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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Why $SU(N)$ and not $U(N)$?
I am going through one example where they introduce the lagrangian density
\begin{equation}
L(x) = \sum_{i = 1}^N \partial^\mu \phi_i^* \partial_\mu \phi_i - m^2 \phi_i^* \phi_i = \partial^\mu \Phi^\...
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Does rotational symmetry imply reflection symmetry for electrostatic interactions?
Consider two charge distributions $\rho_A(\mathbf{x})$ and $\rho_B(\mathbf{x})$. Suppose that the ground state energy of a system of $n$ electrons in a potential generated by the sum of these two ...
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The different choises of a metric with the next form: $ds^{2}=-Adt^2+Bdr^2+Cd\Omega^2$
In some papers the starting point is a metric with the next form:
$$ds^{2}=-A(t,r)^{2}dt^{2}+B(t,r)^{2}dr^{2}+r^{2}d\Omega^{2}.$$
Like in the Schwarzschild metric where they choice A=exp and B=exp ...
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Symmetry Arguments in Radiation Corrections: Gauge vs Custodial and Accidental Symmetries
I'm confused about the symmetry arguments in the radiation correction.
For photon mass, gauge symmetry protects it's mass term. I understand this as a consequence of Ward's identity.
$$\langle\partial^...
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Time dependence of momentum operator in QFT
In QFT in flat spacetime, we want to have a representation of the Poincare algebra on the Hilbert space $\mathscr H$ of states. This is due to the Wightman axioms. So there should exist operators $M_{\...
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Validity of Bertrand's theorem for a self-interacting system
We know that in classical mechanics, a particle of mass $m$ orbiting in a given central-force potential $V$ will satisfy the following eom:
$$\frac{d(m\dot{r})}{dt}-mr\dot{\theta}^2+\frac{\partial V}{\...
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Physical significance of a constant of motion
In problem 21 of Chapter 9 of Goldstein's "Classical Mechanics," 3rd edition, it is given that if the Hamiltonian $H(q, p, t) $ satisfies the scaling condition
$$H (q\lambda, p/\lambda, t\...
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How to demonstrate the form of the FLRW metric from homogeneity and isotropy?
In most introductions to the standard cosmological model, the metric of FLRW is stated to be corresponding to:
$$ds^2 = -c^{2}dt^2+a^{2}\left(t\right)\gamma_{ij}dx^{i}dx^{j}$$
which means (if I'm not ...
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Inversion Symmetry in Second Harmonic Generation (SHG)
I have been reading a paper which uses Second Harmonic Generation (SHG) to probe magnetic order in a magnetic material and as I come from more of a condensed matter background, I was hung up on ...
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Inversion asymmetric little representation in a Inversion symmetric large representation in brillouin zone
I am not am having trouble understanding the little representation which characterize high symmetry points in the brillouin zone of a crystal.
I understand this essentially mean that for some high ...
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Symmetry/Anti-symmetry property of Riemann curvature tensor
One can check that for a Riemann tensor $R_{abcd}$, using the standard definitions of the symmetry bracket "(...)" and the anti-symmetry bracket "[...]",
\begin{align}
R_{a(bcd)}=0 ...
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Which operators do non-invertible symmetries correspond to in quantum mechanics?
In quantum mechanics, symmetry transformations are usually assumed to correspond to unitary operators in the Hilbert space. However, recently there has been an increasing interest in "non-...
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Construct matrices with Altland–Zirnbauer (AZ) symmetry
I came across the 10-fold classification of random matrices: The Altland–Zirnbauer classes.
But having read it all how do I construct a matrix which will belong to a particular AZ class?
As a ...
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How to confirm that a QFT is a conformal field theory (CFT) at the quantum level?
My question is: Given a QFT, what's the usual/reliable/logical way to confirm that it's a CFT at the quantum level? Here are some explanations about why I ask this question.
I have learned a lot about ...
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Definitions of different types of symmetries
I'm a math student and I started studying physics last year. I'm sorry if this question has been asked before but I'm completely confused about it. In page 30 of the book "String theory and M-...
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Mirror symmetry in circuits [closed]
Consider this circuit below. The rectangular resistors have resistance 12 $\Omega$, and the circular water heaters have resistance 6 $\Omega$.
I am looking for a rigorous argument that explains why ...
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Spontaneous symmetry breaking IFF long range order in symmetric state?
