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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How do fundamental symmetries vanish at the macroscopic level?
I recently read an interesting discussion, between Carlo Rovelli and Steven Weinberg which happened at the "Conceptual Foundations of Quantum Field Theory" conference in 1994:
Rovelli: Steve, your ...
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Is Parity really violated? (Even though neutrinos are massive)
The weak force couples only to left-chiral fields, which is expressed mathematically by a chiral projection operator $P_L = \frac{1-\gamma_5}{2}$ in the corresponding coupling terms in the Lagrangian. ...
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Why does a transformation to a rotating reference frame NOT break temporal scale invariance?
Naively, I thought that transforming a scale invariant equation (such as the Navier-Stokes equations for example) to a rotating reference frame (for example the rotating earth) would break the ...
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Is General Relativity based on a Symmetry?
In short: Is there any kind of symmetry one can start with to derive general relativity (GR)?
Longer: Einstein had the opinion that GR was the generalisation of special relativity, because instead of ...
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Why is Planck's constant the same for all particles?
This question came to me while reading Where does de Broglie wavelength $\lambda=h/p$ for massive particles come from? This question has a nice answer that explains that wave number has be ...
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Definition for Chiral Spin Liquid
What is the definition of chiral spin liquid?
Especially what does chiral mean here?
I encounter a lot of terminologies with chiral. It seems they mean differently in different contexts. If you could ...
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Invariance of Functional Integration Measure
Let us consider the functional integral:
\begin{equation}
\int \mathcal{D} A e^{iS[A]}
\end{equation}
where $S[A]$ is the action for $U(1)$ gauge field and
\begin{equation}
\mathcal{D}A\equiv \...
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Quasicrystals - Projections from higher dimensional regular crystal lattices
Why are quasicrystals projections from higher dimensional regular crystal lattices?
See for example wikipedia:
»Mathematically, quasicrystals have been shown to be derivable from a
general ...
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Why does group representation theory look linear?
I'm reading first a few chapters of a physicist's group theory book and one naive question comes into my mind. I feel I probably missed something very basic and got bogged down in the details.
My ...
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Why is the Fourier transform more useful than the Hartley transform in physics?
The Hartley transform is defined as
$$
H(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty
f(t) \, \mbox{cas}(\omega t) \mathrm{d}t,
$$
with $\mbox{cas}(\omega t) = \cos(\omega t) + \sin(\omega t)$...
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Can Coulomb be derived directly from Gauss?
This relates closely to questions such as
Deriving Coulomb's Law from Gauss's Law
and
Does the Coulomb's law include more information than the Gauss's Law in electrostatics?
My ...
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Does Noether's theorem also give rise to quantities conserved over space?
Noether's theorem gives rise to quantities that are conserved over time. But does it also give rise to quantities that are conserved over space?
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Symmetry in quantum mechanics
My professor told us that in quantum mechanics a transformation is a symmetry transformation if $$ UH(\psi) = HU(\psi) $$
Can you give me an easy explanation for this definition?
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What's the exact gravitational force between spherically symmetric masses?
Consider spherical symmetric$^1$ masses of radii $R_1$ and $R_2$, with spherical symmetric density distributions $\rho_1(r_1)$ and $\rho_2(r_2)$, and with a distance between the centers of the spheres ...
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Conserved charges and generators
For the Klein Gordon field, the conserved charge for translation in space is given by:
$$\vec{P}=\frac{1}{2}\int d^{3}k \, \vec{k}\{a^{\dagger}_{k}a_{k}+a_{k}a^{\dagger}_{k}\}$$
If we were to find ...
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The converse of Newton's shell theorem
The shell theorem states that a spherically symmetric body $S$ of mass $m$ has a gravitational field identical to that of a point particle $P$ of mass $m$ located at the center of $S$.
We can ask the ...
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On finite-dimensional unitary representations of non-compact Lie groups
In this thread Lorentz transformations for spinors, V. Moretti made a claim as follows:
"it is possible to prove that no non-trivial finite-dimensional unitary representation exists for a non-...
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Invariance of Action vs. Lagrangian in Noether's theorem?
I have recently started studying classical field theory. Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. But I find ...
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Deriving the action and the Lagrangian for a free massive point particle in Special Relativity
My question relates to
Landau & Lifshitz, Classical Theory of Field, Chapter 2: Relativistic Mechanics, Paragraph 8: The principle of least action.
As stated there, to determine the action ...
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Is the molecular term ${}^1\Sigma^-$ possible in a molecule?
The old question How to understand this symmetry in the wavefunctions of a diatomic molecule? explores how it is possible for a quantum state to have zero angular momentum about a given axis (giving ...
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Symmetry transformations on a quantum system; Definitions
We define a symmetry transformation of a system to be any transformation that, when performed, does not change the outcome of a measurement. Wigner's symmetry theorem says that any symmetry of a ...
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Confusion in a trick in solving an energy eigenfunction
Given a non-relativistic energy eigenfunction for a central potential $\left|\Phi \right>$
In solving relativistic hydrogen atom, one of the terms is
$$
\left<\Phi\middle|\frac{e^2}{r}\middle|\...
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Time reversal symmetry of transverse field Ising model
Is the transverse field Ising model time-reversal invariant? Specifically consider a non-integrable variant:
\begin{equation}
H = -J \sum_i^{L-1} \sigma_i^z \sigma_{i+1}^z + g \sum_i^L \sigma_i^x + h ...
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Why are conformal transformations so prevalent in physics?
What is it about conformal transformations that make them so widely applicable in physics?
These preserve angles, in other words directions (locally), and I can understand that might be useful. Also, ...
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Generator of local symmetries
Let us only consider classical field theories in this discussion.
