Symmetry Definition and 958 Threads

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. T

    MHB Due to a symmetry of the cosine we can just double the integral from 0 to 1

    Hello, ∫|cos(px/2)|dx between [0,2] I encountered this rule. How does this apply to other intervals of say [3,4],[7,9] etc. Are the numbers both halved? so [3,4] becomes [1.5,2] etc? Also, does this rule apply to all symmetrical functions? Thank you, Tim
  2. S

    Conserved charge from Lorentz symmetry

    I am trying to derive the conserved charge from the symmetry of the action under Lorentz transformations, but I am doing something wrong. Noether's theorem states that the current is J^\mu = \frac{\partial \cal L}{\partial(\partial_\mu \phi)} \delta\phi - T^{\mu \nu}\delta x_\nu For an...
  3. ChrisVer

    How Does Chiral Symmetry Breakdown Influence Meson Mass Differences?

    Why is the chiral symmetry breakdown determined for the vector/axial current as: V = \frac{m_{π^{+}}-m_{π^{0}}}{m_{π^{0}}+m_{π^{+}}}\approx 0.01 A= \frac{m_{π^{+}}-m_{f^{0}}}{m_{f^{0}}+m_{π^{+}}}\approx 1 ? why do we choose the difference between the pion+ (~140MeV) and pion0 (~135MeV)...
  4. S

    Kinetic Energy of Rotating Solid disk or cylinder about symmetry axis

    Homework Statement A horizontal 845.0 N merry-go-round with a radius of 4.3 m is started from rest by a constant horizontal force of 74.0 N applied tangentially to the merry-go-round. Find the kinetic energy of the merry-go-round after 2.2 s. Assume it is a solid cylinder. The...
  5. M

    How Do You Prove Symmetry Arguments in E&M?

    Hi everyone. I don't know whether this is an advanced or introductory topic but I I've always wondered how to prove symmetry arguments in electrostatics, magnetostatics etc mathematically. Suppose you have an infinite line charge and you need to calculate the electric field at some distance...
  6. A

    What is the role of symmetry in quantum mechanics?

    I have several questions about symmetry in quantum mechanics. It is often said that the degeneracy is the dimension of irreducible representation. I can understand that if the Hamiltonian has a symmetric group G, then the state space with the same energy eigenvalue will carry a...
  7. E

    Spontaneous symmetry breaking and the photon/baryon ratio

    Hi, I am looking into symmetry breaking and how it (may have) affected the photon/baryon ratio in the primordial universe. I found this wonderful encyclopaedia of cosmology which relates the grand unified theory to an orthorhombic crystal, making analogies for symmetry, spontaneous symmetry...
  8. stevendaryl

    Understanding Inflation: Effects of Spontaneous Symmetry Breaking on Gravity

    I'm trying to understand inflation (in the cosmic sense). I know that ultimately that's a subject that involves both quantum field theory and General Relativity, but I'm wondering to what extent it can be understood from the point of view of classical (non-quantum) GR. If you have a classical...
  9. B

    Symmetry operation in reciprocal space

    For a given lattice, specifically a diamond, it has C3v symmetry and the symmetry operation in real space is easy to see, 2C3 around high symmetry axis and 3sigma_v. My question is, if the lattice is expressed in reciprocal space, (say wavefunction defined in momentum space), then how to define...
  10. J

    Point groups and symmetry: Adding and subtracting operations

    Homework Statement I haven't been assigned these questions, but I'm trying to trudge through them to better understand symmetry. This is for my inorganic class. It's just a series of short questions like: C3 – S56 = ? S4 + i = ? C3 + i = ? Stuff like this. And just looking at the...
  11. R

    Is there any symmetry I can use to find this Fourier sine series?

    Homework Statement I am going over a practice exam, and I need to find the FSS of f(x)=x(\pi^2-x^2) Homework Equations f(x) \sim \sum^\infty_{n=1}a_n sin\left(\frac{n \pi x}{L}\right) a_n=\frac{2}{L}\int^L_0 f(x)sin\left(\frac{n\pi x}{L}\right)dx The Attempt at a Solution I think I...
  12. S

    Exchange symmetry when isospin is concerned?

