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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Comparison between formulations of Noether's theorem

Version 1: An infinitesimal variation on the fields $\phi\mapsto\phi'$ is said to be a symmetry if $\delta \mathcal{L}:=\mathcal{L}(\phi',\partial\phi')-\mathcal{L}(\phi,\partial\phi)$ is a total ...
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Solving Many DOF System with Symmetry (Eigenvector issue)

I'm working on dealing with the simple harmonic oscillations of a benzene atom. We're meant to solve it with symmetry. I can solve it by going the longer way via the Euler-Lagrange method and finding ...
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How can the position operator be displacement invariant?

I am reading chapter 3 of Quantum Mechanics - A Modern Development by Leslie E Ballentine, where he derives the operators for the common dynamical variables from space-time symmetry considerations. At ...
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Time-independent source and quantum field theory

Can anyone explain the fundamental reason of why time-independent sources cannot emit or absorb energy. Does it have to do with time-translation symmetry and Noether's theorem? I was studying the ...
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What is the meaning of the field deformation (delta phi) for continuous symmetries?

I'm taught in class that for a symmetry $\phi \rightarrow \phi + \delta\phi$ (and leaving the spacetime coordinates alone), the Noether current is $$ J^\mu = \frac{\partial L}{\partial (\partial_\mu\...
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Phase of a symmetry

Let's say I have a Hamiltonian $H$ and symmetry of this Hamiltonian such that $[ H, S ]= 0$ It's easy to see that any opertor $\tilde{S} = e^{i\theta} S$ is also a symmetry of $H$ as $[ H, \tilde{S} ]=...
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How to prove symmetry of stress tensor using torque equation? [closed]

** You have to write the angular momentum equation i.e. Torque = (Moment of Inertia)x(Angular Acceleration) for an infinitesimally small 3D element (about any axis of your choice). Finally, you have ...
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Central force problem, Bertrand's theorem [duplicate]

How can you explain the statement that "the only central forces that result in closed orbits for all bound particles are the inverse square law and Hooke's law."
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Why we cannot use Gauss's Law to find the Electric Field of a finite charged sheet?

I could not understand why we can't use Gauss's Law to find the Electric Field of a finite charged sheet?
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Magnetization subspace and Hamiltonian representation

A follow-up question of the subspaces of 4-electrons: assume the magnetization of the system is conserved (the number of total spin-up $(\uparrow)$ particles is conserved), say 1, for example. Then ...
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Why is there an extra term in definition of Noether current for spacetime translations?

I am reading Schwartz's Quantum Field Theory textbook. In chapter 3, Schwartz first defines the conserved current for a symmetry $\phi \rightarrow \phi + \delta \phi$ that depends on a parameter $\...
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Conformal vector fields in $m$-dimensional Euclidean manifold

A vector field $X=X^\mu\partial_\mu\in\mathfrak{X}(M)$, where $M$ is a (pseudo-)riemannian manifold with a generic metric tensor $g_{\mu\nu}$, is a conformal Killing vector field if the conformal ...
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Diagonal and block diagonal matrix

What's the significance of diagonal and block-diagonal matrices in quantum mechanics? For instance, let $S$ be the symmetry operator, since $[S,H]=0$, they have a shared eigenbasis. If I use a basis ...
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Can a perturbation add more symmetry?

Many textbooks of quantum mechanics argues that in presence of the ground degeneracies, at least some of them will be removed if some perturbation reduces the corresponding symmetry. Then, is there ...
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Relationship between symmetries and quantum operators of classical quantities?

I noticed this the other day. I don't really know "what" this means, I'd love to understand. The energy operator is $\hat E = -i \hbar \frac{\partial}{\partial t}$. Conservation of energy ...
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Isometries of $AdS_2$ space

So, in many places it is mentioned that isometries of pure $AdS_2$ space is the group $SL(2,R)$, defined by the transformation, $t = (at+b)/(ct+d)$ where $ad-bc=1$. Here, the boundary of $AdS_2$ is ...
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How is a non-primitive unit cell/lattice helpful?

