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 is the Lie group of gravity?

If the lie group of the three gauge forces are $SU(3)×SU(2)×U (1)$, then what is the symmetry group of gravity? $SL(2,C)$? Just a newbie in Lie groups.
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Are gravitons Goldstone bosons?

This question is inspired by Are Photons Goldstone Bosons?. And recently in Generalized Global Symmetries, photon was interpreted as Goldstone boson for generalized symmetry (see also my answer for ...
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Poisson Bracket of a Quantity Involving a Differential

I am working through Warren Siegel's "Fields" and have come across the following exercise on p. 58 involving an action measure and a symmetry generator: Exercise IA4.1. For general variables ...
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mirror symmetry of spin-1/2 wavefunction: Definition of the reflection operator

I would expect from a reflection operator $\hat{M}$ to leave a wavefunction unchanged if two times applied, thus $\hat{M}^2=1$. However, for a spin-1/2 particle this is not the case when following the ...
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What is the physical idea of isometry of a metric?

I am trying to understand the idea of isometry and its physical meaning. Under a general coordinate transformation (GCT), the covariant metric tensor $g_{ij}$ changes as $$g^\prime_{ij}(x^\prime)=\...
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Full Bondi-Metzner-Sachs (BMS) or asymptotic group are the same and have equal interpretations?

I had red about supertranslations or even superrotations. But I just discovered there are also superboosts and superLorentz ( I suppose this is for superrotations and superboosts). Is the full BMS ...
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Scalar and vector transformation

When we say that a scalar field is invariant under any transformation, why can't we consider vector components as scalar fields ? Let's say we have $O(3)$ acting on a scalar and vector field in a ...
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Lorentz Equation Symmetry

I was going via Lorentz equation & learning the topic on Symmetry, what I couldn't understand is how did they performed this type of substitution & what is the philosophy behind this way of ...
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Correct explanation for generic vectorial, radial and central field

I suppose to have a absolutely general field of the type $\vec{F}(\vec{r})\colon D\subseteq\Bbb R^3\to \Bbb R^3$ where $|\vec{r} |=\vec{OP}$. If I fix the direction of $\vec{r}$, than $\vec{F}(\vec{r}...
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Swapping labels vs. particles in wavefunctions

I have been told that (for example) identical fermion wavefunctions need to be anti-symmetric under exchange of particle. I have also been told that identical fermion wavefunctions need to be anti-...
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Anisotropic vs Isotropic Harmonic Oscillator

IS there any quantum mechanical process which can take over an anisotropic commensurate harmonic oscillation to an isotropic one? Mathematically, this kind of transformation is available. http://dx....
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Does Birkhoff's theorem still apply to an ultra-dense spherical shell of matter falling into a black hole?

In previous posts I discovered that the effects on space-time geometry of an ultra-dense stationary spherical shell yielded pretty simple equations but I imagine some sort of secondary non-linear ...
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How identify symmetries in a Lagrangian and how to derive it using Feymann rules?

I have to find symmetries of this Lagrangian, $$ \mathcal{L} = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}M^2\phi^2 - \frac{\lambda}{4!}\phi^4 +\bar{\psi}(i_0 \partial - m)\psi - g\bar{\...
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The radiative modes of an asymptotically flat spacetime and the symmetries

In Ashtekar's paper, the radiative degrees of freedom of an asymptotic flat spacetime in general relativity are obtained. These degrees of freedom live on the null infinity $\mathscr I$, given by the ...
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Role of isotropy argument in fixing the form of $T^\mu_\nu$ of cosmology (Padmanabhan's book)

The following lines from Gravitation Foundation and Frontiers by T. Padmanabhan (he uses Latin indices for spacetime and Greek indices for space, which is unusual) The assumption of isotropy ...
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Symmetry between gauge fields in the given expression

Is there a symmetry between the gauge fields $A_{\sigma}$ and $A_{\lambda}$ in the expression: $$f^{abm}f^{bcn}f^{cap}\partial_{\rho}C(x-y)A^m_{\lambda}(y)\partial_{\sigma}C(y-z)A^n_{\sigma}(z)\...
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S-matrix elements for Nucleon-Pion decay

