What is Symmetries: Definition and 183 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. Urs Schreiber

    Mathematical Quantum Field Theory - Gauge symmetries - Comments

    Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Gauge symmetries Continue reading the Original PF Insights Post.
  2. 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...
  3. Urs Schreiber

    Mathematical Quantum Field Theory - Symmetries - Comments

    Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Symmetries Continue reading the Original PF Insights Post.
  4. J

    A Any good idea how non-abelian gauge symmetries emerge?

    I think the story where abelian, i.e. U(1), gauge symmetry comes from is pretty straight-forward: We describe massless spin 1 particles, which have only two physical degrees of freedom, with a spin 1 field, which is represented by a four-vector. This four-vector has 4 entries and therefore too...
  5. 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...
  6. 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) =...
  7. F

    I Difference between global and local gauge symmetries

    The mantra in theoretical physics is that global gauge transformations are physical symmetries of a theory, whereas local gauge transformations are simply redundancies (representing redundant degrees of freedom (dof)) of a theory. My question is, what distinguishes them (other than being...
  8. S

    Symmetries of graphs and roots of equations

    Is there a good way to relate the symmetries of the graphs of polynomials to the roots of equations? There's lots of material on the web about teaching students how to determine if the graph of a function has a symmetry of some sort, but, aside from the task of drawing the graph, I don't find...
  9. LarryS

    Mathematical Format of Lagrangian vs Symmetries

    Most of the lectures that I have watched online say that a symmetry exists when the mathematical form of the Lagrangian does not change as a result of some transformation, like a local gauge change. But how does nature "know" the mathematical form of the Lagrangian? Obviously, I am missing the...
  10. M

    MHB Finding Symmetries of a Function

    How would one find the symmetry of the function x^3-3=g(x) ? Or any other symmetry.
  11. Twigg

    Jet Prolongation Formulas for Lie Group Symmetries

    I can never derive the prolongation formulas correctly when I want to prove the Lie group symmetries of PDEs. (If I'm lucky I get the transformed tangent bundle coordinate right and botch the rest.) I've gone through a number of textbooks and such in the past, but I haven't found any clear...
  12. G

    Exploring Physical Symmetries and Automorphisms in Equations

    All the physical simmetries implicate an automorphism in axioms. But ¿all the automorphisms from a physical equation do implicate a simmetry?
  13. Physics Monkey

    BMS symmetries and black hole horizons

    I would like to discuss a bit this paper (http://arxiv.org/abs/1508.06577): BMS invariance and the membrane paradigm Robert F. Penna (Submitted on 26 Aug 2015) We reinterpret the BMS invariance of gravitational scattering using the membrane paradigm. BMS symmetries imply an infinite number of...
  14. mnb96

    Are there other symmetries on the plane besides the wallpaper and point groups?

    Hello, is it so that symmetries on the plane are essentially discrete subgroups of the group of isometries on the plane? If that is true, then why should we think that the only symmetries in the plane are given by the wallpaper group and the point group? Can't we just change the metric of the...
  15. N

    Torsion Scalar and Symmetries of Torsion Tensor

    I've started f(T) theory but I have a simple question like something that i couldn't see straightforwardly. In Teleparallel theories one has the torsion scalar: And if you take the product you should obtain But there seems to be the terms like . How does this one vanish? because we know...
  16. A

    Searching for Symmetries in PDEs with Mathematica(c)

    Hello, I have a problem in the search for symmetries in pde. I would use Mathematica(c), does anyone know how to set up the code to obtain generators and then symmetries? Thanks for all.
  17. P

    Why is Supersymmetry Necessary in String Theory?

    I was recently watching a talk by Witten and he mentioned that one of the magical things about string theory is that it forces us to accept certain symmetries of nature, as opposed to choosing them as we do in QFT. Can anyone give an enlightening explanation of this? I do have very basic...
  18. S

    Understanding Crystal Symmetries & Dielectrics

    Hi, I am currently reading the Feynman Lectures on Physics, and I have just finished the chapter about the geometry and the symmetries of crystals, and there is something I do not quite understand. There are 230 different possible symmetries which are grouped into seven classes (triclinic -->...
  19. B

    Solving Special Function Equations Using Lie Symmetries

    The lie groups + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's equation. Would anyone have a non-dense/explicit/easy reference aimed at physicists for the...
  20. Breo

    How to find Internal Symmetries?

