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. V

    I Why are scalar fields Lorentz invariant?

    Hi. This question most probably shows my lack of understanding on the topic: why are scalar fields Lorentz invariant? Imagine a field T(x) [x is a vector; I just don't know how to write it, sorry] that tells us the temperature in each point of a room. We make a rotation in the room and now...
  2. F

    I Why is energy not Lorentz invariant?

    As I understand it, since space-time is modeled as a four dimensional manifold it is natural to consider 4 vectors to describe physical quantities that have a direction associated with them, since we require that physics should be independent of inertial frame and so we should describe it in...
  3. physicality

    I How to show that Electrodynamics is conformally invariant?

    [Moderator's note: changed thread title to be more descriptive of the actual question.] Consider Maxwell's action ##S=\int L## over Minkovski space, where the Lagrangian density is ##L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}##, and the Electromagnetic tensor is given by ##F^{\mu\nu} = \partial^\mu...
  4. H

    I Prove centre of mass of an arc is rotationally invariant

    Suppose the coordinates ##(\bar{x}, \bar{y})## of the centroid (or the centre of mass) of an arc is defined as follows ##\bar{x}=\frac{1}{L}\int x\,ds## and ##\bar{y}=\frac{1}{L}\int y\,ds##, where ##L## is the arc length. Could you prove that the centroid is invariant under a rotation of...
  5. G

    I Newtonian 4-Momentum Norm Analogue

    Hi. I read that the Lorentz invariance Minkowski norm of the four-momentum $$E^2-c^2\cdot \mathbf{p}^2=m^2\cdot c^4$$ has no analogue in Newtonian physics. But what about $$E-\frac{\mathbf{p}^2}{2m}=0\quad ?$$ It might look trivial by the definition of kinetic energy, but it's still a relation...
  6. D

    I Complex Exponential solutions in time invariant systems

    Hi there! First Post :D In a recent CM module we've been looking at coupled oscillators and the role of time translational invariance in the description of such physical systems. I will present the statement that I am having trouble understanding and then continue to elaborate. In stating that...
  7. bananabandana

    I Energy levels generally invariant under fixed V, changing T

    Why is is true that for a given system, if I fix the volume and increase the temperature, you'd expect the occupancy of the energy levels to change, but not the levels themselves? Can I think of this in terms of the fact that the boundary conditions for the solution of the TISE are fixed, such...
  8. S

    Scale Invariant Classical Field Theory

    Homework Statement A class of interesting theories are invariant under the scaling of all lengths by ##x^{\mu} \rightarrow (x')^{\mu}=\lambda x^{\mu}## and ##\phi(x) \rightarrow \phi'(x) = \lambda^{-D}\phi(\lambda^{-1}x)##. Here ##D## is called the scaling dimension of the field. Consider...
  9. F

    I Third Invariant expressed with Cayley-Hamilton Theorem

    The Cayley-Hamilton Theorem can be used to express the third invariant of the characteristic polynomial obtained from the non-trivial solution of the Eigenvector/Eigenvalue problem. I follow the proof (in Chaves – Notes on Continuum Mechanics) down to the following equation, then get stuck at...
  10. I

    I How can we visualize spin being invariant?

    I have a basic question about what spin invariance means. If we were to use a classical example, if the spin of a basketball was invariant, what would that mean? Would every frame measure the same revs/min? Newtonian mechanics predicts that every frame would measure the same revs/min, but sr...
  11. S

    I Understanding Lorentz-Transformability & Invariant Operations

    An operation is frame-invariant if it maintains the Lorentz-transformability of an input, yes? So for example, the coordinate ##x## transforms according to ##x^\prime = \gamma(x-vt)##, and multiplying the unprimed space and time coordinates by ##c## would give ##cx^\prime = c\gamma(x-vt)##. In...
  12. H

