We say that something is symmetric if there is some transformation we can perform on that object that leaves some property unchanged. The set of symmetry transformations of an object form a group, and the name of this group is used as the name of the symmetry of the object.

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What does “the ${\bf N}$ of a group” mean?

In the context of group theory (in my case, applications to physics), I frequently come across the phrase "the ${\bf N}$ of a group", for example "a ${\bf 24}$ of $\mathrm{SU}(5)$" or "the ${\bf 1}$ ...
0
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0answers
36 views

Questions on the derivation of the Lorentz transform

I began trying to understand how special relativity was developed by reading this. Although the proof is very simplistic and probably gives a somewhat shallow understanding of the theory; it is the ...
27
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4answers
5k views

Are black holes perfect spheroids?

What I know about black holes (correct me if I'm wrong) is that they are the most compact objects in the universe that have been discovered. Due to all that gravity, wouldn't black holes be a perfect ...
19
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1answer
670 views

Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice

Happy holidays, everyone! The following is part question, part visual gallery, and part classical mechanics problem. Inspired by snow over the weekend I began simulating the vibrations of the ...
9
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1answer
643 views

Confusion in a trick in solving an energy eigenfunction

Given a non-relativistic energy eigenfunction for a central potential $\left|\Phi \right>$ In solving relativistic hydrogen atom, one of the terms is $$ \left<\Phi\middle|\frac{e^2}{r}\middle|\...
2
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2answers
84 views

Two particles system

Source: this video For a system with two particles (09:30), why is its wave function a product of each particle's wave function? E.g. $$\psi(x_1,x_2)=\psi_a(x_1)\psi_b(x_2)$$ For indistinguishable ...
2
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0answers
24 views

Sublattice symmetry vs Particle hole symmetry

Sublattice symmetry and particle hole symmetry generally constrain a system's energy spectrum to be symmetric with respect to fermi level. My understanding is that they are both represented by an ...
2
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2answers
58 views

Why is the infinitesimal SUSY variation generated by the sum of a left- and right-chiral generator

I was wondering why in many (all? e.g.https://arxiv.org/abs/hep-ph/9709356) resources on N=1 SUSY the variation of a field in the simplest free susy model is defined as $$\delta_\epsilon \phi = (\...
-1
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0answers
28 views

Angular Momentum of Closed Subshell

Suppose we have a state with $2\ell +1$ fermions all entirely in the subspace of hydrogen eigenfunctions with $n, \ell$ fixed. That is we have a state occupied by $2\ell +1$ fermions involving $\mid \...
2
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3answers
207 views

What symmetry gives you charge conservation?

This is a popular question on this site but I haven't found the answer I'm looking for in other questions. It is often stated that charge conservation in electromagnetism is a consequence of local ...
5
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1answer
57 views

Why does galilean invariance imply that particles that start rest stay on the same line?

I'm reading Arnol'd for self study. I'm struggling with this question: "Show that any system of two particles will remain on the same line that connected them at the initial moment, if they started at ...
3
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1answer
389 views

Derivation of Rashba spin-orbit coupling in tight-binding model

Rashba spin-orbit coupling Hamiltonian in free space can be written as: $H_{\text{so}}=\int d^3r \Psi^{\dagger}(\mathbf{r}) \gamma (p_{x}\sigma _{y}-p_{y}\sigma _{x})\Psi(\mathbf{r})$. I expand $\...
10
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1answer
236 views

Intuition for S-duality

first of all, I need to confess my ignorance with respect to any physics since I'm a mathematician. I'm interested in the physical intuition of the Langlands program, therefore I need to understand ...
6
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1answer
46 views

Why does exchanging coordinates produce a phase of $\pm 1$ in an identical particle wavefunction?

Consider a system of two identical particles described by a wavefunction $\psi(x_1, x_2)$. There are two kinds of exchange operators one can define: Physical exchange $P$, i.e. swap the positions of ...
0
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0answers
39 views

Confusion about symmetry factor

I have some questions about symmetry factor. When we count symmetry factor, we count something like how to permute propagators in Feynman diagram, but I think this counting is already taken care of by ...
3
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2answers
672 views

Invariance of Maxwell's Equations under inverting variables - Reference and use

Some months ago, an ArXiv paper mentioned in passing that Maxwell's Equations were invariant under reciprocating the variables, or at least this results in a dual set of Maxwell Equations. (Actually I ...
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0answers
27 views

Symmetries in quantum systems - Wigner theorem and commutation with Hamiltonian

I was reading about symmetries in quantum systems and found (at least) two different definitions. According to Wigner's theorem symmetry transformation (of a quantum system) is a bijective map between ...
16
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4answers
852 views

What sort of experiment would directly test time reversal invariance?

