Questions tagged [symmetry]

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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Why Lie algebras if what we care about in physics are groups?

In physics, we want irreducible representations of the symmetry group of our system. However, one frequently sees representations of the corresponding Lie algebra being studied instead. Is it that in ...
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Ways to understand spatial symmetry

If you are given a potential, which doesn't change if you change the azimuthal angle, we might call that has spherical symmetry. Which means we must find symmetry of angular momentum $L_Z, L^2$. My ...
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A priori knowledge of the components of the Ricci tensor

Source: Thomas Moore's A General Relativity Workbook In Moore's "diagonal metric worksheet" he doesn't explain his process of determining the "only possible non zero components" of the Ricco tensor, ...
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off shell graviton 3 point vertex

I have a technical doubt regarding the graviton three point vertex from DeWitt's paper of 1967 : https://journals.aps.org/pr/pdf/10.1103/PhysRev.162.1239 Can someone write some of the terms in the ...
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Emergence of rotational symmetry on 2D square lattice

On page 74 of David Tong's Statistical Field Theory lecture notes, it is said that $(\partial_1\phi)^2 + (\partial_2\phi)^2 $ respects both $D_8$ (that includes discrete four-dimensional rotation ...
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Angular momentum and rotation symmetry

In my book, it is written that for any vector $\mathbf{v}$, we have $$\{\mathbf{v},\mathbf{L}\cdot \mathbf{n}\}=\mathbf{n}\times\mathbf{v}.\tag{1}$$ For me it is absurd... For example, if we take $\...
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Is this the Melvin metric (magnetic universe) in disguise?

I'm solving the Einstein equation assuming a cylindrical symmetry and found something interesting which I never saw elsewhere. I now feel that I may have found the Melvin magnetic universe solution ...
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Is Landau&Lifshitz's argument for the classical Lagrangian's symmetries too strict?

I realize that this paragraph has raised more questions on stackexchange, but I wanted to ask this question nevertheless since I want to discuss it in terms of a counter-example. I’ve already ...
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Matching AdS and CFT symmetries

The isometries of AdS in $D+1$ dimensions and the conformal symmetries in $D$ are isomorphic as Lie algebras. However, the generators on each side have a physical interpretation. In the bulk we have ...
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Is a scale-invariant QFT defined by its symmetries?

Do distinct scale-invariant quantum field theories necessarily have different symmetries?
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Cylindrical universe cosmology in general relativity

Is there a compact cylindrical universe solution to the Einstein equation, with space homogeneity, without using "artificial" periodic boundaries? I'm expecting a metric of the following shape: \...
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Does anisotropic stress cancel at the center by symmetry?

Let's define : $$ \sigma_{ij}=\tilde{\sigma_{ij}} - P \delta_{ij} $$ with $\sigma$ the stress tensor, $P$ the pressure and $\tilde{\sigma}$ the anisotropic stress tensor. The balance of force ...
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Symmetry modulo total derivative term in Noether's Theorem

I came across the proof of Noether's Theorem in David Tong's notes (page 14) on QFT. He writes something like, We say that the transformation $$\delta\phi(x) = \chi (\phi) \tag{1.34}$$ is a ...
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Ambiguity in applying Newton's shell theorem in an infinite homogeneous universe

Newton's shell theorem has two corollaries: The gravitational attraction of a spherically symmetric body acts as if all its mass were concentrated at the center, and The gravitational acceleration ...
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Generalized symmetries

I am trying to teach myself about generalized symmetries. I have been attempting to read https://arxiv.org/abs/1412.5148, but am getting a bit confused. I'm hoping that maybe someone could explain the ...
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1answer
63 views

Proof that rotational symmetric potential operators are scalar operators

Defintion: A scalar operator B is an operator on a ket space that transforms under rotations \begin{equation}\left| \xi ' \right >=\exp{(\frac{i}{h} \mathbf{\phi \cdot J})}\left| \xi \right >\...
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Srednicki chapter 22: continuous symmetries and conserved current

In Srednicki's book he says that: The Noether current plays a special role if we can find a set of infinitesimal field transformations that leaves the lagrangian unchanged, or invariant. In this ...
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How to calculate gravity inside of a circle of mass?

