What is Invariant: Definition and 405 Discussions

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.

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1. I Citation needed: Only multivariate rotationally invariant distribution with iid components is a multivariate normal distribution

I need a citation for the following proposition: Assume a random vector ##X=(X_1, ..., X_n)^T## with iid components ##X_i## and mean 0, then the distribution of ##X## is only invariant with respect to orthogonal transformations, if the distribution of the ##X_i## is a normal distribution. Thank...
2. Pendulum attached to a rotating vertical disk

For this problem, I correctly got the same coordinates for the pendulum mass using another coordinate system. The coordinate system I used was the other coordinate system rotated counterclockwise by 90 degrees. Why is the pendulum mass coordinates invariant in my cartesian coordinate system...
3. I Question about full Lorentz transformation

Hello everyone, I recently have learned about space time intervals and how these intervals between two events are invariant across all inertial frames and this can be proven by using the full Lorentz transformation. I wanted to learn more about the full Lorentz transformation and I read the...
4. I Invariant definition of acceleration in Newtonian physics vs proper acceleration in GR

Does it exist an invariant way to define acceleration in Newton physics like the proper acceleration in GR ? In Newton physics if an accelerometer attached to an object reads 0 it does not mean it is actually not accelerating (since gravity is a force). To define inertial motion the concept of...
5. A Relativistically invariant 2-body phase space integral

I encounter a function that I don‘t know in the calculation of Relativistically invariant 2-body phase space integral: in this equation, ##s##is the square of total energy of the system in the center-of-mass frame(I think) I don't know what the function ##\lambda^{\frac{1}{2}}## is. There are...
6. A What is this unitary invariant?

Hi Pfs I am reading this article: https://arxiv.org/abs/0810.2091 It is know that hearing the possible frequencies emitted by a drum are not enough to know its shape. Here the frequencies are the eigenvalues of the Dirac operator. the missing information is the unitary invariant of the title...
7. A Are equations of motion invariant under gauge transformations?

We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations? If yes, can you show a mathematical proof (instead of just saying in words)?
8. B Invariant under rotation: Banal, obvious, or noteworthy?

Given a cartesian coordinate system with a fixed point of origin and three axes, it is a fact, that the coordinates of a point P change, when the coordinate system is rotated around its point of origin. The distance between the origin and point P is of course unaffected by such a rotation. What...
9. B Solving for the Nth divergence in any coordinate system

Preface We know that, in Cartesian Coordinates, $$\nabla f= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}$$ and $$\nabla^2 f= \frac{\partial^2 f}{\partial^2 x} + \frac{\partial^2 f}{\partial^2 y} + \frac{\partial^2 f}{\partial^2 z}$$ Generalizing...
10. I Lorentz invariant phase space and cross section

Can someone please explain to me how can we obtain this integral in eq. 5.27 from eq. 5.26? I quite do not understand how is it possible to make this adjustment and why the (p_(f))^2 appeared there in the numerator and also why a solid angle appeared there suddenly.
11. A Maxwell theory invariant under dual field strength tensor application

Hello everybody! I know in classical field theory adding in the Lagrangian density a term of the form Fαβ (*F)αβ (where by * we denote the dual of the field strength tensor) does not change the EOM, since this corresponds to adding a total derivative term to the action. However when computing...
12. I Equation which is related with the Lorentz invariant quantities

Hi every one.How can i prove the below equation? And then that it's Lorentz invariant quantitude ? Thanks for your help
13. How can l prove that Newton's laws are time invariant?

how can l prove Newton's law is time invariant? if x (t) is a solution of dd/ddx x(t) = f(x(t)) then if l put y(-t) dd/ddt y(t)=dd/ddt x(-t). Now how dd/ddt x(-t) is equal to f(x(-t))?dd/ddt is second derivative with respect to time
14. I Is Acceleration an Invariant? (Taylor & Wheeler)

