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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$T$-invariant Hamiltonians

If $T$ is time-reversal transformation $t\mapsto -t$, Why do $T$-invariant Bloch Hamiltonians obey $$H(-k) = T H(k) T^{-1}$$ and not $$H(k) = T H(k) T^{-1}$$ Somehow I understand the word "invariant" ...
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613 views

Traceless of stress-energy tensor in $d=2$

This is a question regarding Francesco, section 4.3.3. In this section, he considers the two-point function $$ S_{\mu\nu\rho\sigma}(x) = \left< T_{\mu\nu}(x) T_{\rho\sigma}(0)\right> $$ He then ...
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74 views

$B$ field around an infinite wire symmetry argument

Consider the infinity current carrying wire wire drawn below along with the three possible components of the magnetic field at a point a distance $r$ from the centre. I know how to use symmetry to ...
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3answers
136 views

Principle of Sufficient Reason on light travelling in straight line

I was reading a book Laws and Symmetry by Bas C. Van Fraassen I found that there is an argument for arguing that light travel in straight line: Leibniz's reconstruction of these arguments goes ...
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37 views

Non-abelian current transformation

This is my firs post ever here and I just registered on this site but I want to say that this site has helped me a lot and you guys are great! On to the question: I have the equation of motion that ...
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51 views

Why performing axial symmetry, results in the same masses for pion and sigma mesons?

Under axial transformations, $\sigma$ and $\pi$ are rotated into each other: $\vec{\pi} \rightarrow \vec{\pi}+ \vec{\theta} \sigma $, $\sigma \rightarrow \sigma+ \vec{\theta}.\vec{\pi} $. In ...
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92 views

General construction of equations of motion for free particles

I've got a question regarding the different Symmetrie-Lie-Groups of Newtonian Mechanics and special realtivity. Is there a canonical way to obtain the equations of motion for a free particle only by ...
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102 views

Possible mechanics based on the known symmetries in the nature (investigating rumor)

Somewhere I've heard about a relative new mathematical result regarding mechanics. Specifically, there is a list of the known symmetries of mechanics (both Newtonian and relativistic), i.e. different ...
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198 views

Does the projected spin state of the $d+id$ mean-field Hamiltonian on a triangular lattice has time-reversal(TR) symmetry?

Consider the following $d+id$ mean-field Hamiltonian for a spin-1/2 model on a triangular lattice $$H=\sum_{<ij>}(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$$, with $\chi_{ij}=\begin{pmatrix} 0 & ...
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37 views

How can we determine the Hypercharges in a GUT like SO(10)?

I understand how the assignment works for a symmetry breaking like $$SO(10) \rightarrow SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_X$$ The Hypercharge can then easily computed by $$ ...
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62 views

Is the Singlet state for Helium with 2 electrons symmetric rather than anti-symmetric as is meant to be for fermions?

I'm looking at two-electron Helium atoms where one electron is in the ground state (due to if it were in other states, it's de-excitation would simply lead to the ionization of the electron). The ...
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340 views

Quantum symmetries that are not classical symmetries

An anomaly is a symmetry of the classical action that fails to be a symmetry of the path integral, due to non-invariance of the path integral measure. Does it ever occur that the opposite thing ...
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Why should Ward identities only be used with the effective action (as opposed to the generating functional for connected diagrams)?

My question is about the derivation of Ward identities. I will sketch it here in the case of an O(N) symmetric model and point out what it bothering me when I am done. I am being very sloppy with the ...
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2answers
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Can conservation of momentum be violated?

The law of the conservation of momentum has been established for hundred of years. Even in Quantum field theory every particle collision must be momentum-conserving if there is homogenity in space. ...
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Why do the $u$ and $d$ quark not have an associated quantum number?

All the other quarks ($c$,$s$,$b$ and $t$) have quantum numbers of charmness, strangeness, bottomness and topness that are conserved in strong interactions. This allows, among other things, flavour ...
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106 views

Why is $p_\phi$ conserved in a Schwarzschild orbit?

This arises from the question What is the relationship between $a$ and $m$, which I'm afraid I answered just by looking it up in Schutz's book. However Schutz (as he frequently does) glosses over ...
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536 views

Precise statement of Mermin–Wagner theorem

Roughly speaking, Mermin-Wagner theorem states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions ...
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Why aren't orbitals symmetric?

In an hydrogen-like atoms the orbitals are solutions to the Schrodinger equation suitable for the problem. They describe the regions where an electron can be found. So, why don't they have spherical ...
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What is the usefulness of the Wigner-Eckart theorem?

I am doing some self-study in between undergrad and grad school and I came across the beastly Wigner-Eckart theorem in Sakurai's Modern Quantum Mechanics. I was wondering if someone could tell me why ...
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151 views

Questions on the elementary excitations in the resonating-valence-bond(RVB) states?