    Orbit invariant under reflection about apsidal vectors

    The book argues that since substituting ##\theta## by ##-\theta## leaves the orbit equation (3.34) unchanged, the orbit is therefore invariant under reflection about the apsidal vectors (Fig 3.12). If substituting ##\theta## by ##-\theta## leaves the orbit equation (3.34) unchanged, then there...
  13. A

    Weyl Tensor invariant under conformal transformations

    Homework Statement As the title says, I need to show this. A conformal transformation is made by changing the metric: ##g_{\mu\nu}\mapsto\omega(x)^{2}g_{\mu\nu}=\tilde{g}_{\mu\nu}## Homework Equations The Weyl tensor is given in four dimensions as: ##...
  14. Sentosa

    I Does time dilation cause the speed of light to be invariant?

    I'm trying to understand why the speed of light is the same for all observers. I have found different answers on-line. This page claims that it relates to time dilation. But consider the following thought experiment: two ships flying at 98% c. Ship A is moving toward the sun, and ship B is...
  15. PatrickUrania

    I Why are the gamma-matrices invariant?

    Hi, I've been studying Dirac's theory of fermions. A classic topic therein is the proof that the equation is covariant. Invariably authors state that the gamma-matrices have to be considered constants: they do not change under a Lorentz-transformation. I am looking for the reason behind this. It...
  16. philton

    Phonon Lorentz Invariance in Superfluids - Papers?

    It is said phonon(not photon) in superfluid experiments could also produce similar upper-limit speed effect which I'm not sure if that's also Lorentz invariant. Another problem is that I can't dig out those paper that demonstrates this kind of effect. Anyone ever seen any of this paper? Thanks..
  17. T

    Calculating the Z2 Invariant in Kane-Mele Model

    Hi Everyone, As part of my computational project in topological insulators, I wish to calculate the Z2 invariant in my tight binding model of Kane-Mele Graphene. I have so far produced band structure and surface states consistent with literature, and have been looking at the theory of the Z2...
  18. F

    Why does the temperature is invariant on phase transitions?

    Hello there, this is a fact that I can't understand thinking about it... Energy has to be given so that the inter molecular bonds can be broken, and energy goes in when there are formed (even though that sounds very counter intuitive to me), but why is the energy all distributed to potential...
  19. S

    Proof than an equation is Lorentz invariant

    In Peskin and Schroeder page 37, it is written that Using vector and tensor fields, we can write a variety of Lorentz-invariant equations. Criteria for Lorentz invariance: In general, any equation in which each term has the same set of uncontracted Lorentz indices will naturally be invariant...
  20. C

    How Is the Anticommutator Derived in SU(3) Algebra?

    'Using the following normalization in the su(3) algebra ##[\lambda_i, \lambda_j] = 2if_{ijk}\lambda_k##, we see that ##g_{ij} = 4f_{ikl}f_{jkl} = 12 \delta_{ij}## and, by expanding the anticommutator in invariant tensors, we have further that $$\left\{\lambda_i, \lambda_j\right\} =...
  21. D

    Green's functions for translationally invariant systems

    As I understand it a Green's function ##G(x,y)## for a translationally invariant differential equation satisfies $$G(x+a,y+a)=G(x,y)\qquad\Rightarrow\qquad G(x,y)=G(x-y)$$ (where ##a## is an arbitrary constant shift.) My question is, given such a translationally invariant system, how does one...
  22. carllacan

    Relativistic collision and invariant s

    Homework Statement Write the invariant s = (P1+P2)2 as a function of masses amd energies of the process 1+2 → 3+4 in the center of momentum frame and on the lab frame, in which b is at rest. Interpret the result. Homework EquationsThe Attempt at a Solution For the CoM frame I have: s =...
  23. DOTDO

    Lorentz invariant integral measure

    Hi I'm studying electron-muon scattering and now considering the Lorentz invariant integration measure. The textbook introduced it, which use dirac delta function to show that d3p/E is a Lorentz scalar. I understood it but I wanted to find other way and tried like this: I need a hint on the...
  24. S

    Eigenvalues are invariant but eigenvectors are not

    Hi there. How would I show that the eigenvalues of a matrix are an invariant, that is, that they depend only on the linear function the matrix represents and not on the choice of basis vectors. Show also that the eigenvectors of a matrix are not an invariant. Explain why the dependence of the...
  25. D

    What is the Kretschmann Invariant?

