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&...
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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 $$ L^2=\sum_i L_i^2$$ ...
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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 ...
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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 ...
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48 views

Co-ordinate invariance in Lagrangian form of equations

I have read that in his Mecanique Analytique [1788], 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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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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65 views

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 ...
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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 ...
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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?
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52 views

Is angular velocity invariant under SR?

Is angular velocity invariant under special relativity, i.e. do all observers in any relative inertial frames measure same value for angular velocity of a system? If not, what is its expression?
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58 views

Invariance of magnetic moment in slowly-changing magnetic fields (plasma physics)

I am trying to understand how the magnetic moment is invariant in slowly changing magnetic fields. There is a proof in the textbook I am using, but I'm stuck on how $-e\int\frac{\partial B}{\partial t}...
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82 views

Square of Pauli-Lubanski operator

I am following Ashok Das QFT book (pg. 152-153) on the calculation of the Pauli Lubanski operator. The Pauli Lubanski vector operator is defined as $$W^\mu=\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}P_\nu ...
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91 views

Lorentz generators $J_i,K_i$ and Casimir invariants in bispinor representation

Lorentz generators satisfy the Lie algebra $$[J_i,J_j]=i\epsilon_{ij}^kJ_k, ~~~~[J_i,K_j]=i\epsilon_{ij}^kK_k, ~~~~[K_i,K_j]=-i\epsilon_{ij}^kJ_k.$$ Now, define $$A_i=\frac{J_i+iK_i}{2},~~~~B_i=\frac{...
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How do the radius and the center of the circular orbits of a particle depend on an adiabatically changing $B$ field [closed]

Preparing for a classical mechanics qual exam with my CM final from last semester. Problem: Consider the circular orbits in the $xy$ plane with $x > 0$ of a particle mass $m$ and charge $q$ in a ...
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Casimir of $SO(3)$, $SO(2)$, $IO(1,3)$, $T(4)$

It is known that $SO(3)$, a semisimple group of rank 1, has one Casimir $J^2$, and one can use this information to classify its irreps with the eigenvalues of $J^2$ and $J_3$: $(j,m)$. Now, only $j$ ...
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48 views

Invariant rest mass vs Proper velocity

For an object (rest mass $m_0$ ) moving with velocity v, we (observers at rest) say that it has energy $E=m(v)$$c^2$. Where $m(v)$= $m_0$/$\sqrt(1-v^2)$. Thus, for us, we can say it's mass has ...
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Is the density of states Lorentz invariant?

This is something that has been confusing me. A system can have a multitude of quantum states, and the energy of each will change depending on the frame of reference. However, the number of states ...
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163 views

SR parallel of GR: Which physical postulate of GR requires that $ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu$ must be invariant?

Landau, in his book Classical Theory of Fields, exploits the postulate that the velocity of light in a vacuum is the same in all inertial frames, to establish that the spacetime interval between two ...
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Is (net) force invariant in special relativity?

I am aware that acceleration is not invariant under lorenrz transformations, but I was sure that the first postulate of special relativity implied that newton’s second law in its original form, F=dp/...
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126 views

Characterization of variational symmetries

I have the following definition of variational symmetry, this is from Bruce Van Brunt's Calculus of Variations book Suppose we have a functional of the form $$ J(y)= \int_{x_{0}}^{x_{1}} f(x,y,y')dx $$...
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Cartan subalgebra and Casimir invariants

To my understanding, if group $G$ is semisimple, $\mathfrak{g}$ is its Lie algebra, and $\Delta_T=[T,\cdot]$ is the adjoint representation, one can analyze its spectrum with $[T,U]=\lambda U$. A ...
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177 views

On Poincare group’s Casimir operators

We’ve defined Casimir operator for a group as an operator which commutes with all generators of that group. For the Poincare group we’ve found two Casimir operators: $p_\mu p^\mu$ and $W_\mu W^\mu$ ...
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Confusion about Joule expansion for ideal gases

We concern the Joule expansion for ideal gases, where the gas is initially kept in one side of the container and the other side is evacuated. Then the partition between the two side is released, ...
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Using Noethers theorem to show an expression is constant along an extremal of a Lagrangian

I am given the following integral: $$I = \int^{t_2}_{t_1}\dot{x}^2dt $$ And the following 1 - parameter transformation: $$\bar{x} = \alpha x\\ \bar{t}=\alpha^2t$$ I have shown that the given intergral ...
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A question about the Lorentz transformation of "infinitesimals"

Notations conventions: $p$ stands for the momentum (so $d^3p$ is the differential element according to which we integrate, for the $3$ space coordinates). A Lorentz transformation is denoted by $\...
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106 views

Invariant of supersymmetry?

Given two vectors in 3D superspace $(x_1^\mu,\theta_1^\alpha,\overline{\theta}_1^\alpha)$ and $(x_2^\mu,\theta_2^\alpha,\overline{\theta}_2^\alpha)$ I am trying to find a polynomial invariant under ...
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Is temperature a Lorentz scalar [duplicate]

If I see a body at temperature $T$, will I see the same temperature in another frame under a Lorentz boost. And will the internal energy of a body also remain invariant under a Lorentz boost or not..
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What are all objects that are invariant under Lorentz transformations?

I am interested in the following question: What are all objects that are invariant under Lorentz transformations? And, once a list is provided, how to justify that these are indeed ALL such objects? ...
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58 views

Finding if a quantity is a Lorentz scalar

I am new to special relativity, and I am trying to figure out if $\phi(x)=\frac{a \cdot a}{x \cdot x+a \cdot a-x^{0} a^{0}}$ is a Lorentz scalar, where $x$, and $a$ are four-vectors. Since the dot ...
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61 views

Invariance of inner product under Poincare transformation

The Poincare transformation reads, $$x\rightarrow x^\prime=\Lambda x +a $$ The scalar product is preserved under Lorentz transformation. However I do not see how it is preserved under the more general ...

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