It is known that the RVB states can support spin-charge separations and its elementary excitations are spinons and holons. But it seems that there are some different possibilities for the nature of ...
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45 views

Name for the transformation into an accelerated frame?

A transformation into a frame that looks at an experiment from a rotated perspective is called a rotation. A transformation into a frame that moves with a different constant velocity is called a ...
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127 views

Why is the electric field inside a solenoid tangential?

I have been looking at some derivations for the electric field inside a solenoid. I know how to find it, but I don't get the symmetry argument used. This is often of the form: Since if we choose ...
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76 views

Antimatter universe and Noether's theorem

I am studying Feynman's "symmetry in physical laws", where he talks about conservation laws for corresponding symmetries. (I know this is Noether's theorem, I am studying this from David Tong's ...
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62 views

Landau theory of phase transation

In his article http://www.ujp.bitp.kiev.ua/files/journals/53/si/53SI08p.pdf, Landau defines probability distribution $\rho$ which is related to symmetry of crystal. If crystal has certain symmetry ...
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186 views

Spontaneous Symmetry Breaking - struggling with physics based understanding?

Although I am a mathematician by nature, I'm writing an essay in my third year of my undergraduate on Spontaneous Symmetry Breaking in Physics, and as such I've become a little confused by how the ...
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Two puzzles on the Projective Symmetry Group(PSG)?

Recently I'm studying PSG and I felt very puzzled about two statements appeared in Wen's paper. To present the questions clearly, imagine that we use the Shwinger-fermion ...
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245 views

What is the symmetry associated with the local particle number conservation law for fluid?

According to Noether's theorem, every continuous symmetry (of the action) yields a conservation law. In fluid, there is a local particle number conservation law, which is ...
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138 views

Symmetry argument for a toroid?

When using Ampere's law for a toroid (in the toroid and around a circular path) please can someone explain the symmetry argument (or an alternative argument) which allows us to assume the field is ...
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51 views

How to define conserved charges in Euclidean field theory?

In a field theory with signature (1,d), conserved charges are obtained by integrating the time component of a conserved current over a spatial region. What are the corresponding equations and ...
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54 views

Parity transformation is proper orthochronous?

In 3+1 dimensional spacetime the parity transformation is $$P^\mu_{\;\,\nu}=\begin{pmatrix}+1&&&\\&-1&&\\&&-1&\\&&&-1\end{pmatrix}.$$ This is ...
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1answer
47 views

Parameterization of an arbitrary element of $U(2)_L \times U(2)_R$ (Chiral symmetry with two quarks)

When you write down the Lagrangian for two quarks : \begin{equation} \mathcal{L}_\text{QCD}^0 = -\frac{1}{4} G_{\mu\nu}^a G^{a\mu\nu}+ \bar\Psi i \gamma^\mu D_\mu \Psi \end{equation} you find an ...
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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 ...
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117 views

How do states in Hilbert Space act like irreducible representations?

I am reading Georgi's book on group theory and I came across this sentence..." Hilbert space of any parity invariant system can be decomposed into states that behave like irreducible representations". ...
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70 views

Is there a sensible fully-discretized Hamilton's principle?

In computational physics it is common to formulate Hamilton's principle in a semi-discrete way, where space is continuous but time is discrete: in other words the Lagrangian $$L(q, \dot q, t): ...
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253 views

Symmetries in QM and QFT — operator transformation laws

In quantum mechanics, we implement transformations by operators $U$ that map the state $|\psi\rangle$ to the state $U|\psi\rangle$. Alternatively, we could transfer the action of $U$ onto our ...
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Why is the Symmetry Group for the Electroweak force $SU(2) \times U(1)$ and not $U(2)$?

Let me first say that I'm a layman who's trying to understand group theory and gauge theory, so excuse me if my question doesn't make sense. Before symmetry breaking, the Electroweak force has 4 ...
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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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47 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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58 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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Intuition for S-duality

first of all, I need to confess my ignorance with respect to any physics since I'm a mathematician. I'm interested in the physical intuition of the Langlands program, therefore I need to understand ...
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667 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 ...
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1answer
287 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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355 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
51 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
66 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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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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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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Does the similarity of gamma matrices correspond to a conserved quantity?

Gamma matrices have a similarity property, $\gamma^\mu\to S\gamma^\mu S^{-1}$ is a good transformation. Does this transformation correspond to a symmetry of the QED Lagrangian?
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normal degeneracy and the “span” of an irreducible representation

In Tinkham's "Group Theory and Quantum Mechanics", Tinkham defines normal degeneracy so that the span of the action of the Hamiltonian's symmetry group on any energy eigenstate yields all possible ...
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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 ...