    As far as I know identical fermions are antisymmetric under exchange. Identical bosons are symmetric under exchange. Is this fact blurred when we consider isospin? Considering the wavefunction of a proton-neutron system; \psi = \psi_{space} \psi_{spin} \psi_{isospin} I'm told this needs...
  13. S

    Spontaneous symmetry breaking in the standard model

    In the standard model, the Lagrangian contains scalar and spinor and vector fields. But when we consider spontaneous symmetry breaking, we only account for the terms contain only scalar fields, " the scalar potential", in the Lagrangian. And if the scalar fields have vacuum expectation value...
  14. Spinnor

    Fluctuations of the CMB, symmetry between the hot and cold?

    Consider any of the latest maps of the temperature fluctuations of the CMB. Such a map can be considered a 2 dimensional topographical map of the surface of a sphere, high points hotter, low points cooler. Consider the contours that divide such a map into the two regions, fluctuations hotter...
  15. A

    How Does Mean Field Theory Explain Spontaneous Symmetry Breaking?

    I am bit confused by how to approach this concept with mean field theory. As I understand a symmetry break (like a acquiring a finite magnetization) can happen if at low enough temperatures the Free energy decreases when breaking the symmetry. Normally this temperature is found by calculating...
  16. J

    Exploring Symmetry in QED Lagrangian: CPT Transformation Analysis

    I understand that the QED Lagrangian is invariant under the CPT transformation. Is it also invariant under C, P or T separately?
  17. L

    Question about the symmetry of integrals

    O.K. , this question is inspired by a physics class I'm taking where we're working out the expectation values of wave functions, but I think the question really belongs in the math section. Thank you in advance for any help. Here goes nothing... We have a function ψ(x,y,z) = x e\sqrt{}x2 +...
  18. C

    Symmetry (killing vector) preserving diffeomorphisms

    Suppose that on a Riemannian manifold (M,g) there is a killing vector such that ##\mathcal{L}_{\xi} g = 0.## How would one then characterize the group of diffeomorphisms ##f: M \to M## such that $$\mathcal{L}_{f^* \xi} (f^*g) = 0?$$ How would one describe them? Do they have a name...
  19. M

    Quark-Lepton Symmetry: What Happens When Particles Annihilate?

    If I understand correctly, conservation of baryon and lepton numbers imply that quarks and leptons are "basic" i.e. non-interchangeable particles? What happens when one such particle is annihilated, can the energy produced be used to "generate" the other type, or do some additional particles...
  20. J

    Why Is the Fourier Transform's Conversion Factor Split Between Kernels?

    The Fourier transform, wrt to angular frequency, needs of a factor (1/2π) for get f(t) or F(ω), actually, this factor is broken in 2 factors (1/√2pi) and each kernel, direct and inverse, receives one factor for keep the symmetry in equation. F(\omega)=\int_{-\infty }^{+\infty }\frac{e^{-i\omega...
  21. N

    Does a symmetry need to leave the whole action invariant?

    Typically a symmetry is taken to be something that leaves the action invariant. However, on a classical level, isn't that asking way too much? To match what we conceptually mean by symmetry, we only need something that maps solutions to solutions, so something which leaves the action invariant...
  22. T

    How Do 17 Plane Groups Simplify Two-Dimensional Condensed Matter Theory?

    I am working on a condensed matter research project and we have just been introduced to the idea of symmetry. My question is, in two dimensions, how do the 17 plane groups help simplify condensed matter theory problems. If a specific case is needed to narrow the responses than working within...
  23. P

    Symmetry of gravitational field to electric field and Maxwell equation

    Symmetry is an important way to find new physical laws according to Feynman. The equation that describes the electric field and the gravitational field are quite similar. Since the electric and magnetic fields are well defined by the Maxwell equations could it be possible, by symmetry, that...
  24. Saladsamurai

    Can cyclic symmetry be accurately modeled in ANSYS software?

    I am trying to understand a little further how software such as ANSYS implements cyclic symmetry in an analysis. A colleague of mine spoke to a support engineer and I think that he may have misinterpreted what was said. He is now under the impression that when we invoke a cyclic symmetry...
  25. E

    Is every observable the generator of a symmetry?