I've been reading Spacetime Physics by Taylor and Wheeler at @PeterDonis's suggestion. In chapter 3, they say: Ok, so force is not an invariant. Fine. Then I go to http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html, where the term ##a## appears in various equations and is...
15. Subset of the domain for the transformation to be invariant

I found that the a) invariant points are all points on y-axis b) invariant lines are y-axis and ##y=c## where ##c## is real I am confused what the final answer should be. How to state the answer as "subset of domain"? Is it: $$\{x,y \in \mathbb R^2 | (0, y) , x = 0, y=c\}$$ Thanks
16. Invariant mass and energy balance

a) Two particles have energies E1 and E2, and momenta p1 and p2. Write down an expression for the invariant mass of this two-particle system. Leave your answer in terms of E1 and E2, and p1 and p2. b) A typical photon (γ) in the Cosmic Microwave Background (CMB) has an energy of kBTCMB, where...
17. B Why is Light Speed Invariant & Finite?

I think infinite speed is unimaginable. If something is moving at infinite speed, we can't find it at all because it has moved to infinity. Furthermore, when the maximum speed is limited, a reasonable inference should be that observers in different reference frames should find the same one speed...
18. I An equation invariant under change of variable

It's said that the below equation is invariant under a substitution of ##-\theta## for ##\theta## , ##\frac{d^{2} u}{d \theta^{2}}+u=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)## I can't understand this how this is so. It's supposed to be obvious but I can't see it. Please help...
19. I Left Invariant Metric: What I Don't Understand

I haven't learned about Lie Groups yet, but came across this question. What I don't understand: - is the semi-direct product ##R_+ \ltimes R^4## here a matrix group with elements ##\begin{pmatrix} \lambda & x^{\mu} \\ 0 & 1 \end{pmatrix}##? And is the group multiplication then matrix...
20. MHB Find the elementary divisors and invariant factors

Hello I have problems with this exercise Find the elementary divisors and invariant factors of each of the following groups a) $G1= Z_6 \times Z_{12} \times Z_{18}$ , b) $G_2= Z_{10} \times Z_{20} \times Z_{30} \times Z_{40}$Thanks
21. How invariant is digestion of a certain substance?

In around 40% of people, ##C_4H_6O_2S_2## is enzymatically broken down into sulfur-containing compounds such as methanethiol, dimethyl sulfide, or dimethyl disulfide. Is it basically possible to belong to these 40% and later in life to the other 60%? How stable is "enzymatically" and what are...
22. A Understanding that QCD is not CP invariant

In "CP violation" book by Bigi and Sanda (section 8.2.1. QCD), I read that "the QCD Lagrangian is invariant under CP transformations" and wanted to prove it. The QCD Lagrangian is given by \mathscr{L}_{QCD} = \bar \Psi^f [i \gamma^{\mu}D_{\mu} -m_f] \Psi^f - \frac 1 4 G_{i...
23. Proving modified Maxwell action is gauge invariant

I want to show that the action staying the same action after taking ##A^\mu \to A^\mu + \partial ^\mu \chi##, for the first term I suceeded in showing the invariance using the fact ##[\partial ^ \mu , \partial ^\nu]=0## but for the second term I'm getting: ##\epsilon^{\alpha\mu\nu}A_\alpha...
24. I Speed of light not an invariant in GR

Hi all, I need help understanding the light ray bending in the original GR 1916 paper, Die Grundlagen.... First of all, Einstein states the ##c## is not an invariant in GR. In fact, from (70) and (73), it stems that $$\gamma = \sqrt{ -\frac {g_{44}}{g_{22}} },$$ where ##\gamma## is ##|c| <= 1##...
25. Pulley's mass is invariant to the physics problem?

