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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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Symmetry acting on a complex fermion operator

Suppose $S$ is a $\mathbb{Z}_2$ symmetry operator, i.e. $S^2=1$, acting on the fermion $c_{n}$ via $$S c_{n} S^{-1} = \sum_{m} U_{nm} c_{m}$$ and I am interested in $S$ is both linear or anti linear, ...
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Symmetry Operation in Reciprocal Space

I have a set of k points spanning the entire Brillouin zone and I want to reduce it to the irreducible BZ. So for reduction, I use the point group symmetry operation. To verify, I use quantum espresso ...
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Wigner-Eckart theorem: Are expectation values of all vector operators parallel?

The Wigner-Eckart theorem tells us that for any tensor operator, $\mathbf{T}^{(k)}$ that \begin{align} \langle jm|T^{(k)}_q|j'm'\rangle = \langle j'm'kq|jm\rangle \langle j||\mathbf{T}^{(k)}||j'\...
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Why parity required symmetry?

I'm studying parity for the first time but there is something I don't understand. I read that a system conserves parity if every experiment is the same in a mirror that is also $180^{\circ}$ flipped. ...
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Confusion on symmetry and basis transformation

Let {$|a_n\rangle$} and {$|b_n\rangle$} be two basis related by: $|b_n\rangle = \hat{U}|a_n\rangle \forall n$. From my understanding then the unitary operator $\hat{U}$ only transforms the basis {$|...
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Recovering symmetry in coupled oscillators

Consider a pair of LC oscillators, one with capacitance $C_1$ and inductance $L_1$ and the other with capacitance $C_2$ and inductance $L_2$. Suppose they're connected through a capacitor $C_g$. We ...
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Is there a higher dimension analogue of Noether's theorem?

So I have recently read the proof of Noether's theorem from the book variation calculus by Gelfand. Basically, what I have already seen is that for any single integral functional, if we have a ...
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Is there a contradiction between isotropy and the Big Bang? [duplicate]

Disclaimer: I'm not asking whether the Big Bang happened at a point. I'm asking whether the fact that the universe is isotropic and that the Big Bang happened contradict each other. To be honest I am ...
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Difference on the invariance of operators and their transformations under unitary operators

I am confused about, what I believe, refers to passive and active transformations in QM. What I have understood so far is that the matrix elements $\langle \psi| \hat{H}|\phi\rangle$ should remain ...
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Why do atoms repel when they are very close to each other? [duplicate]

Why do atoms repel when they are very close to each other? I know that this is not because of electrostatic force, but I do not know why atoms actually repel? Our lecturer said that it arises from the ...
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Issue showing that the phase of a harmonic wave is invariant under a Galilean transform

The phase $Φ$ of wave is defined as $kx-wt$. It should be the case that all observers moving relative to each other in the non relativistic case will agree on this. So given the transforms $x'=x-vt$ ...
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Why are the two outer charge densities on a system of parallel charged plates identical?

One of the ways examiners torture students is by asking them to calculate charge distributions and potentials for systems of charged parallel plates like this: the ellipsis is meant to indicate any ...
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Connected components of conformal group $ {\rm Conf}(p,q)$ containing $P$, $T$ and conformal inversion are same or different?

As we known (see this post), the global conformal group for $\mathbb{R}^{p,q}$ is $$ {\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}$$ The global conformal group ${\rm Conf}(p,q)$ has 4 ...
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Symmetry arguments on the Berry connection and the polarization charge

Consider the Berry connection $$ A_n(\mathbf{k})=i \langle n(\mathbf{k})|\nabla_{\mathbf{k}}|n(\mathbf{k})\rangle $$ and the polarization charge $$ \mathbf{P}=-\frac{1}{4\pi^2} \int_\mathrm{B.Z.}\...
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Solving differential equation for the Schwarzschild metric with cosmological constant

How do we solve for Einstein's equation in the vacuum with a cosmological constant, in the static spherically symmetric situation? Attempt: Following Sean Carroll's Spacetime and Geometry (p.195), I ...
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How is the point symmetry group of real space related to the reciprocal space?

