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. ...
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 ...
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 ...
20
votes
5answers
84 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 ...
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 ...
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, ...
17
votes
4answers
1k views
Could the Periodic Table have been done using group theory?
These three questions are phrased as alternative-history questions, but my real intent is to understand better how well different modeling approaches fit the phenomena they are used to describe; see 1 ...
16
votes
8answers
2k views
Comprehensive book on group theory for physicists?
I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that ...
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 ...
15
votes
4answers
493 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 ...
14
votes
2answers
52 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 ...
14
votes
0answers
175 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 ...
13
votes
3answers
340 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 ...
10
votes
2answers
517 views
Is the G2 Lie algebra useful for anything?
Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional E8 comes up in string theory. But G2? I've always wondered about that one. ...
8
votes
2answers
361 views
Lie bracket for Lie algebra of $SO(n,m)$
How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by
$$[J_{ab},J_{cd}]
~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$
...
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 ...
8
votes
5answers
671 views
Simple applications of group theory which can be understood by a senior undergrad
I am looking for references (books or web links) which have "simple" examples on the use of group theory in physics or science in general.
I have looked at many books on the subject unfortunately ...
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 ...
8
votes
1answer
332 views
Schwinger representation of operators for n-particle 2-mode symmetric states
A bosonic (i.e. permutation-symmetric) state of $n$ particles in $2$ modes can be written as a homogenous polynomial in the creation operators, that is
$$\left(c_0 \hat{a}^{\dagger n} + c_1 ...
7
votes
2answers
659 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 ...
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 ...
7
votes
2answers
93 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 ...
7
votes
2answers
187 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 ...
7
votes
1answer
334 views
Is this a quaternion representation of the equations of motion of General Relativity?
In The Quaternion Group and Modern Physics by P.R. Girard, the quaternion form of the general relativistic equation of motion is derived from
$du'/ds = (d a / d s ) u {a_c}^* + a u ( d {a_c}^* / ...
6
votes
2answers
163 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 ...
6
votes
3answers
839 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?
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
2answers
297 views
Do Lorentz Boosts in the same direction form a group?
I know that two consecutive Lorentz Boosts in different directions produce a rotation and therefore Lorentz Boosts don't form a group. But, my intuition tells me that, Lorentz Boosts in the same ...
6
votes
1answer
136 views
Group transformations on $H_2O$
In my readings of Mirman (1995), "Group Theory: An Intuitive Approach", on p.35 he asks me to consider a so-called "water group" that has 4 transformations. I'll list them for completeness, but I'm ...
6
votes
2answers
177 views
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 ...
6
votes
1answer
109 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 ...
6
votes
1answer
160 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 ...
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 ...
5
votes
2answers
474 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 ...
5
votes
1answer
106 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 $$
And it is ...
5
votes
1answer
268 views
Wigner-Eckart theorem of SU(3)
I have just come across the Wigner-Eckart theorem and am not sure on how to apply it. How do I find the matrix elements of $\langle u|T_a|v\rangle$ in terms of tensor components and the Gell-Mann ...
5
votes
0answers
154 views
What is the difference between the properties of Electron spin and Photon polarization/helicity?
What is the difference between a photon's polarization/helicity and an electrons spin half? I know that the photon is spin 1 but isn't its polarization analogous to spin half?
This question stems ...
5
votes
0answers
127 views
Decomposing a Tensor Product of $SU(3)$ Representations in Irreps
Can somebody explain in a simple way why, talking about representations, $3\otimes3=3\oplus6$, $3\otimes\bar{3}=1\oplus8$ and $3\otimes3\otimes3=1\oplus8\oplus8\oplus10$?
Here $3$ and $\bar{3}$ are ...
5
votes
0answers
186 views
Coupling Coefficients in SO(4)
I have two equations (from two distinct authors) for the decomposition of a coupling coefficient of SO(4) (i.e. Wigner 3j-symbol for SO(4)). In the first:
...
4
votes
2answers
246 views
Definition of Casimir operator and its properties
I'm not sure which is the exact definition of a Casimir operator.
In some texts it is defined as the product of generators of the form:
$$X^2=\sum X_iX^i$$
But in other parts it is defined as an ...
4
votes
4answers
895 views
How to prove that proper orthochronous Lorentz transformations form a group?
Proper orthochronous Loentz transform are Lorentz transforms that satisfy the conditions (sign convention of Minkowskian metric $+---$)
$$\det \Lambda=+1, \qquad \Lambda^0{}_0 \geq +1.$$
How to prove ...
4
votes
1answer
329 views
representation of SU(2)
The question is regarding SU(2) group and SU(2) algebra.
The SU(2) group can be generated by exponentiating the
generators of SU(2) algebra $X_a$ as $exp(i t_a X_a )$
with $t_a$ being three ...
4
votes
2answers
161 views
Are there any known potentially useful nontrivial irreducible representations of the Lorentz Group $O(3,1)$ of dimension bigger than 4? Examples?
Are there any known potentially useful, nontrivial, irreducible representations of the Lorentz Group $O(3,1)$ of dimension more than $4$? Examples? A $5$-dimensional representation? EDIT: Is there ...
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 ...
4
votes
1answer
203 views
U(1) Charged Fields
I don't quite understand what is actually meant by a field charged under a $U(1)$ symmetry.
Does it mean that when a transformation is applied the field transforms with an additional phase? More ...
4
votes
2answers
254 views
Is there a 1-1 correspondence between symmetry and group theory?
The professor in my class of mathematical physics introduces the definition of groups and said that group theory is the mathematics of symmetry.
He gave also some examples of groups such as the set ...
4
votes
2answers
239 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 ...
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 ...
4
votes
2answers
206 views
Calculating the commutator of Pauli-Lubanski operator and generators of Lorentz group
The Pauli-Lubanski operator is defined as
$${W^\alpha } = \frac{1}{2}{\varepsilon ^{\alpha \beta \mu \nu }}{P_\beta}{M_{\mu \nu }},\qquad ({\varepsilon ^{0123}} = + 1,\;{\varepsilon _{0123}} = - ...
4
votes
1answer
368 views
Why does photon have only two possible eigenvalues of helicity
Photon is a spin-1 particle. Were it massive, its spin projected along some direction would be either 1, -1, or 0. But photons can only be in an eigenstate of $S_z$ with eigenvalue $\pm 1$ (z as the ...
4
votes
2answers
56 views
How to directly calculate the infinitesimal generator of SU(2)
We commonly investigate the properties of SU(2) on the basis of SO(3). However, I want to directly calculte the infinitesimal generator of SU(2) according to the definition $$X_{i}=\frac{\partial ...
