# 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. ### I Uniform Translation of a Lattice

I'm currently reading through this brief review on symmetries, and on page 5, the following statement is made: "Why is there no energy cost for a uniform displacement? Well, there is a translational symmetry: moving all the atoms the same amount doesn’t change their interactions. But haven’t we...
2. ### 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...
3. ### I No symmetries in the Universe at the Big Bang...?

I apologize in advance if this is a stupid question but... According to some scenarios about the beginning of the universe (namely cosmological inflation), in layman's terms, everything was born out of a quantum fluctuation which caused a violent expansion. In this case, since an expanding...
4. ### I Models where all symmetries would be approximate?

I found this interesting discussion here in Physics Forums (https://www.physicsforums.com/threads/are-all-symmetries-in-physics-just-approximations.1005038/) where the topic of all symmetries being approximate is discussed Is there any model (for instance, some type of spacetime metric or...
5. ### I Non-homogeneous and anisotropic metric and laws of physics...?

In this popular science article [1], they say that if our universe resulted to be non-uniform (that is highly anisotropic and inhomogeneous) then the fundamental laws of physics could change from place to place in the entire universe. And according to this paper [2] anisotropy in spacetime could...
6. ### I Are there non-smooth metrics for spacetime (without singularities)?

Are there non-smooth metrics for spacetime (that don't involve singularities)? I found this statement in a discussion about the application of local Lorentz symmetry in spacetime metrics: Lorentz invariance holds locally in GR, but you're right that it no longer applies globally when gravity...
7. ### I Approximate local flatness = Approximate local symmetries?

Pseudo-Riemannian manifolds (such as spacetime) are locally Minkowskian and this is very important for relativity since even in a highly curved spacetime, one could locally approximate the spacetime into a flat minkowski one. However, this would be an approximation. Perhaps this is a naive...
8. ### I Inhomogeneities and topological defects in cosmology...

I have heard that some types of inhomogeneties and topological defects (like cosmic strings) in cosmology have been proposed to be able to break fundamental symmetries of nature such as the Poincaré, Lorentz, diffeomorphism CPT, spatial/time translational...etc symmetries... However, I have not...
9. ### A Symmetries of particle interacting with external fields (Ballentine)

I am following along with Ballentine's (in his *Quantum Mechanics: A Modern Development*) construction/identification of symmetry generators as operators representing the standard observables (observables here being used in the sense of a physical concept which have operators representing them)...
10. ### I What does it mean that symmetries do not hold globally?

Perhaps this is a stupid question but, if Lorentz symmetry and time translational symmetry are not global in an expanding universe, wouldn't that mean that is possible that other Hubble spheres outside our observable universe could have other symmetries or an absence of the Lorentz symmetry? I...
11. ### I Explore Spacetimes, Metrics & Symmetries in Relativity Theory

I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970) In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity...
12. ### I No Symmetry Spacetimes/Metrics in Theory of Relativity

In the context of the Theory of Relativity are there any spacetimes or metrics with a complete absence of symmetries? I mean, consider a type of space or metric where no symmetries would hold (at least not exactly, but approximately). A space or metric where the Poincaré invariance (including...
13. ### I Chiral symmetries in E[SUP]^n[/SUP]

As a biochemist, I deal with chirality of molecules all the time. If you have a tetrahedral molecule, for example a carbon atom, and all 4 vertices are labeled differently, as in different atoms on each one, then that molecule has a mirror-symmetric one that cannot be superimposed on the...
14. ### I What group might represent the symmetries of these carbon rings?

The carbon rings in the upper-middle of this page https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/react3.htm such as corannulene or coronene possess symmetries. But, they are not the typical dihedral arrangements of points, like a single hexagon or single pentagon or single equilateral...
15. ### A Possibility of Discrete Symmetries in GR: Exploring CPT Symmetry

Our current model (FLRW) is clear that the universe has a continuous temporal asymmetry. This is seen as the expansion factor grows with time, and thermodynamically with entropy. A continuous transformation in the current model ##t \rightarrow t + dt## is not the same as ##t \rightarrow t - dt...
16. ### I Would infinite entropy break all symmetries?

If the Universe could somehow reach a state of infinite entropy (or at least a state of extremely high entropy), would all fundamental symmetries of the physical laws (gauge symmetries, Lorentz symmetry, CPT symmetry, symmetries linked to conservation principles...etc) fail to hold or be...
17. ### I Are there types of spacetime where no symmetries are valid?

We derive the most basic laws of physics from several fundamental symmetries (those from Noether's theorems, gauge symmetries, Lorentz symmetry...). But are there any types of spacetime where no symmetries, no matter how fundamental, would be valid? Any special metric, geometry or shape?
18. ### Lagrangian with a charged, massive vector boson coupled to electromagnetism

I need to use hermiticity and electromagnetic gauge invariance to determine the constraints on the constants. Through hermiticity, i found that the coefficients need to be real. However, I am not sure how gauge invariance would come into the picture to give further contraints. I think the...
19. ### I Changing or breaking the most fundamental laws and symmetries?

