What is Symmetry: Definition and 956 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. A

    Symmetry associated with conservation of electric charge.

    What symmetry of nature yields the conservation of electric charge?
  2. F

    Symmetry Groups Algebras Commutators Conserved Quantities

    Symmetry, Groups, Algebras, Commutators, Conserved Quantities OK, maybe this is asking too much, hopefully not. I'm trying to understand the connection between all of these constructions. I wonder if a summary about these interrelationship can be given. If I understand what I'm reading, there...
  3. S

    What are Mirror Planes and Distinct Symmetry Transformations in Mathematics?

    This is in the introduction of the lecture notes I am using to study group theory: "An object is symmetric or has symmetry if there is an operation (e.g. a rotation, reflection or translation) s.t. the object looks the same after the operation as it did originally. An equilateral triangle is...
  4. N

    What does symmetry in GR mean and where does it lead?

    Steve Carlip says this [back in 2007], here,: http://www.2physics.com/2007/06/symmetries-horizons-and-black-hole.html"...Until fairly recently, no one had a clear idea of the microscopic states responsible for black hole entropy. Today, we suffer the opposite problem: we have many...
  5. M

    Power set P(S) with symmetry difference.

    Homework Statement Determine the orders of all the elements of the power set P(S) of a set S with symmetric difference Δ. Homework Equations The Attempt at a Solution If A,b are two elements of the power set the symmetric difference is AΔB = (A-B) U (B - A) How are we...
  6. Y

    What is internal space/internal symmetry?

    Hi, I have already been familiar with that exterior symmetry is the spacetime symmetry. Such kind of symmetry has been depicted by Poincare' group. Then I am still find the concept internal space/internal symmetry ambiguous. And I cannot understand why put Spin in exterior symmetry not...
  7. S

    DFT Symmetry Property: Why Does the Answer Not Void This Property?

    Homework Statement As an example, if we find a DFT of x[n]={1,1,0,1} the result will be X(m)={3,1,-1,1} Homework Equations My Question is that as we know DFT holds symmetry property, why this answer does not void for that property?
  8. F

    What is symmetry breaking in the Standard Model?

    I'm trying to figure out where symmetry breaking occurs in Feynman diagrams. I'm just free wheeling here, correct me where needed. But as I understand it, when you have a Feynman diagram where there is an interaction of one particle which decays to others, the incoming particle may obey one...
  9. E

    Charge Conjugation and Internal Symmetry Representations

    Hi All, I am trying to work through a QFT problem for independent study and I can't quite get my head around it. It is 5.16 from Tom Bank's book (http://www.nucleares.unam.mx/~Alberto/apuntes/banks.pdf) which goes as follows: "Show that charge conjugation symmetry implies that the...
  10. F

    U(1), SU(2), SU(3) are symmetry of what?

    The Standard Model symmetries are U(1), SU(2), and SU(3). But I'm not sure whether these are symmetries of the Action intgral or if they are symmetries of the background spacetime.
  11. D

    What is the physical meaning of the double rotation symmetry of spin-1/2 states?

    Hey, We saw in class that rotating a spin state with an angle of 2pi returns minus the state, and so it has to be rotated 4pi rad in order to return to the original state. However, we also saw that the expected value of the spin DOES return to its original value after a rotation of 2pi rad...
  12. Loren Booda

    How does maximum symmetry compare to its corresponding minimum symmetry?

    In a mathematical system, what similarities are there between the most and least absolute symmetries? Might Noether's theorem explain this?
  13. P

    Axial symmetry, non-rotating

    I'm studying Bondi's work on gravitational radiation. He applied axial symmetry and reflection symmetry to a source of gravity. How is it that axial symmetry does not automatically imply reflection symmetry? The assumption for axial symmetry is \phi \rightarrow \phi'= \phi + constant The...
  14. V

    Understanding Spherical Symmetry in the Hydrogen Atom

    Hi everyone! So we're learning about the Hydrogen atom in QM and I'm having trouble reconciling something in my head. We're looking at potentials that are only radius dependent, like the Coulomb potential. Now, I know the math. I see that we assume the wave function can be separated into the...
  15. Rapier

    How to Find Access of Symmetry in Quadric Surface?