I want to compute nucleon-pion decay rates. I am a bit confused how I can compute the S-matrix. Let's say we have a Nucleon Pion scattering and I want to compute their corresponding S matrix: \begin{...
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$SU(3)$ Clebsch-Gordan Coefficient

I have a problem computing the ratio $$\frac{P(\pi^0 P\rightarrow\Delta^+)}{P(K^- P\rightarrow\Sigma^{*0})}.$$ The problem demands reducing the $S$-matrix first but I really don't see how to get this ...
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What would the shape of orbits of planets be if hypothetically the gravitational force be proportional to the inverse of cube of distance from Sun? [closed]

What would the shape of orbits of planets be if, hypothetically, the gravitational force was proportional to the inverse of cube of distance from the Sun? Please ignore other effects caused due to ...
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Symmetries and Dirac Lagrangian [closed]

For spacetime translation given by Under spacetime translations the spinor transforms as $$\delta\psi=\epsilon^\mu\partial_\mu\psi$$ The Lagrangian depends on $\partial_\mu\psi$, but not $\...
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Scaling transformations, definitions and all that's not mentioned

If we transform the massless scalar field Lagrangian $$\mathcal{L}=\frac{1}{2}(\partial_\mu\varphi)^2-\frac{\alpha}{4!}\varphi^4$$ with the simultaneous transformations $$x\mapsto x^\prime= \lambda x,\...
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Does the space group P63/m (No. 176) have C6 rotation symmetry?

Recently I'm working on a compound with space group P63/m. The top view of its structure is shown below (where only atoms of z=1/4 are shown). From the list of space groups (Wiki: List of Space ...
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Why is the symmetry of measurements in SR no longer valid in GR?

In special relativity, we know that if an inertial observer $A$, having a relative motion WRT an inertial observer $B$, attributes, say, time dilation and length contraction to respectively $B$'s ...
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Dynamics of a linear chain of harmonic oscillators

Let's consider a linear chain of particles with harmonic nearest neighbor interaction: Assuming all particles have the same mass, Equations of motion are (with periodic boundary conditions): $$m\ddot{...
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In 4 spatial dimensions, would motion under a central force law be confined to a plane?

In dimension $3$, we have the angular momentum $\omega = q \times v$, and since $$\frac{d}{dt} \omega = v \times v + q \times (F/m)$$ from the fact the force is central (so $F$ is parallel to $q$) we ...
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Komar mass derivation

The Komar mass, a conserved quantity associated with an asymptotically time-like Killing vector, for stationary spacetime is $$M_K=\frac{-1}{8\pi}\int\int\nabla^{\mu}\xi^{\nu}_{(t)}dS_{\mu\nu},$$ ...
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Why $Q|0\rangle=0$ where $Q$ generates a symmetry?

Quote: "Concepts of Elementary Particle Physics" by Michael E. Peskin In quantum mechanics with a finite number of coordinates, it can be shown that, if $Q$ generates a symmetry of the theory, then ...
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Relation between irreducible representations and eigenstates

When in quantum mechanics the Hamiltonian possesses some symmetry, then knowing the irreducible representations of the group to which the symmetry belongs gives information about the eigenstates. As ...
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Moment of inertia of a solid sphere with a spherical cavity cut out of it [closed]

I am trying to solve the Following question. Consider a sphere of radius R with a cavity of radius r cut out of it. The distance between sphere and cavity center is a such that a < R-r. Find all ...
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Conserved quantity when potential is invariant under transformation

I am very new to Noether's theorem and in our (first!) mechanics class it was proved using generators $X_i$,$X$ of a Lie group. Because I didn't really understand this proof I have trouble solving the ...
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Inverse Square Law and Closed Orbits

The definition of closed in this context is that the body will retrace its orbit. A standard case for closed orbits is the circular case. I know from Bertrand's theorem that the linear restoring ...
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On symmetries in spin systems

I'm reading about concepts related to symmetries in spin systems, and I feel I have some confusions about things that are so obvious that they're never explained. For example, take the AKLT spin chain ...
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If a regularization procedure respects a symmetry, is this symmetry unbroken in perturbation theory?

