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Questions tagged [invariants]

This tag is for questions relating to invariant, a property of a system which remains unchanged under some transformation. In physics, invariance is related to conservation laws.

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On finding a scalar function which leads to a covariant tensor by differentiation

I need to find a scalar (invariant) function $F$ that when differentiated, leads to a covariant tensor function $f$ with one live index as: $$ \frac{\partial F}{\partial \dot x^k} = f_k $$ The ...
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How to find Casimir operator eigenvalues of $SU(N)$? [closed]

The $[f1, f2, f3…fn]$ in the image represent the irreducible representations of $SU[n]$. How to find the irreducible representations of $SU[n]$ that conform to the form $[f1, f2...fn]$. Can you give ...
snow snow's user avatar
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Linearity of Amplitudes and Lorentz Frames

I know that this question may be a bit weird, but I decided to ask. Assume I have an amplitude, say in QED, which depends on a set of four-momenta $\{p_1,\ p_2,\ p_3,\ ...,\ p_n\}$. Further assume ...
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Relation bitween Mandelstam Variables in three-body final state

What is the relation between Mandelstam variables in the three body final state? There are 5 independent Mandelstam variables. What is the relationship between them?
Andrea's user avatar
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Mandelstams for $2\to 2\to 3$ process

2 initial particles become two other particles, from which one decays into 2. Let $p1$ and $p2$ be the momenta of the initial particles, $p3$ the momentum of the decaying particle, $p5$ and p6 the ...
Ville Alanko's user avatar
1 vote
1 answer
69 views

Self-similarity of the diffusion equation

I am going through this book Simulation of Complex Systems. In the chapter on Brownian Dynamics, we considered a "free diffusion" given by the Stochastic differential equation: $$\dot{x}(t)=...
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Followup question to "Invariant symbol, group representation"

There is a 2 year old answer by Cosmas Zachos which is very helpful regarding invariant symbols here. Aside this context, I have never encountered these and thus I have 3 questions: Why is it ...
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Clarification question about definition Lorentz covariance for an arbitrary higher rank tensor

What does it mean for a tensor of rank N to be Lorentz covariant. For a scalar (rank 0) it simply means that it is Lorentz invariant, in other words it remains unchanged under Lorentz transformations. ...
Мікалас Кaрыбутоў's user avatar
4 votes
1 answer
376 views

Why do physicists refer to irreducible representations as "charges" or "charge sectors"?

My question is in the title: Why do physicists refer to irreducible representations (irreps) as "charges" or "charge sectors"? For concrete examples, irreps are referred to as &...
Maple's user avatar
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The little group of the connected Lorentz group [duplicate]

For $n\geq 2$, the $n$-dimensional connected Lorentz group is defined as $$SO(n-1,1)^{\uparrow}=\{f\in M_n (\mathbb{R}):f^T\eta f=\eta, \det (f)=1,f^0_0>0\}$$ where $$\eta=\begin{bmatrix}-1 & ...
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Challenging Cauchy's Stress Tensor: Objectivity and Generalization of Divergence Theorem

I'm investigating the limitations of the Cauchy stress tensor model in classical continuum mechanics, specifically focusing on its compliance with the principle of material frame indifference (MFI) ...
Foad's user avatar
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How to show that $G_p=SO(D-1)$?

Let $G=SO(D-1,1)^{‎\uparrow‎}$ be the connected Lorentz group. Let $p$ be a timelike momentum with $p_0>0$. I want to show that $G_p=SO(D-1)$, the little group of $p=(M,0,\ldots,0)$ where $M>0$....
Mahtab's user avatar
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Kretschmann Scalar for the FLRW Metric

I am trying to understand the Wikipedia definition of the Kretschmann scalar for a cosmological solution. The metric is given by the standard FLRW metric \begin{equation*} ds^2 = -dt^2 +a^2(t)\left(\...
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Self-similar solution of the second kind

I have a problem trying to understand the procedure for using self-similar solution of the second kind. More specifically, I was reading about an equation of this form, $$\partial_t{d} + \frac{1}{r} \...
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Fundamental invariants of a Lorentz tensor

As answered in this question, an antisymmetric tensor on 4D Minkowski space has two Lorentz-invariant degrees of freedom. These are the two scalar combinations of the electromagnetic tensor (as proven ...
Spinoro's user avatar
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What isn't the metric invariant under translation with killing vectors?