There are some theoretical processes (like vacuum decay in quantum field theory) that could change the physical constants of the universe. Similarly, in inflation theory, various models predict that multiple regions that would stop inflating would become "bubble universes" perhaps with different...
20. ### I Problems involving combinatorics of lattice with certain symmetries

I was reading about numerical methods in statistical physics, and some examples got me thinking about what seems to be combinatorics, an area of math I hardly understand at all beyond the very basics. In particular, I was thinking about how one would go about directly summing the partition...
21. ### I Minkowski Spacetime KVF Symmetries

Hi, reading Carrol chapter 5 (More Geometry), he claims that a maximal symmetric space such as Minkowski spacetime has got ##4(4+1)/2 = 10## indipendent Killing Vector Fields (KVFs). Indeed we can just count the isometries of such spacetime in terms of translations (4) and rotations (6). By...
22. ### I What are the 230 space groups in solid state physics?

It's written in one book I've got on solid state physics the following: Can someone please explain how to get this number of 230 combinatorially? Thanks!
23. ### A Why does D(1,1) representation of SU(3) give baryon octet?

The question may be ambiguous but it's really simple. One says that the baryon octet is the D(1,1) representation of SU(3), but then uses the same one for mesons. D(1,1) means one quark and one antiquark, which corresponds perfectly to mesons. But how can it explain baryons? My information and...
24. ### I Are all symmetries in physics just approximations?

I came across this video of Leonard Susskind saying that all symmetries in physics are approximations. Unfortunately, I don't have the links on hand, but I have come across other sources of physicists claiming that all symmetries are approximations. My confusion though is that it was my...
25. ### Symmetries in Lagrangian Mechanics

In Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws (starting on page 291 here), the authors introduce the concept of a generator function G, where the transformation generated by G is given by (equation 12.29 on page 292 in the text)...
26. ### Why do we see symmetry everywhere in science and nature?

In your own words, why do humans look for and prefer symmetries in mathematics, physics, philosophy and in general?
27. ### I Discrete symmetries and conserved quantities

Hi, please correct me if I use a wrong jargon. If I have discrete symmetries (like for example in a crystal lattice) can I find some conserved quantity ? For example crystal momentum is conserved up to a multiple of the reciprocal lattice constant and it is linked (I think) to the periodicity...
28. ### I Polarization mode symmetries of massless particles

I am just reading Carroll's textbook on GR, where at the end of chapter 7.4 Gravitational Wave Solutions he discuss how rotational symmetries in polarization modes are related to spin of massless particles. He then explains that we could expect associated spin-2 particles to gravity - gravitons...
29. ### Showing that the Weyl tensor is invariant under conformal symmetries

The Weyl tensor is given by (Carroll's EQ 3.147) \begin{align*} C_{\rho \sigma \mu \nu} &= R_{\rho \sigma \mu \nu} - \frac{2}{n-2}\left(g_{\rho [\mu}R_{\nu]\sigma} - g_{\sigma [\mu}R_{\nu]\rho}\right) \\ &+ \frac{2}{(n-1)(n-2)}g_{\rho [\mu}g_{\nu]\sigma}R \end{align*} Where ##n## are...
30. ### MHB Symmetry Operations of a Cube: Geometric Descriptions and Matrix Representations

Hey! :giggle: Let $\displaystyle{W:=\left \{\begin{pmatrix}x\\ y\\ z\end{pmatrix}\in \mathbb{R}^3\mid x,y,z\in \{-1,1\}\right \}}$. Draw the set $W$ in a coordinate system. Let $v=\neq w$ and $v,w\in W$. If they differ only at one coordinate connect these points by a line. With this...
31. ### A Different symmetries or no symmetries in string theory?

I was reading the book "A Fortunate Universe" by Geraint Lewis and Luke Barnes and something caught my attention: At page 195 the authors say that universes with different symmetries could be modeled and they would have dramatic results like having different conservation laws. I asked Mr...
32. ### A What would it mean if symmetries in physics would not be fundamental?

Physicist Joseph Polchinski wrote an article (https://arxiv.org/pdf/1412.5704.pdf) where he considered the possibility that all symmetries in nature may not be fundamental. He says at page 36: "From more theoretical points of view, string theory appears to allow no exact global symmetries, and...
33. ### I Physicists who propose that symmetries are emergent?

I know of some physicists (e.g Holger B Nielsen, Grigory Volovik or Edward Witten) who have proposed that all symmetries (Local gauge symmetries associated with forces and dynamics and global symmetries associated with conservation laws) are emergent rather than fundamental. Are there any other...
34. ### How can I find conserved current for a Lagrangian involving vector fields?

