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. Garlic

    Gauge Symmetry Breaking: Exploring Interactions at Low Temperatures

    Can there be interactions that are symmetric under low temperatures but exhibit spontaneous symmetry breaking under extremely low temperatures? (Maybe that symmetry breaking temperature is so low that it couldn't be discovered in experiments) Does electromagnetism split into electricity and...
  2. D

    Green's functions for translationally invariant systems

    As I understand it a Green's function ##G(x,y)## for a translationally invariant differential equation satisfies $$G(x+a,y+a)=G(x,y)\qquad\Rightarrow\qquad G(x,y)=G(x-y)$$ (where ##a## is an arbitrary constant shift.) My question is, given such a translationally invariant system, how does one...
  3. davidbenari

    Does time translational symmetry imply H'=0 or E'=0?

    The Hamiltonian is not always equal to the total energy. In fact the Hamiltonian for a system of particles could be defined as ##H=L-\sum \dot{q_i}\frac{\partial L}{\partial \dot{q_i}}## Which is the total energy only if the potential energy is a function of ##q_i## and if the kinetic energy...
  4. Kevin McHugh

    Linear Algebra Exploring Groups and Symmetry: A Beginner's Guide

    Hi folks, I know nothing about groups or symmetry. Could anyone recommend a good introduction that defines groups and explains their notation and operations? I am particularly interested in general, orthogonal, and unitary groups. Thanks in advance. Kevin
  5. davidbenari

    Can ##\mathbf{D}## field only exploit symmetry for linear?

    There are problems in classical electromagnetism where they ask you to find the electrical displacement given some geometry (like a sphere or a cylinder) and the dielectric constant ##\epsilon_r##. The solution to these problems typically employs symmetry arguments along with Gauss' laws for...
  6. J

    Correlation funciton of a system with translational symmetry

    Hi guys, i have been confused by one statement on the spatial correlation funciton in the statistical physics textbook. They say for a spatial correlation function f(x1,x2), where x1 and x2 are the coordinate of particle 1 and 2, if the system has translational symmetry, then f depends only...
  7. fricke

    Time-Reversal Symmetry Explained

    I completely have no idea what time-reversal mean. Why does, by substituting -t into an equation and if the result is the same as the original equation, then the equation is said to be time-reversal symmetry? Also, what does that 'symmetry' mean there? An even function?
  8. F

    Does relativistic QM obey rotational symmetry?

    SO(3) is subgroup of Poicare group.Does Relativistic Quantum Mechanics obey rotational symmetry.If it is,why we do not still keep the non-relativistic concept of angular momentum(orbit angular momentum plus spin) for relativistic concept of angular momentum,but we instead replace the concept by...
  9. S

    Proving Symmetry and Finding Conserved Quantities for Lagrangian Functions

    Homework Statement Given L (q, dq/dt, t). translation: q ---> q + e (e is infinitesimal constant) show that if ∂L/∂q = 0, then L is symmetry under the above translation. then find conserved quantity. Homework Equations S = ∫ L dt The Attempt at a Solution My attempt is nothing... because I...
  10. H

    Is band structure in semiconductors symmetrical with respect to k?

    Do any band structure (in absence of any external field) in general, is symmetric with respect to k? In other words, do we always have E(k)=E(-k).
  11. arivero

    Puzzling Chiral Symmetry Breaking

    Thinking aloud. Most descriptions of chiral symmetry breaking nowadays present it as something happening in QCD. But it was defined well before of the quark theory, and then it was something related to isospin symmetry. It is a bit puzzling because it seems as if pion mass were originated...
  12. P

    What is the relationship between dynamical symmetry and Noether's theorem?

    Hi, I am learning classical mechanics right now, Particularly Noether's theorem. What I understood was that those kinds of transformations under which the the Hamiltonian framework remains unchanged, were the key to finding constants of motion. But here are my Questions: 1. What is...
  13. genxium

    Why is stress tensor (in this derivation) symmetric?