    Homework Statement Construct a hyperboloid of one sheet whose axis of symmetry is the y-axis. Homework Equations Hyperboloid of One Sheet --> x^2 + y^2 - z^2 = c The Attempt at a Solution The relevant equation is the one given in the book and in my notes. Obviously I can have...
  16. N

    Why CP symmetry is violated in weak interaction?

    Please teach me this: Why is CP symmetry violated in electro-weak interaction? Is it correct that because of the mass of neutrino is zero(so that the left and righ neutrinos do not mix with each other)? Thank you very much in advance.
  17. A

    Guage symmetry - invariance under arbitrary phase change

    As I understand it the heart of gauge symmetry is that I can change the phase at different points different amounts and the Lagrangian/action is unchanged. What I am not clear on is whether the changes I can make are completely arbitrary - I can make any change I want at any point - or whether...
  18. C

    How does U(1) guage symmetry lead to Maxwell's equations?

    Here are some questions that have been puzzling me about symmetry and charge. Any answers to any of these questions would be very helpful. Thank you. What does U(1) gauge symmetry mean? Does anyone have a simple explanation? Can Maxwell's equations be derived from the premise of U(1)...
  19. S

    ElectroWeak symmetry breaking.

    Hello, new member here. I've been fascinated reading some of the threads and decided I had to register to ask a question that's always been a bit confusing to me. From what I've learned The Big Bang theory seems the most likely explanation of the start of the universe but there's one thing in...
  20. alemsalem

    In comparing different renormalization procedures why do we care about symmetry?

    Why do we care about Lorentz or gauge invariance if we're going to remove the cutoff at the end. our physical answers are independent of the procedure,, why do we care about preserving the symmetry "during" the calculation? Thanks for your time!
  21. L

    BRST Symmetry and unphysical polarizations

    Hi, I am reading the BRST Symmetry section of Peskin and Schroeder but I can't find anywhere why the BRST transformation for the gauge vector, \delta A_\mu^a = \varepsilon \partial_\mu c^a implies that only forward polarized states can create ghosts by applying Q, Q being define by...
  22. alemsalem

    Symmetry breaking and the ground state of a quantum field theory.

    when a continuous symmetry is broken, we say that the ground state is just one of the possible ground states, and there is no energy cost in moving from one to the other.. why doesn't the state keep changing with the slightest perturbation (production of goldstone boson). why don't we have a...
  23. N

    How many ''charges'' are there in SU(2) and SU(3) symmetry?

    Please teach me this: How many conserved observations(''charges'') are there in SU(2) and SU(3) symmetries?I know that U(1) has only one charge that is electric charge. Thank you very much for your kind helping.
  24. N

    Why symplectic symmetry has N(N+1)/2 generators?

    Please teach me this: Why Sp(N) symmetry has N(N+1)/2 generators?(QFT of Peskin and Schroeder). Thank you very much for your kind helping.
  25. H

    Proof of Symmetry for x^2 + 4y^2 = 1 Curve without Graph Drawing

    Homework Statement Show that the curve is symmetrical about the x-axis (without drawing the graph) eq of the curve is : x^2 + 4y^2 = 1 also show that the curve is symmetric about the y axis Homework Equations The Attempt at a Solution To prove that the curve was symmetric abou...
  26. L

    Chiral Lagrangian symmetry

    Hi, If I have the Lagrangian L=i\chi^{\dagger\alpha i}\bar{\sigma}^{\mu}(D_{\mu})_{\alpha}^{\beta}\chi_{\beta i}+i\xi^{\dagger}_{\bar{i}\alpha}\bar{\sigma}^{\mu}(\bar{D}_{\mu})^{\alpha}_{\beta}\xi^{\beta i}-1/4 F^{a\mu\nu}F_{\mu\nu}^{a} where \alpha,\beta are colour indices, and i=1,2 is a...
  27. F

    Is electroweak symmetry breaking a reversible process?

