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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Conserved current for a constant translation of a free massless scalar field

In Zinn-Justin's Quantum Field Theory and Critical Phenomena they start with an action for a free massless scalar field: $$S(\varphi) = \frac{1}{2}\int ...
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Examples of manifolds (not) being: flat, homogeneous and isotropic

I am looking for (at least) one example of the following manifolds: Flat, homogeneous and isotropic Curved, homogeneous and isotropic Flat, non-homogeneous and isotropic Flat, homogeneous and ...
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73 views

Notation in the book Symmetry by Hermann Weyl

I'm having troubles understanding a notation of the symmetry groups in a book "Symmetry" by Hermann Weyl. On the page 80 of the 1952 Princeton University Press edition of the book, Weyl lists the ...
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146 views

The role of SO(3) and SU(2) in quantum mechanics [duplicate]

When studying the irreducible representations of SO(3) one usually looks at the irreps of the infinitesimal rotations instead, i.e. the ones of so(3), the Lie Algebra of SO(3). The Irreps of so(3) can ...
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202 views

Why do we look at the representations of $SO(3)$ in QM?

I have a bit of an understanding issue why the representations of $SO(3)$ are so important for Quantum Mechanics. When looking at its Irreps one gets the Spin and Angular Momentum operators and thus ...
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74 views

Symmetric eigenfunctions?

So a symmetric eigenfunction / wavefunction is defined as: $$P_{ij} ψ_a (r_1,r_2,…,r_i,…,r_j,…,r_N )=ψ_a(r_1,r_2,…,r_i,…,r_j,…,r_N )$$ But for it to be symmetric does this have to be true for all $ij$ ...
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302 views

Quantum symmetries that are not classical symmetries

An anomaly is a symmetry of the classical action that fails to be a symmetry of the path integral, due to non-invariance of the path integral measure. Does it ever occur that the opposite thing ...
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106 views

Why is there a 'loophole' in Mermin Wagner for rotations?

I'm just starting out in my mathematics career by looking at some simple stuff on broken symmetries in statistical mechanics. Since 3D is 'hard' it would be very nice to look at 2D toy models of ...
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123 views

Could the universe have evolved WITHOUT the non-determinism of quantum mechanics? [closed]

(I'm going to make a few conjectures here - please answer the question in light of them as if they were true, even though of course they may be overly simplistic or wrong) Assuming that: the ...
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227 views

Do the standard cosmology models spontaneously break Lorentz symmetry?

In standard cosmology models (Friedmann equations which your favorite choice of DM and DE), there exists a frame in which the total momenta of any sufficiently large sphere, centered at any point in ...
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79 views

Time Energy symmetry in General Relativity (not asking about energy conservation)

In General Relativity is there a TE symmetry similar to CPT symmetry in the Standard Model ? It's pretty easy to understand that by flipping charge and parity you merely get a time reversed equivalent ...
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98 views

Quasicrystals - Projections from higher dimensional regular crystal lattices

Why are quasicrystals projections from higher dimensional regular crystal lattices? See for example wikipedia: »Mathematically, quasicrystals have been shown to be derivable from a general ...
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67 views

What's the corresponding symmetry of enstrophy conservation?

In fluid mechanics, especially 2D turbulence study, people talk about conservation of enstrophy. But I can't really understand enstrophy very well, and what's the corresponding symmetry of enstrophy ...
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140 views

I want to decompose a tensor product using Littlewood-Richardson rule, How do I find the component of this in each irreducible space?

Let me set up the notation I am using. $(abc,de)$ denotes the standard Young tableau where the first row is $abc$ and the second row is $de$. Each young tableau corresponds to the young symmetriser, ...
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1answer
132 views

Why is the inertia ellipsoid of a higher symmetry than the rigid body?

I was always puzzled by this fact. A uniform cube has a sphere-shaped inertia ellipsoid. The sphere has a higher symmetry then the cube. Is there any deep reason or implication behind it?
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129 views

Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3)$ isometry group?

