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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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The transformations of Lorentz as a general case of the transformations of Galileo

Starting from the transformations of Lorentz, $$ \left\{\begin{aligned} x&=\gamma (x'+\beta ct)\\ y&=y'\\ z&=z\\ ct&=\gamma (ct'+\beta x')\\ \end{aligned}\right. \quad \tag{*}$$ I ...
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How can I prove this relationship between the S-matrix and the Gamma matrices?

For an infinitesimal Lorentz transformation: $$ S(\Lambda)=1+i\epsilon_{\rho\sigma}s^{\rho\sigma} $$ $$ S(\Lambda)^{-1}=1-i\epsilon_{\rho\sigma}s^{\rho\sigma} $$ and apparently if we just plug that ...
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Invariance of $ds^2$ from invariance of all null intervals

Is this linear algebra statement true? Let $\eta= \begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}$. If $x^T (\Lambda^T\eta\Lambda) x$...
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Integral calculation: finding the correct ansatz

I have already asket this question, but I would like to reask it and add some details. Let us consider the following integral $$I^{\mu\nu}=\int_{0}^{\pi}d\theta\,\sin\theta\frac{f^{\mu}f^{\nu}}{1-\cos\...
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How to show that $\overline{\psi} \gamma^{\mu}\partial_{\mu}\psi$ transforms as a scalar? [closed]

I'm having a bit of difficulty with this problem. Here's what I've got: $$\delta(\overline\psi \gamma^{\mu}\partial_{\mu}\gamma) = \delta(\psi^{\dagger} \gamma^0\gamma^{\mu}\partial_{\mu}\gamma)\\ =\...
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Computing correlation functions sandwiched between momentum eigenstates

Suppose I have a free scalar theory, and I want to compute the following correlation function $<p|\phi(\zeta)\phi(0)|p>$ where $|p>$ is a momentum eigenstate in my quantum field theory. My ...
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Geometric derivation of Lorentz boosts

In two dimensions a very nice parametrization of the rotation group is obtained by the following line of arguments: The group of rotations is connected and compact. Therefore the exponential is ...
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Is Lorentz transform a tensor?

I am confused whether Lorentz transform is a tensor or not, since it is a linear transform. If yes how can I verify that?
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What does Lorentz index structure say about a full-fledged correlator?

I have a probably dumb question. Consider the following position space correlation function in a YM-theory (with or without matter fields): $$f_{\mu_1\cdots \mu_n}^{a_1\cdots a_n}(x_1,\ldots,x_n)=\...
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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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Solution to invariance equation when deriving Lorentz transformation

Requiring that the spacetime interval (1 spatial dimension) between the origin and an event $(t,\, x)$ stays constant under a transformation between reference frames: $$c^2t^2-x^2= c^2t'^2- x'^2$$ we ...
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Transformation of $4-$velocity

Notation: a greek index indicates four labels; spacetime coordinates $\mu = (0,1,2,3)$. A latin index indicates three labels; spatial coordinates $i = (1,2,3)$. $$* * *$$ A quantity, to be ...
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Weyl spinor representations and the Lorentz group

I'm currently trying to read up on the Lorentz-group and it's representations. I've found a couple of posts here on stack-exchange that I find helpful and confusing at the same time, so I would be ...
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Invariance of measure under Lorentz transformations without the four dimensional expression

Ticciati's book Quantum Field Theory for Mathematicians states that the invariance of the measure $$\int\frac{\text{d}^3\vec{k}}{2\sqrt{\left\|\vec{k}\right\|^2+\mu^2}}$$ can be shown by direct ...
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Measuring the lorentz transform generators $J$, $K$, and providing evidence that photons have no internal continuous d.o.f

I am reading Weinberg's first QFT book. We looked for (and I suppose found) unitary representations of the Lorentz group: $$U(\Lambda) = 1 - i (\vec{\theta}\cdot\vec{J}-\vec{\eta}\cdot \vec{K})$$ ...
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Are pseudospinors valid or useful?

We all know that in addition to scalars and vectors, there are pseudoscalars and pseudovectors, which have an additional sign flip under parity. These are useful and necessary when constructing ...
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Are the so-called representations of the Lorentz group actually all representations of it?

Fermionic fields change sign under a rotation by $2\pi$. However, in $SO^+\left(1,3\right)$ a rotation by $2\pi$ is the identity. For any representation $R$ of $SO\left(1,3\right)$ then we have $$R\...
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What does $U^{-1}(\Lambda)\phi(\Lambda y) U(\Lambda) = \phi(y)$ physically mean?

In QFT, let $U(\Lambda)$ denote a unitary representation of the Lorentz group. Let $\phi(x^{\mu})$ be scalar field operators in the Hilbert space; in other words: $$U^{-1}(\Lambda)\phi(x^{\mu}) U(\...
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Einstein's Simultaneous Lighning and the Lorentz Transformation

I am reading Einstein's book, "Relativity: the Special and General Theory" translated by Robert W. Lawson, and I have some questions concerning Special Relativity -- specifically, the thought ...
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Are representations of $\text{SL}(2,\mathbb{C})$ indexed by one half-integer or two?

I am very confused by this. In Hall's book on Lie theory, he states that the representations of $\text{sl}(2,\mathbb{C})$ are indexed by a half-integer. This is the usual result for $\text{su}(2)$ in ...
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Spin (helicity) and polarizations of photons: are they secretly related?

Edit Circularly polarized photons have $$\textbf{S}\cdot\hat{\textbf{p}}=\pm \hbar\tag{1}$$ and it also satisfies $$\boldsymbol{\epsilon}\cdot\hat{\textbf{p}}=0\tag{2}$$ where $\textbf{S}$ is the spin,...
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Deriving special relativity free particle Lagrangian using infinitesimal boost?

At the very beginning of Landau and Lifshitz Mechanics they derive the form of the Lagrangian for a free particle in Newtonian mechanics. I want to see how to do the analogous derivation in special ...
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Matrix elements of stress-energy tensor $\langle q | T^{\mu\nu} |q\rangle$ in QFT?

In many QFTs we can define a stress tensor $T^{\mu\nu}$. What is the matrix element of $T^{\mu\nu}$ in momentum eigenstates? For instance, consider $$\langle q | T^{\mu\nu} |q\rangle$$ in QCD, ...
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Derivation of Inverse Lorentz Transformation in Index Notation

To review my special relativity I tried to work out the inverse lorentz transformation explicitly. This led to a lot of confusion; I would like to ask what the issue was with the assumptions I made in ...
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$(1,0)$ representation of $\text{SL}(2,\mathbb{C})$ and selfdual antisymmetric tensors

The $(1,0)$ representation of $\text{SL}(2,\mathbb{C})$ is realized on two indexed symmetric spinors $\psi^{ab}$ transforming like $$D^{(1,0)}(A)\psi^{ab}=\sum_{c,d=1}^2A^a_cA^b_d\psi^{cd}$$ for all $...
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Relativity from a basic assumption

Is it possible to derive Lorentz transformations just by assuming that if two spaceships are moving away from each other with a constant speed, it is impossible for them to tell who is moving, even if ...
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Proving two space-time intervals are equivalent with matrix algebra

η=$ \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $ v=$ \begin{bmatrix} ct\\ x\\ y\\ z \end{...
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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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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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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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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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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 $\...