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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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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0answers
53 views

Any importance of $i$ before the time reversal operator for spin-1/2 system?

I've read about that: For systems with spin 1/2, time-reversal symmetry has the operator $\mathcal{T}=i\sigma_y K$. I wonder if the imaginary unit $i$ has any importance. Without $i$, ...
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1answer
60 views

Does invariance under infinite small transformation imply invariance to the finite one?

Let's say that I have finite chiral transform and I would like to show invariance of Dirac's Lagrangian when $m=0$ under it. The chiral transform is defined as: $$\psi(x) \rightarrow \psi'(x) =e^{i ...
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2answers
734 views

How are anyons possible?

If $|ψ\rangle$ is the state of a system of two indistinguishable particles, then we have an exchange operator $P$ which switches the states of the two particles. Since the two particles are ...
5
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1answer
345 views

Why is a hexagon such a stable shape for materials?

A hexagonal lattice is famously the shape of graphene, the source of the 2010 Nobel prize. The shape also shows up in beehives and in the basalt columns of Giant's Causeway in County Antrim. ...
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358 views

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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1answer
56 views

Conserved current in a complex relativistic scalar field

For my field theory class I have the following Lagrangian density $$\mathscr{L}=\frac{1}{2}\eta^{\mu\nu}\partial_\mu\phi^*\partial_\nu\phi-\frac{1}{2}m^2\phi^*\phi$$ Where $\eta^{\mu\nu}$ is the ...
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1answer
67 views

Fine Structure Correction

The fine structure correction is composed of the relativistic correction and spin-orbit coupling. The lowest-order relativistic correction to the Hamiltonian is $$ H_r' = -\frac{p^4}{8m^3c^2}$$ ...
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2answers
288 views

Is internal symmetry the same as gauge symmetry?

This is more a terminology question. I have seen that some people differentiate between the two types of symmetry: internal symmetry and gauge symmetry (of a field theory). Is there a difference (in ...
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1answer
92 views

Given potentials, how does one find conserved quantities using Noether's theorem?

I've been asked to find the conserved quantities of the following 3D potentials: $U(\vec{r}) = U(x^2)$, $U(\vec{r}) = U(x^2 + y^2)$ and $U(\vec{r}) = U(x^2 + y^2 + z^2)$. For the first one, ...
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1answer
131 views

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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1answer
125 views

Why Levi-Civita term signal the breaking of parity and time reversal?

For example, referring to Zee's QFT book, in Chern-Simons matter theory, after writing a term $$\gamma {\varepsilon ^{\mu \nu \lambda }}{a_\mu }{\partial _\nu}{a_\lambda }$$ he said The ...
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1answer
148 views

Does a symmetry necessarily leave the action invariant?

A symmetry maps a configuration with stationary action to another configuration with stationary action. However, does it necessarily preserve the value of the action exactly? It seems that it should ...
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1answer
136 views

Does an on-shell symmetry necessarily change the Lagrangian by a total derivative?

This is a follow-up question to: Does a symmetry necessarily leave the action invariant? Qmechanic writes here: Here the word off-shell means that the Lagrangian eqs. of motion are not assumed to ...
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1answer
58 views

How to show OPE coefficients are symmetric in three indices ?

May it is very trivial, but I am stuck here, given (I have suppressed the conjugate coordinates) $$ \phi_i(x) \phi_j(y) \sim \sum_{k} c_{ijk} (x-y)^{h_k - h_i - h_j} \phi_k(y) $$ $$ \langle ...
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1answer
133 views

Infinitesimal transformations and Poisson brackets [duplicate]

I want to understand how bracket operations in general are related to symmetry and infinitesimal transformations (in hindsight of quantumfieldtheory), so I calculated an example with a particle that ...
3
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1answer
77 views

How to quantify translational symmetry?

