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. J

    Model with SU(2) gauge symmetry and SO(3) global symmetry

    1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##. The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...
  2. L

    A potential well with 3-fold reflection symmetry

    When we are talking about Bloch's theorem and also the tight-binding approximation, we can use them to help finding eigenstates of a system. However, I am so confused how to apply it in this case (below is my homework) and don't even know how to start it... All I understand about the Bloch's...
  3. G

    I Antimatter Susy Sparticles: Exploring Symmetry

    In supersymmetry, are there corresponding antimatter particles to the Susy sparticles similar to the corresponding standard model antimatter particles, e.g., antiselectron, etc.?
  4. Demystifier

    A Gauge invariance confusions: symmetry vs redundancy, active vs passive

    Symmetry transformations in physics can be either passive or active. Symmetries in field theory can be either global or local. But only the local ones, the so called gauge symmetries, are fundamental. Except that local transformations cannot be active (despite the fact that diffeomorphisms are...
  5. Q

    I Unlocking the Ultimate Symmetry of a Mirror Universe

    After countless google searches, I decided to ask the experts. The multiverse idea seems to be popular among physicists, but the idea of a mirror universe seems much more plausible. Like quantum theory, it’s symmetrical. For every quantum fluctuation in our universe, there would be an equal and...
  6. H

    A Is there a symmetry in the Unruh effect?

    Hi Pf I read that the accelerated Unruh observer is in a thermal bath of particles. For him the mean value of his occupation number N of the inertial vacuum V0 is not null. <V0 N V0> is greater than 0. He is not in his vacuum V'. Suppose that he travels with a box. he cools it and leaves it...
  7. ScruffyNerf

    B Black Hole Observation: Outside Observer & Spherical Symmetry

    I know that for the infalling observer the horizon is a fake singularity that can be removed via the Eddington-Finkelstein co-ordinates but wouldn't the classic Swartsheild co-ordinates still apply for the outside observer? So, while for the infaller it takes a finite time, the outside...
  8. A

    Symmetry associated with current conservation

    As I understand it for every symmetry there is associated a conserved quantity - so for time symmetry there is energy conservation. I understand as well that charge conservation is associated with a 'mathematical' local symmetry - something turning in a mathematical space at a point so to...
  9. Arne

    I Z2 symmetry in the hydrogen molecule when mapping to qubits

    Hello! When using a Jordan-Wigner-mapping or parity-mapping to map the hydrogen molecule \mathrm{H}_2 with two electrons and 4 spin-orbitals to 4 qubits, it is possible to reduce the number of qubits down to two [1,2,3]. The reason is apparently that the molecule has a discrete...
  10. 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...
  11. M

    Check Inversion Symmetry: Honeycomb Kitaev Model

  12. J

    Is there a conflict between CPT symmetry and the 2nd law of thermodynamics?

    Naively there is a conflict between CPT symmetry being at heart of fundamental physics models like QFT, and 2nd law of thermodynamics: saying that entropy grows toward future. Is there really a conflict here - so is physics symmetric or not? How to understand it? Personally I disagree with that...
  13. M

    A New ideas about symmetry groups

    https://arxiv.org/abs/2009.14613 A group-theorist's perspective on symmetry groups in physics Robert Arnott Wilson [Submitted on 29 Sep 2020 (v1), last revised 12 Nov 2020 (this version, v3)] R.A. Wilson (physics blog) worked on finite simple groups such as the famous "Monster". He writes...
  14. joneall

    I Gauge theory symmetry breaking in L&B

    I’m reading Lancaster & Blundell, Quantum field theory for the gifted amateur (even tho I”m only an amateur...) and have a problem with their explanation of symmetry breaking from page 242. They start with this Lagrangian: ## \mathcal{L} = (\partial_{\mu} \psi^{\dagger} - iq...
  15. 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}...
  16. phywithAK

    How can I find conserved current for a Lagrangian involving vector fields?

    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...
  17. S

    A High energy symmetry breaking and laws of physics?

