Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. ...
6
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2answers
155 views
+50
Coherent $U(N)$ intertwiners in Loop Quantum Gravity (LQG) and a measure on the Grassmannian
This is a detailed question about $U(N)$ intertwiners in LQG, and it comes from the the paper by Freidel and Livine (2011 - archive). It is very specific but related to finding a measure on a quotient ...
1
vote
1answer
105 views
Does anyone know the difference and relation between $k\cdot p$ method and tight binding (TB) method?
Among the methods of calculating energy bands for crystals, first-principles method is the most accurate. Besides first principles, two commonly used modeling methods are the $k\cdot p$ method and ...
3
votes
2answers
97 views
Do generators belong to the Lie group or the Lie algebra?
In Physics papers, would it be correct to say that when there is mention of generators, they really mean the generators of the Lie algebra rather than generators of the Lie group? For example I've ...
26
votes
7answers
2k views
Is there something similar to Noether's theorem for discrete symmetries?
Noether's theorem states that, for every continuous symmetry of a system, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any ...
5
votes
0answers
147 views
What is the difference between the properties of Electron spin and Photon polarization/helicity?
What is the difference between a photon's polarization/helicity and an electrons spin half? I know that the photon is spin 1 but isn't its polarization analogous to spin half?
This question stems ...
4
votes
2answers
220 views
Definition of Casimir operator and its properties
I'm not sure which is the exact definition of a Casimir operator.
In some texts it is defined as the product of generators of the form:
$$X^2=\sum X_iX^i$$
But in other parts it is defined as an ...
3
votes
1answer
87 views
Different representations of the Lorentz algebra
I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
13
votes
3answers
338 views
Homotopy $\pi_4(SU(2))=\mathbb{Z}_2$
Recently I read a paper using $$\pi_4(SU(2))=\mathbb{Z}_2.$$ Do you have any visualization or explanation of this result?
More generally, how do physicists understand or calculate high dimension ...
1
vote
1answer
77 views
Spin(n) group SO(n) relation
Is it correct to state that the elements of Spin(n) fulfill a Clifford algebra and that the Lie group generators of Spin(n) is given by the commutator of the elements?
If not, then what is the ...
6
votes
1answer
158 views
A Game Of The Number Of Space-Time Dimensions
Holger Bech Nielsen, one of the founders of string theory, has apparently just played some sort of game between different potential dimensions for space-time and reached the conclusion that D4 wins in ...
4
votes
0answers
190 views
Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry
Background: Classical Mechanics is based on the Poincare-Cartan two-form
$$\omega_2=dx\wedge dp$$
where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, ...
7
votes
2answers
658 views
Modern and complete references for the $k\cdot p$ method?
I've recently started studying the $k\cdot p$ method for describing electronic bandstructures near the centre of the Brillouin zone and I've been finding it hard to find any pedagogical references on ...
5
votes
1answer
99 views
Vector and Spinor Representation in Ramond-Neveu-Schwarz Superstring Theory
I am learning Ramnond-Neveu-Schwarz Superstring theory (RNS theory). I often find the following notation, especially in the closed string spectrum etc.:
$$\mathbf{8}_s,\mathbf{8}_v $$
And it is ...
5
votes
0answers
122 views
Decomposing a Tensor Product of $SU(3)$ Representations in Irreps
Can somebody explain in a simple way why, talking about representations, $3\otimes3=3\oplus6$, $3\otimes\bar{3}=1\oplus8$ and $3\otimes3\otimes3=1\oplus8\oplus8\oplus10$?
Here $3$ and $\bar{3}$ are ...
21
votes
4answers
2k views
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 ...
2
votes
3answers
118 views
How to judge whether a symmetry will be spontaneously broken while only given a Hamiltonian preserving this symmety
As asked in the title, is Hamiltonian containing enough information to judge the existence of spontaneously symmetry breaking?
Any examples?
20
votes
3answers
782 views
Lie theory, Representations and particle physics
This is a question that has been posted at many different forums, I thought maybe someone here would have a better or more conceptual answer than I have seen before:
Why do physicists care about ...
4
votes
0answers
75 views
Is the search for a Simple-group-based Electro-Weak theory over?
