Questions tagged [group-theory]

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. Groups are used in physics to describe symmetry operations of physical systems.

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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 ...
17
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456 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, ...
11
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0answers
237 views

Holonomy group of Schwarzschild spacetime, other interesting examples?

I'm teaching myself a little about holonomy groups in the context of general relativity. This paper by Hall and Lonie classifies a lot of the possibilities for simply connected spacetimes in 3+1 ...
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300 views

Is the QCD Lagrangian without a $\theta$-term invariant under large gauge transformations?

In his book "Quantum field theory", Kerson Huang states that we need to add the term $$\frac{i\theta}{32\pi^2}G_{\mu\nu}^a \tilde{G}_{\mu\nu}^a$$ to the Lagrangian, to make it invariant under large ...
8
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136 views

What is the difficulty in extending geometrodynamics to non-abelian fields?

In an attempt to widen my own horizons I've decided to educate myself in Wheeler's Geometrodynamics. In the so-called "already unified theory" one can essentially reproduce an electromagnetic field ...
7
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109 views

Group Theory of Superconducting Order Parameters?

In crystalline superconductors, the order parameter $\Delta(\mathbf{k})$ (aka gap, or Cooper pair wavefunction) can be classified by its symmetry according to the representations of the symmetry group ...
7
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151 views

Representation Theory of $SL(2,\mathbb R)$

The representation theory regarding the finite-dimensional representations of $SL(2,\mathbb C)$ is well-understood; namely, they all decompose into irreducibles $V_n$, $\dim(V) = n > 0$. ...
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139 views

Completely positive maps and symmetric states

Let $\mathcal{N}$ be a completetely positive trace preserving map (aka a quantum channel) acting on a finite dimensional system $\mathrm{A}$, and let $\pi$ denote the maximally mixed state on $\mathrm{...
6
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2answers
205 views

Is there any qualitative difference between the WZW $SO(2)_1$ and the WZW $SU(2)_1$ CFT?

Consider the anisotropic spin-$\frac{1}{2}$ Heisenberg chain $$H = \sum_{n=1}^N S^x_n S^x_{n+1}+S^y_n S^y_{n+1} + \Delta S^z_n S^z_{n+1}$$ which for $\Delta = 0$ realizes the Wess-Zumino-Witten (WZW) $...
6
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440 views

Is there a Virasoro group?

On page 14 of the survey article Kac-Moody and Virasoro algebras in relation to quantum physics by Goddard and Olive, the authors show that smooth selfmaps of the circle form a Lie group corresponding ...
6
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1answer
348 views

Massive states of the closed bosonic string fitting into a representation of $SO(D-1)$

It is usually shown in the literature that massive light-cone gauge states for a closed bosonic string combine at mass level $N=\tilde{N}=2$ into representations of the little group $SO(D-1)$ and then ...
6
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282 views

Where in fundamental physics are Lie groups actually important (and not just Lie algebras)?

I was wondering where in fundamental physics the global structure of a Lie group actually makes a difference. Most of the time physicists are sloppy and don't distinguish groups and algebras ...
6
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339 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
5
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1answer
349 views

Truncation of $D=5$, ${\cal N}=8$ Supergravity by $\mathbb Z_2^3$

The scalar manifold of $D=5, \mathcal N=8$ SUGRA is $$\mathcal M = \frac{E_{6(6)}}{Usp(8)}$$ where $USp(8)$ is a maximal compact subgroup of $E_{6(6)}$ and the 42 scalars of the theory correspond ...
5
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417 views

Monstrous Moonshine outside of String Theory

My question concerns applications of monstrous moonshine, which is the connection between the $j$-function and the monster group. Recently, physicists have applied it to string theory and, ultimately, ...
5
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0answers
169 views

From $U(3)$ to $SU(3)\times U(1)$ Color symmetry. There is a “gluon” photon-like?

Suppose that $U(3)$ was the gauge group. We can decompose this as $U(3)=U(1)\times SU(3)$, which implies that in addition to the $SU(3)$ that has eight generators corresponding to eight gluons, there ...
5
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154 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 ...
5
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117 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 ...
5
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136 views

Calabi Yau compactification based on U(1) charges

In Green-Schwarz-Witten Volume 2, chapter 15, it is argued (roughly) that we need 6-dimensional manifolds of $SU(3)$ holonomy in order to receive 1 covariantly constant spinor field. And it turns out ...
4
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2answers
154 views

Why would a spinor transform under Lorentz transformations?

From my understanding of spinors, they arise as projective representations of $SO_0(1,3)$ that do not correspond to representations of $SO_0(1,3)$. But still one says here - and virtually everywhere - ...
4
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1answer
161 views

What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
4
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120 views

Physical/geometrical interpretations of spinors?

Physically, a scalar is a quantity invariant with reference frame, a vector is a quantity associated with a direction, tensors are higher relationships between vectors - what are spinors? I thought I ...
4
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168 views

Infinite-dimensionality of unitary representations of non-compact simple Lie Groups

I have a question about the argument given in On finite-dimensional unitary representations of non-compact Lie groups. I have been looking for a good proof for this claim for a little while now. I ...
4
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365 views

Minkowski as a Quotient Space

I've read many times, in many articles or books that one can obtain the four dimensional Minkowski space $\mathbb{M}^4$ as the quotient space $$ ISO(3,1)/SO(3,1), $$ and equivalently de Sitter and ...
4
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347 views

Understanding the Monster CFT

I try to understand what the Monster CFT and its possible connection to 3 dimensional gravity at ($c=24$) is about (see https://arxiv.org/abs/0706.3359) To my best understanding (and please correct ...
4
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130 views

Global $SU(N)$ on the gravity side in AdS/CFT

For AdS/CFT to make sense, symmetries must match between the AdS side and the CFT side. Gauge symmetries are redundancies, not symmetries, therefore the CFT can have a (large) gauge symmetry, say $SU(...
4
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154 views

What Lie supergroup does the super-Poincare algebra generate?