    We know that observables correspond to hermitian operators on the Hilbert space of physical states of the system. We also know, via Wigner theorem, that for each symmetry there is a linear unitary operator (or anti-linear and anti-unitary). In the case of a continuous symmetry, that is in the...
  26. C

    MHB Even Functions, Symmetry, Inverse Functions

    Can someone explain why the answer is D a < 0 because it finishes downwards e < O because the y-intercept is in the negatives. b, & d = zero (but i don't get this) c is supposedly > 0 (nor do i get this) According to the solutions the graph is an even function, and symmetrical about the...
  27. P

    What is the Symmetry of the Ricci Tensor?

    Hey, I have been doing a few proofs and stumbled across this little problem. Trying to show the symmetry of the Ricci tensor by using the Riemann tensor definition ##R^m_{\ ikp} = \partial_k \Gamma^m_{\ ip} - \partial_p \Gamma^m_{\ ki} + \Gamma^a_{\ ip} \Gamma^m_{\ ak} - \Gamma^a_{\ ik}...
  28. L

    What Symmetry Group Does the Quantum Harmonic Oscillator Exhibit?

    ##H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac{1}{2}m\omega^2x^2## Parity ##Px=-x## end ##e## neutral are group of symmetry of Hamiltonian. ## PH=H## ##eH=H## so I said it is group of symmetry because don't change Hamiltonian? And ##e## and ##P## form a group under multiplication. Is there...
  29. T

    Molecular States, Symmetry and Allowed Transitions

    I have a question about allowed transitions and molecular states. For an electric dipole transition between two states (say molecular or atomic) to have a non-zero probability of occurring, the transition dipole moment \langle \psi_{f}|\textbf{μ}\left|\psi_{i} \right \rangle must be non-zero...
  30. T

    Is This the Correct Way to Apply Symmetry Transform to the Action?

    Homework Statement Hi I'm trying to understand how symmetry transform works. Suppose a lagrangian L = q^{-2} (actually it had another kinetic member, but I don't need it for my question here) The matching action S = \int dt q^{-2} We were told that it has the next symmetry t...
  31. Q

    Symmetry Groups of the Standard Model: SU(3) x SU(2) x U(1)

    I have a question regarding symmetry groups. I've often heard that the Standard Model is a SU(3) x SU(2) x U(1) theory. From what I understand these groups contain the symmetries under which the Lagrangian function is invariant. If so, what does every one of the 3 groups above contain (what...
  32. bananabandana

    Symmetry in a Triangular Circuit

    Homework Statement Find the resistance across AB. Use symmetry to determine currents. Homework Equations Please see diagram. The Attempt at a Solution I really don't seem to understand symmetry in circuits. What is the link between the geometrical symmetry of the situation and...
  33. H

    Stress components on cantilever beam loaded off symmetry

    Homework Statement An element is taken from the wall end of the cantilever beam that is loaded as per the diagram. What are the stress components of the element, taken at x=0, y=h/2, z=0 Homework Equations σ=My/IThe Attempt at a Solution So from the shear moment diagram, the moment and shear...
  34. R

    Symmetry energy in nuclear physics

    I have a question about symmetry energy in semi-empirical mass formula, According to semi-empirical mass formula as follows: E=avA-asA2/3-acZ(Z-1)/A1/3-asym(N-Z)2/A why in the symmetry energy only squared parameter symmetry are exist and there is not the first power of asymmetry parameter?
  35. A

    Forced to use symmetry to solve this double integral?

    Homework Statement http://i.imgur.com/d4ViHux.png Homework Equations The Attempt at a Solution The author writes: "Now, using symmetry, we have..." But what symmetry does the author use? Also, I got the integral as shown in the remark but why is it wrong?
  36. L

    Symmetry of an Integral: Is it Zero?

    If function is ##f(-x,-y)=f(x,y)##, is then ##\int^{a}_{-a}\int^{a}_{-a}f(x,y)dxdy=0##? Thanks for answer.
  37. J

    Electroweak Symmetry: Exploring the Unified Force

    Hello, I have read that above a certain temperature, the weak bosons become massless and become indistinguishable to the photon. Is the idea simply that at high enough energies, the Higgs field can sit on top of the peak in the mexican hat potential? I.e. at high enough energies, it's vacuum...
  38. Superposed_Cat

    Temporal symmetry of nature violation according to nobel prize winner

    "..his equations indicated that atoms could indeed form a regularly repeating lattice in time, returning to their initial arrangement only after discrete (rather than continuous) intervals, thereby breaking time symmetry..." I was wondering about the theory's validity and if you had heard of...
  39. D

    U(1) Gauge Symmetry: What Informs Its Choice?