Just wondering:
We know that, in its current form of the $SU(2)_L\times U(1)$, the electroweak theroy rides a wave of huge success. However, is it not possible that the correct simple group ...
2
votes
2answers
133 views
Two ways to form SU(2) singlets?
I am trying to reconcile the two ways of forming SU(2) singlets out of a pair of doublets.
Method (1):
If $v=\begin{pmatrix}v^1\\ v^2\end{pmatrix}$ and $w=\begin{pmatrix}w^1\\ w^2\end{pmatrix}$ are ...
4
votes
3answers
373 views
Must all symmetries have consequences?
Must all symmetries have consequences?
We know that transnational invariance, for example, leads to momentum conservation, etc, cf. Noether's Theorem.
Is it possible for a theory or a model to have ...
6
votes
1answer
103 views
What is the meaning of non-compactness in the context of $U(1)$ in gauge theories?
In John Preskill's review of monopoles he states
Nowadays, we have another way of understanding why electric charge is
quantized. Charge is quantized if the electromagnetic U(l)em gauge
group ...
3
votes
1answer
146 views
An odd relation with the epsilon/delta invariant tensors of SO(3)
The rotation group SO(3) can be viewed as the group that preserves our old friends the delta tensor $\delta^{ab}$ and $\epsilon^{abc}$ (the totally antisymmetric tensor). In equations, this says:
...
6
votes
2answers
158 views
Is this a simple Lie algebra?
This question comes from Georgi, Lie Alegbras in Particle Physics. Consider the algebra generated by $\sigma_a\otimes1$ and $\sigma_a\otimes \eta_1$ where $\sigma_a$ and $\eta_1$ are Pauli matrices ...
4
votes
1answer
35 views
Convexity — reference request
I've been reading a few papers on generalized probabilistic theories, and have been struggling through proofs of some results that involve use of convexity and group theory, e.g. this paper on bit ...
15
votes
4answers
489 views
Elegant approaches to quantum field theory
I have been reading Quantum Mechanics: A Modern Development by L. Ballentine. I like the way everything is deduced starting from symmetry principles. I was wondering if anyone familiar with the book ...
6
votes
1answer
66 views
what compactifications of the Poincare group have been studied?
as we know the Poincare group is non-compact. Poincare invariance have been observed in velocities and energies up to $10^{20}$ eV in cosmic rays. The other day i was thinking in how $SU(2)$ ...
6
votes
3answers
833 views
How is it that angular velocities are vectors, while rotations aren't?
Does anyone have an intuitive explanation of why this is the case?
3
votes
1answer
57 views
Isometry group from information about the center of the group
I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
3
votes
0answers
61 views
Finding symmetry of a part of an equation, given the group transformation property of another part
I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
19
votes
1answer
66 views
Any use for $F_4$ in hep-th?
In high energy physics, the use of the classical Lie groups are common place, and in the Grand Unification the use of $E_{6,7,8}$ is also common place.
In string theory $G_2$ is sometimes utilized, ...
5
votes
3answers
245 views
Could general relativity and gauge theories in principle be covered in one course?
It's always nice to point out the structural similarieties between (semi-)Riemannian geometry and gauge field theories alla Classical yang Mills theories. Nevertheless, I feel the relation between the ...
7
votes
1answer
34 views
Are lens spaces classified via a Weinberg angle?
I am thinking about Kaluza Klein theory in the 3 dimensional lens spaces. These have an isometry group SU(2)xU(1), generically, and in some way interpolate between the extreme cases of manifolds $S^2 ...
20
votes
5answers
82 views
Which symmetric pure qudit states can be reached within local operations?
There are two pure symmetric states $|\psi\rangle$ and $|\phi\rangle$ of $n$ qudits. Is there any known set of invariants $\{I_i:i\in\{1,\ldots,k\}\}$ which is equal for both states iff ...
8
votes
2answers
131 views
Wilson Loops in Chern-Simons theory with non-compact gauge groups
VEVs of Wilson loops in Chern-Simons theory with compact gauge groups give us colored Jones, HOMFLY and Kauffman polynomials. I have not seen the computation for Wilson loops in Chern-Simons theory ...
