What is Symmetry: Definition and 955 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. H

    I Spherical Symmetry & Electron Spin: An Exploration

    Can an electron in a spherically symmetrical potential energy function have non-zero spin angular momentum?
  2. Alfredo Tifi

    I The Symmetry of Angular Momentum Conservation

    I suppose that the principle of conservation of angular momentum holds also for a cloud of particles weekly interacting at low pressure, density and temperature. And it should be still applicable when the particles or the atoms would start condensing and forming fusion products or simply solid...
  3. M

    A Observables when the symmetry is not broken?

    Hi, Let be a scalar field φ that permeates all space. The quantum of the field has a mass m. The field is at the minimum of its potential. When this minimum is for φ≠0 (a broken symmetry), the quantum may be observed by exciting the field, as with the Higgs boson. But if the symmetry is not...
  4. 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...
  5. QuasarBoy543298

    Exploring Gauge Symmetry in Classical Field Theory

    hi, I'm currently taking a classical field theory class (electromagnetism in the language of tensors and actions and etc) and we have just encountered the gauge symmetry, that is for the 4 vector potential we can add a gradient of some smooth function and get the same physics (if we take Aμ →...
  6. L

    I Definition of a symmetry transformations in quantum mechanics

    By the Wigner theorem, symmetries transformations are implemented by operators ##\hat{U}## that are unitary or antiunitary. This is what is written in most books. But I have read somewhere that, to ##\hat{U}## represent a symmetrie, it's necessary that ##\hat{U}^{\dagger} \hat{H} \hat{U} =...
  7. G

    I Symmetry of an exponential result from a Fourier transform

    I used a matrix to calculate the Fourier transform of a lorentzian and it did generate a decaying exponential but that was followed by the mirror image of the exponential going up. I am referring to the real part of the exponential. If I use an fft instead I also see this. Shouldn't the...
  8. 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...
  9. N

    Find an equation of a line of symmetry in the form px+qy = r

    Homework Statement ABC is an isosceles triangle such that AB = AC A has coordinates (4, 37) B and C lie on the line with equation 3y = 2x + 12 Find an equation of the line of symmetry of triangle ABC. Give your answer in the form px + qy = r where p, q and r are integers. Show clear...
  10. H

    I Symmetry transformation in Heisenberg vs Schrödinger Picture

    Symmetry transformations are changes in our point of view that preserve the possible outcomes of experiment: $$\Psi \rightarrow U(\Lambda) \Psi$$ In the Heisenberg picture, observables in a fixed reference frame evolve according to: $$P(t) = U^\dagger (t)PU(t)$$ while in the Schrodinger...
  11. Eugene Ulrich

    Is "Broken Symmetry" truly foundational?

    How certain can we be that the four fundamental forces as identified in modern cosmology, owe their existence to broken symmetry?
  12. D

    MHB Find equation of a quadratic with line of symmetry

  13. Pushoam

    Using symmetry of action to find the constant of motion

    Homework Statement Homework EquationsThe Attempt at a Solution I need help in solving second part of this question. I put ## e^{i \alpha }\psi ## instead of ##\psi ## and got to see that the integrand doesn't change which means the given transformation is a symmetry of the given action. But...
  14. D

    I Noether's theorem for discrete symmetry

    I am wondering if it existes some discret version of the Noether symmetry for potential with discrete symmetry (like $C_n$ ). The purpose is to describe the possible evolution of the phase space over the time without having to solve equations numerically (since even if the potential may have...
  15. E

    I Derivation of the Onsager symmetry

    Derivation of the Onsager symmetry in many textbooks and papers is as follows: First, assume that the correlation function of two state variables,##a_i## and ##a_j## satifsies for sufficiently small time interval ##t## that $$\langle a_i(t) a_j(0) \rangle = \langle a_i(-t) a_j(0) \rangle =...
  16. Hiero

    Conservation law associated with the symmetry of a helix

    In a problem in Landau’s mechanics (end of section 9) he asks for the quantity conserved in the field of “an infinite homogenous cylindrical helix.” The solution is that the Lagrangian is unchanged by a rotation of dΦ together with a translation of hdφ/(2π) (about and along the symmetry axis)...
  17. Q

    Understand Logic of Wald & Zoupas' Expression on 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?
  18. M

    A How can modular forms be used in particle physics mass models?