Noether's theorem states that for every global symmetry, there exists a conserved current and a conserved charge. The charge is the ...
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Explaining chirality for spin 1/2 particle
I found the following explanation for chirality for spin 1/2 particles here
What happens when you rotate a left- vs right-chiral fermion 360
degree about its direction of motion. Both ...
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How to arrive at the Dirac Equation from Poincaré Algebra?
For the case of Galilean group, the time translation is given by the generator $H$. Hence,
$$\mid\psi(t)\rangle\to \mid\psi(t+s)\rangle =e^{-iHs}\mid\psi(t)\rangle$$
Which immediately is the ...
user29978
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Intuitive argument for symmetry of Lorentz boosts
The Lorentz boosts are represented by symmetric $4\times4$ matrices. Though the most general Lorentz transformations has no obvious symmetry property, can the symmetry (under transpose) of the Lorentz ...
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Why is a theory Lorentz invariant if the Lagrangian is Lorentz invariant?
For if I started by trying to make the Hamiltonian Lorentz invariant, I would have failed. Indeed, the Hamiltonian is part of a covariant tensor. But how do I know that the Lagrangian is not a part of ...
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Noether currents in QFT
I am trying to organize my knowledge of Noether's theorem in QFT. There are several questions I would like to have an answer to.
In classical field theory, Noether's theorem states that for each ...
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Confusion about two definitions of anomalies
As I am currently studying for an exam about quantum field theory and string theory, I got confused about the notion of "anomalies" and how they are actually defined. Similar questions have already ...
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More general invariance of the action functional
I will formulate my question in the classical case, where things are simplest.
Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the ...
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What is replica symmetry breaking, and what is a good resource for learning it?
M. Mezard, G. Parisi and coworkers have written about replica symmetry and its breaking in spin glasses, structural glasses, and hard computational problems.
I am just getting acquainted with this ...
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Low energy description of Symmetry Enriched Topological phases
Prelude: low energy description of Symmetry Protected Topological (SPT) phases
It is known [1] that the low energy effective description of SPT phases, protected by a group $G$ is an invertible ...
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Understanding the zero current in a simple circuit
I've simulated the very simple circuit shown below:
As you can see, it has two batteries of 1.5 V, and three identical resistors. The electric currents passing through each wire are also indicated. I'...
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Is hydrogen the same everywhere?
Silly thought. Feel free to shoot it down
Does a hydrogen atom undergo any kind of change subject to it's environment?
If one were to study a hydrogen atom on the surface of Mercury, another above ...
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From Manifold to Manifold?
Tensor equations are supposed to stay invariant in form wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor ...
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Why there is a flat band for Kagome lattice?
For the nearest neighbor hopping model on the Kagome lattice, there is a flat band among the three energy bands.
Is there some reason, such as symmetry or the special structure of the model, to ...
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Particle number conservation equals $U(1)$-symmetry?
If have by now frequently read the above but never really understood it. It is said that the particle number conservations is related to the phase of the wave function, but how?
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Why $3$-dim isotropic harmonic oscillator's symmetry is not $O(6)$ but $U(3)$?
3D harmonic oscillator's Hamiltonian is
$$H=\sum_{i=1}^3p_i^2+q_i^2$$
Why all textbooks say that its symmetry is $U(3)$. But I think it's $O(6)$. Because the rotation of $6$ coordinates in phase ...
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Electric field on the surface of a charged sphere
We know that the electric field for a point charge is
$$
E = \frac{KQ}{R^2}.
$$
If $R$, i.e. distance from the electric field producer to the point where we want to find the electric field becomes ...
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What does it mean to be "gauging" a symmetry?
I read this and other similar questions, but they all address the problem of gauging a global symmetry (implying that one could also gauge a local one).
This confused me a lot: in my mind gauge and ...
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What is the invariant associated with the symmetry of boosts? [duplicate]
Noether's Theorem states that if a Lagrangian is symmetric for a certain transformation, this leads to an invariant: Symmetry of translation gives momentum conservation, Symmetry of time gives Energy ...
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Is a tensor which is symmetric in two indices still symmetric after raising/lowering one index?
I have had this question for a while. I have yet to find information on this online or use this property in any calculation. I believe myself to have proven that it will still be symmetric but I am ...
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In a fluid, why are the shear stresses $\tau_{xy}$ and $\tau_{yx}$ equal?
In solid bodies, $\tau_{xy}=\tau_{yx}$ makes sense to me because the volume elements "hold together" and can not spin against each other and therefore the resulting torque from the shear stresses has ...
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Why we call the ground state of Kitaev model a Spin Liquid?
Now we always talk about the so-called Kitaev spin liquid. One important property of spin liquid is global spin rotation symmetry. Let $\Psi$ represents a spin ground state, if $\Psi$ has global spin ...
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What is the minimal symmetry required for a spin Hamiltonian to describe a spin-liquid ground state?
Let's restrict to the case of spin-1/2 system. As we know, a spin-liquid (SL) state is the ground state of a lattice spin Hamiltonian with no spontaneous broken symmetries (sometime it may ...
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Noether's theorem for arbitrary conformal coordinate transformations
I have been reading Introduction to Conformal Field Theory by Blumenhagen and Plauschinn. Equation (2.19) on page 19 states that if our theory is invariant under a general conformal transformation $x^\...
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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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Anomaly is due to the noninvariance of the path-integral under a symmetry. Is the noninvariance reflected on 1PI effective action?
When a symmetry is anomalous, the path integral $Z=\int\mathcal{D}\phi e^{iS[\phi]}$ is not invariant under that group of symmetry transformations $G$. This is because though the classical action $S[\...
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