3
votes
2answers
210 views
Why does $\mathcal L = -\frac14 F^{\mu\nu} F_{\mu\nu}$ imply Photons are massless?
The Lagrangian $\mathcal L = -\frac14 F^{\mu\nu} F_{\mu\nu}$ with $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ results in the four-potential's equation of motion
$$ \underbrace{\partial^\mu ...
1
vote
0answers
69 views
Deriving term symbols from electron configuration using Young tableaux
Can somebody explain me how to derive all term symbols using Young tableaux? Our lecturer showed us but I couldn't quite understand it without any background on group theory. I have some vague ...
16
votes
2answers
88 views
Can symmetry generators be used for quantization?
Take the Poincaré group for example. The conservation of rest-mass $m_0$ is generated by the invariance with respect to $p^2 = -\partial_\mu\partial^\mu$. Now if one simply claims
The state where ...
1
vote
1answer
114 views
Action of the Lorentz group on scalar fields
The Lorentz groups act on the scalar fields as:
$\phi'(x)=\phi(\Lambda^{-1} x)$
The conditions for an action of a group on a set are that the identity does nothing and that
$(g_1g_2)s=g_1(g_2s)$. ...
7
votes
2answers
92 views
Group of symmetries of Lagrange's equations
Consider the following statements, for a classical system whose configuration space has dimension $d$:
Lagrange equations admit a smaller group of "symmetries" (coordinate change under which ...
8
votes
1answer
279 views
Representations of Lorentz Group
I'd be grateful if someone could check that my exposition here is correct, and then venture an answer to the question at the end!
$SO(3)$ has a fundamental representation (spin-1), and tensor product ...
4
votes
2answers
238 views
Number of Components of a Spinor
I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral ...
7
votes
2answers
184 views
How to model a symmetry using Lie Groups?
I have been reading lately about Lie groups, and although all books keep listing the groups, and talk about Lie algebras and all that, one thing I still don't know how is it made, and I guess it's the ...
14
votes
0answers
174 views
Orbits of maximally entangled mixed states
It is well known (Please, see for example Geometry of quantum states by Bengtsson and Życzkowski ) that the set of $N-$dimensional density matrices is stratified by the adjoint action of $U(N)$, where ...
14
votes
2answers
51 views
Counting complete sets of mutually unbiased bases composed of stabilizer states
Consider $N$ qubits. There are many complete sets of $2^N+1$ mutually unbiased bases formed exclusively of stabilizer states. How many?
Each complete set can be constructed as follows: partition the ...
1
vote
1answer
98 views
Commutation relation of $J^2$ and $R(\alpha,\beta,\gamma)$
If $R(\alpha,\beta,\gamma)$ is the Rotation operator and $\alpha,\beta,\gamma$ are Euler angles and $J$ is the total angular momentum then how to get to this:
$$[J^2,R]~=~0?$$
This is stated in ...
4
votes
4answers
867 views
How to prove that proper orthochronous Lorentz transformations form a group?
Proper orthochronous Loentz transform are Lorentz transforms that satisfy the conditions (sign convention of Minkowskian metric $+---$)
$$\det \Lambda=+1, \qquad \Lambda^0{}_0 \geq +1.$$
How to prove ...
1
vote
2answers
164 views
high spin atoms SU(2) representation
I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry.
Why not $SU(N)$?
5
votes
2answers
473 views
Is this a quaternion Lorentz Boost?
The quaternion Lorentz boost $v'=hvh^*+ 1/2( (hhv)^*-(h^*h^*v)^*)$ where $h$ is $(\cosh(x),\sinh(x),0,0)$ was derived by substituting the hyperbolic sine and cosine for the sine and cosine in the ...
1
vote
0answers
46 views
Wigner $3j$ symbols
I am trying to determine the expansion that requires using $3j$ symbols; however, I am running into some conceptual snags. First, the expansion produces symbols that have m's that do not agree with ...
1
vote
2answers
177 views
What does “the N of a group” mean?
In the context of group theory (in my case, applications to physics), I frequently come across the phrase "the N" of a group, for example "a 24 of SU(5)" or "the 1" (the integer is usually typeset in ...