I have this system of masses and the goal is to find the velocity of $m_1$ at the ground. But it gives me the moment of inertia of the pulley as well which is $xMR^2$. I know how to solve a pulley problem but since it gives me the moment of inertia of the pulley maybe it has something to do...
26. I Equations of Spacetime Invariant - Understanding the Difference

The distance/difference between two points in spacetime can be written in two forms (as shown in attachment). Can anyone explain the difference in the two equations? I have read that the two equations are the same, but i don't understand the change in sign. Why is it written in two forms...
27. I Variant and Invariant Physical Quantities....

Hello, In non-relativistic physics (where things move slower than the speed of light), the following physical quantities are invariant and variant (or relative) i.e. vary in value depending on the chosen frame of reference: Variant quantities: time ##t##, velocity ##v##, momentum ##p##...
28. Showing that the Weyl tensor is invariant under conformal symmetries

The Weyl tensor is given by (Carroll's EQ 3.147) \begin{align*} C_{\rho \sigma \mu \nu} &= R_{\rho \sigma \mu \nu} - \frac{2}{n-2}\left(g_{\rho [\mu}R_{\nu]\sigma} - g_{\sigma [\mu}R_{\nu]\rho}\right) \\ &+ \frac{2}{(n-1)(n-2)}g_{\rho [\mu}g_{\nu]\sigma}R \end{align*} Where ##n## are...
29. How Do I Perform a Galilei and Lorentz Transform?

c^2 * t^2 = x^2 + y^2 + z^2 = r^2
30. A Invariance of ##SO(3)## Lie group when expressed via Euler angles

I am trying to understand the properties of the ##SO(3)## Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions. I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of ##SO(3)## dynamics ##\mathbf{\dot{R}} =...
31. I Different invariant tori in the case of a 2D harmonic oscillator

Hi everyone! Both sources I'm currently reading (page 291 of Mathematical Methods of Classical Mechanics by Arnol'd - get it here - and page 202 of Classical Mechanics by Shapiro - here) say that, in the case of the planar harmonic oscillator, using polar or cartesian coordinate systems leads...
32. I How to call an invariant quantity?

When referring to an invariant quantity, which one is better "coordinate independent" or "frame independent"?
33. A Exists ? : Invariant geodesic equation

Does there exist a form of the geodesic equation which is invariant under coordinates change ?
34. B Nearly Scale Invariant Power Spectrum in inflation

I recall hearing once a very intuitive explanation as to why inflation is thought to lead to a nearly scale invariant power spectrum but i can't recall it. Can anyone offer an explanation that might help me? Why is it nearly scale invariant and not perfectly scale invariant? many thanks
35. E

B Showing that formulae are invariant for all values in the domain

I was just thinking about this and couldn't decide whether it was a silly question or not, so naturally I thought I might ask. It was partly prompted by one of the questions asked in the homework section. Every physical law I can think of is "self-correcting" if you substitute negative values...
36. Find a transformation that leaves the given Lagrangian invariant

The given Lagrangian is: ##L = \frac 1 2 m_1 ( \dot x_1^2 + \dot y_1^2 + \dot z_1^2) + \frac 1 2 m_2 ( \dot x_2^2 + \dot y_2^2 + \dot z_2^2) + G \frac{m_1 m_2}{\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}}##Please note: I have been inspired by the post...
37. I Is the four current in Relativity an invariant quantity?

Is the four current in relativity an invariant quantity? I know the divergence is zero for the four gradient, i.e. the continuity equation. But is the four current a vector in the sense that it has invariant properties?
38. I What is the Invariant Mass of Two Perpendicular Photons?

I think since Esystem=(PsystemC)^2 + (Minvariant C^2)^2. Then the invariant mass of the system should be zero, but I am hesitated with this is it always the case that photon that travels perpendicular to each other have zero invariant mass
39. B Is Bell's inequality Lorentz invariant?