    For instance, how to find what internal symmetries a free relativistic lagrangian density has for complex scalar field?
  21. Breo

    Internal Symmetries (Prove Lagrangian is invariant)

    Homework Statement Note: There is an undertilde under every $$\phi$$ Imagine $$ \phi ^t M \phi $$ . M is a symmetric, real and positive matrix. Prove L is invariant: $$ \mathcal{L} = \phi ^t M \phi + \frac{1}{2} \partial_\mu \phi ^t \partial ^\mu \phi $$ Trick: Counting parameters. Homework...
  22. arivero

    Continuous symmetries as particles

    I am not sure if I recall all the ways for a symmetry to appear as some particle in a Quantum Field Theory. - The Lagrangian and the vacuum is invariant under the generators of a global symmetry/gauge group. Then the particles in the theory are classified according representations of such...
  23. E

    Riemann Curvature Tensor Symmetries Proof

    I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies $$R_{{abcd}} = - R_{{bacd}}$$ Antisymmetry last pair of indicies $$R_{{abcd}} = - R_{{abdc}}$$...
  24. shounakbhatta

    SU(2), SU(3) and other symmetries

    Hello, I am trying to understand the concept of symmetries, SU(2), SU(3), unitary group, orthogonal group SO(1)...so on. I don't know from where to start and what would be the first group to study and then move on step by step into the other. Also, I need to have a basic (theoretical)...
  25. F

    Local gauge symmetries Lagrangians and equations of motion

    Hey gang, I'm re-working my way through gauge theory, and I've what may be a silly question. Promotion of global to local symmetries in order to 'reveal' gauge fields (i.e. local phase invariance + Dirac equation -> EM gauge field) is, as far as i can tell, always done on the Lagrangian...
  26. M

    Symmetries of the Standard Model: exact, anomalous, spontaneously brok

    There are a number of possible symmetries in fundamental physics, such as: Lorentz invariance (or actually, Poincaré invariance, which can itself be broken down into translation invariance and Lorentz invariance proper), conformal invariance (i.e., scale invariance, invariance by...
  27. Q

    Group, Symmetries and Representation

    I'm starting to learn about particle physics but I really want to see the whole picture before going deep. Here is what I know: - There are symmetries in quantum physics, which are symmetry operators commute with the Hamiltonian (translation operators, rotation operators...) which act on a...
  28. B

    What symmetries are in the following action:

    S=\int d^4x\frac{m}{12}A_μ ε^{μ \nu ρσ} H_{\nu ρσ} + \frac{1}{8} m^2A^μA_μ Where H_{\nu ρσ} = \partial_\nu B_{ρσ} + \partial_ρ B_{σ\nu} + \partial_σ B_{\nu ρ} And B^{μ \nu} is an antisymmetric tensor. What are the global symmetries and what are the local symmetries? p.s how...
  29. R

    EM Field Theory (Action Symmetries)

    Homework Statement I uploaded a picture with the question Homework Equations my problem is : How should I find all the symmetries of the action ? Is there an easy way to recognize those symmetries or should I try all the symmetries I know and see if the action doesn't change...
  30. F

    Lattice systems and group symmetries

    Dear all, In Marder's Condensed matter physics, it uses matrix operations to explain how to justify two different lattice systems as listed in attachment. However, I cannot understand why the two groups are equivalent if there exists a single matrix S satisfying S-1RS-1+S-1a=R'+a'...
  31. Matterwave

    What is the relationship between Lie algebras and Lie groups on a manifold?

    Hello, I'm reading the book Geometrical methods of mathematial physics by Brian Schutz. In chapter 3, on Lie groups, he states and proves that the vector fields on a manifold over which a particular tensor is invariant (i.e. has 0 Lie derivative over) form a Lie algebra. And associated with...
  32. P

    Spacetime symmetries vs. diffeomorphism invariance

    This is a very basic question, but I cannot get my head around the following: Any physical system should be invariant under changes of coordinates, because these are just a way of parametrizing the manifold/space in which my physical system is embedded. Now, let us consider a system that...
  33. N

    Why do we know the particle/quantum field theory is phys of symmetries

    Why do we know that particle physics/quantum field theory is a physics of symmetries?What leads we to the gauge symmetries of all interactions?.Why we can not assume a physics without symmetry?
  34. ChrisVer

    What are the consequences of breaking the Baryon and Lepton U(1) symmetries?

    I am trying to understand the concept of anomalies and the global or gauge symmetry breaking, i.e. the Lepton and Baryon number symmetries. Could someone explain me, what we mean by saying that the Baryon or Lepton U(1) symmetries are broken? and how is that done? I've seen the loop diagram...
  35. michael879

    Euler-Lagrange equations and symmetries

    I'm hoping this is a really simple question, but I can't seem to find a definitive answer anywhere! If the action is invariant under some symmetry transformation, do the equations of motion need to be invariant as well? The trouble I'm having relates to SU(N) yang-mills theories where I'm...
  36. michael879

    Symmetries + Conserved currents of SM

    Can anyone give or point me to a list of ALL continuous symmetries in the standard model, and the conserved currents associated with them? I've spent a lot of time looking and for the most part everything I find is very abstract, where as I want the specific details to the SM (i.e. SU(N) gauge...
  37. N

