. This 4-vector can be seen as 4-displacement of a ...
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Gamma factor in special relativity

I try to derive the Lorentz transformation of a Lorenzt transformation frame an inertial frame $O$ to the frame $O'$ of a moving particle at constant speed v. We have four vectors $\textbf{x}'=\Lambda ...
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How to build an antisymmetric selfdual tensor out of two 4-vectors?

In problem C of section 1.4 of Ramon's Field Theory: A Modern Primer, we are asked to build a field bilinear in $\chi_L$ and $\psi_L$, two left-handed weyl spinors, which transforms as the (1,0) ...
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Invariance of coincidence

I am reading the book "Space and Time in Special Relativity" by David Mermin. In chapter 13, at page 128 in my print, he says the following (screenshot): I'm referring specifically to the sentence "...
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What kind of average could give a Lorentz invariant energy-momentum tensor?

The electromagnetic (EM) radiation energy-momentum tensor is of the following shape, in the case of incoherent superposition of EM plane waves (I'm using $c = 1$ to simplify things, and metric ...
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When do Lorentz transformations take straight field lines to straight field lines?

If you look at elementary examples, it seems like a Lorentz transformation takes a field pattern with a lot of straight field lines to another field pattern with a lot of straight lines. Examples: an ...
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(Lorentz etc) invariant vector fields

(Background: I know some but not much differential geometry, hopefully enough to formulate this post.) I want to ask about what physicists mean when they say scalar, vector, etc. The answer in ...
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Clebsch Gordan coefficients and spin sum

I read that question (A,B)-Representation of Lorentz Group: Coefficient functions of fields and Weinberg book. Why $u(a,b)=\frac{C_{ab}}{\sqrt{2m}}$, where $C_{ab}$ is Clebsch-Gordan coefficients,...
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Invariance of the relativistic interval

Imagine we have two events, $E_1, E_2$ in the coordinate systems $K, K'$ (with coordinates $(x,y,z,t),\ (x',y',z',t')$), whilst $K'$ ist moving with the speed $\vec v$ in regard to $K$. From the ...
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How is $\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}$ manifestly Lorentz-Invariant?

When writing integrals that look like $$ \int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}} \ = \int \frac{d^4p}{(2\pi)^4}\ 2\pi\ \delta(p^2+m^2)\Theta(p^0) $$ it is often said ...
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Wigner proof of the non-existence of finite unitary representation of the Lorentz group

I am reading Wigner's paper ”On unitary representations of the inhomogenous Lorentz group” (Annals of Mathematics, Vol. 40, No.1, p. 149) found here: https://www.maths.ed.ac.uk/~jmf/Teaching/Projects/...
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Wigner-Eckard theorem in 3+1 Minkowski dimensions

From this source, I have: I cannot find much (if not any) information online for Wigner-Eckard in 4D, hyperboiloid harmonics etc. And there are many facts just states in this source that I would ...
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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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Theory for free, non-interacting anyons?

This link suggests that one cannot make a free theory out of anyons, because of its Lorentz representation. How exactly does the $SO(2,1)$ representation enforce the $\pm1$ eigenvalues? How can one ...
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How can zero point energy vacuum be Lorentz invariant?

What distribution of electromagnetic radiation is Lorentz invariant? How can radiation look the same regardless of inertial frame? According to Marshall and Boyer a cubic distribution like $ \rho(\...
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Direct Product vs Tensor Product

I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product: ...
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Lack of invariance of relativistic doppler effect for inertial frames?

The equation for the relativistic doppler effect in a medium where waves travel at speed $c_m$ is $$\frac{f_r}{f_s} = \frac{c_m - v_r}{c_m-v_s}\frac{\gamma_r}{\gamma_s}$$ My question is whether this ...
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Why is it important that the combination of charge, parity & time reversal symmetry not violated?

If looking for more particles or decays that violate CP symmetry can explain why there is so few antimatter in the known universe, I guess finding things that violate CPT symmetry might helps clear up ...
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Why do the $\gamma$ matrices behave like vectors (tensors)?

In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\...
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$(1,1)$ representation of $SL(2,\mathbb{C})$

How do you prove that the $(1,1)$ representation of the $SL(2,\mathbb{C})$ group acts on symmetric, traceless tensors of rank 2?
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Is a tetrad equivalent to a local (position-dependent) Lorentz transformation $e_{\mu}^{a}=\Lambda(x)_{\mu}^{a}$?