    I can only assume it is, if a Higgs can be found anyway. I learned about hysteresis in certain physical processes. I don't have the math to guess nor Google skills to find a clear answer. If it were, would a sufficiently large and hot enough black hole be in danger of losing its mass terms...
  28. K

    Axisymmetric vs cyclic symmetry boundary conditions

    Hi, Can anyone explain the difference between axisymmetric and cyclic symmetry boundary conditions? Isn't it the same i.e. bith cyclic symmetry and axisymmetric?
  29. I

    Using symmetry in electrostatic problem

    here is what I am doing. I am trying to argue that the electric field is zero at the center of the sheet using only symmetry arguments. consider an insulator in the shape of infinite sheet of thickness 2a and with uniform charge density \rho. Now let me specify the coordinate system. The...
  30. C

    [Holography] Global symmetry in boundary corresponds to gauge symmetry in bulk?

    I hear the statement that global symmetries in the boundary field theory corresponds to gauge symmetries in the bulk. 1) Is this a generic statement that is expected to hold for all holography pairs? (Maldacena states this towards the end of his first lecture at PiTP2010, which was supposed to...
  31. F

    What is nuclear symmetry energy?

    in a nutshell
  32. S

    Vibration analysis of a structure with cyclic symmetry

    Lets us say we are doing a vibration analysis of a structure with cyclic symmetry In very brief (as pointed out by AlephZero in one of his excellent reply) the whole motion can be represented by complex numbers which describe the motion of one segment. Now, my question is: 1)Is it...
  33. N

    Does a symmetry of Lagrangian reserve in each Feynman diagram?

    Please teach me this: Does a symmetry of Lagrangian be reserved in each Feynman diagram of perturbative QFT,because even Ward Identity still deduces from U(1) symmetry that we consider each diagram has?. By the way, does effective action reserve the symmetry that Lagrangian has?. Thank...
  34. J

    Reissner-Nordström black hole: Spherical symmetry of EM field stregth tensor

    The setup: I am reading the review: arXiv:hep-th/0004098 (page 9-10). In Einstein-Maxwell theory, the gravitational field equations read: \begin{equation} R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = \kappa^2 \left( F_{\mu \rho} F^{\rho}_{\;\;\nu} - \frac{1}{4} g_{\mu \nu} F_{\rho \sigma}...
  35. Q

    Time Reversal Symmetry & Magnetic Field: Charges Explained

    Talking about charges. If someone claims that in his work time reversal symmetry is conserved, does that equal to say he/she is not imposing a magnetic field?
  36. N

    Why we only consider ''group'' symmetry but not general symmetry?

    Please teach me this: Why do we only consider symmetry group(Lie group and Lie algebras) but not general symmetry(the transform that keeps Lagrangian invarian) in QFT?Is it because the symmetry group is more simple and more beautiful and in reality the forces of Nature obey the symmetry...
  37. S

    Is Electroweak Symmetry Breaking a Requirement of Universal Law?

    What law of nature says that electroweak symmetry must be broken? Is it possible that in other parallel Superstrings (or others) universes.. electroweak symmetry were not broken and even after temperature of the Big Bang decreased to what is like ours, electroweak symmetry still existed in that...
  38. S

    Exploring Cyclic Symmetry: A Jet Engine Primer

    Hi, Please can anyone explian what is cyclic symmetry? I'm new to this term and have encountered this in jet engine example Vishal
  39. L

    Gauge symmetry and renormalization

    Here and then I read gauge symmetry makes theories renormalizable. Unfortunately I could not find a satisfactory explanation why that so is. Could someone shed some light? thanks
  40. P

