Symmetry Definition and 64 Discussions

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music.This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music.
The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry.

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  1. physics2023

    A What is meant when a phase is said to have "symmetry protected"?

    What is meant when a phase is said to have "symmetry protected"?
  2. M

    I The Feynman way of explaining Symmetry in Physical laws

    So on this page under heading 11-2 Translations first he tries to proof that there is no origin in space. Joe writes newtons laws after measuring quantities from some origin. $$m(d^2x/dt^2)=F_x$$ $$m(d^2y/dt^2)=F_y$$ $$m(d^2z/dt^2)=F_z$$ We need...
  3. Rzbs

    I Space group symmetry

    What does mean spinel structure has F d3m space group? I know F is for face centred cubic, 3 is 3-fold symmetry and m is mirror, but I don't know what means "d"?
  4. B

    Help with Space Inversion Symmetry Problem

    {a} P = identity Matrix w/ -1 on diagonals {b} eigenvalues = +/- 1
  5. I

    I Symmetry and Finite Coupled Oscillators

    For an infinite system of coupled oscillators of identical mass and spring constant k. The matrix equation of motion is \ddot{X}=M^{-1}KX The eigenvectors of the solutions are those of the translation operator (since the translation operator and M^{-1}K commute). My question is, for the...
  6. joneall

    A Symmetry of QED interaction Lagrangian

    I am trying to get a foothold on QFT using several books (Lancaster & Blundell, Klauber, Schwichtenberg, Jeevanjee), but sometimes have trouble seeing the forest for all the trees. My problem concerns the equation of QED in the form $$ \mathcal{L}_{Dirac+Proca+int} = \bar{\Psi} ( i \gamma_{\mu}...
  7. phywithAK

    Vector Field Symmetries

    Untill now i have only been able to derive the equations of motion for this lagrangian when the field $$\phi$$ in the Euler-Lagrange equation is the covariant field $$A_{\nu}$$, which came out to be : $$-M^2A^{\nu} = \partial^{\mu}\partial_{\mu}A^{\nu}$$ I have seen examples based on the...
  8. M

    Weird condition describing symmetry transformation

    I'm a bit confused about the condition given in the description of the symmetry transformation of the filed. Usually, given any symmetry transformation ##x^\mu \mapsto \bar{x}^\mu##, we require $$\bar\phi (\bar x) = \phi(x),$$ i.e. we want the transformed field at the transformed coordinates to...
  9. HaoBoJiang

    A There is a problem about the translation symmetry in the LT

    As we all know, for the reference frame S' and S of relative motion, according to Lorentz transformation, we can get As we all know, for the reference frame S' and S of relative motion, according to Lorentz transformation, we can get As we all know, for the reference frame S' and S of relative...
  10. F

    B field between the plates of a charging capacitor (Ampere's law)

    A standard example consider a capacitor whose parallel plates have a circular shape, of radius R, so that the system has a cylindrical symmetry. The magnetic field at a given distance r from the common axis of the plates is calculated via Ampere's law: \oint_\gamma {\mathbf B} \cdot d{\mathbf...
  11. W

    I What are the global symmetries before and after symmetry breaking?

  12. gregorspv

    Removing connections between equipotential points in solving circuits?

    A sketch of the setup and the equivalent circuit are attached. I believe the correct way to solve this is to redraw the circuit as shown in Fig. 3 and then remove the connections between evidently equipotential points, which reduces the problem to a familiar setup of in parallel and in series...
  13. W

    I Time-reversal symmetry in QM

    I've bumped into a few interesting papers talking about time-reversal symmetry in QM (eg: but I can't seem to wrap my head around the concept. 1) What does it mean for one to say that standard QM isn't time-reversal symmetric? Does this have to do with the...
  14. Danny Boy

    A Query about an article on quantum synchronization

    I am currently studying this paper on quantum synchronization. The first page gives an introduction to synchronization and the basic setup of the ensembles in the cavity. My query is on the second page where the following statements are made. Can anyone see why the implication is that all...
  15. R

    B How to explain "the right hand rule" to an alien universe

    Suppose we are in communication with aliens who live in a different universe. I know, that's impossible, communication requires the exchange of mass or energy, which implies that we live in the same universe. But suppose it is true. I am wondering, can we and the aliens, via this communication...
  16. Q

    A question about Wald's paper on the conserved quantities

    Wald and Zoupas discussed the general definition of ``conserved quantities" in a diffeomorphism invariant theory in this work. In Section IV, they gave one expression (33) in the linked article. I cannot really understand the logic of this expression. Would you please help me with this?
  17. G

    I What Does Gauge Invariance Tell Us About Reality?

