Questions tagged [symmetry]
Symmetries play a big role in modern physics and have been a source of powerful tools and techniques for understanding theories and their dynamics. 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 forms a group, and the name of this group is used as the name of the symmetry of the object.
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Are group representations possible when the solution space is not a vector space?
As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector ...
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Demonstrate why conservation of momentum stems from invariance to translation - in Newtonian mechanics [duplicate]
I know that conservation of momentum is caused by invariance (symmetry) of the laws of mechanics to translation - this is an example of Noether's theorem.
However, I'm looking for an explanation, ...
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Energy-Momentum Tensor for the Electromagnetic Feild
Question
When calculating the hamiltonian for the free Electromagnetic Field with Lagrangian density
$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$
Using Noether's theorem I found the answer to be
$...
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How a symmetry transformation acts on quantum fields
I study particle physics and am finally tired of pushing through QFT with annoying doubts which seem to be both very simple and fundamentally important, and to which several professors of mine couldn'...
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Killing tensor in the Kerr metric
It was famously shown by Carter that the Kerr metric possesses a 4th non-obvious constant of the motion, derived from the separability of the Hamiltonian. This constant is related to a Killing tensor.
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How to determine the Lagrangian's "true" explicit dependence on time?
If your Lagrangian satisfies
$$
\frac{\partial \mathcal L}{\partial t} = 0
$$
then you're happy, energy is conserved, etc. However, if the above doesn't hold, that doesn't necessarily mean energy ...
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Topology and Superconductor Symmetry Breaking
I read an article today on stack exchange titled "Superconductor Symmetry Breaking". The 2016 Nobel Prize was awarded for research on topological phase transitions in the study of superconductors and ...
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What does it mean for a topological phase to be "symmetry protected"?
I have seen some very nice and enlightening awnsers to questions related to topological order and insulators, such as here, or here. However, I'm still puzzled by the concept of "symmetry protection" ...
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A question on the existence of Dirac points in graphene?
As we know, there are two distinct Dirac points for the free electrons in graphene. Which means that the energy spectrum of the 2$\times$2 Hermitian matrix $H(k_x,k_y)$ has two degenerate points $K$ ...
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Why a gravitational lense makes multiple distinct pictures of a distant object rather than making a symmetric ring?
I cannot imagine how a group of galaxies may produce pictures of a distant object on a ring-like region that is not symmetric. Why there are empty parts of that ring where there are no pictures of the ...
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Counterexample to spherical symmetry definition in general relativity
In practical terms we say a spacetime is spherically symmetric in GR when we have coordinates in which the spacetime metric takes the form:
$$ds^2 = -f(r,t)dt^2 +g(r,t)dr^2+h(r,t)d\Omega^2 \tag{*}$$
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Many Body Physics: Hamiltonian block structure and Symmetries
Consider a many body problem of a small cluster, e.g. the 'Hubbard-Cluster' (albeit the question may be of relevance for other Hamiltonians as well):
$$\mathcal{H}=\sum_{<ij>\sigma} t_{ij} (c^\...
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A simple model that exhibits emergent symmetry?
In a previous question Emergent symmetries I asked, Prof.Luboš Motl said that emergent symmetries are never exact. But I wonder whether the following example is an counterexample that has exact ...
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What is topological about topological (Dirac or Weyl) semimetals?
The following is my rough understanding of topological phases of matter (please let me know if it is incorrect.) Topologically ordered phases of matter are topological in the sense that they are ...
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Why is the symmetry variation $\delta_s q$ different from the ordinary variation $\delta q$?
I was reading about symmetry of action when I came before the symmetry variation in Particles and Quantum Fields by H. Kleinert; there he wrote:
Symmetry variations must not be confused with ...
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What is the definition of an Asymptotic Symmetry Group (ASG) of a spacetime?
What is the definition of an Asymptotic Symmetry Group (ASG) of a spacetime?
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How to find points with same potential while solving an equivalent resistance problem?
Lately, I've been reading about techniques to reduce networks and find their equivalent resistance/capacitance. While doing this, I came across the cube resistance problem and many other problems (eg. ...
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Is the spin-rotation symmetry of Kitaev model $D_2$ or $Q_8$?
It is known that the Kitaev Hamiltonian and its spin-liquid ground state both break the $SO(3)$ spin-rotation symmetry. So what's the spin-rotation-symmetry group for the Kitaev model?
It's obvious ...
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Peccei-Quinn-symmetry and effective Lagrangian for the Axion field
To solve the strong CP-problem Peccei and Quinn suggested the use of a new $U(1)$-symmetry called the PQ-symmetry. For this symmetry they constructed an effective Lagrangian involving the Nambu-...
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Translations and Noether's Theorem
I'm fine with $U(1)$ symmetry and Noether's Theorem, but struggling with the translations of the field; namely
$$\phi'(x^{\mu})=\phi(x^{\mu}-a^{\mu}),$$ where $a^{\mu}$ constant four-vector
$$x^{\...
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What is the reason for turning global symmetries into local symmetries?
For example a simple complex scalar field theory has a global $ U(1) $ symmetry where the field $ \psi $ can be replaced by $ e^{ i \alpha } \psi $, where $ \alpha $ is just some real constant, ...
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How do we know that renormalization doesn't change the form of the ghost action in Yang-Mills theory?
In field theory, we typically construct a Lagrangian by only specifying its (global or gauge) symmetries, then writing down all renormalizable terms that respect those symmetries with arbitrary ...
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What symmetry does conserved $L^2$ imply?
According to Noether's theorem, if $[L,H]=0$ the system has rotational invariance. Does $[L^2,H]=0$ also imply some symmetry for the system?