I am learning about Killing Vectors in GR class, and I'm testing my knowledge of them as a start with the Minkowski metric. I used the simple 2d Minkowski metric: $$ds^2 = -dt^2 + dx^2$$ and got 3 ...
Habouz's user avatar
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Energy of circle orbit in Schwarzschild metric

For Schwarzschild metric we have invariants: \begin{equation} \begin{aligned} & r^2 \frac{d \varphi}{d \tau}=\frac{L}{m} \\ & \left(1-\frac{r_s}{r}\right) \frac{d t}{d \tau}=\frac{E}{m c^2}...
Aslan Monahov's user avatar
2 votes
3 answers
289 views

Is the metric distance invariant under any coordinate transformation?

Consider an infinitesimal coordinate transformation, $$ x^{\mu} \rightarrow x'^{\mu} = x^{\mu} + \epsilon^{\mu}(x). $$ We can show that the metric tensor under such a transformation, up to first order ...
ratchet411's user avatar
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Partition function for a $SO(3, 1)$ invariant "Hamiltonian"

Suppose, I look at the $SO(3, 1)$ generalization of $H = \frac{p^2}{2m}$, i.e. $$H = \lambda P^{\mu}P_{\mu}$$ where $P^{\mu}P_{\mu}$ is a $SO(3, 1)$ invariant object and $\lambda$ is some dimensionful ...
Dr. user44690's user avatar
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2 answers
188 views

The speed of light is constant?

In the proof that the speed of light is a constant we make the assumption that space at large scales is homogeneous, but there are patches of space where the density is higher and there are patches ...
Euler-Masceroni's user avatar
1 vote
0 answers
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What is the difference between the center of mass energy of partons, and the invariant mass of the particles after a collision?

In a paper about the Atlas experiment (https://arxiv.org/abs/2103.01918) differential cross sections of $pp→ZZ→4\ell$ are being presented. However these cross-sections are functions of the invariant ...
Ozzy's user avatar
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1 answer
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Proof that a scalar field invariant under rotations only depends on norm

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a real valued scalar field and $\mathbf{r}\in\mathbb{R}^3$ a vector with $r = \sqrt{\mathbf{r}\cdot\mathbf{r} }$ its norm. Let's say that $f$ is ...
Pere Rosselló's user avatar
3 votes
0 answers
68 views

Counting independent components of Lorentz tensor

Say I have Lorentz tensors $A^{\mu\nu}$ and say this Lorentz tensor is symmetric under $\mu \Leftrightarrow \nu$ and there are only $p^\mu$ and $q^\mu$ as the physical Lorentz vectors involved. If so, ...
Quantization's user avatar
2 votes
1 answer
66 views

How to obtain the relation of eta invariant of the trivial gauge field and Chern-Simons invariant of the flat connection?

In Quantum Field Theory and the Jones Polynomial by Edward Witten(1989), how does $\eta(0)$ come from in this equation? $$\frac{1}{2}(\eta(A^{(\alpha)})-\eta(0))=\frac{c_2}{2\pi}I(A^{(\alpha)})$$ $c_2(...
Jimi's user avatar
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2 answers
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Inner product not invariant in QM?

For simplicity, consider 2D space. Let $\{|a\rangle, |b\rangle\}$ and $\{|a'\rangle, |b'\rangle\}$ be two sets of basis kets. Now the kets $|a\rangle, |b\rangle$ can be represented in its basis $\{|a\...
Deep's user avatar
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4 votes
1 answer
142 views

How many relativistically invariant degrees of freedom in $n$-particle scattering?

