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. nomadreid

    I Oddity of a functional equation for the R zeta function

    In https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/fnleqn.htm the equation ζ(s)=ζ(1-s) is used, where ζ is the Riemann zeta function, which I find curious, for the following reasons this indicates a symmetry around Re(s)=1/2, which seems to be what the diagram at 20:27 of seems to...
  2. arivero

    A New symmetry principles and anomalies

    I noticed recently the content of TASI 2023 lectures, a lot about "generalised symmetries" and a particular one, "non invertible symmetries", that seems to propose a new way to understand the ABJ anomaly. Have you guys read it?
  3. K

    Quantum Well Centred at the Origin Doubled in Size at time t=0. Symmetry seems to render the question incorrect

    Firstly I have found the eigenstates for both the original well and the new well as the following $$\psi_{n,\frac{L}{2}} = \begin{cases} \sqrt{\frac{2}{L}} \cos{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ odd} \\ \sqrt{\frac{2}{L}} \sin{\frac{n \pi x}{L}} \; \; \; \; \; n \text{ even}...
  4. Gunter1977

    How to Calculate Forces on Sun Wheels in a Planetary Gear System?

    Hello everyone, I hope somebody can assist me with this. I am working on a calculation for a planet gear with 4 sun wheels and 1 planet wheel. The aim is to determine the forces on the sun wheel. Initially, I have 3 equations and 4 unknowns. By making an assumption based on symmetry, I now have...
  5. DaTario

    I On the necessary conditions to form the hydrogen molecule

    Hi All, Is it true that in order to form the ##H_2## molecule from two hydrogen atoms one needs a third body so that the energy difference between initial and final states can be released? Best Regards, DaTario
  6. C

    B What does symmetry of time dilation really mean?

    Hello everyone, I've been studying Morin's book and the case for time dilation makes sense, a clock in the rest frame of the moving body counts to ##T_A##, and a clock in the lab frame counts to ##T_B## and we find ##T_B = {\gamma}{T_A}## What I might be failing to do is understand what the...
  7. C

    I Question about time dilation symmetry

    Hello everyone, Suppose there is an airplane that is taking off from an airport but before it takes off it synchronizes it's clock to zero with the clock at the airport. In the rest frame of the plane the airport is moving, so you could argue ##T_{plane} = \gamma*T_{airport}## In the rest...
  8. haruspex

    B Symmetry regarding induced potentials?

    A homework thread, https://www.physicsforums.com/threads/point-charge-with-very-thin-metal-sheet-along-a-spherical-surface.1057702/, references https://arxiv.org/pdf/1007.2175.pdf. There is an uncharged conductor and a point charge. In the paper referenced, ##\bar\phi_y(x)## is defined as the...
  9. Z

    A Symmetry in terms of Lagrangian

    Hi, as we know SM is symmetric under SU(3) X SU(2) X U(1) , But my question is , how can we check the invariance of terms in Lagrangian under these symmetries - thanks
  10. chwala

    I Cylindrical coordinates -Curvilinear

    Why are the coordinates seemingly used when the symmetry is around ##z## axis? Any particular reason why not ##x## or ##y##. In transforming from Cartesian to cylindrical form; I can see that ##z## is not considered when determining ##r##. Can we also use ##x## and ##z## assuming that the...
  11. G

    I Lagrangian symmetry for the bullet and gun

    I have been learning emma noether’s theorem where every symmetry results in conservstion law of something, sometimes momentum, sometimes angular momentum, sometimes energy or sometimes some kind of quantity. Every explanation that I have listened to or learned from talks about homogeneity of...
  12. Feynstein100

    I Does mass conservation correspond to a pseudo-symmetry?

    Emmy Nöther proved that mathematically, if a certain quantity is conserved, there must be a corresponding symmetry somewhere. Momentum conservation stems from spatial symmetry, charge conservation stems from complex phase symmetry at the quantum level and energy conservation stems from time...
  13. Junaidjami

    A Real vs complex spherical harmonics for hexagonal symmetry

    Are real spherical harmonics better than complex spherical harmonics for hexagonal symmetry, which are directly associated to a finite Lz?
  14. S

    I Consequences of the absence of global symmetries...?