I guess the title says it all: how could/would you experimentally test whether our universe is truly time reversal invariant, without relying on the CPT theorem? What experiments have been proposed to ...
7
votes
1answer
241 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
2
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1answer
183 views

How symmetry is related to the degeneracy?

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 ...
2
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2answers
228 views

Symmetries of non-parallel infinite conducting planes

Suppose I have semi-infinite conducting planes that intersect at some angle $\theta_0$ and have a potential difference of $V$ (the axis of intersection is somehow insulated so they are not actually in ...
0
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0answers
49 views

Matching symmetry factor when a heavy vector field is integrated out

Let us consider the lagrangian $$ \mathcal{L} = \alpha \bar{u}\gamma^\mu u V_\mu + \frac{\beta^2}{2}V_\mu V^\mu $$ there $V_\mu$ is a heavy vector field and $u$ is a massless SU(3)-colored quark. If ...
1
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1answer
173 views

Point group symmetries and unit cell

I was wondering if the unit cell (of a given lattice) had to have every point group symmetries of the lattice it defines ? I guess there is no unique way to define a unit cell and that it may not have ...
4
votes
2answers
161 views

Does the conservation of the Wronskian follow from Noether's principle?

Noether's principle is the paradigm that symmetries of Hamiltonian and Lagrangian systems correspond to conservation laws of various kinds. Consider a one-dimensional harmonic oscillator $$\tag{*} \...
2
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1answer
167 views

Para and ortho hydrogen angular momentum values

In Wikipedia, it is said that: Orthohydrogen, with symmetric nuclear spin functions, can only have rotational wavefunctions that are antisymmetric with respect to permutation of the two protons. ...
5
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3answers
63 views

Why is there a matter-dark matter asymmetry?

It is said generally that nature is symmetric. For example if light behaves as both a particle and a wave, then matter must also do so, which turns out to be true. But we find that the Universe ...
0
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3answers
56 views

Could dark matter possibly be anti-matter?

Considering the broken symmetry after the big bang - what I understand as there being a huge surplus of matter and a lesser presence of anti matter - is it possible that dark matter could be anti-...
6
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1answer
151 views

Derivations of Newton's laws?

I feel convinced that the mathematics behind newtons laws can be derived from Noether's symmetry theorems. The fact that displacement s can be described by a cartesian coordinate system with a ...
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0answers
30 views

Hexagonal shape of snow flakes [duplicate]

As we know snowflakes has hexagonal shape. My question is why is that? and Is there any mathematical model which can explain that particular geometric shape of the snowflakes?
1
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2answers
274 views

Mechanics Landau Galilean Principle

I started reading Landau's Mechanics book and was having some trouble understanding the Galilean Relativity Principle. What does Landau mean by saying space to be homogenous and isotropic and time is ...
1
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0answers
91 views

The Mechanism Behind Massless Particles Acting As One Massive Particle

I am reading a historical account of the development of the Higgs Field theory by Sean Carroll. In it, he states that the 1963 paper by Anderson postulated that "the massless Nambu-Goldstone bosons ...
1
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0answers
26 views

Can an azimuthally symmetric perturbation lift the 2l+1 degeneracy of angular momentum eigenstates?

Assume the initial Hamiltonian of a spinless, non relativistic particle is $$H_0(r,\theta,\phi)=\frac{{\bf p}^2}{2m}+V_0(r)$$ Such that the eigenstates are angular momentum eigenstates $|n,l,m>$, ...
0
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0answers
29 views

Why is charge conjugation multiplicative?

I'm reading Mann's book on the standard model and particle physics and he doesn't explain why C symmetry is multiplicative other than saying it's discrete which isn't very convincing to me. In ...
0
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1answer
51 views

What does it mean to say “internal symmetry”?