Suppose we have a scattering process with $n$ external legs with four-momenta $p_1, \cdots, p_n$. Naively there are $4n$ degrees of freedom, however most of these putative degrees of freedom are not ...
Panopticon's user avatar
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0 answers
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How to derive the Lorentz Invariant bilinear form in the $(1/2, 1/2)$ representation?

We can represent the complexified proper Lorentz group Lie algebra as the direct sum $\mathfrak{sl}(2, \mathbb{C}) \oplus \mathfrak{sl}(2, \mathbb{C})$. The representation nomenclature is $(n,m)$, two ...
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Spin tensor $S^{\mu \nu}$ and its invariants [closed]

I am studying a theory in which the spin tensor $S^{\mu \nu}$ has been used for the description of a particle endowed with spin. I saw that $S^{\mu \nu}$ is a 2-tensor and has the same form of the ...
Gyro's user avatar
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Understanding and proving the Lorentz invariance of the volume element in momentum space

By performing the relevant four-dimensional Lorentz transformations explicitly, demonstrate the Lorentz invariance of the volume element $$\frac{1}{(2\pi)^3}\frac{d^3p}{2E_{\vec p}},\,\text{with}\,E_{\...
some_math_guy's user avatar
9 votes
2 answers
477 views

Speed of light postulate in special relativity in inertial vs. non-inertial frames

I'm trying to learn special relativity by myself. I've been following this series of videos, plus some other articles I've managed to find online. At this point I'm already quite far into the theory, ...
Luke__'s user avatar
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5 votes
3 answers
608 views

Where does the negative signature case come from in the Pythagorean derivation of distances in spacetime?

I am reading Why does $E=mc^2$ (and why should we care?) by Brian Cox and Jeff Forshaw. I want to understand these three sentences (from page 76/77): Once we follow Occam and make these two ...
sleep's user avatar
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2 votes
1 answer
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Why do the eigenvalues of the 4-momentum operator organize themselves into hyperboloids?

Specifically I'm asking for the motivation behind figure 7.1 in page 213 of the QFT textbook by Peskin and Schroeder. In that section they just consider eigenstates of the 4-momentum operator $P^\mu=(...
Function's user avatar
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3 votes
1 answer
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Why intuitively do the eigenvalues of the Casimir operators classify irreps?

Distinguishing between distinct unitary irreducible representations is important from the point of view of distinguishing between different sorts of particles; the eigenvalues of the Casimir operators ...
Panopticon's user avatar
2 votes
2 answers
117 views

Why does the inner product of a scalar operator vanish?

A scalar operator $\mathscr{Y}$ is defined as an operator where the inner product $\langle\phi|\mathscr{Y}|\phi\rangle$ is unaltered by a rotation of the coordinate system. Then for an operator to be ...
Rasmus Andersen's user avatar
1 vote
1 answer
215 views

Do the Casimir operators for $su(3)$ algebra of particle physics carry any physical meaning?

For the $su(2) $ algebra of angular momentum, the eigenvalues of the Casimir operator, $\hat{J^2}$, represents the square of the total angular momentum of a system. The $su(3)$ algebra has rank 2 and ...
Solidification's user avatar
1 vote
1 answer
59 views

Lorentz group from Einstein relativity principle

Einstein's relativity principle states that: All the laws of physics are the same in every inertial frame of reference The speed of light is the same in every reference frame. Why these two ...
Antonio19932806's user avatar
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0 answers
44 views

D Alembert Wave Equation is not Gallean Invariant but Why y=Asin(wt-kx) is Gallean Invariant? [duplicate]