    This is not a technical question. I'd like to have a more conceptual discussion about what - if anything - gauge invariance tells us about reality. If we could, please try to keep the discussion at the level of undergrad or beginning grad. To focus my questions and keep things elementary, I'd...
  18. C

    Symmetry in Electrical Circuit Analysis

    How is symmetry used to solve electrical circuits? I have seen several problems in books in which currents in two resistors are said to be equal due to 'symmetry'. That is a concept that I fail to understand and thus cannot apply. In class, we were shown a few circuit diagrams which were...
  19. Ibix

    I Coordinates for diagonal metric tensors

    In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page ( produces an ansatz for the metric as follows (I had to re-type the LaTeX -...
  20. entropy1

    B Symmetry in nature

    Is symmetry a general aspect of physical nature?
  21. Ventrella

    A Differences between Gaussian integers with norm 25

    I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors: (1) Four of them...
  22. J

    Crystal Symmetry Problem

    Hello guys! I have to solve a problem about crystal symmetry, but I am very lost, so I wonder if anyone could guide me. The problem is the following: Using semiclassical transport theory the conductivity tensor can be defined as: σ(k)=e^2·t·v_a(k)·v_b(k) Where e is the electron charge, t...
  23. NatanijelVasic

    I "Unexpected" Symmetry in Elliptical Orbit

    Hello everyone :) Not too long ago, I was thinking about planetary motion around a sun, both with circular orbits and elliptic orbits. However, when thinking a little longer about these two cases in a broader sense, I spotted a big difference which I found quite odd (assume purely classical...
  24. amjad-sh

    A Inversion symmetry in solids

    Hello Can somebody explain for me what is the meaning of inversion symmetry in solids? and why does it breaks at the surface? and also why this inversion symmetry breaking leads to SOC(spin orbit coupling)? If somebody also know a document that explain this in full details(from A to Z) please...
  25. Ventrella

    A Binary fractal tree with equidistant leaves on a circle

    Does there exist a binary fractal tree… (reference: ) …whose leaves (endpoints) lie on a circle and are equidistant? Consider a binary fractal tree with branches decreasing in length by a scaling factor r (0 < r < 1) for...
  26. L

    A Tensor symmetries and the symmetric groups

    In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...
  27. A

    Contractions of the Euclidean Group ISO(3) = E(3)

    Homework Statement Consider the contractions of the 3D Euclidean symmetry while preserving the SO(2) subgroup. In the physics point of view, explain the resulting symmetries G(2) (Galilean symmetry group) and H(3) (Heisenberg-Weyl group for quantum mechanics) and give their Lie algebras...
  28. F

    Magnetic field outside a conducting hollow cylinder

    Homework Statement A current I flows along the surface of a hollow conducting cylinder. The radius of the cylinder section is r. By using Ampere's law, show that the magnetic field B outside the cylinder is B=\frac{\mu_0}{2 \pi} \frac{I}{r} Homework Equations Ampere's law...
  29. S

    A Symmetries in particle physics

    We often use SO(N) and SU(N) to describe symmetries in particle physics. I am not clear which one to choose when I try to discuss a symmetry. For example, why do we use SU(3) but not SO(3) to describe the symmetry of the three colors of quarks? Similarly, why do we use SU(2) but not SO(2) to...
  30. S

    I Can a Hermitian matrix have complex eigenvalues?

    Hi, I have a matrix which gives the same determinant wether it is transposed or not, however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian? If so, is there any other name to classify it, as it is not...
  31. J

    A Breaking of a local symmetry is impossible, so what about global symmetry....

    Breaking of a local symmetry is impossible. It is often said that therefore the role of the Higgs mechanism in the standard model is a different one. Namely, Once a gauge is fixed, however, to remove the redundant degrees of freedom, the remaining (discrete!) global symmetry may undergo...
  32. S

    A Symmetry/Conservation Violated in Quantum Anomaly

    String theorists have apparently applied String Theory to expose a Quantum Anomaly in a physical analog system: electrons flowing in a Niobium Phosphide crystal. The electrons were found to violate symmetry in relation to Spin...
  33. Eslam100

    Intro Physics Textbooks on symmetries and physics

    I just started to develop an interest in symmetries after taking an introductory course in electromagnetism . The instructor explained to us how physical laws can be obtained by considering the symmetries of the physical system. It was really amazing how we can obtain such information just by...
  34. ShaddollDa9u

    Electrostatics and symmetry

    Homework Statement "Find the direction and the variables which the electrostatic field depends on at all points of the plane (xOy) uniformly charged with the density of positive charges ϱ" The Attempt at a Solution So first of all, I have to study the Invariances of symmetry. I tried to...
  35. Marcus95