As the title says, is it possible to have a Riemannian Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3) $ isometry group?
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117 views

Argument for symmetry of potential of a spherical capacitor filled with dielectric

A spherical capacitor with inner radius $r_1$ and outer radius $r_2$ is filled with dielectric material with permittivity $\epsilon=\epsilon_0+\epsilon_1\cos^2\theta $. $\theta$ is the polar angle. ...
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170 views

Noether's Theorem: Foundations

I'm wondering on what principles Noether's theorem foots. More precisely: The action is a functional on the fields only. Why do we consider then variations of the space time too? In principle careful ...
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385 views

Hamiltonian Noether's theorem in classical mechanics

How does one think about, and apply, Noether's theorem in the classical mechanical Hamiltonian formalism? From the Lagrangian perspective, Noether's theorem (in 1-D) states that the quantity ...
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2answers
94 views

Gravitational force due to asymmetric Earth shell

http://en.wikipedia.org/wiki/Shell_theorem According to above link the gravitational force inside a symmetrical shell due to itself is zero. Is it also true for an asymmetrical shell?
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105 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 ...
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1answer
105 views

Pions as a superposition of quark states

in Isospin space there are two fundamental states called up and down quarks, which satisfy the following eigenvalue equations: $I u = (1/2) u$, $I d = (1/2) d$ and $I_3 u = (1/2) u, I_3 d = (-1/2) ...
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128 views

Conceptual question about field transformation

(c.f Conformal Field Theory by Di Francesco et al, p39) From another source, I understand the mathematical derivation that leads to eqn (2.126) in Di Francesco et al, however conceptually I do not ...
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60 views

Local symmetry and General Relativity

First I want to consider an example of 1D motion. Lagrange equation: $$ \frac{d}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0 $$ If we transform $ L \rightarrow L+a $ ...
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1answer
85 views

Reversing Noether's theorem [duplicate]

Noether's theorem states: any differentiable symmetry of the action of a physical system has a corresponding conservation law. Is this statement invertible? I mean, if a conservation law exists, this ...
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100 views

Why is the $SO(4)$ symmetry of the Hydrogen atom called dynamical?

Why dynamical? My previous quantum mechanics teacher could not answer it.
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34 views

Association of financial phenomena/indications with the conservation laws of Black Scholes equation

For a while I've been doing research on methods of obtaining conservation laws via the symmetries of differential equations (DEs). I'm presently doing research on identifying financial ...
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Why is the electric field of an infinite insulated plane of charge perpendicular to the plane?

I'm studying Gauss' Law, and I came across a section where we're supposed to find the electric field of various shapes (like an infinite line of charges, etc), and for an infinite plane with a uniform ...
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99 views

Kac-Moody algebras in 5 dimensional Kaluza-Klein theory

I am trying to make sense to the issue of how does the Kac-Moody algebra encode the symmetries of the non-truncated theory. Let's contextualize a little bit. Ok, so in the 5 dimensional Kaluza-Klein ...
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66 views

Charge inside a sphere

Suppose I have a sphere of radius $r$ with all the charge residing on the surface, distributed uniformly i.e. charge density $\sigma$ is constant. I want to find the electric field created by this ...
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92 views

A Subtle Connection Between Time Dilation in SR and GR - Why is this so?

I've been reading a book on General Relativity lately (Gravitation and Cosmology, Weinberg), and I was reading about the weak field approximation. It derived the time dilation in a weak gravitational ...
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2answers
105 views

Gauss's Law for a Uniformly Charged Solid Sphere [duplicate]

We want to calculate $\vec{E}$ at a distance $r$ from the center $O$ of a spherical polar coordinate system. Let the point on the Gaussian surface at which we want to calculate $\vec{E}$ is ...
2
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165 views

Are diffeomorphisms a proper subgroup of conformal transformations?

The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not ...
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26 views

Computing Parity by numerical tables of characters

I have a table of the characters of a set of wavefunctions for different points in reciprocal space and for different band indices (this is for a solid). For the case of a single irreducible ...
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133 views

Quick question on degeneracy of harmonic oscillator states

I'm currently learning about symmetry between particles. For a simple case of two non-interacting particles at $x_1$ and $x_2$, we know that the wavefunction can be written as $\psi_{n_1, n_2} = ...
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108 views

Why is the projective symmetry group (PSG) called projective?