    So, I have a basic/general question here. I understand that, for example, the QED Langrangian has U(1) gauge symmetry. I also understand that this means (when you have written the Lagrangian with the covariant derivative) that if you transform the wavefunction (\psi \rightarrow e^{i \theta (x)}...
  40. C

    Point Group & Symmetry of Urea Molecule: Solution Explained

    Problem: Consider the planar urea molecule. i) Determine the point Group, as well as the symmetry of the x,y and z coordinates. ii) Consider the following atomic orbitals: C : 2s; 2p N : 2s; 2p O : 2s; 2p H : 1s Determine the equivalent sets of orbitals and the symmetry of each set...
  41. PsychonautQQ

    Use symmetry in double integral.

    Homework Statement Evaluate the double integral of (2+xy^2) over dA (dxdy) using symmetry where R = [0,1] x [-1,1] Homework Equations The Attempt at a Solution I don't know how to use symmetry to evaluate this.. However if I integrate this integral normally i first get...
  42. A

    Standard model + symmetry questions

    Homework Statement 1) Which U(1), SU(2) and SU(3) gauge invariances are implemented in nature according to the Standard Model? What are the related quantum numbers? 2) The SU(2) symmetry is referred to as a non-abelian symmetry. What does this imply for the interactions between the force...
  43. C

    Conserved quantities as symmetry generators

    Suppose we have a Lagrangian \mathcal{L(\phi, \partial_\mu \phi)} over a field \phi, and some variation on the field \delta \phi. If this variation induces a variation \delta \mathcal{L} = \partial_\mu F^\mu for some function F^\mu, then Noether's Theorem tells us that if we construct the...
  44. L

    Do spin-1 particles also have phase symmetry?

    In almost every QFT or particle textbook we learn that complex scalar fields or spinor fields (or even multiplets of spinor fields) have a phase symmetry (global gauge symmetry.) You can append to these fields an exponential with a complex phase in the Lagrangian and the dynamics remain the...
  45. C

    Relation R symmetry for x= +- y

    Hi. I'm working through one of my first problems on sets and relations, and I need some help understanding if I'm getting this right. Any help/suggestions on my through process is greatly appreciated. The question is: Determine whether the relation R on the set of all real numbers is...
  46. K

    MHB Is the Function f a Homomorphism for Symmetry Group of a Circle?

    I have a question that I have approached, but want to check if I'm on the right track. Let G denote the group of symmetries of a circle. There are infinitely many reflections and rotations. There are no elements besides reflections and rotations. The identity element is the rotation by zero...
  47. J

    CPT symmetry analogue of free electron laser?

    Quantum unitary evolution is time symmetric, what leads to retrocausality phenomenas like delayed choice quantum erasure or Wheeler's experiment. However we understand why they don't allow to send information back in time - I would like to ask for help with understanding why another experiment...
  48. V

    Gauss's Law for Spherical Symmetry

    Find the electric field for a non-conducting sphere of radius R = 1 meter that is surrounded by air in the region r > 1. The interior of the sphere has a charge density of ρ(r) = r. The answer is k(pi)/r^2, but I can't seem to get that. My problem is with finding the enclosed charge. I've tried...
  49. L

    High symmetry points and lines in Brillioun Zone

    Hi, I've seen pictures like this one: http://www.lcst-cn.org/Solid%20State%20Physics/Ch25.files/image002.gif Is there any good explanation somewhere on this subject? I'm using Kittel's book but there's nothing in there on this.
  50. W

    Spin-half mass term with symmetry breaking

    I've been thinking about chapter 11 of Griffiths' Introduction to Elementary Particles. In section 11.7, he gives the Lagrangian density \mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)+\frac{1}{2}\mu^{2}\phi^{2}-\frac{1}{4}\lambda^{2}\phi^{4} and shows that the minimum...
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