    Fourier Series Coefficient Symmetries

    Homework Statement Let ## f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx) ## What can be said about the coefficients ##a_n## and ##b_n## in the following cases? a) f(x) = f(-x) b) f(x) = - f(-x) c) f(x) = f(π/2+x) d) f(x) = f(π/2-x) e) f(x) = f(2x) f) f(x) = f(-x) =...
  36. P

    I The Symmetry of the Liénard–Wiechert potentials

    According to responses at: The Lorentz contraction of the electric field of a charge with uniform velocity is supposed to be symmetric across the plane pi/2 radians from the velocity vector of the...
  37. J

    Quantum Theoretical Physics course textbooks

    Hello, I will be attending an undergraduate course called "Theoretical Physics" and I want to borrow some books from the library that cover the material of this course. I would appreciate any suggestions. The syllabus of the course is the following(I will be translating so I am sorry If...
  38. S

    A Symmetry of hamiltonian under renormalization

    Hi everyone, Currently, I am self-learning Renormalization and its application to PDEs, nonequilibrium statistical mechanics and also condensed matter. One particularly problem I face is on the conservation of symmetry of hamiltonian during renormalization. Normally renormalization of...
  39. Y

    Time Inversion Symmetry and Angular Momentum

    Homework Statement Let ##\left|\psi\right\rangle## be a non-degenerate stationary state, i.e. an eigenstate of the Hamiltonian. Suppose the system exhibits symmetry for time inversion, but not necessarily for rotations. Show that the expectation value for the angular momentum operator is zero...
  40. S

    I Singlet state meaning

    Hello! I am a bit confused about the relation between the singlet configuration and symmetry of the system. So in the spin case, for 2, 1/2 particles, the singlet configuration is antisymmetric. But I read that the quarks are always in a singlet configuration, which means that they are symmetric...
  41. K

    B Matrices of su(3) and sphere symmetry

    i used to get pauli matrices by the following steps it uses the symmetry of a complex plane sphere i guess so..? however i can't get the 8 gell mann matrices please help ! method*: (x y) * (a b / c d ) = (x' y') use |x|^2 + |y|^2 = |x'|^2 + |y'|^2 and |x| = x * x(complex conjugate) this way...
  42. P

    A SU groups in QCD

    I am trying to learn about the various SU groups related to QCD. I have about 5 QFT and Particle physics books from my student library and written down about 20 pages of handwritten notes about specific parts of say generators, matrices, group properties etc. - but i don't really feel that I...
  43. Elvis 123456789

    B How do you know a force if a force is radially symmetric?

    If a force only depends on a radial distance "r" and it only has a radial component in the "er" then is it radially symmetric? This pertains to some homework problem I have, but part of the problem is that I'm not exactly sure what is meant by "radially symmetric". I assume its asking if the...
  44. Robin04

    I Time translation symmetry and the Big Bang

    Hi, As I know we now think that time translation is not a symmetry of spacetime because of the Big Bang, so we cannot say that our physical laws are applicable at every point in time. But then isn't the developing of the Big Bang theory against this asymmetry?
  45. M

    Quantum Mechanics-Spin State for Identical Particles

    Consider a system of two identical spin-1 particles. Find the spin states for this system that are symmetric or antisymmetric with respect to exchange of the two particles. (Problem 13.3, QUANTUM MECHANICS, David H. McIntyre) I know that for bosons, the total wavefunction should be symmetric...
  46. S

    I Properties of body with spherical symmetry

    I'm studing Gauss law for gravitational field flux for a mass that has spherical symmetry. Maybe it is an obvious question but what are exactly the propreties of a spherical simmetric body? Firstly does this imply that the body in question must be a sphere? Secondly is it correct to...
  47. A. Neumaier

    A Is Poincare symmetry the real thing?

    This is a continuation of a side issue from another thread.
  48. W

    The elasticity/stiffness tensor for an isotropic materials

    Hi PF, As you may know, is the the elasticity/stiffness tensor for isotropic and homogeneous materials characterized by two independant material parameters (λ and μ) and is given by the bellow representation. C_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu(\delta_{ik}\delta_{jl} +...
  49. C

    In an infinite quantum well, why Δn=0?

    I've been reading up a bit on semiconductor quantum wells, and came across a selection rule for an infinite quantum well that says that "Δn = n' - n = 0", where n' is the quantum well index of an excited electron state in the conduction band, and n is the index of the valence band state where...
  50. S

    'Symmetry argument' for eigenstate superposition

    Homework Statement For an infinite potential well of length [0 ; L], I am asked to write the following function ##\Psi## (at t=0) as a superposition of eigenstates (##\psi_n##): $$\Psi (x, t=0)=Ax(L-x) $$ for ## 0<x<L##, and ##0## everywhere else. The attempt at a solution I have first...