The equation in the title won't make since till halfway down. Let us suppose we are looking at some nonflat spacetime equipped with a metric $g_{\mu\nu}$. Furthermore our manifold admits a covering ...
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Special relativity: I arrive at a contradiction regarding the Lorentz invariance of certain quantity

I want to show the Lorentz invariance of $d^3 p/E$ (Eq. 8.11 of Mandl-Shaw), where $E$ is the relativistic energy. Peskin-Schroeder gives sort-of, a proof in section 2.3 which I am convinced of. But ...
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What is the transpose of Lorentz transformation under spinor representation?

Let $S$ be the Lorentz transfortmation under spinor representation, and from any quantum field theory textbooks, we know that $$ S^\dagger=\gamma^0S^{-1}\gamma^0 \\ S^{-1}=\gamma^0S^\dagger\gamma^0 $$ ...
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Rindler Coordinates Derivation

In my GR lectures we've derived Rindler coordinates by first showing that the four velocity, which we defined as $$u^{\mu} = (\gamma c, 0, 0, \gamma u),$$ as a function of proper time can be written ...
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Operators, gamma matrices and Lorentz invariance

In class, we have define the following operator: $$\Pi_{\pm} = \frac{1 \pm \gamma^0}{2} \tag1$$ Where, $\gamma^0$ is the usual first gamma matrix in Weyl representation. Applying it to a 4-...
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Why the speed of light is independent of the relative motion of observer? [duplicate]

As we use Lorentz transformation equation to relate velocity of particle measured by observer which is in frame s this frame is in relative motion having some velocity to that particle so Why the ...
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The Lorentz-invariant particle spectrum

Before the question, I need to mention some necessary definitions. The rapidity is defined as: $$y=\frac{1}{2}\ln\frac{E+p_z}{E-p_z}=\frac{1}{2}\ln\frac{1+v_z}{1-v_z}=\tanh^{-1}(v_z)$$ where $v_z=...
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How do gravitational waves agree with Lorentz invariance?

Following is a simple but incorrect explanation for gravitational waves. My question is what is wrong with it? I'd like to say that a gravitational wave is a periodic variation in the local ...
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Non-Euclidean geometry of a rotating cylinder

I am reading Ta-Pei Cheng's book "Relativity, Gravitation and Cosmology" and having some difficulties with question 6.3. It basically asks to work out the spatial distance in a rotating cylinder to ...
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Why is the scalar product of two four-vectors Lorentz-invariant?

Why is the scalar product of two four-vectors Lorentz-invariant? For instance, given two four-vector $A^\mu$ and $B^\mu$, so their scalar product is $A\cdot B=A^\mu B_\mu=A^\mu g_{\mu\nu}B^{\nu}$. ...
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Why do we use the $\left(\tfrac{1}{2}, \tfrac{1}{2}\right)$ rep for spin $1$ particles and not $(0, 1)$? [duplicate]

The spin 1 $A^\mu$ field transforms under the $\left(\tfrac{1}{2}, \tfrac{1}{2}\right)$ representation of the Lorentz field. When restricted to the $SO(3)$ subgroup, it decomposes into the $0 \oplus 1$...
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What causes $A^{\mu\nu}_{\pm}=F^{\mu\nu}\pm i \tilde{F}^{\mu\nu}$ to have three independent components rather than six?

Both the elctromagnetic field strength tensor $F^{\mu\nu}$ and its dual $\tilde{F}^{\mu\nu}$ defined as $\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}F_{\lambda\rho}$ are examples of ...
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Should non-relativistic Navier Stokes Equations be modified so that they become pseudo-Lorentz invariant?

Choking mass flow seems to reflect the fact that fluid momentum density has a maximum value (in stationary conditions) equal to $\rho_* c_*$ where $\rho_*$ is the critical mass density and $c_*$ is ...
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Could there be a pseudovector kinetic term for fermions?

Could there be a kinetic term of the form $\bar{\Psi} \gamma_5 \gamma^\mu \partial_\mu \Psi $ in addition to the usual one? Or is this forbidden by a symmetry?
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Why is $4=3\oplus 1$? What are propagating modes? Etc

In Schwartz's QFT book, he said that the vector representation of the Lorentz group, $V_\mu$ that is four-dimensional, is the direct sum of two irreducible representations of $SO(3)$: a spin-0 ...
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The derivation of the Lorentz transformation: addition of distances

In the derivation of the Lorentz transformation, one has a reference frame, $S$, at rest and another, $S'$, moving away at constant speed $v$. At time $t$ there is an event at a point $x$ in $S$. The ...
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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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Lorentz invariance from Dirac spinor

I have a really naive question that I didn't manage to explain to myself. If I consider SUSY theory without R-parity conservation there exist an operator that mediates proton decay. This operator is $...
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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 ...