    In some models of the beginning of the universe, like for example in chaotic inflation, space would stop expanding in some points, creating Hubble volumes that could experience different spontaneous symmetry breaking, which would result in different properties, such as different physical...
  18. JD_PM

    Modifying photon states to get a gauge symmetry

    I see that this procedure helps to get rid of the two extra degrees of freedom (due to the scalar and longitudinal photons) one firstly encounters while writing the electromagnetic field theory in a Lorentz-covariant way; it indeed shows that modifying the allowed admixtures of longitudinal and...
  19. peguerosdc

    I Understanding crossing symmetry: inverse beta decay

    Hi! This is a very very noob question, but I am starting to get into particle physics and I don't understand the application of crossing symmetry in the inverse beta decay. Crossing symmetry says (from Griffiths) that, in a reaction "any of these particles can be 'crossed' over to the other...
  20. TempleSweeper

    A Exploring Symmetry Breaking Phenomenon in a Physical System

    In some experiments of a complex physical mathematical system, we found some symmetry phenomenon, very similar as symmetry breaking phenomenon, say, as translatable sysmmetry etc. These symmetry (breaking) phenomenon occurs in most of the parts of the system or some parts of the system. Can you...
  21. Frigus

    What is the logic behind the mirror symmetry method of solving circuits?

    I recently encountered mirror symmetry method of solving circuits and by it solving circuits became very easy but problem I am facing with it is that I can't figure out logic behind it. For example if we try to simplify this circuit Then we say that if ##I## current flows from Point A to C then...
  22. A

    Derive Newtonian gravity from symmetry?

    Is it possible to derive laws of Newtonian gravity from some symmetries?
  23. J

    I Correlation between Symmetry number & Total wavefunction

    Some rotational quantum states are not allowed for a rotating particle. At quantum level, these "forbidden" quantum states is based on the requirement of the total wavefunction being either symmetrical or anti-symmetrical, depending on whether the particle is a fermion or boson. The particle's...
  24. S

    B Symmetry in Higher Dimensions: Sean Carroll's Video & Physics

    I'm watching Sean Carroll's video on symmetry [relevant section at around 8:05] He talks about 120 degree rotations of triangles that leave them invariant. Then he proceeds to talk about flipping them with an interesting (at least to me) remark - "there's nothing that says I'm confined to...
  25. 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...
  26. K

    I Spontaneous symmetry breaking from a QM perspective

    Hello! This questions might not make sense and I am sorry if that is the case (I am asking from a QM class perspective). I am a bit confused about the idea of spontaneously symmetry breaking (SSB), from the point of view of QM. I am talking here about the energy plot looking like a mexican hat...
  27. sophiatev

    I Invariance of a system under symmetry operations

    I'm trying to understand the precise reason we claim that a value being conserved means that the system in question is invariant under the corresponding symmetry transformation. Take parity for example. If the parity operator satisfies the commutation relation ##[P, H] = 0## for a given...
  28. LCSphysicist

    Prove the symmetry of an orbit from the given equation

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  29. H

    I Time Reversal Symmetry in Classical Physics

    I try to justify time-reversal symmetry in a very simple classical problem; Free Fall. The position, ##x##, and the velocity, ##v## are obtained versus time from the equation ##-g=\ddot x##. So, if we consider the primary conditions as ##t_0,x_0,v_0## it is clear that...
  30. HaoBoJiang

    A LT & Translation Symmetry: Problem Analysis

    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...
  31. H

    I Does Galilean symmetry imply that all systems are monogenic?

    The equations of motions for a closed system consisting of ##N## particles are: $$m_i \vec x_i'' = \sum_{j \neq i}^N \vec F(\vec x_i, \vec x_i', \vec x_j, \vec x_j')$$ $$ i = 1,..., N$$ Now if we impose the requirement that this closed system be symmetric under Galilean transformations, do we...
  32. A

    I Is it true that isotropic biaxial strain does not lower C2 symmetry?

    Hello, My question is simple. I have read that isotropic biaxial strain does not lower C2 symmetry, but no proof whatsoever was provided. I would like to know if it is actually true and have a solid proof. If someone can provide it, that would be wonderful. But also explaining me how to start...
  33. R

    B Scale of Measurement vs Equipment Sensitivity: Does Symmetry Always Hold?