    Hi. I have a couple more questions in my quest to self-study GR. 1 - I have some notes where the Kretschmann Invariant is defined as Rk = RabuvRabuv and is given in Schwarzschild coordinates as Rk = 48u2/r6 . My notes say this is an invariant field so that its value at any point as evaluated in...
  26. T

    Exploring the Role of Z2 Invariant and Edge States in Topological Insulators

    I have been reading about Z2 topological invariant recently. However, after some literature survey, I still cannot understand Z2 invariant in language of time reversal polarization. Basically, my struggle includes the following two questions: As the ref paper says(see the picture below): On...
  27. Bakali Thendo

    Proving Maxwell's Equations are Lorentz Invariant

    I want to know how can i prove that Maxwell's equations for the propagation of electromagnetic wave are Lorentz invariant.
  28. O

    MHB What is the relationship between $f$ and $X$?

    let f:X to X be and f(X) C X...then f is invariant..if f is invariant, then f is self map on X ? is it true ?
  29. Kairos

    Time dilation, length contraction, but velocity invariant

    If a frame is moving at constant velocity relative to an observer, this observer perceives a time dilation and a length contraction. But in this case how the velocity (length/time) can appear constant ? It is expected to be contracted.. Thank you in advance for the explanation
  30. B

    Would magnetic charge be Lorentz invariant?

    Would magnetic charge be Lorentz invariant (the way electric charge is) if magnetic charge existed?
  31. Clueless

    When is the momenergy vector invariant?

    This has been confusing me for an entire hour or so. The momenergy vector is defined as mass*((spacetime displacement)/(proper time)). I understand this as much, but I don't know how to apply it to situations. The following snapshot is from a question about a photon colliding with a stationary...
  32. Imager

    Invariant mass of a photon changes - from Wiki

    I'm reading the Wiki article below to say the invariant mass of photons in an expanding volume of space will decrease. I thought invariant mass of a photon was always zero and the energy of photon changed due to the expansion of space. So where did I go wrong? Quote from Wiki General...
  33. MTd2

    LQG is not proved to be locally lorentz invariant. (Bee)

    Sabine Hossenfelder said... Arun: This has never been proved. These deformations are problematic for other reasons, but they don't suffer from the density problem that I alluded to here, if that is what you mean, yes. LQG itself isn't actually based on a space-time network so the argument...
  34. ognik

    MHB Antisymmetry Invariant Under Similarity Orthogonal Transforms

    Hi - the text is very brief on similarity transforms and wiki etc. a bit beyond where I am. In fact I think I am muddling a few things up, so I have a few questions around this topic please: 1) I'd appreciate a 'beginners' explanation of similarity transforms, what they really are and what they...
  35. M

    Making Lagrangian gauge invariant

    Homework Statement [/B] The Lagrangian ##\mathcal{L}\frac{1}{2}(\partial_\mu\phi^\nu)^2+\frac{1}{2}(\partial_\mu\phi^\mu)^2+\frac{m^2}{2}(\phi_\mu\phi^\mu)^2## for the vector field ##\phi^\mu## is not invariant with respect to the gauge transformation ##\phi^\mu\rightarrow...
  36. Primroses

    Why are invariant tensors also Clebsch-Gordan coefficients?

    On one hand, in reading Georgi's book in group theory, I comprehend the invariant tensor as a special "tensor", which is unchanged under the action of any generators. On the other hand, CG decomposition is to decompose the product of two irreps into different irreps. Now it is claimed that...
  37. F

    Is the wave function the invariant thing?