I browsed the net and found : https://arxiv.org/abs/quant-ph/0408127 It is said the value of Bell's operator depends on the speed, so how can it be Lorentz invariant ?
40. I 2 and 3 dimensional invariant subspaces of R4

I am looking at the representation of D4 in ℝ4 consisting of the eight 4 x 4 matrices acting on the 4 vertices of the square a ≡ 1, b ≡ 2, c ≡ 3 and d ≡ 4. I have proven that the 1-dimensional subspace of D4 in ℝ2 has no proper invariant subspaces and therefore is reducible. I did this in 2...
41. How to Calculate Invariant Mass for Negligible Particle Mass?

So I know that since we are ignoring the mass of the electron, and the proton starts at rest, the proton has no KE and the electron has no rest energy. So the initial total energy of the system would be rest energy of proton + KE of electron = 2GeV + .938GeV = 2.938 GeV and since energy is...
42. [Special Relativity] Scalar Invariant under a Lorentz-transformation

"My" Attempted Solution To begin, please note that a lot - if not all - of the "solution" is largely based off of @eranreches's solution from the following website: https://physics.stackexchange.com/questions/369352/scalar-invariance-under-lorentz-transformation. With that said, below is my...
43. A Towards formulating an invariant Lagrangian

Assuming a Lagrangian proportional to the following terms: ##L \sim ( \partial_\mu \sigma) (\partial^\mu \sigma) - g^{m\bar{n}} g^{r\bar{p}} (\partial_\mu g_{mr} ) ( \partial^\mu g_{\bar{n}\bar{p}} ) ~~~~~ \to (1) ## Where ##\mu =0,1,2,3,4## and m, n,r, p and ##\bar{n}, \bar{p}, \bar{m}## and...
44. A Understanding time invariant of EM

on the conquering the physics gre book it says e.g. for time invariant "if you can see someones eyes in a mirror, they can see yours as well" so what the hell does that mean? isnt person A sending photons to person B and person B sending different sets of photons to person A? how does that...
45. A Showing Delta^3(p-q) is Not Lorentz Invariant

From page 22 of P&S we want to show that ##\delta^{3}(\vec{p}-\vec{q})## is not Lorentz invariant. Boosting in the 3-direction gives ##p_{3}' = \gamma(p_{3}+\beta E)## and ##E' = \gamma(E+\beta p_{3})##. Using the delta function identity ##\delta(f(x)-f(x_{0})) =...
46. Trying to use polar coordinates to find the distance between two points

##{dx}^2+{dy}^2=3^2+3^2=18## ##{dr}^2+r^2{d\theta}^2=0^2+3^2*(\theta/2)^2\neq18## I have a feeling that what I'm doing wrong is just plugging numbers into the polar coordinate formula instead of treating it as a curve. For example, I naively plugged in 3 for r even though I know the radius...
47. I Invariant Mass in Gravitational Fields: Special Relativity

In Special Relativity, you learn that invariant mass is computed by taking the difference between energy squared and momentum squared. (For simplicity, I'm saying c = 1). m^2 = E^2 - \vec{p}^2 This can also be written with the Minkowski metric as: m^2 = \eta_{\mu\nu} p^\mu p^\nu More...
48. I Invariant properties of metric tensor

Which properties of metric tensor are invariant of basevectors transforms? I know that metric tensor depends of basevectors, but are there properties of metric tensor, that are basevector invariant and describe space itself?
49. I Invariant symbol implies existence of singlet representation

I don't understand what the last paragraph of the attached page means. Why does the Kronecker delta being an invariant symbol mean that the product of a representation R and its complex conjugate representation has the singlet representation with all matrices being zero? Doesn't the number...
50. Prove that these terms are Lorentz invariant

Homework Statement Prove that \begin{align*}\mathfrak{T}_L(x) &= \frac{1}{2}\psi_L^\dagger (x)\sigma^\mu i\partial_\mu\psi_L(x) - \frac{1}{2}i\partial_\mu \psi_L^\dagger (x) \sigma^\mu\psi_L(x) \\ \mathfrak{T}_R(x) &= \frac{1}{2}\psi_R^\dagger (x)\bar{\sigma}^\mu i\partial_\mu\psi_R(x) -...