I've been reading about the Shell Theorem and how the gravitational force inside of a sphere is equal to zero. I was wondering if the same was true for a circle, but couldn't find any definitive ...
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Does the conservation of $\frac{\partial L}{\partial\dot{q}_i}$ necessarily require $q_i$ to be cyclic?

If a generalized coordinate $q_i$ is cyclic, the conjugate momentum $p_i=\frac{\partial L}{\partial\dot{q}_i}$ is conserved. Is the converse also true? To state more explicitly, if a conjugate ...
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Noether's theorem for arbitrary conformal coordinate transformations

I have been reading Introduction to Conformal Field Theory by Blumenhagen and Plauschinn. Equation (2.19) on page 19 states that if our theory is invariant under a general conformal transformation $x^\...
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Does technical naturalness hold only for global symmetries, or also gauge symmetries?

Suppose you have an action $S(\epsilon) = S_1 + S_2 + \epsilon\, S_\mathrm{int}$. Assume that $S_1$ is gauge invariant under the action of the group $G$ and $S_2$ is gauge invariant under the action ...
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What are the transformation equations for deviatoric planes?

What are the transformation equations $$r=r(x,y,c),$$ $$\phi=\phi(x,y,c)$$ for a Matsuoka-Nakai deviatoric plane? Variables $x,y$ are Cartesian coordinates, $r,\phi$ are "polar" coordinates, $c$ ...
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Is the vacuum of a local ${\rm U(1)}$ gauge theory unique?

Consider a spontaneously broken scalar field theory with a global ${\rm U(1)}$ symmetry described by the Lagrangian $$\mathscr{L}=(\partial_\mu\phi^*)(\partial^\mu\phi)-\mathcal{V}(\phi),\\ \mathcal{V}...
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Is the interval $ds^2$ NOT invariant under translation in an inhomogenous space?

In the Chapter 9 Symmetries, Section 9.1 The Killing vectors (page 101) are Killing vectors defined such that an infinitesimal translation along the vector keep the line element invariant. It means ...
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Obstruction in quantization. Weyl Ordering

What is an obstruction in quantization? I've found that obstructions object of the study of a mathematical theory, previously concerned with homotopy. The problem is that to explain what an ...
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Why do we use a cylinder as a Gaussian surface for infinitely long charged wire?

Why do we use a cylinder as a Gaussian surface for infinitely long charged wire and not some other shape like cube?
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Are the electron and positron doublets in any mathematical sense?

I know left handed electrons and electron neutrinos are isospin doublets. Is there any analogous way in which an electron and a positron are related? My first thought is some sort time-parity doublet.
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Why is the $\vec E$ field inside a sphere = 0? [closed]

I was taught Electric Field inside a sphere is 0, because of Gauss Law. But inside a uniformly charged sphere, if I go at a distance of $r$ from the centre, I will be closer to the +ve charge and ...
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Spin rotation symmetry in triplet superconductors

Triplet superconductors can be described using the $d$ vector formalism and many properties of the superconducting states can be checked via operations on the $d$ vector itself. I want to know how can ...
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What exactly is the problem that Inflation solves?

There's copious documents about how Inflation solves the problem that General Relativity predicts a lumpy CMB. That influation 'smooths' out the curvature fluctuations and, so, predicts a CMB at ...
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Electric field inside infinite charged hollow cylinder

Gauss's Law says that electric field inside an infinite hollow cylinder is zero. My question however is that an infinite hollow cylinder can be constructed by taking rings as element and the field ...
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2answers
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Einstein equations in the spherically symmetric, static case

This question is not about the solutions but much rather about the equations we write in GR for a spherically symmetric, static vacuum 4D spacetime. The Einstein equations are $$G_{\mu\nu}=0\;\;\;\...
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1answer
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Free fermion Lagrangian invariance under chiral symmetry

I want to apply this transformation to a free-fermion lagrangian: $$ L=\bar{\psi}(\gamma^\mu{\partial_\mu \,- m)\,\psi}$$ $$ \psi ' =\psi\; e^{i \alpha \gamma_5} $$ $$ \bar{\psi}'=\bar{\psi} \;e^{-i \...
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Shell model label symmetry

I'm working through the quark model Hadron wavefunctions, and trying to figure out the flavour-spin part of the wave functions. This is easy enough for S shell spatial states, since the space wave ...
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Does it make sense to say that the action is even or odd under time reversal?