As discussed by Prof.Wen in the context of the quantum orders of spin liquids, PSG is defined as all the transformations that leave the mean-field ansatz invariant, IGG is the so-called invariant ...
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96 views

Euclidean functional Integrals

In the chapter "Uses of Instantons" from the book "aspects of symmetry" by Sidney Coleman I have come across the euclidean version of the path integral in semi-classical approximation. To evaluate the ...
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2answers
98 views

Possible mechanics based on the known symmetries in the nature (investigating rumor)

Somewhere I've heard about a relative new mathematical result regarding mechanics. Specifically, there is a list of the known symmetries of mechanics (both Newtonian and relativistic), i.e. different ...
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73 views

What is the minimal symmetry required for a spin Hamiltonian to describe a spin-liquid ground state?

Let's restrict to the case of spin-1/2 system. As we know, a spin-liquid (SL) state is the ground state of a lattice spin Hamiltonian with no spontaneous broken symmetries (sometime it may ...
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347 views

Symmetry arguments in solving problems

There was a question which involved calculation of final charges on two spheres when one uncharged and the other having charge $Q$ were brought in contact with each other. (radius same). If potential ...
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141 views

Translations and Noether's Theorem

I'm fine with $U(1)$ symmetry and Noether's Theorem, but struggling with the translations of the field; namely $$\phi'(x^{\mu})=\phi(x^{\mu}-a^{\mu}),$$ where $a^{\mu}$ constant four-vector ...
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Symmetry factor of tree diagram

In Mark Srednicki's Quantum field theory(page 89) it says This is a general result for tree diagrams (those with no closed loops): once the sources have been stripped off and the endpoints ...
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56 views

How to easily calculate lengths (or relative lengths) of paths between symmetry points in BZ

I am trying to easily calculate the length between special kpoints within the BZ of the 32 point groups in a crystal system. I am calculating the lengths in order to scale k point sampling along these ...
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1answer
242 views

Plotting a bandstructure along High-symmetry points when kx,ky,kz is known

Suppose you know kx,ky,kz points along with the corresponding energies. Basically, you know about the 4-D E(k) dispersion. How you do then convert that data into the bandstructure plots you commonly ...
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36 views

Behavior of Pion-Mediated Nuclear Force before Electroweak Symmetry Breaking?

When Chiral Symmetry was exact, as it was before EWSB due to the lack of mass terms for quarks, would the residual strong force have infinite range? Related to this, does the Negative Beta Function ...
3
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250 views

Why are there gapless excitations in the anti-ferromagnetic Heisenberg model while the true ground state is a singlet?

The true ground state of the anti ferromagnetic quantum Heisenberg Model (nearest neighbor only)is known to be a singlet (I think this is Liebs theorem.) Since a singlet is invariant under rotations, ...
2
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214 views

Spatial part of wavefunction of helium, both electrons are in ground state

If both electrons are in ground state, characterized by $(1s)^2$, this means that n =1, l = 0. The two electrons occupy the same energy and space (why doesn't pauli exclusion principle prevent this?), ...
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511 views

A question about Feynman diagram and symmetry factor

Consider a $\varphi^3$ theory: $$ Z_1(J) \propto \exp\left[\frac{i}{6} Z_g g\int \mathrm{d}^4 x \left(\frac{1}{i}\frac{\delta}{\delta J}\right)^3\right] Z_0(J), $$ where $$ Z_0(J) = ...
2
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1answer
59 views

Unit determinant for relevant symmetry groups in QFT

When treating QFT we want our theory to be invariant under different symmetry groups, for example, the Standard Model is a non-abelian gauge theory with the symmetry group $U(1)×SU(2)×SU(3)$. ...
2
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215 views

Intuition Behind Conservation of Angular Momentum

I'm having a fairly hard time understanding the intuition behind Noether's derivation of the conservation of angular momentum from the rotational invariance of the Lagrangian, though I do understand ...