    One of the interesting new ideas in BSM model building began with this paper: https://arxiv.org/abs/1706.08749 Are neutrino masses modular forms? Ferruccio Feruglio A modular form is a kind of two-dimensional analogue of the periodic trigonometric functions. Just as trigonometric functions can...
  19. Ventrella

    I Iterating powers of complex integers along axes of symmetry

    I am exploring the behaviors of complex integers (Gaussian and Eisenstein integers). My understanding is that when a complex integer z with norm >1 is multiplied by itself repeatedly, it creates a series of perfect powers. For instance, the Gaussian integer 1+i generates the series 2i, -2+2i...
  20. N

    A Symmetry of the permittivity tensor of lossless media

    I read in various sources (such as page 8 of these notes) that the dielectric permittivity tensor of a lossless medium is always symmetric. I am wondering how this can be the case, when: Phase accumulation in the medium could in theory depend on direction Coordinate system may be rotated to...
  21. A

    Implications of Symmetry on Displacement Field

    Consider the figure attached. Assuming that the coordinate system is placed at the centre of the cube and Plane x-y is the plane of material symmetry, x_i represents the position vector of ith point, and u(x_i) represents the displacement of ith point. Due to symmetry, the displacement vector...
  22. 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...
  23. Nik_2213

    A Infinite-dimensional symmetry....

    https://phys.org/news/2018-11-infinite-dimensional-symmetry-possibility-physicsand-particles.html quote: In a recent paper in Physical Review Letters, Prof. Krzysztof Meissner from the Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, and Prof. Hermann Nicolai from the...
  24. S

    A Stressing Over Stress Tensor Symmetry in Navier-Stokes

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  25. C

    Symmetry in Electrical Circuit Analysis

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  26. Milsomonk

    I Lorentz Symmetry at Black Hole Event Horizon

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  27. 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 (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...
  28. I

    I Symmetry in geometry vs physics

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  29. K

    I Symmetry Group Freedom: Choosing How Groups Act on Coordinates

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  30. S

    I Time reversal symmetry in physics

    It is said that Newton's laws of motion or laws of Quantum Mechanics posses time reversal symmetry but the second law of Thermodynamics does not. What I understand by the first part of the sentence is the following. The dynamical state of a system changes with the increase of time. The state at...
  31. fluidistic

    Why is it said that magnetism breaks time reversal symmetry?

    It is known that Maxwell equations have the time reversal symmetry. I.e. by changing t by -t, J by -J (which can be understood as the charges going in the opposite direction when time is reversed, which makes sense), E to E and B to -B, the equations are still satisfied. However, it is also...
  32. F

    I Calculating E Field Inside Cylinder of Finite Height

    I have a cylinder with radius r and height h and a uniform charge density. Now I am supposed to calculate the E field inside the cylinder. If the cylinder was infinitely long the symmetry would dictate that the field can only point radially outward, so gausses law would give me a field of zero...
  33. G

    A Charged particles mass before symmetry breaking

    How could electrically charged particles be massless before the symmetry breaking? Wouldn't the energy stored in the electric field contribute to particles mass?
  34. Robin04

    I Translational Symmetry: Effects on Laws & World

    What would the world look like if there was no translational symmetry? How would certain laws be different and what wouldn't change?
  35. entropy1

    B Symmetry in Nature: Is Physical Nature Symmetrical?

