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Questions tagged [invariants]

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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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4answers
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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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2answers
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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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3answers
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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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1answer
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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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1answer
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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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4answers
363 views

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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1answer
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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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1answer
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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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2answers
801 views

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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0answers
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SU(3) adjoint representation's invariant tensors

Considering a complex scalar field $\varphi^a$ that transform in the adjoint representation (8) of SU(3). A quartic interaction term SU(3) invariant is $$\lambda C^{abcd}\varphi^{\dagger a} \varphi^...
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2answers
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energy threshold of proton-proton collision

Could you please help me understand what's wrong with the following calculation? When a proton moves toward a fixed proton with very high speed and collides with it, then the following interaction ...
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1answer
534 views

What is the definition of invariance under Lorentz transformation?

I want to learn how to check if any scalar is a Lorentz scalar, so what is the definition of being invariant under Lorentz transformation? Is it correct to say that $\phi$ is invarant under Lorentz ...
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3answers
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Providing an intuitive description of scalar and vector quantities in physics [closed]

Often the standard introduction to the concept of scalars and vectors in physics is something along the lines of: A scalar is a quantity that is completely described by a single number (it has no ...
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2answers
344 views

Scalar fields and general coordinate transformations

In classical mechanics, a scalar field is characterised by the fact that its value at a particular point must be invariant under rotations and reflections of coordinates. That is, one requires that $\...
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2answers
126 views

Given a metric of a torus can we measure it's thickness?

What I mean is, if you have a metric of a torus $T_2$, and you want to distinguish between say very thin stringy tori and very thick tori with the same surface area, is there a nice formula for this ...
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2answers
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What is the difference between a physical constant, a scalar, an invariant, and a conserved quantity?

I don't really know how to properly articulate this question. This question popped into my mind when pondering why the fact that a physical constant like the speed of light doesn't have an associated ...
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2answers
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How to show invariance using the Maxwell tensor?

I want to show the invariance of $E^2-c^2B^2$ under the Lorentz transformations. The obvious way to do this is to show that $$E^2-c^2B^2=E'^2-c^2B'^2,$$ where $E'$ and $B'$ are the Lorentz ...
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1answer
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Confusion about supergravity multiplet

I have a bit of confusion with the following consideration. Generally, to impose some BPS condition (i.e. to check if some multiplet preserves some SUSY charges) one imposes to zero the SUSY ...
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4answers
1k views

What do observers in relative motion agree on?

What are the measurements on which two observer in relative motion will agree? Other than the speed of light.
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1answer
103 views

Why does length contraction seem to conflict with invariance of intervals?

Suppose we have two simultaneous events, $A$ and $B$, separated by a distance $L$ (the simultaneity is in frame of reference $S$). Now suppose we have a second frame of reference $S'$ moving with ...
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3answers
690 views

Is the “number of photons” of a system a Lorentz invariant?

I'm wondering whether the number of photons of a system is a Lorentz invariant. Google returns a paper that seems to indicate that yes it's invariant at least when the system is a superconducting ...
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1answer
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Invariant tensors in a general representation and their physical meaning

I'm trying to use tensor methods to find invariant elements of representations. Specifically I'm looking at representations of $SU(5)$. I can show that the invariant element in $5\otimes\bar{5}$ (or ...
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2answers
758 views

Invariance and conservation

Why in a collision between particles is the four-momentum conserved within a frame of reference but not invariant between frames of reference?
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1answer
1k views

Difference between symmetry and invariance

I'm wondering what's the real difference between symmetry and invariance in Physics? I believe that sometimes the two words are given the same meaning and some other times they are used in a different ...
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1answer
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Invariance in general relativity, university in problems question

From Problem #5 here, Free falling particles' worldlines in General Relativity are geodesics of the spacetime, i.e the curves $x^\mu(\lambda)$ with tangent vector $u^\mu=dx^\mu/d\lambda$, such that ...
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1answer
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Minimal set of invariants to specify a Kepler orbit

In the Kepler problem, we know that there are various invariants, including: Energy Angular momentum vector Runge-Lenz vector All together these consist of 7 parameters. On the other hand, the ...
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3answers
692 views

Why invariance is important?

The concept of invariance seems to have a great importance. Indeed, the fact that the laws of Electrodynamics are not invariant in every inertial reference frame led to the theory of Special ...
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2answers
428 views

Identifying a scalar function

We know that a scalar is invariant under rotations. What about a scalar function? Should it also be invariant under rotations? Therefore, under rotation $\phi(x,y,z)$ must be equal to $\phi^\prime(x^\...
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0answers
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Finding Casimir operators for the Poincare group $ISO(1,2)$

I was asked to write the generators for translations and Lorentz-transforms in 1+2 dimensions and then to find the Casimir operators. For the generators I can take the same ones as in 1+3 case $$P_\...
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2answers
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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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2answers
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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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1answer
597 views

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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2answers
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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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1answer
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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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1answer
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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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4answers
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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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4answers
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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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1answer
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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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1answer
250 views

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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2answers
963 views

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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1answer
865 views

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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2answers
289 views

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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1answer
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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 ...
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5answers
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Fundamental invariants of the electromagnetic field

It is a standard exercise in relativistic electrodynamics to show that the electromagnetic field tensor $F_{\mu\nu}$, whose components equal the electric $E^i=cF^{i0}$ and magnetic $B_i=-\frac12\...
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1answer
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Lorentz covariance of the Noether charge

The invariance under translation leads to the conserved energy-momentum tensor $\Theta_{\mu\nu}$ satisfying $\partial^\mu\Theta_{\mu\nu}=0$, from which we get the conserved quantity$$P^\nu=\int d^3\...
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1answer
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Principle of relativity - a second, equivalent form, using invariants

Most people state the principle of relativity like this: "The rules of physics must take the same form in all inertial frames." Question: is this an equivalent way of saying the same thing: "...
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1answer
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Rank of the Poincare group

There are two Casimirs of the Poincare group: $$ C_1 = P^\mu P_\mu, \quad C_2 = W^\mu W_\mu $$ with the Pauli-Lubanski vector $W_\mu$. This implies the Poincare group has rank 2. Is there a way to ...
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2answers
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Invariance, covariance and symmetry

Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from quantum field theory?