# 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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### Causality: Invariant Under Lorentz Transformation

I'll begin by stating that I have only studied a very small bit of special relativity: a couple of lectures or so around the end of the Physics course I took, intended just to "give us a taste&...
35 views

### Different Casimirs and Casimirs of $E_6$

I am a bit confused by the notion of Casimirs (maybe it is related to terminology). In the simplest example of $su(2)$ with generators $L_i$, we get the Casimir operator $$L^2=\sum_i L_i^2$$ ...
206 views

### What is the intuition behind the spacetime interval?

In an article that I am currently reading (under the Lorentz Invariants sub-heading), it explains that, just as the distance between two points on a Cartesian plane are obviously invariant of the ...
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### What are the invariant structures in special relativity?

What geometric structures in Minkowski spacetime are Lorentz invariant?
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### What do these Casimir invariants of the Galilean group physically represent?

There exist Casimir invariants of the Galilean group which commute with all the generators of the group. They are, of course, Galilean scalars (i.e., scalars under space and time translations, ...
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### Can we write the mass $M$, a Casimir invariant of the Galilean group, as a function of its generators?

According to Wikipedia, the mass $M$ is one of the Casimir invariants of the Galilean group. Casimir invariants of a group are made out of the generators, and they commute with all the generators of ...
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### What physical properties are invariant under relativistic transformation?

Most of the familiar physical properties vary according to the relativistic observer's reference frame - speed, mass, energy, time, length, etc. Which properties remain invariant, so everybody will ...
449 views

### Landau Classical Fields theory argument for invariance of $ds^2$

In Landau's classical field theory, chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from Einstein's postulate of the invariance of the speed of light. Landau says that ...
48 views

### Co-ordinate invariance in Lagrangian form of equations

I have read that in his Mecanique Analytique , Lagrange sought a “coordinate invariant expression for mass times acceleration”. The discussion regarding this is given in 'Marsden and Ratiu [15, ...
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### What are some of the uses of the invariant spacetime interval? [closed]

I am currently researching special relativity and I have come across the invariant interval. So far, I know that all observers will measure the same interval regardless of position or velocity, and ...
76 views

### How can you make mechanics invariant under inversion of length? [closed]

I read that in some string theories they are invariant under a change of distance from $r$ to $1/r$. I wondered what is the simplest theory or lagrangian in 4 dimensions that could have this property. ...
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### Is the speed of light through a medium invariant, just like the speed of light through vacuum is invariant? Also, do time dilation etc. still occur?

So, we know the speed of light through a vacuum is $c$. Let us say that both our observers are moving past each other at speed $v$ in a medium in which the speed of light is $c'$. So, does the usual ...
68 views

### What are the physical state invariants of loop quantum gravity?

What are the "physical state" invariants for Loop Quantum Gravity? The Wikipedia page talks about "physical states" being invariant, diffeomorphism invariance, "quantities ...
53 views

### Coordinate invariance in Physics

Let us consider a classical field theory on flat background spacetime. The action is $$S[\Phi] = \int d^nx \mathcal{L}(\Phi,\partial_\mu\Phi).$$ Why shouldn't this action be independent of the chosen ...
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### Analyzing the Schrodinger equation under Lorentz and Galilean transformation

I'm trying to see what happens to the Schrodinger equation under Lorentz and Galilean transformation. So I assumed a free Schrödinger wave with the form: $$\psi = A e^{i(\omega t - kx)} ~.$$ A Galilei ...
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### Condition for choosing a wavelet

Can we choose a mother wavelet that is not Lorentz invariant in practical applications of wavelet transform in Minkowski space?
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### Lattice Gas Automata and Galilean Invariance

I have been studying Lattice Gas Automata methods (also this), and every time I read up on their drawbacks, I see that they are not Galilean invariant and that the simulations have statistical noise. ...
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### Can we derive the momentum of photon from special relativity?

I don't really have strong backgrounds studying quantum physics, but I did learn special and general relativity, and I have now a question how to get the momentum of photon. For my understanding, ...
211 views

### Is mass still a scalar in special relativity?

In Euclidean space is the space of classical Mechanics, A scalar is the same for all observers that are to say remain invariant under the change of coordinate systems. A Vector $\mathbf{V}$ is a ...
61 views

### Is the rest mass of a system invariant even if it is a function of time?

If we imagine a system of particles, and we consider only $N$ of them, then the rest mass of these $N$ particles is given by $$\left(\sum_{i=1}^N P_i\right)^2=P_T^2=m^2$$ Where $P_i$ is the 4-momentum ...
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### What does invariance under time translation exactly mean?

Even after a google search I didn't find a definition of the concept "invariance under time translation" used (and not defined) in my lecture notes. Consider for example the differential ...
69 views

### Why do we integrate up to the invariances

Following Witten's essay What every physicist should know about string theory I understood that in the Hilbert-Einstein action is invariant under diffeomorphism in 1D and under conformal mapping in 2D....
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### Why does 'spherically symmetrical' function $\phi$ consist of $r^2$, $p^2$, $\vec{r}\cdot \vec{p}$?

Why does the spherically symmetric function $\phi$ can only depend on $r^2$, $p^2$, and $\vec{r}\cdot \vec{p}$, but not on $r$? I thought $\phi$ could also depend on $r$ because $r$ is always bigger ...
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### Pair Production invariant

I am having trouble understanding what the fixed quantity in pair production is. A photon hits a nucleus and based on the photon's minimum energy relative to the nucleus, we get pair production. Is ...
60 views

### Adiabatic Invariant when forcing is at the natural frequency of a classical simple harmonic oscillator

Consider a simple harmonic oscillator of unit mass, natural frequency $\omega_0$, given by the Hamiltonian \begin{align} H_0(q,p)=\frac{1}{2} \left[ p^2 + \omega_0^2 q^2 \right] \ . \end{align} Now ...
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### How to know if a measure is Lorentz invariant? [duplicate]

In a Pr. Badis Ydri's book, 'A Modern Course on QFT', the following integral is said to imply that $d^3p/2E_p$ is indeed a Lorentz invariant. How so ? Does it have to do anything with the notion of &...
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### What is the role of the mass Casimir invariant in Galilean and what it's actual role in special relativity? [closed]

What is the role of the (mass) Casimir invariant of the algebra of relativistic symmetries in Galilean and what it's actual role in special relativity?
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### Why is the special form of space-time interval chosen out of the invariance of the speed of light?

On page 4 of Landau & Lifshitz's The Classical Theory of Fields, the interval$$ds^2=c^2dt^2-dx^2-dy^2-dz^2$$ is introduced after the invariance of the velocity of light is stated, and later, after ...
482 views

### Can Galilean transformation be derived from length invariance?

Given length invariance in Euclidean 3D space between two inertial frames:$$ds^2=ds'^2$$ Can Galilean transformation be derived like Lorentz transformation derived from space-time interval invariance?