I am interested in the 'tower-of-states' type continuous spontaneous symmetry breaking (SSB) in quantum systems. For concreteness, assume a $d$ dimensional lattice Hilbert space. That is, suppose a ...
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Fermions can be described by both symmetric and antisymmetric wavefunctions?
I stumbled upon a lecture note on MIT OpenCourseWare, as I was trying to read up on antisymmetric and symmetric wavefunctions of two particle Fermions and Bosons, respectively.
Interestingly, I came ...
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Deriving an explicit form of the boundary term in the symmetry variation of action
In many treatments of Noether theorem [e.g. Srednicki's book eq. (22.27), Banados' review eq. (2.65), David Tong's notes on QFT eq. (1.38), Peskin & Schroeder eq. (2.12), etc.] an almost explicit ...
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Can Lorentz invariance turn out to be an approximate symmetry in string theory?
I don't know whether I need to ask much but the essential question is at scales either approaching or at $M_{pl}$, can Lorentz invariance be broken? I am especially interested in the context of string ...
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Is Sakurai's Modern Quantum Mechanics explanation that the oscillation period between $|S\rangle$ and $|A\rangle$ is infinite wrong?
[Errors that persist in the 3rd edition of Sakurai's textbook?]
The content dealing with the symmetry double-well potential contains an error in the coefficient of $|A\rangle$ in $|R,\,t\rangle$, ...
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What is the displacement of E$_g$ mode of SrTiO$_3$ in tetragonal phase? [closed]
In the tetragonal phase, SrTiO$_3$ belongs to the D$_{4h}$ point group and possesses two Raman-active modes with the symmetry rep of A$_{1g}$ and E$_g$. However, in most books only 7 modes (normal ...
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Edge Modes and SPT Order
I am trying to understand the notion of edge modes in the context of symmetry protected topological order (SPTO) and its relation (if there is any) with the virtual quantum register that sits at the ...
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How do we know that only symmetrical and antisymmetrical states occur in nature?
I have a question concerning exactly how we get to the following two conclusions in quantum mechanics (which are both experimentally obtained as I understand it):
Identical particles are ...
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Does Birkhoff's theorem work just as well in the non-geometric approach to general relativity?
Does Birkhoff's theorem work just as well in the non-geometric approach to general relativity?
My guess is, that it should due to the equivalence with the geometric approach. On the other hand, ...
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The proof of conservation of momentum in Mechanics by Landau and Lifshitz
In reading the first chapter of Mechanics by Landau and Lifshitz, there is one point on which I consistently get stuck. This regards the proof in $\S 7$ that space homogeneity implies conservation ...
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Symmetry of Green's function inside a rectangular cavity with Perfect Electric Conductor (Boundary) and metallic scatterers within them
An elementary dipole source generates electromagnetic waves at a specific frequency inside a rectangular cavity whose sides are Perfect Electric Conductors. Few scatters are present inside the cavity, ...
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Why electric field inside a hollow sphere on points except centre is 0? [duplicate]
Why electric field inside a hollow sphere on points except centre is 0
I got the explation through the gauss law but I can mannually see at points except centre the field is non zero
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Angular momentum conservation and isotropy of space
If we consider a spin-less particle at some state $|\psi\rangle$ belonging to the state space $\mathcal{H}_r$ then the following holds true:
Invariance of the quantum mechanical system under rotation ...
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Confusion of the Proof of Goldstone Theorem in QFT
Context
I found many proof of the symmetry breaking is the following:
Let $\langle \delta \phi(x) \rangle$ be the corresponding symmetry transformation w.r.t. the charge $Q = \int d^Dx\; J^0(x)$.
$\...
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Confusion between trigonal and hexagonal systems
I'm studying space groups.
It's quite clear (I think) why trigonal and hexagonal systems collapse in the same primitive Bravais lattice, while are different when we introduce non-primitive unit cells, ...
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Spontaneous symmetry breaking versus degenerate excitations in gapless quantum systems
I am trying to understand the difference between ground-state degeneracy induced by spontaneous breaking of a symmetry in the thermodynamic limit (system size $N\rightarrow \infty$) for a gapless ...
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$(1/2,1/2)$ Representation transformation laws in Schwichtenberg has extra transpose
I'm reading Schwichtenberg's Physics from Symmetry. I have a question about the derivation of the $(1/2,1/2)$ representation of the lorentz group.
The issue is that Schwichtenberg adds an extra ...