I just watched this video from MIT 8.04 Quantum Physics I by Barton Zwiebach, explaining Galilean transformation of y=Asin(wt-kx). I have a confusion, are ordinary waves Galilean invariant or not? ...
Dibyajit Bhattacharyya's user avatar
1 vote
2 answers
113 views

Trying understand the argument on Special Theory of relativity on Goldstein book

I am reading Goldstein and trying to understand the special theory of relativity. I am not sure how did he make this following argument. The books explain that $ds^2$ is invariant in spacetime. He ...
ran singh's user avatar
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1 answer
64 views

Relation between $SL(2,R)$ and $U(1)$ symmetry

I have an action that I have proven to be invariant under an $SL(2,R)$ symmetry. But I actually want my action to be invariant under an $U(1)$ symmetry (because i know that for the system I am ...
Charlotte Myin's user avatar
3 votes
2 answers
648 views

Physical meaning of the invariance of the dot product of $\vec{E}$ and $\vec{B}$

I recently learned about the fact that $\vec{E} \cdot \vec{B}$ is invariant under Lorentz transformations, which seems like a really nice and useful result. Is there a physical meaning similar to the ...
Elias K.'s user avatar
1 vote
1 answer
157 views

Invariants from the covariant derivatives of a scalar field

I am reading Theoretical minimum: Special Relativity and Classical Field Theory where you construct a Lagrangian for the field by the argument that it would be invariant under the Lorentz ...
Ajaykrishnan R's user avatar
23 votes
4 answers
6k views

Why is Noether's theorem not guaranteed by calculus?

The action of a system, say a scalar field is $$ S = \int_{\mathcal{M}} {\rm d}^4 x ~ \mathcal{L}(\phi(x),\partial \phi(x)). $$ Now, if one does a variable transformation $x \to x'$, then $$ S' = \...
Faber Bosch's user avatar
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1 answer
54 views

How one shows a law is invariant/variant under a transformation in general case?

Say we have a law of physics in a form of a differential equation: $$ F(x,y,\dot{y},\dots) = 0. $$ What is the general algorithm to show that this law is invariant under a certain transformation. The ...
Kid A's user avatar
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1 vote
1 answer
81 views

Invariant nature of mass and particle annihilation [closed]

Since mass is a Lorentz invariant, it can never change to preserve the vectorial nature of the four-momentum and the other four vectors. Thus the only interpretation of the energy-mass equation that I ...
GedankenExperimentalist's user avatar
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1 answer
162 views

Confusion about Lorentz invariance of scalar product

I am a bit confused about the way that Lorentz invariance of the scalar product $A^\mu g_{\mu\nu}B^\nu$ is proved. Usually, the proof would go like this (see also e.g. this Physics SE question). The ...
Quercus Robur's user avatar
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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1 answer
197 views

Forces that are invariant under Galilean spacetime rescaling $\mathbf x' = \lambda \mathbf x$, $t' = \lambda^2 t$

Consider a force of the form $$ m \ddot{\mathbf x}(t) = -k\frac{\mathbf x(t) - \mathbf x_0}{|\mathbf x(t) - \mathbf x_0|^d}. $$ For what values of $d$ is this force invariant under the Galilean ...
Chris Yang's user avatar
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1 answer
217 views

Meaning of rotational invariance breaking and lifting of degeneracy

The $2p$ level of hydrogenic atoms are $3-$fold degenerate. That is $\psi_{210}$ and $\psi_{21\pm1}$ all have the same energy $E_{n=2}$. And the level $n=2$ as a whole is $4-$fold degenerate, since $...
QuestionTheAnswer's user avatar
2 votes
1 answer
127 views

Invariance and covariance [duplicate]

What exactly do we mean when we say that something is covariant? How is it different from being invariant? Am i right if I say that the word invariant is used when we're talking about a physical ...
iota's user avatar
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0 answers
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Topological Invariants of Floquet Topological Insulators

I've been working through the following sources: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.155118 https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.195303 where they derive new ...
Fred's user avatar
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