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.
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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...
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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!
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...
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...
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)...
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...
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...
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...
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...
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...
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...
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...
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 &...
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...
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...
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...
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 -...
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...
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...
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)...
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...
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."...
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...
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.
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...
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...
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...