    First by "this derivation" I'm referring to an online tutorial: http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node9.html It's said in the above tutorial that the ##i-th## component of the total torque acting on a fluid element is ##\tau_i = \int_V \epsilon_{ijk} \cdot x_{j} \cdot F_{k}...
  14. S

    Finite Element Symmetry Problem

    Hi, I am running a finite element on a cylinder with that converges at the bottom for a opening, which is symmetrical in both directions so i modeled one quarter but the problem is my stresses are the same with when i compare with a full model that i also done but the deflections are different...
  15. J

    Symmetry about x-axis, y-axis or origin

    Homework Statement determine whether the graph of the function has symmetry about the x-axis, the y-axis or the origin. Check work by graphing: x^(2/3) + y^(2/3) = 1 Homework EquationsThe Attempt at a Solution [ x^(2/3) + y^(2/3) = 1 ] ^(3/2) y = -x+1 Its a straight line with y-intercept at...
  16. Vinci128

    Hi all I'm italian, i'm graduating

    Hi everyone, my name is Vincenzo, i come from puglia in Italy and I'm studying physics at th university La Sapienza in rome. I'm writing a dissertation about the discovery of the violation of CP symmetry in the neutral K mesons system. I 'd love to discuss about these themes, and i hope i'll...
  17. R

    PT symmetry breaking in optical systems

    There are very results published about PT symmetry breaking in optical systems, with effects like anysotropical transmission resonance in waveguides. But if PT symmetry is broken in a optical system and CPT symmetry must always be respected, then what C symmetry is broken in a optical system?
  18. P

    Are heat radiation and absorption symmetric?

    A friend of mine heard a popular science show on the radio. A caller asked what is better to wear on a hot day, white clothes or black clothes. The answer given was that it did not matter because although black absorbs more readily it also radiates it more readily. My friend said of course that...
  19. SoumiGhosh

    Topological phase and spontaneous symmetry breaking coexist?

    As we know topological phases cannot be explained using spontaneous symmetry breaking and order parameter. But can they coexist? Suppose there is a system which is undergoing quantum phase transition to a anti-ferromagnetic phase from a disordered phase. So in the anti-ferromagnetic phase...
  20. T

    Why does gauge fixing break gauge symmetry?

    By fixing a gauge (thus breaking orspending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the...
  21. 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.
  22. G

    A question about symmetry in the phi^4 theory

    Homework Statement Why does the symmetry ##\phi\rightarrow-\phi## mean that an amplitude can be written as ##\alpha + \beta p^2 + \gamma p^4 + ...## without the odd terms in ##p##? Homework Equations I understand that, due to this symmetry, any diagram in ##\phi^4## has an even number of...
  23. S

    G- & Isospin Symmetry for Γ(ρ0→π0γ)

    Homework Statement Show Γ(ρ0→π0γ) = Γ(ρ+→π+γ) Using G- and isospin symmetries, without exact calculating the matrix elements using additive quark model. Homework Equations L = jμAμ G = CR1802 Mif ≅ <π|jμ|ρ>eμ jμ=2/3 * (anti u)γμu - 1/3 * (andi d)γμd) The Attempt at a Solution Mif ≅...
  24. A

    Motion vs Rest: How Does Relativistic Time Differ?

    There are two separate clocks, each set in a plane and on the ground. Assuming inertial reference frames, how can this be? Perspective of the observer in motion: The observer in motion on the plane will have recorded some time duration. Since the the world outside the plane is moving at a...
  25. T

    What is the global and local symmetry in gauge theory?

    For details and to try to explain the above, the complex scalar field is a useful example. The complex scalar field has an action S=∫d4x(∂μϕ∗)(∂μϕ)−V(|ϕ|). There is a global, continuous symmetry to this action -- an overall phase. That is, if one replaces ϕ→eiαϕ, then the action does not change...
  26. A

    Wilson Loop and spontaneous symmetry breaking.