    I found some interesting discussions in this site (e.g: https://www.physicsforums.com/threads/smolin-lessons-from-einsteins-discovery.849464/; https://www.physicsforums.com/threads/relatismo-to-the-max.83885/) which are related to Lee Smolin's ideas that laws are not immutable and can therefore...
  15. H

    I Was Electroweak Symmetry Breaking in the Early Universe Controversial?

    According to Sabine Hossenfelder in the extremely early Universe nothing had mass because the electroweak symmetry was not yet broken so there was no Higgs field. Am I correct in thinking this is not controversial? https://youtu.be/9-jIplX6Wjw?t=638
  16. H

    I Symmetry and two electron wave function

    In the picture below we have two identical orbitals A and B and the system has left-right symmetry. I use the notation ##|n_{A \uparrow}, n_{A \downarrow},n_{B \uparrow},n_{B \downarrow}>## which for example ##n_{A \uparrow}## indicates the number of spin-up electrons in the orbital A. I would...
  17. physics2023

    A What is meant when a phase is said to have "symmetry protected"?

    What is meant when a phase is said to have "symmetry protected"?
  18. C

    I Lack of Symmetry in X & T: Explained for Beginners

    Hello everyone, I'm reading "Special Relativity For the Enthusiastic Beginner" (by David Morin) and so far I really like the book, but I have a little bit of a hiccup in understanding terminology regarding stated symmetries or the lack there of in reference to the Galilean transformation when...
  19. E

    I The implications of symmetry + uniqueness in electromagnetism

    I have tried to follow "Symmetry, Uniqueness, and the Coulomb Law of Force" by Shaw (1965) in both asking and solving this question, but to no avail. Some of the mathematical arguments there are a bit too quick for me but, it suffices to say, the paper tries to make the "by symmetry" arguments...
  20. E

    A Ballentine on construction of the (Galilean) symmetry generators

    In Problem 3.7, Ballentine says: The unitary operator ##U(v) = exp(iv·G)## describes the instantaneous ##(t = 0)## effect of a transformation to a frame of reference moving at the velocity ##v## with respect to the original reference frame. Its effects on the velocity and position operators are...
  21. M

    I The Feynman way of explaining Symmetry in Physical laws

    So on this page https://www.feynmanlectures.caltech.edu/I_11.html under heading 11-2 Translations first he tries to proof that there is no origin in space. Joe writes newtons laws after measuring quantities from some origin. $$m(d^2x/dt^2)=F_x$$ $$m(d^2y/dt^2)=F_y$$ $$m(d^2z/dt^2)=F_z$$ We need...
  22. M

    Symmetry behind charged spring-mass system in Electric field

    For this problem, If we assume that x = 0 is where the spring connects to the wall, then the rest position of the mass-spring-electric field position is x = EQ/k and the max position is x = 2EQ/k. Is there a reason for the symmetry between the rest position and max position? (The symmetry...
  23. H

    I The left-right symmetry and forbidden triplet spin state

    Suppose that we have a system including two single (and identical) orbitals as shown in the attached image. Then, consider the states that we have two different spins on the two orbitals. It seems that because of the left right symmetry of the system the special part of teh wave function can not...
  24. Q

    A Does CPT Symmetry Apply to a Mirror World with Reversed Properties?

    Say we have a copy of our world, but with the "parity" and "time" reversed. (For example, the mirror world from the novel "Alice Through the Looking Glass".) If the "charge" is also reversed, then due to the CPT symmetry, the same physics law would apply to the mirror world, right? If we...
  25. G

    I Can Spherical Symmetry Be Achieved Without Varying Line Element?

    "Spherical symmetry requires that the line element does not vary when##\theta## and##\phi## are varied,so that ##\theta##and ##\phi##only occur in the line element in the form(##d\theta^2+\sin^{2}\theta d\phi^2)##" I wonder why: "the line element does not vary when##\theta## and##\phi## are...
  26. T

    I Why is K an anti-unitary operator in (26)?