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How are rotations invariant by time-reversal?
The title summarizes it. I'm confused as to how the anti-unitary operators work. If it inverts the signal of the angular momentum and conjugates the rotation operator, I can see why the rotation ...
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Do all magnetic monopole models always imply higher symmetry than regular EM?
(Isn't this question asked anywhere before)
First, Maxwell equations do not say that magnetic monopoles do not exist. The equations can easily be generalized to include magnetic monopoles.
What I want ...
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Which symmetries lead to the ladder operators of the harmonic oscillator?
It seems like symmetries usually lead to ladder operators. For example in a central potential problems the conservation of angular momentum leads to angular momentum ladder operators being used in the ...
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Moment of Inertia of cube about the axis along one of its diagonal [closed]
How can we find Moment of Inertia of a solid as well as a hollow cube about the axis passing from one of its diagonal?
I tried really hard solving it, please guide me to the solution.
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Symmetry group and dualities
Let's say two quantum models $M_1$ and $M_2$ are dual to each other and let their symmetry groups be $S_1$ and $S_2$ respectively. Is it necessary for $S_1$ to be isomorphic to $S_2$? (I thought so ...
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In equation (20) from lecture 10 in Leonard Susskind’s ‘Classical Mechanics’, why is there a summation involved?
Here is the equation
$$\{x_i,L_j\}=\sum\limits_{k}ϵ_{ijk} x_k.$$
Is this equation generalised for any number of dimensions? In which case, would the following example be correct assuming 4 dimensions?
...
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Why Consider Only Triplet States for Spin in $2$-Electron Systems?
I have a question regarding systems of 2 electrons and their spin properties. When the Hamiltonian of a system of 2 electrons can be written as a sum of two single-particle Hamiltonians that are ...
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Does quasi-symmetry preserve the solution of the equation of motion?
In some field theory textbooks, such as the CFT Yellow Book (P40), the authors claim that a theory has a certain symmetry, which means that the action of the theory does not change under the symmetry ...
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Two contradictory derivations of Killing equation
In David Tongs lecture notes he derives the Killing equation by showing that the charge $Q=\xi_\mu \frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}$ is conserved
$$ 0=\frac{\mathrm{d}Q}{\mathrm{d}\tau}=\frac{\...
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Parity transformation of the $\pi^{0}\rightarrow\gamma\gamma$ process
I want to prove that the amplitude
$$\mathcal{M}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}q_{1\alpha}q_{2\beta}$$
is violating parity. Here $q_{i=1,2}$ are the external momenta of the photons. The total ...
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Potential of Monolayer Graphene as a High-Precision Cutting Material
"I am exploring the use of monolayer graphene as a cutting material for high-precision applications. We know that graphene has exceptional mechanical properties, such as high strength and ...
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How does inserting an operator in the path integral change the equation of motion?
I am reading this review paper "Introduction to Generalized Global Symmetries in QFT and Particle Physics". In equation (2.43)-(2.47), the paper tried to prove that when
$$U_g(\Sigma_2)=\exp\...
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Using particle-hole symmetry of the Hubbard model to study the model at different densities
In Condensed Matter Field Theory by Altland and Simons, they state that the Hubbard Hamiltonian
$$
H = \sum_{\text{nearest neighbors } ij \text{ and spin } \sigma} a^\dagger_{i\sigma} a_{j\sigma} + U \...
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Is the FRW metric, based on spatial homogeneity and isotropy, rotationally and translationally invariant? If so, how?
The spatial part of the Minkowski metric, written in the Cartesian coordinates, $$d\vec{ x}^2=dx^2+dy^2+dz^2,$$ is invariant under spatial translations: $\vec{x}\to \vec{x}+\vec{a}$, where $\vec{a}$ ...
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Designing a thought experiment on Noether's Theorem [closed]
By Noether's theorem, in classical physics, conservation of total momentum of a system is result of invariance of physical evolution by translation.
So logic says "if" there exists closed ...
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Does all symmetry breaking have corresponding unitary group?
In high energy physics. Symmetry breaking like electroweak's has corresponding $SU(2)\times U(1)$ unitary gauge group broken down to $U(1)$. Does it mean all kinds of symmetry breaking (even low ...
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Probabilistic behavior of quantum mechanics [closed]
In a hypothetical scenario, if I were to measure the quantum spin of an electron and it showed "up," and then I traveled back in time without changing the initial conditions, would measuring ...
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