    Why can we apply the symmetries of S-Matrix to part of Feynman diagram

    How can we demonstrate that the symmetries of S-Matrix can be applyed to parts of Feynman diagrams?The S-Matrix is the sum of infinite diagrams,why we know each or part of each diagram has the same symmetries as the symmetries of S-Matrix?
  38. K

    MHB Decomposition formulas for rotational symmetries of a cube

    I have a problem that I would like to check my work on. I am also stuck on the verifications for $E$ and $F$. Any help would be greatly appreciated. Thanks in advance. **Problem statement:** Let $G$ be the group of rotational symmetries of a cube, let $G_v, G_e, G_f$ be the stabilizers of a...
  39. K

    MHB Orders of elements for rotational symmetries of cube

    I am having lots of trouble doing this problem because I have particularly poor visualization skills. (Or maybe haven't developed them well yet). I would appreciate any help on this math problem. Here is the question: Suppose a cube is oriented before you so that from your point of view there...
  40. K

    Gauge forces and internal symmetries

    gauge forces like electromagnetic, weak and strong forces have local gauge symmetry invariance in terms of u(1), su(2), su(3) because the em for example can't have the same global phase or global symmetry at all points of space. but is there no corresponding gauge forces for global symmetry?
  41. J

    Figuring symmetries of a differential operator from its eigenfunctions

    So, I understand that the derivative operator, D=\frac{d}{dx} has translational invariance, that is: x \rightarrow x - x_0, and its eigenfunctions are e^{\lambda t}. Analogously, the theta operator \theta=x\frac{d}{dx} is invariant under scalings, that is x \rightarrow \alpha x, and its...
  42. C

    Lagrangian explicitly preserves symmetries of a theory?

    How are the Hamiltonian and Lagrangian different as far as preserving symmetries of a theory? Peskin and Schroeder write that the path integral formalism is nice because since it's based on the action and Lagrangian it explicitly preserves all the symmetries, but I'm wondering how/why the...
  43. H

    Conservation Laws and Symmetries Particle Physics

    2) A π0 of kinetic energy 350 MeV decays in flight into 2 γ rays of equal energies. Determine the angles of the γ rays from the incident π0 direction. Not sure where I am going wrong but my answer is not correct. Energy of π0 meson E = 350 MeV Rest mass Energy of π0 meson E0 = 135...
  44. M

    Symmetries of a Tetrahedron video

    Hello, I made this video with my iphone depicting the Symmetries of a Tetrahedron for a presentation I did recently: I have been searching and trying to figure out if I have presented it correctly that a Tetrahedron has full S4 symmetry if we could reflect it in a "higher dimension." I was...
  45. W

    Identifying Symmetries of a Curve

    Homework Statement Identify the symmetries of the curve. Homework Equations r = 8 + 7sinθ The Attempt at a Solution x-axis: (r,-θ) ---> 8 + 7sin(-θ) = 8-sinθ (not equal to ) 8+7sinθ; not symmetric about x-axis. y-axis: (-r,-θ) ---> -8+7sinθ (not equal to) 8+7sinθ; not symmetric...
  46. R

    Global Symmetries: Understanding ##T^a##s and (1.10)

    I don't know how (1.10) pops up and why the ##T^a##s satisfy the Lie algebra. Is there any physical intuition? Any comment would be very appreciated!
  47. C

    Symmetries of 6-Gon: Proving D_6 isomorphic to D_3 x Z_2

    Homework Statement We were given the following argument showing that ##D_6 \cong D_3 \times \mathbb{Z}_2##. What we did was take two subgroups H and K of G such that their intersection was trivial and that ##hk = kh## for all h and k. H = {symmetries of 6-gon that preserved a triangle inside}...
  48. S

    Symmetries in Sidney Colemans QFT script

    Hello Everybody, I am learning QFT using Sidney Colemans lecture notes (and the Peskin Schroeder book). They can be found here: http://arxiv.org/abs/1110.5013 Now, in page 40, he introduces in the paragraph Symmetries and Conservation laws some definitions which I don't quite...
  49. C

    Symmetries of graphs/Graph theory

    Homework Statement Determine the number of symmetries of the graph attached. The Attempt at a Solution I know the answer, I would just like someone to look over my argument to make sure I haven't missed out anything important. I also have some questions along the way. Vertex 1 is the...
  50. M

    Notes on symmetries of the KdV equation

    I am having trouble understanding a section in http://www.mathstat.dal.ca/~francisv/publications/XXV-ICGTMP-Proceeding/dkdv.pdf: . It is on page 3. Section 3 -- Discretization of the Korteweg-de Vries equation. I don't understand why V_4=x∂_x+3t∂_t-2u∂_u generates a symmetry group of the KdV. I...
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