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Galilean transformation in non-relativistic quantum mechanics
I'm reading Weinberg's Lectures on Quantum Mechanics and in chapter 3 he discusses invariance under Galilean transformations in the general context of non-relativistic quantum mechanics. Being a ...
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Symmetry and group theory book [duplicate]
I would like to start learning about symmetries in physics and how they affect physical quantities. As far as I know, the mathematical language that describes symmetries is the Group Theory. So, I ...
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Can you gauge a $U(1)_L$ symmetry?
I was recently calculating the one loop correction for the propagator of a gauge boson,
$\hspace{5cm}$
I assumed arbitrary left and right couplings, $ g _L $ and $ g _R $. I found that the one loop ...
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Relativistic center of mass
Recently I realized the concept of center of mass makes sense in special relativity. Maybe it's explained in the textbooks, but I missed it. However, there's a puzzle regarding the zero mass case
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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 conserved quantity causes degeneracy of spectrum in rational polygonal billiards?
Apart from Laplace-Runge-Lenz vector conservation in Coulombic and something similar in harmonic and other central potentials, something leads to existence of periodic trajectories in such systems as ...
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Angular momentum of vacuum solution in Einstein gravity
In Strominger's "Lecture Notes on Infrared Structure of Gravity", page 38, he mentioned about how part of this whole mess about "vacuum degeneracy" (classically, i.e. in the sense ...
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Symmetries in QFT
There are very famous Coleman–Mandula theorem and Haag–Łopuszański–Sohnius theorem , see also this and this.
It states that "space-time and internal symmetries cannot be combined in any but a ...
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What is Noether's Second Theorem?
I have been unable to find a short statement of Noether's second theorem. It would be helpful to have the following:
A short mathematical statement of the theorem. Does it imply a conservation law ...
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Completely positive maps and symmetric states
Let $\mathcal{N}$ be a completetely positive trace preserving map (aka a quantum channel) acting on a finite dimensional system $\mathrm{A}$, and let $\pi$ denote the maximally mixed state on $\mathrm{...
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Questions on Penrose's paper - Conformal Treatment of Infinity
I have several questions. Perhaps it would be better to separate them into different posts. However, given their relative closeness to each other, I think putting it all in one place would be better. ...
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Can any body be uniform in the universe?
If I take any body in the shape of a rod and stretch that, after it reaches breaking stress it breaks at one point.
Even though we apply the same the stress on each and every part of the rod it broke ...
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Why does it take a projectile as long to get to its apex as it does to hit the ground?
I was once asked the following question by a student I was tutoring; and I was stumped by it:
When one throws a stone why does it take the same amount of time for a stone to rise to its peak and then ...
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Doesn't Newton's equation of motion have a bigger invariance group than the Galilean group?
Newton's equation ${F}^i=m\frac{d^2x^i}{dt^2}$ is unchanged in form, under the Galilean group:
(i) under a translation of the origin of coordinates,
(ii) rotation of coordinates, and
(iii) Galilean ...
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Very precisely explaining when phase plays a role or doesn't play a role in QM
My question is probably basic at first view but I would like to really understand this in details.
The way I understand the role of the phase in quantum mechanics is that as soon as we have a physical ...
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Noether's theorem for space translational symmetry
Imagine a ramp potential of the form $U(x) = a*x + b$ in 1D space. This corresponds to a constant force field over $x$. If I do a classical mechanics experiment with a particle, the particle behaves ...
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Killing Vectors in Schwarzschild Metric
Given the Schwarzschild metric with $(-,+,+,+)$ signature,
$$\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$
the lack of ...
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Justification of form $L(v^2)$ of Lagrangian for a free particle in Landau-Lifshitz vol 1
See the screenshot below for Landau's argument on the form of a free particle lagrangian.
My question is regarding whether the Lagrangian $L$ of a free particle must only be dependent on $v^2$. In my ...
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What does scale invariance or non-invariance of electromagnetism physically imply?
According to Wikipedia,
classical electromagnetism is scale-invariant.
I understand what it means mathematically as explained in Wikipedia. But what does it really imply physically?
Next, here it ...
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Special Relativity - Reference Frames $S$ and $S'$
Consider the standard arrangement in special relativity.
Let $S'$ move in the +ve $x$-axis with a velocity $v$ with respect to $S$.
This implies that $S$ moves with a velocity $-v$ with respect to $S'$...
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Intuitive explanation for why time symmetry implies conservation of energy?
According to Noether's Theorem, every physical symmetry leads to a conservation law. For example, time-translation symmetry (the laws of physics don't change over time) implies conservation of energy,...
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The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices
The Pauli spin matrices
$$
\sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}),
\qquad\qquad
\sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 \end{...
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Where do symmetries in atomic orbitals come from?
It is well established that:
'In quantum mechanics, the behavior of an electron in an atom is described by an orbital, which is a probability distribution rather than an orbit.
There are also many ...
user98038
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Does isotropy imply homogeneity?
This question comes from exercise 27.1 in Gravitation by Misner, Thorne and Wheeler. They required the following:
Use elementary thought experiments to show that isotropy of the universe implies ...
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Equation in Noether's paper
When I was reading the paper by Emmy Noether about the famous Noether Theorem, there is an equation I don't know its meaning and why it holds on page 5.
$$\phi\frac{\partial^{\sigma} p(x)}{\partial x^{...
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Why discrete gauge fields must be flat?
I found in some papers, for example "Generalized Global Symmetries" and "Generalized Symmetries in
Condensed Matter", that the gauge field of a discrete symmetry must be flat, i.e. ...
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Symmetries in quantum mechanics
I have been studying symmetries in quantum mechanics and I have come across two types:
Given some transformation on a Hilbert space of states $\mathcal{H}$, an operator $U: \mathcal{H} \rightarrow \...
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