    I would like to ask about the case of: ##SU(2)\otimes U(1) \rightarrow U(1)\otimes U(1),## spontaneous symmetry breaking. It is given that the Wilson Loop: ##W \equiv exp[ig \oint dy H T^1]= diag(−1,−1,1).## Where ##y## is the ##S^1/Z^2## fifth/extra dimension, ##H = \frac{1}{g R}## and...
  27. w3dnesday

    Gauss' law, spherical symmetry

    Homework Statement Assume that a ball of charged particles has a uniformly distributed negative charge density except for a narrow radial tunnel through its center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton anywhere along the...
  28. K

    Exploring Lorentz Factor Symmetry: A 125-Year Mystery

    Hi All, Recently, in my earlier thread, I asked about the symmetry of Lorentz Factor (where +V == -V). I had several pointers about how to handle this (Thank you all those). However, it seemed unanimous that there was no explanation beyond the equation itself (math or physical). Also, I...
  29. Stephanus

    Does the Twin Paradox Break Symmetry in the Zig-Zag Scenario?

    Dear PF Forum, Sorry if I ask (again) about twin paradox, after so many question about this topic here. Supposed T is a star 100 ly from earth. If B travels to T from earth A. Is the symmetry broken? B. If B watches A's clock at Earth and A watches B's clock at T, do they see the other clock...
  30. J

    Conformal symmetry of FRW spacetime

    The standard spatially flat FRW metric in Cartesian co-moving co-ordinates is given by: $$ds^2=dt^2-a(t)^2(dx^2+dy^2+dz^2)$$ As far as I understand it the fact that the metric can be written in a form that is independent of ##x,y,z## implies that the Universe has the physical qualities of being...
  31. A

    SU(N) symmetry breaking by non-trivial parity.

    I would like to prove the following: Suppose we have the diagonal matrix ##P = diag(1,\ldots,1, -1,\ldots, 1)##, with ##N_+## elements of ##1## and ##N_-## elements of ##-1## such as ##N_+ + N_- = N## and ##N_+, N_- \geq 1##. This matrix is a non trivial parity matrix since it is not...
  32. F

    Why does anomaly mean symmetry is broken?

    Is the expection value of expession in left hand side of motion equation of field(example: Klein-Gordon,Dirac...equations) equal zero or not?(left hand side of the equation equals zero when we put condition of mimimizing the action).If not,why we can say when expectation value of divergence of...
  33. TrickyDicky

    About time symmetry in quantum physics

    What is more fundamental and why, the postulated time symmetry of QM tie evolution or the time asymmetry of the CPT theorem?
  34. R

    Is Aijkl a Symmetric Rank 4 Tensor? Proof Needed!

    Homework Statement Let Aijkl be a rank 4 square tensor with the following symmetries: A_{ijkl} = -A_{jikl}, \qquad A_{ijkl} = - A_{ijlk}, \qquad A_{ijkl} + A_{iklj} + A_{iljk} = 0, Prove that A_{ijkl} = A_{klij} Homework EquationsThe Attempt at a Solution From the first two properties...
  35. stevendaryl

    Is there a connection between EPR and crossing symmetry in Feynman diagrams?

    This is sort of a nebulous question, but I'm wondering if anyone knows of analysis of the EPR experiment from the point of view of crossing symmetry for Feynman diagrams? In the above diagram, I've drawn a very simple particle interaction diagram. The same diagram can be interpreted in two...
  36. D

    Symmetry of Orthogonally diagonalizable matrix

    Can someone confirm or refute my thinking regarding the diagonalizability of an orthogonal matrix and whether it's symmetrical? A = [b1, b2, ..., bn] | H = Span {b1, b2, ..., bn}. Based on the definition of the span, we can conclude that all of vectors within A are linearly independent...
  37. P