I am starting with the basics of X-ray crystallography, and I have encountered something I'm not able to rationalize. As I understand it, the unit cell is the smallest parallelepiped enclosing the (a?)...
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Topological Quantum Field Theory with Symmetries and Knot Quandles

It is well known that Chern-Simons theory provides an intrinsically three dimensional way to compute knot invariants like the Jones Polynomial. 3D TQFTs also have an algebraic description in terms of ...
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Confusion about chiral anomaly (Fujikawa's method)

I am reading Fujikawa's method for calculating chiral anomaly, see this wiki page. The method can be described as follows. It starts with the path integral \begin{equation} Z=\int\mathcal{D}\psi\...
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Use of Symmetry Arguments [migrated]

I am trying to solve the problem $$\int d^3x\,e^{i{\bf a}\cdot{\bf x}}e^{-br^2}$$ using symmetry arguments. Could someone direct me to a similar question or guide me through a similar problem so I can ...
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The groups and symmetries of superstrings/SUGRA/M-theory

When giving talks to Laymen, we find out that the M-theory paradigm says that there are 5 (only 5) superstring theory types, 11d SUGRA and M-theory. Curiously, we read regularly the symmetry groups of ...
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Subspace and symmetry of 4 electrons

For 2 coupled electrons, the possible spin wavefunctions are: \begin{align*} \chi_{1,1} = \upuparrows,\quad \chi_{1,-1}=\downdownarrows\ \end{align*} \begin{align*} \chi_{1,0}=\frac{1}{\sqrt2}(\...
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Change of sign of angular momentum operator in Weinberg's QFT

In Chapter 2 of Weinberg's QFT, in order to represent a "boost" $L(p)$ , $$ L(p)=R(\hat{\mathbf{p}}) B(|\mathbf{p}| / \kappa), \quad(2.5 .44) $$ he tries to define a rotation $R(\hat{\mathbf{...
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Why is there no $H_1H_1\bar{e}^c$ term in the most general renormalisable superpotential?

I am these notes by Jeff Asaf Dror on Supersymmetry. In the section on the MSSM, it is stated that the most general renormalisable superpotential is given by $$W=y_1QH_2\bar{u}^c+y_2QH_1\bar{d}^c+...
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BETRANDS THEOREM [duplicate]

For a force that behaves like $$F=k r^n,$$ the stability condition requires $n > -3$. How this leads to the Bertrand’s theorem: The only central forces that result in closed orbits for all bound ...
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4 spin-1/2 particles representation [duplicate]

A follow-up question of $\frac{1}{2}\otimes\frac{1}{2}=0⊕1 $ : If I have 4 spin-1/2 particles in my system, how can I use a series of direct sums to represent $\frac{1}{2}\otimes\frac{1}{2}\otimes\...
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A question about the equation $\frac{1}{2}\otimes\frac{1}{2}=1\oplus0$

I have a question about this equation: $$\frac{1}{2}\otimes\frac{1}{2}=1\oplus0.$$ I'm a bit confused by the right-hand side. Should '1' and '0' be interpreted as the total spin? If so, if there're ...
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Fermion masses and $SU(2)$ symmetry

Why is the standard mass (Dirac) not compatible with $SU(2)$ symmetry? I consider the standard mass this $$ m\bar{e}e = m(\bar{e}_Le_R+ \bar{e}_Re_L)$$
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What does a broken symmetry mean for the Lagrangian?

I am a little confused about symmetry breaking - in particular, what I see to be too different interpretations of it. First, what I have seen taken to be the definition of a broken symmetry - we start ...
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The definition of the weaker notion of symmetry in the sense of Wigner's theorem

The weaker notion of symmetry, in the sense of Wigner's theorem, is a transformation on the states that leave all quantum mechanical amplitudes invariant. This tells that such transformations are ...
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The Eigenstates of a Symmetric Operator

Good Afternoon, By definition, an observable $O$ for a system of N identical particles is symmetric just in case $\langle\psi|O|\psi\rangle = \langle\psi|P^{\dagger}OP|\psi\rangle$ for any permutation ...
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Application of Noether Theorem

I attempt to understand one of the examples of the application of Noether theorem given in Peskin & Schroeder's An Introduction to Quantum Field Theory (Page no. 18, Student Economy Edition). The ...
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Proof of Wigner's theorem in Weinberg’s book

I was following the proof of Wigner's theorem from Weinberg’s book Quantum Theory of Fields, volume 1, pp.91-94 and got stuck in the middle: the proof proceeds as follows for arbitrary state vector: ...
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Are broken symmetry states non-stationary?