I'm trying to study phase transitions and I'm trying to find a way to classify regions of space based on their "crystallinity". I'm working with 3D coordinates, but I'll present the problem in 2D ...
6
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1answer
484 views

Noether's Theorem: Lie algebra, Lie groups

I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard ...
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1answer
531 views

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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0answers
83 views

Conserved charge of a conformal transformation

From Becker, Becker and Schwarz String Theory and M-Theory: For the infinitesimal conformal transformation $$\tag{3.25}\delta z=\varepsilon(z)\quad\text{and}\quad \delta\bar ...
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0answers
128 views

How can gauge invariance be unphysical?

Gauge symmetry is said to be "unphysical" because the transformations - unlike changes of reference frame - do not correspond to real physical operations. But the consequences of gauge symmetries are ...
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1answer
87 views

Silicon: conduction band minima

Why do the energetic minima of the silicon conduction band lie not in a high-symmetry point like a $X$-point, but somewhere in $\Delta$-direction between points $\Gamma$ and $X$? What is the physical ...
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1answer
125 views

Schrodinger equation, commutative operators, and Symmetry

When solving Schrodinger's equation in 3D with a spherical laplacian you reach a point at which you introduce a separation constant and can see that the same eigenvalue satisfies the radial and ...
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3answers
367 views

Ideal, isotropic fluid and stress tensor

An ideal fluid is the one which cannot support any shearing stress. It also doesn't have viscosity. My question is what does it mean by a fluid to be isotropic? Is an ideal fluid necessarily isotropic ...
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3answers
508 views

How to understand this symmetry in the wavefunctions of a diatomic molecule?

In Wikipedia (and elsewhere), a particular symmetry of the quantum system of a diatomic molecule is mentioned: symmetry under reflection along a plane containing the internuclear axis. The ...
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0answers
205 views

What is the symmetry group of this Hamiltonian?

Consider a Hamiltonian $$\hat H=-\partial_x^2-\partial_y^2+(x-y)Q,$$ where $x,y\in[0,a]$ (homogeneous Dirichlet boundary conditions assumed), and $Q$ is some real parameter. When $Q=0$, the ...
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1answer
86 views

Relation between gauge symmetry and mass difference

Usually (like in Georgi's Lie Algebra book) people argue the reason why Gellmann $SU(3)$ flavor symmetry (u,d,s) can't extend to $SU(4)$ (u,d,c,s) or higher flavour symmetry is the their mass ...
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0answers
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What is the difference between the groups $PSU(N)$ and $SU(N)$? [closed]

What is the difference between the groups $PSU(N)$ and $SU(N)$? For example how is $PSU(2,2|4)$ different than $SU(2,2|4)$?
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7answers
931 views

What does the statement “the laws of physics are invariant” mean?

In the first paragraph of Wikipedia's article on special relativity, it states one of the assumptions of special relativity is the laws of physics are invariant (i.e., identical) in all inertial ...
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0answers
138 views

Does point group symmetry also act within “spin space” for a lattice spin system?

As an example, let's consider a quantum spin system on a 2D square lattice. The lattice point group symmetries include $C_4$ rotation, parities, etc.... And let's take $C_2$ symmetry (2-fold rotation) ...
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3answers
155 views

$SO(3)$, $SU(2)$ and symmetries in quantum mechanics [duplicate]

A rotation in the vector space $\mathbb{R}^3$ is represented by the known 3x3-matrices. But at this point I'm really confused how to get from there to Quantum Mechanics. The group of ...
3
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2answers
213 views

Changing vector basis in AdS$_3$

I have AdS${}_3$ given as a surface embedded in a 4 dimensional pseudo-Riemannian space $$x^2+y^2-u^2-y^2=-l^2$$ With metric: $$ds^2=dx^2+dy^2-du^2-dv^2$$ I have Killing vectors of that space ...
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2answers
474 views

What are spin and valley symmetries in graphene?

I have been assigned a presentation on a part of a paper (http://arxiv.org/abs/1303.6942). My task is to present on the spin and valley symmetries in graphene, and relate it back to the paper above. ...
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2answers
120 views

Definition of Duality (opposed to Symmetry)

I'm learning basic string theory right now and we came across T-duality which was presented as a symmetry of the formula for the mass of a string in the context of compactification. There was a remark ...
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1answer
307 views

What exactly do we mean by symmetry in physics?