    Is the wave function of quantum mechanics considered to be the quint-essential invariant object? Is it the wave function that must not change with space, time, gravitational field, etc? It would seem to me that the relative probabilities that things happen is the thing that can not change with...
  38. c3po

    Find matrix representation for rotating/reflecting hexagon

    Homework Statement Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves...
  39. I

    Interplay of space and time in Spacetime invariant interval

    According to Special Relativity, the same event could have a different time duration and a different space extension for different observers, depending on their frame of reference. Relativity subsequently introduced the the notions of Spacetime as a continuum ( as opposed to the classical...
  40. V

    Violation in diffraction? Lagrange (Optical) invariant

    It says you can not change with lenses the value L - radiance. Below I have an example where it proves that you can or where am I wrong? (I made L for 2D case, in 3D case everything the same - L2>L1)
  41. binbagsss

    Ds^2 Invariant Interval: Sign Dependence?

    Some sources have ##ds^{2}=d\tau^{2} ##, and others have ##ds^{2}=-d\tau^{2}##, Does the sign depend on the signature chosen for the metric? Thanks in advance.
  42. C

    Why is Normal Area to Light Rays Invariant?

    It is a fact that all inertial observers would measure the same area normal to a beam of light rays in relativity. You can prove this by considering the displacement vector connecting a light ray to its neighbouring light rays. But I wondered if there were some intuitive explanation of why this...
  43. A

    Proving Newton's third law invariant with Galilean tranfrom

    Homework Statement Consider Newton’s force law for two particles interact through a central force F12(r1',r2',u1,u2), where by Newton’s third law F12 = -F21. m1(d^2r1/dt^2) = F12(r1,r2,u1,u2) m2(d^2r2/dt^2) = F21(r1,r2,u1,u2) A. Show that Newtonian mechanics is form invariant with respect to...
  44. Z

    Some questions on Invariant Mass Spectrum

    What can we learn from Invariant Mass Spectrum?How to measure it?So,how to read it? Mass measurement is converted into energy measurement,but how could we make the quantity change continuously in order to form the horizontal axis? How to divide different particles to measure them...
  45. binbagsss

    Meaning of tensor invariant, covariant differentiation

    E.g - considering co variant differentiation, The issue with the normal differentiation is it varies with coordinate system change. Covariant differentiation fixes this as it is in tensor form and so is invariant under coordinate transformations.'If a tensor is zero in one coordinate system...
  46. J

    Spacetime Interval & Metric: Equivalent?

    This may seem an odd question but it will clear something up for me. Are "The spacetime interval is invariant." and the "The spacetime metric is a tensor." exactly equivalent statements? Does one imply more or less information than the other? Thanks!
  47. T

    Is there a difference between rest mass and invariant mass?

    1. is there a difference between 'rest mass' and 'invariant mass'? I thought there wasn't... To put it another way (or maybe this next question is a different question): 2. Is there a difference between the rest mass of a positron/electron pair, and the rest massa of a system containing two...
  48. C

    Lagrangian is invariant under the transformation

    I should mention that I'm self-studying this material, not taking it as part of a course, but since this is still a homework-style problem I figured it'd be best to post here. Homework Statement In Peskin and Schroeder problem #11.2, they ask us to consider the Lagrangian: $$\mathcal{L} =...
  49. V

    Solving the D3x D3p Invariant Puzzle

    Homework Statement Hello, I have probably quit easy task, but I don't know how show that d3x d3p is a Lorentz invariant.Homework EquationsThe Attempt at a Solution I mean I have to show that d3x d3p = d3x' d3p', where ' marks other system. I can prove ds2=ds'2, but I am not sure what with p...
  50. D

    Showing that the real Klein-gordon lagrangian is Lorentz invariant

    Homework Statement Hey guys! So this question should be simple apparently but I got no idea how to do it. Basically I have the following Lagrangian density \mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)-\frac{m}{2}\phi^{2} which should be invariant under Lorentz...
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