What does it mean to say "internal symmetry"? Let me try to express the way I see it, so you can have it as a starting point. There are spacetime symmetries, which are global since any Lorentz ...
18
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1answer
1k views

Gauge redundancies and global symmetries [closed]

It is often said that local (gauge) transformation is only redundancy of description of spin one massless particles, to make the number degrees of freedom from three to two. It is often said that ...
0
votes
1answer
100 views

Is the Potential Energy just a bookkeeping device?

It is said that if the space is homogeneous then momentum is conserved. But I've been thinking in the following situation: Consider a parallel plates capacitor. In between the plates there is a ...
0
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1answer
40 views

Homogenity and Isotropicity of space

In school it is given that law of conservation of momentum is a result of homogeneity of space and law of conservation of angular momentum is a result of isotropicity of space but what is isotropicity ...
8
votes
1answer
285 views

Symmetries of AdS$_3$, $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$

Basically, I want to know how one can see the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry of AdS$_3$ explicitly. AdS$_3$ can be defined as hyperboloid in $\mathbb{R}^{2,2}$ as $$ X_{-1}^2+X_0^...
1
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0answers
32 views

Generalise Noether's theorem [closed]

I'm not sure how to generalise Noether's theorem. For this L, I think $B\cdot\dot{x}$ is conserved so I tried to relate F and K to this and try to show that that was conserved but got no where. any ...
2
votes
0answers
102 views

How to arrive at the Dirac Equation from Poincare Algebra?

For the case of Galilean group, the time translation is given by the generator $H$. Hence, $$\mid\psi(t)\rangle\to \mid\psi(t+s)\rangle =e^{-iHs}\mid\psi(t)\rangle$$ Which immediately is the ...
3
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1answer
196 views

Local translations in curved spacetime

A global Poincare transformation on a scalar field induces $$\delta(a, \lambda)\phi(x) = [a^{\mu}+\lambda^{\mu\nu}x_{\nu}]\partial_{\mu}\phi(x). \tag{11.46}$$ In curved spacetime we replace $a^{\mu} ...
2
votes
1answer
76 views

Relationship between zero modes and symmetry in a simple system of coupled springs

This Wikipedia page states that "zero modes appear whenever a physical system possesses a certain symmetry," and gives the example of a ring of beads connected by springs having a zero mode associated ...
2
votes
2answers
699 views

Killing Vectors in Schwarzschild Metric

Given the Schwarzschild metric with $(-,+,+,+)$ signature, $$\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$ the lack of ...
1
vote
1answer
37 views

Why are numbers not consistent? (Identifying phonon's frequency)

I am trying to connect several facts between each other to acquire a consistent picture. Fact 1. Monolayers of tungsten disulphide $WS_2$ have hexagonal crystal lattice, and the first Brillouin zone ...
0
votes
1answer
48 views

Lorentz transformation and symmetries of the Lagrangian [duplicate]

Since the Lagrangian of our quantum field theories is covariant under Lorentz transformations I'm asking myself if there is any link to some symmetries (like that we get from gauge transformations ...
0
votes
2answers
54 views

Energy of central potential in QM

A hydrogen atom (Coulomb potential) has energy that only depends on $n$ (if we ignore other effects like spin-orbit coupling). In general (not necessarily Coulomb, can be any V), does $E$ depend on ...
2
votes
1answer
125 views

Are all maximally symmetric spacetimes constant curvature spacetimes?

A $d$ dimensional maximally symmetric spacetime is a spacetime with the maximum allowed number of Killing vectors. This number is $\frac{d(d+1)}{2}$. Constant curvature spacetimes are spacetimes ...
1
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1answer
35 views

What is the gravitational/electric/inverse-square field inside a cylinder?

I've read from the shell theorem that an inverse-square potential has zero field inside a spherical shell. What about the field inside a cylinder? Are objects inside a long cylinder attracted to the ...
1
vote
2answers
86 views

Prove energy conservation using Noether's theorem

I wonder how you prove that energy is conserved under a time translation using Noether's theorem. I've tried myself but without success. What I've come up with so far is that I start by inducing the ...
7
votes
2answers
257 views

Damped oscillator: time-reversal, time-translation and dissipation

The equation of motion of a damped oscillator $$\frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x=0$$ which is invariant under time-translation $t\rightarrow t+a$, but not under time reversal $t\...