I'm referring here to invariance of the Lagrangian under Lorentz transformations. There are two possibilities: Physics does not depend on the way we describe it (passive symmetry). We can choose ...
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2answers
151 views

Global symmetry and particle multiplets

In chapter 20, of Peskin and Schroeder's quantum field theory book, they start with a comment that a global symmetry that is manifest lead to particle multiplets with restricted interactions. Can ...
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265 views

Is there a mathematical reason for the Lagrangian to be Lorentz invariant?

The Hamiltonian is the energy, which is just one component of a four-vector and therefore not Lorentz invariant. The Lagrangian is the Legendre transform of the Hamiltonian and I was wondering if ...
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2answers
127 views

How can I prove that $\langle\Omega\vert \phi(x) \vert\Omega\rangle \langle\Omega\vert\phi(y)\vert\Omega\rangle=0$ for a scalar field?

From Peskin-Schroeder, p.212: The term $$ \langle \Omega | \phi(x) | \Omega \rangle \langle \Omega |\phi(y) | \Omega \rangle$$ is usually zero by symmetry; for higher-spin fields, it is zero ...
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Symmetrizing the Canonical Energy-Momentum Tensor

The Canonical energy momentum tensor is given by $$T_{\mu\nu} = \frac{\partial {\cal L}}{\partial (\partial^\mu \phi_s)} \partial_\nu \phi_s - g_{\mu\nu} {\cal L} $$ A priori, there is no reason to ...
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1answer
54 views

What conserved quantities does a one-dimensional non-symmetric lattice have?

When I asked what leads to degeneracy of eigenstates of free particle, the answer was parity. But it appears that even if we consider a lattice with non-symmetric cell, so the potential looks as shown ...
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1answer
241 views

Why does renormalization need an unbroken symmetry?

Common wisdom is that for a QFT to be renormalizable it must be invariant under a symmetry transformation. Why does renormalization need an unbroken symmetry? Which is the first publication that ...
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1answer
110 views

How to diagonalise the Lagrangian mass term with SU(4) symmetry and self-dual tensors

I should write the mass term of the Lagrangian with global SO(4) symmetry in tensor representation with anti-symmetric tensors and then diagonalise this term with defining a new set of tensors ...
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1answer
44 views

Finding a basis for minimal representation of a wavefunction (extracting symmetries)

I asked something like this on Math StackExchange, but now that I think about it, this probably belongs better over here. I want to find all linear operators (non necessarily hermitian) $\{\hat{A}\}$ ...
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3answers
172 views

Do algorithms have an intrinsic time direction?

This article says There is no intrinsic time direction in Newton's mechanics nor in the differential equations of the new physics. My question is, do other types of mathematics, say a cellular ...
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1answer
320 views

Why do we need spontaneous symmetry breaking in Lagrangian formalism?

I have always struggled with the concept of spontaneous symmetry breaking. It seems to me that many others don't find it very intuitive as well, but that could be just me having difficulties with the ...
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1answer
206 views

Why are large scale structures isotropic in the Ising model?

I have at least a qualitative understanding of why the critical state of the Ising model is scale invariant, by arguments to do with renormalisation, which I understand only very roughly. However, in ...
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1answer
474 views

Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
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Explain materials with 4 fold symmetry having same reflectance when shone with LCP and RCP

This is my first post here. I am currently reading "Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation" by Do-Hoon Kwon, Pingjuan L. Werner, and ...
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1answer
90 views

Question about global internal $SO(n)$ symmetry

I have the following Lagrangian (density) for bosons $$L = \partial_{\mu} \phi^i \partial^{\mu}\phi^i+ m^2\phi^i \phi^i$$ and I am trying to understand why this Lagrangian is invariant under ...
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1answer
144 views

Is the weak interaction Lagrangian invariant under parity transformations?

The weak interaction term in the Lagrangian reads $$ \bar \Psi \gamma_\mu P_L \Psi W^\mu. $$ Under parity transformations, because of $\Psi \rightarrow \gamma_0 \Psi$ and $\gamma_5 \rightarrow ...