The action of a system in mechanics is an integral over time defined as $$S[x(t)]=\int\limits_{t_1}^{t_2}L(x,\dot{x},t)dt.$$ Here, the time $t$ is integrated making the left hand side depend only on ...
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Electric field inside and outside a hollow spherical shell

If a charge(+q) is placed at distance away from a hollow spherical conducting shell , would the net electric field inside the hollow portion remain zero? If the +q charge was placed anywhere inside ...
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2answers
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Color symmetries in variant QCD

Suppose we only have two colors, for example, red (R) and blue (B) to construct the wave functions of baryons and mesons and that the color symmetry is SU (2) and not SU (3). In this situation, ...
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Magnetic field on a super conductor cylinder

Suppose that i have a cylinder of the z axis. I'll put this cylinder on a uniform magnetic field with magnitude of B0 along de y axis. Since we have B = 0 inside the superconductor, this will ...
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Resistors equivalent using symmetry and equipotential line

I propose the following statement: In a circuit, while determining equivalent resistance across two points, if the circuit is symmetric wrt the line joining the two points , we may fold the circuit ...
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1answer
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Rotation as an example of symmetry in classical mechanics

I modified the question because it was confused. On my book there is this mathematical definition of symmetry transformation: "The equations of motion have a symmetry, if the solutions of the ...
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Why is field action not a pseduo-scalar in 4D?

If the Lagrangian density is a scalar and the 4-volume is a pseudo-scalar (w.r. to proper orthochronous LT), how is then action not a pseudo-scalar? If it is a pseudo-scalar (i.e. the above reasoning ...
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Parity transformation on annihilation operator for the Dirac field $U(p)^{+} c_{r}(\vec k)U(p)=i c_{r}(- \vec k)$

I want to verify the discrete parity transformation action on annihilation operator for the Dirac field. Given the dirac field: $$ \psi(x) = \frac{1}{(2\pi)^{\frac{3}{2}}} \int \frac{d^3k}{\sqrt{2\...
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Proof of Schur's Theorem

On Pg. 123 of Schaum's Tensor Calculus: At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$ for any ...
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Symmetries of solution

I have a system of coupled nonlinear differential-difference equations as model of particles with harmonic interaction in some potential, of the form: $$ \dot{x}_{i}=x_{i+1}+x_{i-1}-2x_{i}-\sin(2\pi ...
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Why doesn't shell theorem work here?

I think I have a misunderstanding about the shell theorem regarding electrostatics here. Since we have a conducting spherical shell, and we are looking for the $E$ field, shouldn't it be 0 within ...
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Proof of first Bianchi identity

The proof is often simplified by using the following theorem: "If the metric tensor $(g_{ij})$ is positive definite, then, at the origin of a Riemannian coordinate system $(y^i)$, all $\partial g_{...
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Masslessness of Goldstone modes

Suppose we have a $G$-invariant action $S$ of a field $\phi$, where $G$ is a Lie group; let then exist a non-zero value $v$ of $\langle\phi\rangle$ such that the $G$-symmetry of the action is broken, ...
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Why are marginal eigenvalues of Jacobian of a periodic orbit related to the symmetry?

In ChaosBook, at page 61 of the unstable version of the book, it is stated that $$J_p (x) \mu (x) = \mu (x,)$$ i.e the velocity vector is an eigenvector of the Jacobian along periodic orbit $p$ ...
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Eigenstates of the scattering matrix?

Consider a single-particle non-relativistic problem. Consider a 3D spherically symmetric potential. What are the eigenstates of the $S$-matrix? Are they spherically symmetric? And what are the ...