    Hey all, I just wanted to double check my understanding of (26) in the following notes: https://arxiv.org/pdf/1512.08882.pdf. Is the reason that ##(U_{T}\cdot K) \cdot (U_{C}\cdot K) = U_{T}\cdot U_{C}^{*}## because ##K## is a unitary operators and thus ##(K\cdot U_{C}\cdot K) = U_{C}^{*}## as...
  27. Rafaelmado

    A Symmetry & Invariance of Pions: π+, π0, π− and Other Mesons Explained

    Pions are particles with spin 0 and they form an isospin triplet: π+, π0, π− (with the superscript indicating the electric charge). Their intrinsic parity is −1 and they are pseudoscalar mesons. In nature we also find other kind of mesons, like the ρ mesons, ρ+, ρ0 and ρ−. As pions, they also...
  28. P

    A Symmetry of the Green-Keldysh-Nambu function

    Hello, I would like to understand a relation of this article by Volkov (eq. 4). Let's define the Green function $$ G^{ij}_{ab} (1,2) = -i \langle T_c \Psi_a (1_i) \Psi_b (2_j) \rangle $$ where ##a,b = (1,2)## are the spin indices and ##i,j = (1,2) ## are the indices for the Keldysh contour ...
  29. Alpha Roy

    A Mapping High Symmetry Points from Primitive to Conventional Cell

    We usually plot electronic bands with the help of high symmetry points of the irreducible zone of primitive cell of particular material. But if we want to plot bands with conventional cell, we have to map the high symmetry points from primitive cell to conventional cell. so how can we map the...
  30. Orodruin

    Insights Symmetry Arguments and the Infinite Wire with a Current

    Continue reading...
  31. S

    A Vacuum Transitions and Lorentz Symmetry Breaking

    There are several "bumblebee" models [1], [2] where Lorentz invariance is violated usually resulting from a local vector or tensor field acquiring a nonzero vacuum expectation value We do not know whether we are in the true vacuum state or in a "false"/metastable vacuum state that could decay...
  32. S

    I Could the Lorentz symmetry be theoretically broken in vacuum?

    In this paper [1] which considers the possibility that the Lorentz symmetry could be broken, at page 4-5 the author says: "We now introduce a Higgs sector into the Lagrangian density such that the gravitational vacuum symmetry, which we set equal to the Lagrangian symmetry at low temperatures...
  33. Rzbs

    I Space Group Symmetry: What is F d3m Spinel Structure?

    What does mean spinel structure has F d3m space group? I know F is for face centred cubic, 3 is 3-fold symmetry and m is mirror, but I don't know what means "d"?
  34. A

    A A good quantum number for Cnv symmetry?

    Hello, I was wondering if it was possible to define good quantum numbers in solid state physics or chemistry when systems posses a discrete cylindrical symmetry Cnv. I know that in terms of angular momentum, L and L_z will be good quantum numbers for spherical symmetry, then only L_z is a good...
  35. M

    A MTW Gravitation: Exercise 5.1 | Beginner GR Stress-Energy Tensor Symmetry

    I am a beginner in GR, working my through Gravitation by the above authors. If there is a better place to ask this question, please let me know. I understand (from section 5.7) that the stress-energy tensor is symmetric, and from equation 5.23 (p. 141), it is explicitly symmetric. But...
  36. greg_rack

    Calculating the Line Integral of F over C: Stokes' Theorem and Symmetry

    From Stokes we know that ##\iint_{\textbf{S}}^{}curl \textbf{F}\cdot d\textbf{S}=\int_{C}^{}\textbf{F}\cdot d\textbf{r}##. Now, we can calculate the surface integral of the curl of F by calculating the line integral of F over the curve C. The latter ends up being 0(I calculated it parametrizing...
  37. chwala

    Find order of rotational symmetry for the given shape

    Ok for (1) I would say that the order of rotational symmetry is ##8##. Would that be correct? What about ##4##? For (2) The number of lines of symmetry is ##4##. And if one would say infinity for both (1) and (2) would that be correct?And if you consider a kite. Would the order of...
  38. P

    A Global vs. Local (gauge) Symmetry

    Gauge symmetry is highly confusing, partly because many definitions differ in the literature. Strictly speaking gauge symmetry should be called gauge redundancy since you are mapping multiple representations to the same physical state. What is your favourite definition of what "large" gauge...
  39. dRic2

