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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What exactly is an invariant quantity?

I have a bit of confusion regarding an invariant quantity. It is something which doesn't change on switching from one inertial frame to other like $\Delta$$\mu$$J$$\mu$ is an invariant. I read ...
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Explicit Quadratic Casimir for $sp(2N)$

We know that $so(3)$ has the explicit quadratic Casimir $$L^2=\sum L_{i}^2.$$ Are there analogs to this in other simple lie algebras? I know that for a simple lie algebra I can always use the ...
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Interval Preserving in Minkowski Space

The squared line element in any spacetime is given as $$ds^{2}=g_{ab}dx^{a}dx^{b}.$$ The use of tensors helps us to infer that the line element in some other frame would be $$ds'^{2}=g'_{ab}dx'^{a}dx'^...
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Use of step function to show that total charge is Lorentz invariant if the four-gradient of current density is zero

I am reading the chapter on Special Relativity in Steven Weinberg's 'Gravitation and Cosmology'. It is stated in the book that total charge can be written as $Q=\int d^4xJ^{\alpha}(x)\partial_{\alpha}\...
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What Lorentz covariance has to do with Lorentz invariance? [duplicate]

Does saying that the Dirac equation is invariant under Lorentz transformations is the same as saying that it is Lorentz covariant?
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467 views

Distance formula in Euclidean space vs. Spacetime Interval - why is one Pythagorean and one not?

I appreciate if this question has been posited before and easily findable by Google searching, but as of yet I haven't found anything to answer this. I'm sure I'm making an incorrect assumption in the ...
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Why is the Ricci scalar the only independent scalar constructed from products of the metric and its first and second derivatives? [duplicate]

In Sean Carroll's book, last paragraph of page 160, this statement is found: "The Riemann tensor is of course made from the second derivatives of the metric, and we argued earlier that the only ...
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Why are the metric and the Levi-Civita tensor the only invariant tensors?

The only numerical tensors that are invariant under some relevant symmetry group (the Euclidean group in Newtonian mechanics, the Poincare group in special relativity, and the diffeomorphism group in ...
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importance of invariant tensors

while studying representations of SL(2,C), for raising and lowering indices of spinors invariant tensor $\epsilon$ was constructed analogous to $\eta$ in SO(1,3).What is the importance of invariant ...
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3answers
428 views

The implications of Einstein's first law

I'm struggling with the physical meaning/consequence of Einstein's first postulate of Special Relativity, which states that all physical laws are the same (invariant) in all inertial frames. Any ...
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645 views

Has the energy-momentum invariant any meaning?

For a single particle system we have: $$E^2 - (pc)^2 = (mc^2)^2.$$ In my lecture notes it has also been stated that for a system of several particles: $$\left(\sum E\right)^2 - \left(\sum p\right)^2c^...
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Why do we have $\phi'(x')=\phi(x)$ for a field satisfying Klein-Gordon equation?

I would like to know why we have $\phi'(x')=\phi(x)$ for a field satisfying Klein-Gordon equation. Is it an assumption or can it be proved? The $'$ means a Lorentz transformation: $\phi'$ is the ...
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1answer
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Three questions and explanations for the Lorentz invariant $E^2-c^2B^2$

It is demonstrated that the square trace of the electromagnetic tensor is nothing and it is valid: $$ \mathrm{Tr}\,{F}^2_{\mu\nu}=\frac{2}{c^{2}}(E^2-c^2B^2). $$ Proof: $F_{\mu\nu}=-F_{\nu\mu}$, hence ...
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rotational invariance

I have been treating this Hamiltonian: $$H=v\vec{p}\vec{\sigma}+\vec{A}\vec{\sigma}$$ where $\vec{\sigma}=(\sigma_x , \sigma_y)$. It is relevant for 2D graphene quantum dots with some vector ...
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Why $x^2_0-x^2_1-x^2_2-x^2_3$ is invariant under $O(1,3)$?

What exactly means that a certain mathematical statement is invariant under a group? For Ex:$$O(1,3)$$ for $$x^2_0-x^2_1-x^2_2-x^2_3$$ and how do you check it?
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1answer
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Srednicki QFT ch34: invariants of the Lorentz group

In chapter 34 of his Quantum Field theory handbook, Srednicki discusses invariants of the Lorentz group and how they appear in the decomposition in irreducible representations of Lorentz tensors. As ...
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Prove that $\mathbf{E}^2-\mathbf{B}^2$ and $\mathbf{E}\cdot\mathbf{B}$ are the only two independent Lorentz invariant quantities [duplicate]

How to prove that $\mathbf{E}^2-\mathbf{B}^2$ and $\mathbf{E}\cdot\mathbf{B}$ are the only two independent Lorentz invariant quantities that are constructed by $\mathbf{E}$ and $\mathbf{B}$? It's ...
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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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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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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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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
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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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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
783 views

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
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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
549 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
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Given a metric of a torus can we measure its 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
374 views

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
200 views

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
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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
110 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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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
943 views

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
931 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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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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842 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
497 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
617 views

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
1k views

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
742 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 ...