    Many times when i ask about test theories of SR, i am reminded by forum members, that equipment sensitivity, is equivalent to producing more extreme physical values. For example, you don't necessarily have to go faster in speed, in order to have a better measurement of time dilation, if you have...
  34. 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...
  35. mgkii

    B Entanglement - lack of "symmetry" in no-hidden variables explanation

    Hi. Over the years I've read LOT of "popular science" (i.e. non-textbook) books on entanglement, and on the explanations / objections / arguments Einsten, Bohr, Bohm and others had that still remain today. There's one aspect which never seems to get covered in these books and I wondered if...
  36. L

    Dipole and Quadrupole moments and axial symmetry

    Hello, I would like to find a mathematical demonstration of this problem. While I have always used it, I could never prove it. Given a charge or current distribution with axial symmetry the electric and magnetic dipole moment are null: (electric dipole moment) (magnetic dipole moment)and...
  37. Ranku

    I Energy of massless particles before electroweak symmetry breaking

    Would it be correct to represent the energy of massless particles before electroweak symmetry breaking as ##E = cp##, just as we do with photons post-symmetry breaking?
  38. M

    MHB Understanding Symmetry Maps and Rotations in the Plane

    Hey! :o Let $F\subseteq \mathbb{R}^2$. A map $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ is called symmetry map of $F$, if $\pi(F)=F$. A symmetry map of $F$ is a map where the figures $F$ and $\pi (F)$ are congruent. Let $\pi_1:\mathbb{R}^2\rightarrow \mathbb{R}^2$, $x\mapsto \begin{pmatrix}0...
  39. M

    Stiffness Matrix Method: Symmetry vs Introducing a new node / joint

    Hi, In the question outlined in the images (apologies for the poor quality of the scans), the chosen solution has opted to use a symmetry argument and proceed from there. Question is from "Structures: theory and analysis" by Williams & Todd My question is: How could we approach the same...
  40. M

    MHB Can we determine the symmetry of a distribution without creating a diagram?

    Hey! :o We are given a list of $300$ data which are the square meters of houses. I have calculated the mean value and the median. After that we have to say something about the symmetry of the distribution. For that do we have to make a diagram from the given data? Is there a program to do that...
  41. W

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

  42. G

    A Vacuum energy density after spontaneous symmetry breaking

    If we start with the Lagrangian \begin{equation} \begin{split} \mathcal{L} = & \frac{1}{2}(\partial_\mu \phi)^2 + \frac{1}{2}\mu^2 \phi^2 - \frac{1}{4}\lambda^2 \phi^4\\ \end{split} \end{equation} and give the scalar field a VEV so that we can define the field ##\eta##, where $$\eta = \phi...
  43. M

    Nother current of given symmetry

    This is my first time dealing with scaling symmetry, so I'm sorry if the following is fundamental wrong. My approach was the same as if I was trying to show the same for translation or Lorentz symmetry. We have $$\delta\phi(x)= \phi'(x')-\phi(x)=...
  44. 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...
  45. M

    I Christoffel's symbols: symmetry in the two (lower) indices

    I have a technical problem. 1. Accordingly to historical E.B. Christoffel’s work (I think year 1869), (Christoffel’s) symbols are symmetric in the two (today writing) lower indices. 2. These symbols have been introduced when studying the preservation of differential forms of degree two. The...
  46. Vishesh Jain

    Chemistry Optical isomerism in spite of plane of symmetry ....?

    Shown above is an octahedral coordination complex and it's mirror image. The complex has a plane of symmetry (shown in the left diagram, a diagonal plane passing through bottom left and top right corner, perpendicular to the square plane) ... but still it's mirror image is non-superimposable ...
  47. PeterDonis

    A Implications of Symmetry and Pressure in Friedmann Cosmology

    The thread title is the title of a recently published paper: https://iopscience.iop.org/article/10.3847/1538-4357/ab32da The paper claims to resolve an ambiguity in "cosmological backreaction" models, which are models that take into account spatial inhomogeneity to derive correction terms to...
  48. x-vision

    B Reference frame symmetry in Special Relativity

    Hello, I have a couple of questions related to reference frames in Special Relativity. Let's consider a rocket that is inertially moving towards a star with a relative velocity 0.9c. I'd like to look at this example from both the rocket's and the star's perspectives. In the reference frame of...
  49. W

    I Time-reversal symmetry in QM

    I've bumped into a few interesting papers talking about time-reversal symmetry in QM (eg: https://arxiv.org/abs/1507.07745) 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...
  50. H

    A Quantum Mechanics: Exploring Symmetry and Continuation

    Does quantum mechanics follow causality, phase transition, critical point, symmetry, asymmetry, order, disorder, continuation, discontinuation, limitation, without limitation, convergence, divergence, similarity, hierarchical structure, singularity, plurality?
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