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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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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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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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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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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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 ...
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Why Electrical conductivity tensor is symmetric? Or is it not always symmetric?

How to show that the electrical conductivity tensor is symmetric? (or it's not always symmetric?)
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Conventional unit cells and Bravais lattices

Conventional unit cell is defined in the following: A definition of a conventional unit cell of a lattice is one that contains the same point group symmetries as the overall lattice and is the ...
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What is the hermitian of the time reversal operator?

So, I was reading about the time reversal operator, and I came to know that it can be expressed as: $$T = KU$$ where, $U$ is an unitary operator and $K$ is the complex conjugation operator. Now, if I ...
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What is a local operator in quantum mechanics?

In quantum mechanics, what exactly is meant by "local" operator? What about a "global" or a "non-local" operator? Are these the same? Can you also also help me understand what exactly is a local ...
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Symmetries of Wigner $3j$-symbols by exchange

I know that Wigner $3j$-symbols have certain symmetry factors arising by exchange of two columns within one symbol. But what happens if you have two 3j symbols and do an exchange like this: $ \left(\...
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Why is $L_z$ operator more important the $L_x$ or $L_y$ operators?

When we talk about orbital angular momentum, we always use L_z but never talk about L_x or L-y. Why is that?
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Symmetry relation of Wigner-Eckart

I saw a symmetry relation following from the the Wigner-Eckart Theorem looking like this $$(\xi j|| T_L || \xi'j') = (-1)^{j-j'} (\xi' j'|| T_L || \xi j)^*$$ I know that it must come somehow under ...
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Why are Goldstone bosons associated to translation symmetry breaking corresponding to the coordinates transverse to the D-brane?

In some lecture notes, in the presence of a Dp-brane, translational symmetry is broken in the direction transverse to the brane. However, they identify the coordinates, $$\Phi^a,\quad a=p+1,\cdots,D-1$...
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$SU(2)$ symmetry of $\mathcal{L}=\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - \Phi^{\dagger}M\Phi$

I'm considering a Lagrangian of two complex scalar field: $$\mathcal{L}=\partial_{\mu}\phi_1^{*}\partial^{\mu}\phi_1-m_1^2\phi_1^{*}\phi_1+\partial_{\mu}\phi_2^{*}\partial^{\mu}\phi_2-m_2^2\phi_2^{*}\...
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Parametrizing $SU(2)$ with Hermitian matrices

There is something that is not clear to me Here is what I know: Pauli matrices are $\sigma_1 = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$, $\sigma_2 = \begin{pmatrix}0 & -i \\ i & 0\...
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Difference continous - discrete symmetry

I am trying to understand the difference between the two types of symmetries.Wiki Wikipedia says that Translation in time : $t \rightarrow t + a$ is a $\textbf{continuous}$ symmetry, for any real $...
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Custodial Symmetry in SM Higgs

this excercise is about the custodial $SU(2)_R$ symmetry in the SM Higgs Mechanism. Step by step I have to develop the theory. I get the generel idea of the global custodial symmetry and why we need ...
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How to know the symmetries of a coupling in the Hamiltonian

Suppose you have an interaction term in your hamiltonian that looks like \begin{equation} H=\sum_{ijkl}U_{ijkl}c^\dagger_ic^\dagger_jc_kc_l \end{equation} where $U$ is the coupling and $c$, $c^\...
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Mcintyre Quantum Mechanics - Angular Momentum Conservation

I have two questions regarding this topic. 1. I captured the part of the section I'm referring to. If I didn't my question would probably not make sense. My first question is to the second to last ...
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Masslessness of Goldstone Bosons

I have a question about the origin of the massless nature of Goldstone bosons. This is what I understood: we know thanks to Goldstone's theorem that from the breaking of continuous symmetries emerge ...
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Legendre transformation and correspondance between Noether charges and quasi-symmetries

I have been trying to understand the Legendre transformation (in mechanics, in the hyperregular case: when the Legendre transformation is one-to-one) and the correspondence between symmetry $\to$ ...
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253 views

Even and odd solutions to time independent Schrodinger equation on symmetric potential

I have to solve the following problem: Consider the potential well: $$ V(x)=-V_0, \hspace{10px} |x|<a/2 $$ and $0$ everywhere else. $a$ is also a positive constant and so is $V_0$. Find the ...
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Why expression of energy occurs in triplet?

I have basically two questions in mind,which are Why expression of energy occurs in triplet? Why the expressions are somewhat symmetrical? Coming to elaborated form of 1st part, we have $$U=\frac{...
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Adding a total derivative to the Lagrangian does not preserve $\int\mathrm{d}^3\mathbf{x}~ T^{00}$

In problem 3.3 of Schwartz's QFT, the first two questions ask us to prove that if we add a total derivative to the Lagrangian: $$ \mathcal{L}\mapsto\mathcal{L}+\partial_\mu X^\mu\tag{1} $$ then $$ \...
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Misconceptions in spontaneous symmetry breaking

Spontaneous symmetry breaking occurs when we have a potential like a mexican hat as shown in figure (right) and is unbroken for the potential shape as shown in left figure. Under the Symmetry ...
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What is meant by cubic symmetry with regard to thin films growth?

I am reading a paper on epitaxial thin film growth of an alloy and it mentions that for one conditions the films grow with a cubic symmetry and for another they have an in-plane anisotropy. I would ...
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Inertial frame definition in Rindler Introduction to STR vs Landau' & Lifshitz Mechanics

Juxtaposing Rindler's Introduction to STR (page 7) vs Landau's Mechanics (page 5) inertial frame definition,I get that rindler assumes frame moving uniformly w.r.t inertial frame as an inertial frame ...

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