I am taking a course of solid state physics. My teacher asked me that. I have been looking for the answer in the book Solid State Physics by Ashcroft & Mermin, and also in the book of Ibach & ...
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Can there be conservation laws without a corresponding symmetry? [duplicate]

Noether's theorem implies that if there is a continuous symmetry in the Lagrangian of a system, this necessarily implies a corresponding conservation law. But does the theorem also imply the reverse? ...
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What's the reason for this symmetry argument? [migrated]

Okay so I've learnt about various methods to find equivalent resistance in circuit. But there's one particular argument which I don't understand and I would be grateful if someone will prove it using ...
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How to place the axis so that I can calculate the center of mass for the two instances?

I've got this: A wagon of mass $M$, initially at rest, can move horizontally along a frictionless track. When $t = 0$, a force $F$ is applied to the cart. During the acceleration of M by the force $F$...
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Representative Volume Element for crystals

I am trying to average the tensor properties of crystals with different types of unit cells. Due to symmetry elements for different crystals, the integration over a limited volume should be enough, ...
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Real symmetric matrix in Wigner's theorem

A consequence of Wigner's theorem is that if a Hamiltonian matrix obeys time reversal symmetry then it is real-symmetric. It seems to me that for this to make sense then "real symmetric" should be a ...
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Symmetry operations on an infinite uniform sheet of charge

My book has a section on symmetry operations. It says, (if the plane of charge is the yz plane) translation symmetry along the y-axis and z-axis implies that the electric field is constant if one ...
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Inversion applicable to three-dimensional lattices only

I have just started my first course in solid state physics and while studying symmetries, Inversion is defined as "A point operation which is applicable to three-dimensional lattices only. This ...
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Lagrange's equation for system not having time translation

While we are deriving Lagrange's equation from D'Alembert's principle, when we argues as; $$\delta \vec r_\alpha = \sum_i \frac{\partial \vec r_\alpha}{\partial q_i }\delta q_i + \frac{ \partial \...
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Bosonic commutation relations for force carriers?

Why are force carriers bosons? The easiest answer that I can give myself is that the gauge field $A_\mu$ is introduced like this: $$ \partial_\mu \rightarrow D_\mu = \partial_\mu+ieA_\mu, $$ so it ...
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Canonical transformations preserve Hamilton's equations. Which transformation preserve the Euler-Lagrange equations?

An important aspect in the Hamiltonian formulation of Classical Mechanics are canonical transformations which provide maps between different sets of canonical coordinates. These canonical coordinates ...
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Why intuitively, do we define symmetries as transformations that map solutions of the equations of motion into other solutions?

Of course, strictly speaking, a symmetry is always a transformation that leaves a given object unchanged. But I'm curious why observable symmetries of physical systems are exactly those ...
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Are symmetries necessarily canonical transformations?

A canonical transformation is defined as a transformation such that afterwards Hamilton's equations still hold. It can then be shown that this requirement implies that canonical transformations are ...
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Global $U(1)$ Symmetry in GSW Model

Consider the theory of a single generation of $e, \nu_L$ matter content. The initial lagrangian is $$ \mathcal{L} = i\bar{\ell}\not \partial\ell + i\bar{e}_R\not \partial e_R \tag{1} $$ where $$ ...
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Why are symmetries in phase space generated by functions that leave the Hamiltonian invariant?

Hamilton's equation reads $$ \frac{d}{dt} F = \{ F,H\} \, .$$ In words this means that $H$ acts on $T$ via the natural phase space product (the Poisson bracket) and the result is the correct time ...
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Active Diffeomorphism

I am a little confused about active diffeomorphism $f:M\to M$. Let us focus on translation. When we say that we are doing an active translations does that means that all the particles $\gamma$,and ...
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what do you mean by five fold coordination of the Fe site?

I got this paper to study, it says that in FeS2 (100) there is a five-fold coordination of the Fe sites and a threefold coordination of the S sites. what does that mean? is it inferring to the ...
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How groups act on fields in QFT?