    Symmetry in special relativity

    I have read Einstein's 1905 paper and a number of other explanations and have a question I can't resolve. The basic problem was that the mathematics differed between the case of a conductor moving relative to a magnet vs. a magnet moving relative to a conductor. Einstein used the Lorentz...
  41. tom.stoer

    SU(N) symmetry in harmonic oscillator

    Starting with the D-dim. harmonic oscillator and generators of SU(D) T^a;\quad [T^a,T^b] = if^{abc}T^c one can construct conserved charges Q^a = a^\dagger_i\,(T^a)_{ik}\,a_k;\quad [Q^a,Q^b] = if^{abc}Q^c satisfying the same algebra and commuting with the Hamiltonian H =...
  42. N

    Does Ward Identity in QCD has origin of U(1) or SU(3) symmetry?

    Please teach me this: Can we deduce Ward Identity in QCD from U(1) symmetry of QED?Because QCD is a theory of quarks and quarks have electric charge.So we need not deduce the Ward Identity from SU(3) symmetry,but we can be able to demontrate the Ward Identity( considering gluons)with U(1)...
  43. N

    Is the Function f(x) = (2x^2-x)/(x^2+x) Even, Odd, or Neither?

    Homework Statement is the function f(x) = (2x^2-x)/(x^2+x) even, odd, or neither?Homework Equations f(-x)=f(x) = even f(-x)=-f(x) = odd f(-x)≠f(x)≠ -f(x) The Attempt at a Solution f(x) = (2x^2-x)/(x^2+x) f(-x)=(2(-x)^2+x)/((-x)^2+(-x)) f(-x) = (2x^2+x)/(x^2-x) i think that's the right way to...
  44. S

    Relation on X: Symmetry, Reflexivity & Transitivity

    Let X = { a, b, c } X x X = { (a,a), (b,b), (c,c) } { (a,b), (b,a), (a,c), (c,a) } { (b,c), (c,b) } 1. Symmetric but not reflexive or transitive: R = { (a,b), (b,a), (a,a), (b,c), (c,b) } How come this is right? Isn't aRb, bRa imply...
  45. N

    Ward-Takahashi identities from symmetry

    Does anyone know how to derive the Ward-Takahashi identity for a field starting from a known conserved Noether current (or equally helpful, from a known symmetry transformation of the Lagrangian)? It'll probably be enough to allow me to do it for myself if you could explain quantitatively what...
  46. C

    Quantum mechanics, symmetry and degeneracy

    Homework Statement I'm struggling to understand the concept of symmetry in quantum mechanics. My notes state "In general if the probability density has lower symmetry than the hamiltonian, the wavefunction will be degenerate". I don't really get the connection with the hamiltonian. It...
  47. marcus

    Can Lorentz Symmetry be Emergent?

    http://pirsa.org/11100056/ Can Lorentz Symmetry be Emergent? Speaker(s): Ted Jacobson Abstract: I will begin by discussing some of the strongest observational evidence for Lorentz symmetry, and the essential role that Lorentz symmetry appears to play in the consistency of black hole...
  48. zonde

    Symmetry assumption in derivation of SR

    I would like to ask if anybody knows about some analysis of this part in Einstein's derivation of SR where he gets rid of unknown scaling function φ(v): "From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measured in the stationary...
  49. J

    Why should there be CPT symmetry?

    A few of the books and magazines I have read talk about maintaining CPT symmetry, but I can't see why it should be the case. What is the evidence, theoretical or experimental, of CPT symmetry?
  50. sweetser

    Gauge symmetry in EM by inspection

    Hello: I was under the impression that gauge symmetry was a property of the Lagrange density. Here is the Lagrangian for EM written out in its components: \begin{align*} \mathcal{L}_{EM} &= J\cdot A +\frac{1}{2}\left(B^2-E^2\right) \quad eq.~1\\ &=\rho \phi - Jx Ax - Jy Ay - Jz Az \\...
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