    Is symmetry a general aspect of physical nature?
  36. H

    I Lorentz Invarience and Spontaneous Symmetry Breaking

    Spontaneous symmetry breakingI’m not sure if I understand spontaneous symmetry breaking.In the context of the Mexican hat (and marble) example, wouldn’t the actual path of the marble down the Mexican hat from the top be determined by several small factors that one would normally not consider...
  37. twist.1995

    A Symmetry Considerations in Hartree-Fock-Roothaan Method

    Hello, dear Quantum Chemists or Physicists. I have been extensively researching the restricted closed-shell Hartree-Fock method, and wrote a code that is working for diatomic molecules, like H2 or He2++. I, however, have a few unresolved questions that do not allow me to proceed further in...
  38. 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...
  39. W

    Proving the symmetry property of Riemann curvature tensor

    Homework Statement Hi everyone! Just wondering if there's a way to prove the symmetry property of the Riemann curvature tensor $$ R_{abcd} = R_{cdab}$$ without using the anti-symmetry property $$ R_{abcd} = -R_{bacd} = -R_{abdc} $$? I'm only able to prove it with the anti-symmetry property and...
  40. P

    I Lie Derivatives and Symmetry

    I'm trying to better understand how people refer to symmetry in Physics and Differential Geometry. In "Exterior Differential Systems and Euler Lagrange Partial Differential Equations," by Bryant, Griffiths and Grossman, it seems a vector field is a symmetry of a Lagrangian if the Lie derivative...
  41. It's me

    Dirac hydrogen atom vs spin symmetry

    Homework Statement Exact spin symmetry in the Dirac equation occurs when there is both a scalar and a vector potential, and they are equal to each other. What physical effect is absent in this case, that does exist in the Dirac solution for the hydrogen atom (vector potential = Coulomb and...
  42. ShayanJ

    A Conformal symmetry, qed and qcd

    1-As you know classical electrodynamics has conformal symmetry. But does this symmetry survive renormalization? if not, can anyone give an explanation on why? 2-What is the situation with QCD? Does it have conformal symmetry at a classical level? If yes, does it survive renormalization? Thanks
  43. W

    A Defining a Symmetry Statistic to test for Normality?

    Hi, Just curious. One of the requirements for a data set to be normally-distributed is that of symmetry. Question: Is there or could one define a statistic to this effect, meaning to determine the level of symmetry on the data set and use that to determine at certain confidence level if the...
  44. TheQuestionGuy14

    B What Is Lorentz Symmetry? Understanding & Differences

    I'm curious to know what Lorentz symmetry is, and how it works. The wiki description is a bit too difficult for me to understand. And how does it differ from a computer's cubic symmetry? I understand that in cubic symmetry, the laws of physics or physical things look different depending on what...
  45. J

    A Time reversal symmetry and Bloch states

    Hello! The time reversal operator, ##\hat{\Theta}## transforms a Bloch state as follows: ##\hat{\Theta} \psi_{nk}=\psi^*_{nk}##. How does one proceed to prove the condition that ##\psi_{nk}## and ##\psi^*_{nk}## must satisfy in order for our system to be time reversal invariant? Thanks in advance!
  46. J

    How Can Crystal Symmetry Affect Conductivity Tensor Components?

    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...
  47. J

    Effective resistance between two points using symmetry

    Homework Statement Homework EquationsThe Attempt at a Solution I am trying to solve this question by symmetry .If I draw a line perpendicular to the line joining AB then points C and D are symmetric . Now I am unsure how to proceed further .
  48. J

    B Relation between spin and symmetry of wave function

    Why is it that bosons (particles having symmetric wave functions) have integral spins and fermions (particles having antisymmetric wave functions) have half integral spins? A lot of books state this without specifying the reason. I was wondering if this is a theoretical deduction. Or is it an...
  49. 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...
  50. P

    Blocking the symmetry of motion and the second law

    Hi, The article below states that they made a material that blocks the motion of molecules in only one way but passing the other way is possible. See the article. https://physicsworld.com/a/blocking-the-symmetry-of-motion/ This seems to me in contrast with the second law. You can for example...
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