Untill now i have only been able to derive the equations of motion for this lagrangian when the field $$\phi$$ in the Euler-Lagrange equation is the covariant field $$A_{\nu}$$, which came out to be : $$-M^2A^{\nu} = \partial^{\mu}\partial_{\mu}A^{\nu}$$ I have seen examples based on the...
35. ### I Seeking Symmetry: Lie Group & Algebra Models

I wonder whether there is a physical theory / model / example whatever, that uses one of the (Lie) groups ##\begin{bmatrix}1&a_2&\ldots&a_n\\0&1&\ldots&0\\ \vdots & \vdots &&\vdots \\0&0&\ldots&1 \end{bmatrix}## or the Lie algebra ##\begin{bmatrix}a_1&a_2&\ldots&a_n\\0&0&\ldots&0\\ \vdots &...
36. ### Other Classical and Field books on symmetries

Hi. I am interested in finding books dealing with symmetries. Specifically books that make me understand assertions like, and I quote Orodruin's #10 and #16 here https://www.physicsforums.com/threads/find-a-transformation-that-leaves-the-given-lagrangian-invariant.984601/, 'a rotation in the...

38. ### A G-parity - where does the minus sign come from?

Hi all, I have a question on G-parity. I know it's defined as ## G = exp(-i\pi I_{y})C ##, with ##I_y## being the second component of the isospin and ##C## is the C-parity. In other words, the G-parity should be the C-parity followed by a 180° rotation around the second axis of the isospin...
39. ### I Symmetries in quantum mechanics and the change of operators

When we make a symmetrie transformation in a quantum system, the state ##|\psi \rangle## change to ## |\psi' \rangle = U|\psi \rangle##, where ##U## is a unitary or antiunitary operator, and the operator ##A## change to ##A'##. If we require that the expections values of operators don't change...
40. ### I Coordinates for diagonal metric tensors

In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...
41. ### I Groups as symmetries of objects

So it's said that every group is a symmetry group of some tangible object. For example, ##S_3## is the symmetry group of ##\{1,2,3 \}##, and ##D_{2n}## is the symmetry group of an n-gon. But what is ##GL_{10} (\mathbb{R})## the symmetry group of? What about ##\mathbb{Z}##? I have found two...
42. ### I Minimum requisite to generalize Proca action

Hello guys, In 90% of the papers I've read about diferent ways to achieve generalizations of the Proca action I've found there's a common condition that has to be satisfied, i.e: The number of degrees of freedom allowed to be propagated by the theory has to be three at most (two if the fields...
43. ### A Symmetries of a Diamond Unit Cell - Point Group Confusion

Dear All, I've been recently reading the very clear text of Burns and Glazer entitled Space Groups for Solid State Scientists in the context of my thesis which requires understanding of symmetries of crystals, more specifically symmetries of (approximate triply periodic minimal surfaces)...
44. ### I ##A_\mu^a=0## in global gauge symmetries ?

Hi, this question is related to global and local SU(n) gauge theories. First of all, some notation: ##A## will be the gauge field of the theory (i.e: the 'vector potential' in the case of electromagnetic interactions) also known as 'connection form'. In components: ##A_\mu## can be expanded in...
45. ### Symmetries and Conservation Laws

I've been caught by a quite interesting statement of Berkeley physics Course Vol.1 (Chap. 5), that says "In the physical world there exist a number of conservation laws, some exact and some approximate. A conservation law is usually the consequence of some underlying symmetry in the universe."...
46. ### The crystal structure for Ba2TiO explained with symmetries

Hello! I would love to get help on this particular question that I find extremely difficult to answer. -"Bariumtitanate can under a certain set of conditions crystallize in the given structure. With the help of the compounds symmetries, explain how it is possible for it to achieve such a...
47. ### A What Other Symmetries Exist in Superfluids?

I am reading "Introduction to superfluidity" by Andreas Schmitt. He mentions the global symmetry U(1). What other symmetries are there in superfluids? Thank you.
48. ### A Tensor symmetries and the symmetric groups

In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...
49. ### Conserved quantity for a particle in a homogeneous and static magnetic field

The equation of motion for a charged particle with mass ##m## and charge ##q## in a static magnetic field is: ##\frac{d}{dt}[m{\dot{\vec{r}}}]=q\ \dot{\vec{r}}\times \vec{B}## From this, we can see that ##\frac{d}{dt}[m\dot{\vec{r}}-q \vec{r}\times \vec{B}]=0## and so the following quantity is...
50. ### A Why do space translations satisfy the Wigner's Theorem?

Some books argue that typical coordinate transformations such as space translations and rotations are represented in quantum mechanics by unitary operators because the Wigner's theorem. However I do not find any clear proof of this. For instance, suppose 1D for the sake of simplicity, by...