Questions tagged [invariants]

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106 questions
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How to identify cosntants of motion/ which constants of motion are independent of mass?

I was asked to identify a constant of motion which does not depend on the mass of the object whilst in contact with the surface. I found the equation of motion of the material point but I don't know ...
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Definition of the Spacetime Interval

The spacetime interval is defined as follows: $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$ or in tensor notation: $$\Delta s^2 = \eta_{\mu\nu} \Delta x^\mu \Delta x^\nu$$ ...
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Defining invariant spacetime interval

So, my textbook goes about defining the invariant spacetime in the following way: Consider two frames of references, S and S', with a relative speed to each other, coinciding at t=t'=0. At t=0, a ...
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Strange result from braking energy in different reference frames. Spot the error [closed]

Assume a truck of mass 1 ton braking from 60mph to 0 on a road surface. From the reference frame of the road, the following energy is transferred to the brakes: 1 ton * 60mph^2 / 2 = 1800 ton mph. ...
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Proving an invariant relationship [closed]

Two particles, moving at relativistic speeds in the x direction, are observed to have energies E1 and E2, and momenta p1 and p2 in frame A. Frame B moves at relativistic speed v relative to frame A, ...
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Does the Kosterlitz–Thouless transition connect phases with different topological quantum numbers?

The Kosterlitz-Thouless transition is often described as a "topological phase transition." I understand why it isn't a conventional Landau-symmetry-breaking phase transition: there is no local ...
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Is the Invariant interval S between two points independent of the path taken?

Due to a misunderstanding of what I was asking, I'm re-asking this question (Is the Invariant interval S between the singularity and the present, the same for any point in space in an FLRW universe?) ...
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Is the Invariant interval S between the singularity and the present, the same for any point in space in an FLRW universe?

I've mostly been considering the closed FLRW universe in this. It seems that any point on the FLRW universe at a given cosmological time would have the same invariant interval between it and the ...
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Mandelstam variables for 2 to 3 particle scattering

I'm trying to work out the mandelstam variables for 2 particles scattering to produce 3 particles. Also each particle is massless. I think there must be 5 because all possible scalar products of the ...
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What exactly does it mean for a scalar function to be Lorentz invariant?

If I have a function $\ f(x)$, what does it mean for it to be Lorentz invariant? I believe it is that $\ f( \Lambda^{-1}x ) = f(x)$, but I think I'm missing something here. Furthermore, if $g(x,y)$ ...
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Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
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Why scalar function of vector can only depend on norm of vector?

In Field Quantization by Greiner and Reinhardt as well as The Qunatum Theory of Fields by Weinberg, concerning the spectral function, the authors say a scalar function of the four-vector $p^\mu$ can ...
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Is metric tensor invariant under rotation?

It is said that metric tensor depend on the local coordinate system and therefore are not intrinsic to the surface of an 3d-object? How is it possible, kindly provide any proof or discussion. Also is ...
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Scalar operators In Quantum Field Theory

I am trying to learn Quantum Field Theory and I am stuck in a basic point. What is the definition of a scalar operator in QFT? That is, how does it transform under a Poincare transformation? Why do ...
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Invariance of action $\Rightarrow$ covariance of field equations?

Invariance of action $\Rightarrow$ covariance of field equations? Is this statement true? I have only seen examples of this, like the invariance of Electromagnetic action under Lorentz ...
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Calculate minimum energy of incident neutrino using Mandelstam variables

I am studying the following nuclear reaction: $v + \tilde{v}\rightarrow Z^0$ where the antineutrino is motionless and has a given mass. The $Z^0$ boson has also a known mass. I'm trying to calculate ...
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A common definition of a scalar

Some dictionaries define a scalar as follows: A quantity, such as mass, length, or speed, that is completely specified by its magnitude and has no direction. -- The Free Dictionary However, it is ...
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Is relative velocity invariant under special relativity?

If a metre stick passes an observer at speed $v$, would all observers in any inertial frame of reference say the speed of the meter stick relative to the observer is exactly $v$? If so what is it ...
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Action of the Poincare Group on a Scalar Function

Let $F(x^\mu)$ is a scalar function; i.e. $F(x^\mu): \mathbb{R}^{1,3} \rightarrow \mathbb{R}$. How the Poincare Group $P(1,3)$ will act on it; i.e., by which formula I can calculate it for a specific ...
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Heat energy in special theory of relativity [duplicate]

Is heat energy invariant under Lorentz transformation? If so then how?
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All possible electromagnetic Lorentz invariants that can be built into the electromagnetic Lagrangian?

Given the electromagnetic Lagrangian density $$\mathcal{L}~=~-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}~=~\frac{1}{2}(E^2-B^2)$$ is a Lorentz invariant, how many other electromagnetic invariants exists that ...
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Reason why $F^{\mu\nu}F_{\mu\nu}$ and $\tilde{F}_{\mu\nu}F^{\mu\nu}$ are Lorentz invariant

I'm trying to think of an intuitive reasoning for why $F^{\mu\nu}F_{\mu\nu}$ and $\tilde{F}_{\mu\nu}F^{\mu\nu}$ are Lorentz invariant. By this I mean that I don't simply want to show that they remain ...
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Is Lagrangian a scalar?

I may be wrong: Lagrangian are scalars. They are NOT invariant under coordinate transformations. The simplest example is when you have a gravitational potential ($V=mgz$) and you translate $z$ by $a$ ...
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Showing the Poincare invariance of a term

I know that this is a simple question! But I would like to know the details. How we can show that the term $$A_\mu(x)\dot{x}^\mu$$ is global and local Poincare invariant? Where $A_\mu(x)$ is ...
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Modular invariance of CFT

I am looking at the Cardy formula for entropy in CFT, and in the article 'Kerr/CFT correspondence and its Extensions' there is a sentence: In any unitary and modular invariant CFT, the asymptotic ...