I read in this paper the statement that a proof that SUSY is preserved in perturbation theory would be the existence of a regularization procedure which respects SUSY (for a particular theory). Is ...
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Shift symmetry and vacuum expectation values

I have come across theories with a Nambu-Goldstone boson $\phi$ originating from a broken $U(1)$ symmetry where there is a leftover shift symmetry $\phi \rightarrow \phi + \alpha$ (and possibly other ...
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What is the physical significance/interpretation of a vanishing Lie Derivative?

In my lectures, an isometry of the metric is introduced as follows: A flow on a manifold $M$ is a one-parameter family of differomorphisms $\sigma_t:M \to M$. The flow is said to be an isometry if ...
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Why is there a relationship between symmetries and conservation laws?

I am reading through my professor's notes and I am unsure as to what the intimate relationship between the symmetry property of a physical system and the conservation laws of energy, momentum, and ...
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Isometry Group of spacetime acting on states of system

I read the following statement in Wald's book and it is a bit unclear to me. "The isometry group of spacetime acts in a natural way on the states of a physical system." In what sense the isometry ...
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Isotropic moments of inertia

Explicit integration can show that the moment of inertia of a Platonic solid (i.e., tetrahedron, cube, octahedron, dodecahedron, or icosahedron) of uniform density is the same around any axis passing ...
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What space group describes a 1-dimensional crystal with reflection symmetry along axis?

I'm trying to understand the symmetry of an effectively 1-dimensional system, but I'm confused about how the 1-dimensional ``line groups'' are classified. If you have a system along the $z$-axis which ...
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Why is elastic scattering of photons (largely) non-isotropic, but inelastic scattering is isotropic?

I'm working on an experiment regarding Raman spectroscopy, and i'd like to fully understand the reasoning behind this fact. I assume it is related to the photons momentum. My apparatus has a '...
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Curvature and Symmetries of spacetime

Is there any relation between symmetries of spacetime and the curvature invariants? For example is spherical symmetric spacetimes, necessarily have positive curvature? Could we define any spherical ...
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Is the Kerr metric more symmetric than a normal type D spacetime?

The Kerr spacetime is of Petrov type D (see here for the Petrov classification of spacetimes). In the Newman-Penrose formalism, from the Goldberg-Sachs theorem we can conclude that there is a choice ...
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Show Lagrangian is invariant under infinitesimal $SO(3)$ transformation

Suppose we have the Lagrangian density for a triplet of real scalar fields, $$ L = \sum_{a=1}^3 \left[ \frac{1}{2}\partial_\mu\phi_a\partial^\mu\phi_a - \frac{1}{2}\phi_a\phi_a \right]. $$ How do ...
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How do we know that the actual universe has no Killing vector fields?

This article states the following: The infinitude of conserved energies constructed via Noether’s theorem suffers a startling reversal as soon as Special Relativity is superseded by General ...
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Showing rotation is a symmetry of given Lagrangian

I have the Lagrangian $L = \frac{1}{2}m(\dot{x}^2+\dot{y}^2) - ax^2 -by^2 -cy^3$. I am trying to work out the conditions that $a,b,c\in\mathbb{R}$ must satisfy so that rotations around the origin, i.e....
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maximally symmetric spacetime

An empty spacetime has zero or constant Ricci Scalar (depending on the cosmological constant). Is there a theorem which guarantees that such a spacetime should be Minkowski or dS/AdS? In other words, ...
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Does Special Relativity Set a Canonical Zero of Energy?

In special relativity, one has the equation $$ E^2 = m^2 + p^2 $$ It seems like this is saying that there is an absolute zero of energy: the energy of a massless, momentumless particle. On the other ...
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What is the $R$-symmetry group for ${\cal N}=6$ supergravity in $D=4$ dimensions?

What is the $R$-symmetry group for ${\cal N}=6$ supergravity in $D=4$ dimensions?
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Gauge group R and U(1) and a global symmetry

In the beautiful paper by Harlow et Ooguri, they write in section 2.1 about this action $$ S=-\frac12 \int_M F_a \wedge \star F_b \delta^{ab}\;, $$ with index $a=1,2$. They say that if the gauge ...
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(Coming from Wigner's Theorem): What is a Symmetry in QFT?

In classical mechanics, classical field theory and QM, I was introduced to the concept of "Symmetry" as some kind of active transformation of either spacetime / time or configuration space (or of the ...

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