    I Action of a symmetry operation on a Bloch state

    Generally speaking, if the Hamiltonian has a specific symmetry defined by an operator M, that is ##[H,M]=0##, when I apply such symmetry operator to a Bloch state I would expect the state to be left unchanged up to a phase: $$M\ket{ψ_{\mathbf k}}=e^{iϕ(\mathbf k)}\ket{ψ_{\mathbf k}}$$ For the...
  40. P

    A Einstein Field Equations: Spherical Symmetry Solution

    [Moderator's Note: Thread spin off due to topic and level change.] For a spherically symmetric solution, if SET components were written in terms a single one of 4 coordinates, in a way plausible for a radial coordinate, the I believe solving the EFE would require spherical symmetry of the...
  41. P

    A Order parameter, symmetry breaking Landau style

    Hi all, I am somewhat familiar the Landau Ginzburg paradigm for phase transition. My understanding is that it is a phenomological model of 2nd order phase transitions by "guessing" that the free energy can be expanded a configuration integral (path integral) of a functional of a local order...
  42. Antarres

    A Question about S(3) spontaneous symmetry breaking in Peskin & Schroeder

    In chapter 20 of Peskin&Schroeder about spontaneous symmetry breaking, he considers and example on page 696 of spontaneous symmetry breaking of SU(3) gauge group with generators taken in adjoint representation. Covariant derivative is defined with: $$D_\mu\phi_a = \partial\phi_a +...
  43. S

    Engineering Applying symmetry to an asymmetrical circuit

    Hello everyone, I need to solve a problem applying symmetry and superposition theorem but the problem is that the circuit is almost symetrical but has little differences in both sides (the source in the left side and the unknown impedance in the right side). I couldn't solve the problem using...
  44. M

    B Could Spontaneous symmetry breaking cause momentum change in an atom?

    If you were to fire a single atom from a fixed point into a chamber of perfect vacuum and measure where it collides with the opposite wall. Could Spontaneous symmetry breaking in the sub atomic particles cause momentum change in the atom, changing the part of the wall the atom interacted with?
  45. P

    How does mirror symmetry for resistance work?

    The question is to find the resistance between AF (top 2 points). Let the far right unlabelled vertex of the pentagon be B. Why can we say that points O and C are at the same potential? I get that both points O and C appear to be at 'half the path' if you consider AOF and ABCDF, and so potential...
  46. V

    Understanding Spherical Symmetry of Electron Clouds in External Fields

    The given diagram looks something like this: Electric force on nucleus from external field must balance attraction force from electron cloud and electric force from external field. $$e\vec{E}=\frac{k(\frac{L^3}{R^3}e)}{L^2}\hat{L}$$ where ##\vec{L}## is from center of electron cloud to...
  47. ergospherical

    A Fluid Flow Symmetry: Showing $\pi_b G^b$ is Constant

    I don't know where to start with this problem. If ##\pi_a = (\mu + TS) u_a## then show that \begin{align*} u^a \nabla_{a} (\pi_b G^b) = 0 \end{align*}where the field ##G^a## is a symmetry generator. [##S## is entropy/baryon, ##T## is temperature, ##u_a## is a one-form field corresponding to a...
  48. yucheng

    How does metal sphere determine the potential of the plane by symmetry?

    The answer given states that: The entire x-y plane is obviously at the same potential since all the fields are strictly perpendicular to it (draw a diagram if youre confused). Since we choose the sphere to be at potential zero, the point on the sphere which cuts the x-y plane is also at zero...
  49. Q

    A How to systematically find the symmetry operator given a Hamiltonian?

    For instance,how to systematically derive the equns 2.2 & 2.5 given a Hamiltonian on the article below?; arxiv.org/pdf/0904.2771.pdf .
  50. W

    B Symmetry of parity: Mistake in the experiment?

    Recently I saw this YouTube video from Veritassium about CPT -Symmetry: In this video an experiment of Prof. Chien-Shiung Wu is presented, which has proven that parity is not symmetric, by observing the emmition of electrons from Co60 atoms with synchronised spin. After thinking about this...
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