In his famous paper, More is Different (link), Philip W. Anderson states that in the context of quantum mechanics : [...] the state of the system, if it is to be stationary, must always have the same ...
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How to find cyclic coordinates for a given Lagrangian?

If you are given a Lagrangian in coordinates which are not cyclic, is there a rule for finding a transformation of the coordinates to another set of coordinates where one of them is cyclic?
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Origin of the form of symmetry operation

It is well known that the symmetry operations $U$acting on the operators could be written as $$U AU^{-1}$$ Now I want to know the logical origin or motivation of this form of operation, my thought ...
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Weinberg’s proof of Wigner's theorem

I'm working through the proof of Wigner's theorem in Weinberg's The Quantum Theory of Fields Volume 1 Chapter 2 Appendix A pp. 91. Consider the following steps of the proof Let $\left\{\psi_{k} \...
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Representation of homogeneous Lorentz transformation

In Page 63, Section 2.5 of Weinberg's QFT Volume 1, on "One-particle states", he considers the representation of homogeneous Lorentz transformation, $U(\Lambda, 0) \equiv U(\Lambda)$ $$ U(\...
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What is meant by the symmetry group of a crystal?

When we talk about the symmetry group of a crystal, does it mean the set of transformation under which a unit cell is taken to itself? For example, if we have a two-dimensional hexagonal lattice, can ...
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Why does gravity act at the centre? [closed]

Why does gravity act at the centre of earth and how does that happen?
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Commutation relations of angular momentum operators

On page 54 of Weinberg's QFT I, he says that an element $T(\theta)$ of a connected Lie group can be represented by a unitary operator $U(T(\theta))$ acting on the physical Hilbert space. Near the ...
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Hermitian operators in the expansion of symmetry operators in Weinberg's QFT

This is related to Taylor series for unitary operator in Weinberg and Weinberg derivation of Lie Algebra. $\textbf{The first question}$ On page 54 of Weinberg's QFT I, he says that an element $T(\...
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Question in Derivation of Lie algebra

In Weinberg's QFT Volume 1 Chapter2, he "derives" the Lie algebra from the Lie group as follows [...] a connected Lie group [...is a...] group of transformations 𝑇($\theta$) that are ...
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How to apply anti-unitary symmetry operators?

It is know that the symmetry operators can be applied to operators like $$ \hat{O} \stackrel{g}{\rightarrow} \widehat{g O} $$ demand the matrix element to be invariant under symmetry, we have $$ \...
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Symmetry group in physics [closed]

When we say like “spatial translation/rotation” symmetry in physics context, do we really mean the representation of the symmetry group in coordinate basis rather than the “symmetry group” itself?
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Plugging a global phase into an operator

Cheers to everyone. I' ve got a serious doubt about the following: consider the annihilation operator $\hat a$. For practical reasons, I sometimes find useful redefining it in the following way : $\...
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Construction of symmetry group algebra

In Kitaev's reasoning of constructing the algebra of symmetry group, he said, "considering the symmetry group G of a fermionic system and a map $\alpha$ $$ \alpha: G \rightarrow \mathbb{Z}_2 $$ ...
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Getting wrong number of Wick contractions

Consider this lagrangian: $$\mathcal{L} = \dfrac{1}{2} (\partial_{\mu}\phi_{1})^2 + \dfrac{1}{2} (\partial_{\mu}\phi_{2})^2 + \dfrac{m^2}{2}(\phi_{1}^2 + \phi_{2}^2) + \dfrac{g}{4!}(\phi_{1}^4 + \phi_{...
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Conserved charges as generators of symmetries in Hamiltonian mechanics

I’ve been trying to understand the relationship between conserved charges and symmetry transformations; in particular how the conserved charges act as generators for the symmetry in the Hamiltonian ...
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Particle Hole Symmetry of BdG Hamiltonians

It is straight-forward to verify that any Hermitian BdG Hamiltonian of the form $$ \mathcal{H} = (c_1^\dagger, c_1, c_2^\dagger, c_2,...) \begin{pmatrix} H_{11} & H_{12} & \cdots \\ H_{21} &...

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