I read a lot a posts on how to verify what are the symmetries of a given Lagrangian but I really can't find what I need and can't even get it by myself, this because I don't actually understand how ...
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second-harmonic generation and inversion centers confined to a planar defect

Let's say that we have a phase which breaks inversion symmetry. Now lets say two domains of this phase meet at a plane such that the whole system now has inversion symmetry, with the inversion ...
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What is “Rotational Invariance” in the context of qubits

In this question the state, $\frac{1}{\sqrt{3}}\left|00\right\rangle +\frac{1}{\sqrt{3}}\left|01\right\rangle +\frac{1}{\sqrt{3}}\left|11\right\rangle$, has been said in the answers to not be able to ...
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Why is charge conjugation $\hat C | \alpha \psi \rangle = C_\alpha |\alpha\psi \rangle$?

Charge conjugation replaces all particles by antiparticles in the same state, so that momenta, positions, etc are unchanged. It can be represented by $$\hat C | \alpha \psi \rangle = C_\alpha |\alpha\...
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Why isn't $SO(n)/SO(n\!-\!1)$ a symmetric space?

It's my understanding that one way to define a symmetric space $G/H$ is by the commutation relations $$ [T^a, T^b] = f^{abc} T^c, \qquad [T^a, X^{\hat{b}}] = f^{a\hat{b}\hat{c}}X^{\hat{c}}, \qquad [X^{...
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Weinberg's Coleman-Mandula theorem proof sufficient condition for isomorphism?

In Weinberg's QFT Volume 3 book on Supersymmetry, he presents his own proof of the Coleman-Mandula theorem. As part of the proof, he proves that the only possible internal symmetry generators must ...
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Symmetries of a differential equation, its solutions and hydrogen atom

A symmetry of a differential equation need not be shared by its solutions. However, under that symmetry, the one solution goes to another. For example, consider the time-independent Schrodinger ...
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Classical angular momentum components are numbers. Can they be generators of some symmetry group?

In Quantum Mechanics (QM), angular momentum turn out to be the generator of rotational symmetry. This is trivial to see because in QM, angular momenta are defined by the commutation relations $$[J_j,...
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How does the hydrogen atom actually “look like”? [duplicate]

When deriving the solutions for the "simple" quantum mechanical hydrogen problem, one normally uses spherical coordinates $(r,\theta,\phi)$, since the problem has rotational symmetry. The solution has ...
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The connection between symmetry and classifying spaces of a group

I recently read the following statement: "For any type of mathematical object, an object of that type with $G$ symmetry “is” a map from [its classifying space] $BG$ to the space of all objects ...
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“Killing leaves” in General Relativity?

I now about Killing vector fields in GR but recently stumbled upon the notion of "Killing leaves" and couldn't find any simple explanation of this notion. For example, this paper writes: "integral ...
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Possible maximally symmetric 3D spaces

I was watching Neil Turok's lectures on General Relativity. After introducing the Einstein equation, he tries cosmology and postulates "The space is assumed to be isometric and homogenous." Then he ...
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Time translation invariance of Hamiltonian

I am learning about the time translation invariance of the Hamiltonian. I read that the time translation invariance is already manifest in the fact that our Hamiltonian is chosen an ...
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Symmetry group of two complex scalar fields with different masses

Which is the symmetry group of the following Lagrangian: $$ \mathcal{L} = (\partial^\mu \phi_1^\dagger)(\partial_\mu \phi_1) + (\partial^\mu \phi_2^\dagger)(\partial_\mu \phi_2) - m_1^2\phi_1^\dagger\...
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Does gravity get stronger when you climb a mountain?

As stated in the question title, what happens to the strength of the gravitational field (or equivalently, your weight) as you climb a hill or mountain? Would a weighing scale show that you were ...
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How does time-translation symmetry morph into evolution in time?

I am reading Ballentine's textbook "Quantum Mechanics: A Modern Development". In it he transitions from discussing time-symmetry to discussing evolution (of the state) in time. I'm finding it ...
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Antiparticles, CPT and leptogenesis

When people are being careful they'll tell you that antiparticles are the CPT conjugates of particles. You can't say that they are C conjugates or CP because these, while they do reverse the charge, ...
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Beyond Kerr Carter constant?

What are the most symmetrical black hole spacetimes whose motion is completely integrable with a Carter constant-like and hidden symmetry superintegrability condition? Do type D-spacetimes have a ...