    Symmetry in Integrals: Peskin's Equation 6.43 & 6.44

    In peskin p. 192, they says that the denominator (that is equation 6.43) is symmetric under x<--> y. Thay all so say that you can see it in equation 6.44. But one of the terms in the denominetor is y*q which dose not have that symmetry! Looking at (6.43) and removing the summetric parts leave...
  38. A

    Electromagnetic boundary conditions for symmetric model

    I stumbled upon this article: http://www.comsol.com/blogs/exploiting-symmetry-simplify-magnetic-field-modeling/ Since the article does not contain any mathematical formulations, I was wondering how the boundary conditions can be expressed in terms of magnetic vector potential. From what I...
  39. powerof

    Symmetry in second order partial derivatives and chain rule

    When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...
  40. c3po

    Find matrix representation for rotating/reflecting hexagon

    Homework Statement Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves...
  41. Mentz114

    Understanding Gauge Symmetry in Massive Gravity: Analysis of Fierz-Pauli Action

    Refering to this paper "Theoretical Aspects of Massive Gravity" (http://arxiv.org/abs/1105.3735) about the spin-2 boson field and GR. The author uses the Fierz-Pauli action ( I quote the massless part) ##-\frac{1}{2}\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu} + \partial_\mu...
  42. V

    Understanding symmetry in electric field calculations

    Homework Statement Consider the specific case of a point above the axis of a circular ring of charge, how do the calculations follow to cancel the radial components? I understand the concept of the symmetry but don't understand how to express it in the expression without just removing the term...
  43. K

    Questions on electroweak symmetry breaking

    I have some questions or thoughts about EW symmetry breaking. (1) Higgs mechanism gives mass to SM particles after the background higgs field rolls from ##h=0## to ##h=v## and symmetry is broken. We are talking about pole mass, aren't we? So pole mass changes continuously from ##0## to ##m##...
  44. Qturtle

    Expanding the Hamiltonian around a point of symmetry

    Hey I have a tight binding Hamiltonian of a BCC lattice which is a 4x4 matrix in k space (the 4 elements correspond to 4 atoms that are in a unit cell) I want to expand it for small k's around the symmetry points P or Gamma or H. I'm looking at a paper by J. L. Ma˜nes, PHYSICAL REVIEW B 85...
  45. L

    Do FCC Structures only have a CN of 12?

    Hi all, Without first delving too deeply into the literature I wanted to ask if its is only permissible for FCC structures to have a coordination number of 12. In the case of lattice vacancies and/or distortion: wherein the atomic sites are slightly displaced; is there a kind of tolerence for...
  46. N

    Finding the Axis of Symmetry in a Parabola: How to Solve the Problem

    Homework Statement The Attempt at a Solution I would guess it is -b/2a but I have gotten this incorrect. Maybe I have to figure out what the parabola equation is and apply it into the axis of symmetry. Assuming that is correct how would I find that? Thanks[/B]
  47. JonnyMaddox

    Infinitesimal transformations and Poisson brackets

    Hello, I want to understand how bracket operations in general are related to symmetry and infinitesimal transformations (in hindsight of quantumfieldtheory), so I calculated an example with a particle that is moving on a circle with a generic potential. (I used simple polar coordinates in two...
  48. M

    Gauss' Law: Spherical Symmetry

    Homework Statement Figure 23.52 gives the magnitude of the electric field inside and outside a sphere with a positive charge distributed uniformly throughout its volume. The scale of the vertical axis is set by Es = 5.0 x 10e7 N/C. What is the charge on the sphere? Homework Equations Net Flux...
  49. Digitalism

    Scale Symmetry: Resources, People, History

    Hello, I am looking for resources on scale symmetry including the persons who are at the forefront, arxiv preprints, history of the development of the idea, and any other sources you can think of. Thank you.
  50. K

    Gauge symmetry of cylindrical rod

    A twisted cylindrial rod has the cross sectional symmetry so that it's not posible to tell whether it is twisted or not without knowing if there is any torsional energy. now drawing a line on the surface of it can tell us whether or not it's twisted. It might not be a straight line.. there are...
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