Every Lie supergroup has an associated Lie superalgebra of generators (in general, some of which are bosonic and some fermionic). Which Lie supergroup(s) are generated by the Super-Poincare algebra ...
4
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89 views

Highest weight unitary representations of $psl(2|2)$

I'm having some trouble understanding how to extend representation theory from Lie algebras to super Lie algebras, in particular with $psl(2|2)$. Ultimately I'm interested in 2D quantum sigma models ...
4
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223 views

Any examples of commensurable subgroups appearing in physics?

I am a mathematician. I am studying and working on Hecke pairs which I am going to give the related definitions in the following. But first let me explain what I am looking for to learn by asking this ...
4
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335 views

Fields with SO(3) diagonal subgroup symmetry

I read about a Higgs field $\vec{\phi}=\frac{1}{2}a\hat{r}\cdot \vec{\sigma}$ (in the context of 't Hooft-Polyakov monopole) with SO(3) diagonal subgroup symmetry consisting of simultaneous and equal ...
3
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51 views

Relation of Wigner $d$-matrix $d^l_{m',m} = d^l_{-m,-m'}$

How do you derive the symmetry relation of the Wigner $d$-matrix, i.e., $$ d^l_{m',m} = d^l_{-m,-m'} $$ I know how Wikipedia proves this using the fact that $(Y_l^m)^* = (-1)^m Y_l^{-m}$ (basically ...
3
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1answer
71 views

Is electrodynamics associated with $O(3)$?

Let $\mathbf{q}$ be a complex vector of three elements defined as: $$ \mathbf{q}:=\pmatrix{ E_x + iB_x\\ E_y + i B_y\\ E_z +i B_z } $$ I define the function $f(\mathbf{q})$: $$ \begin{align} f(\...
3
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51 views

Has this group something to do with the cone of light?

Consider the group $V=(-1,1)$ with addition $+_{rel}:V\times V\to V$ defined as: $$v+_{rel}w=\frac{v+w}{1+vw}$$ This group is analogous to the relativistic velocities where the speed of light equals ...
3
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0answers
56 views

Quantum representation of a system of identical particles

I'm studying mathematics and I began a course in quantum statistics, in which I got to the discussion related to indistinguishibility of particles. My professor's notes are not very clear and ...
3
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1answer
61 views

Connected components of conformal group $ {\rm Conf}(p,q)$ containing $P$, $T$ and conformal inversion are same or different?

As we known (see this post), the global conformal group for $\mathbb{R}^{p,q}$ is $$ {\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}$$ The global conformal group ${\rm Conf}(p,q)$ has 4 ...
3
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0answers
57 views

How is Inönü-wigner contraction done?

I have read that little group for the massive particles is $SO(3)$ and for the massless particles is $E(2)$ in 4 dimensions. How does one take zero mass limits for the representations and show that it ...
3
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0answers
72 views

Fermionizing the Gell-Mann Algebra

In condensed matter physics one often solves a spin Hamiltonian by transcribing the Pauli matrices into fermionic operators. For instance, in the Kitaev model you can introduce four Majorana modes for ...
3
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1answer
128 views

Why are all transformations of quantum operators inner automorphisms?

Operators in quantum mechanics are basically related to each other through their Lie algebra i.e. the commutator $\times \frac{1}{i\hbar}$. This is then connected to the state space i.e. the Hilbert ...
3
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1answer
79 views

Scattering matrix symmetries and standard model

I am not able to get around the following question (if it make sense): Suppose I can derive the scattering matrix S for any particle scattering process. Suppose that the standard model is actually ...
3
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0answers
64 views

Can you do gauge theories over topological groups?

Quantum gauge theories involve (functional) integration over a Lie group. Is there any meaningful generalisation to (non-manifold) topological groups? Consider for example the Whitehead tower $$ \...
3
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0answers
222 views

How to determine the degree of how high a symmetry of high-symmetry points in the first Brillouin zone?

For exmple, we have a hexagonal lattice with hexagonal Brillouin zone, shown in the picture The points $\Gamma$, K, M and $\Lambda$ are high symmetry points. Now, $\Gamma$ point is the highest-...
3
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0answers
74 views

Does topological mass imply preservation of global symmetry whose current is topological?

This question is general but the motivation for it lies within the paper "A Duality Web in 2+1 Dimensions and Condensed Matter Physics". On pages 20-23, they consider a system which has four phases ...
3
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0answers
84 views

Comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of: Chern class (1st, 2nd), and ...
3
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0answers
50 views

Mapping from spinor to tetrad

I am reading the journel by Patrick l. Nash: mapping from tetrad to Dirac spinor. While reading this ,I came across the term concrete real 4*4 irreducible representation of SO(3,3). I know SO(3) is ...
3
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0answers
92 views

What is the physical meaning of Lie congruence classes?

Any weight $\lambda$ characterising a representation of $\mathfrak{su}(N)$ is an element of one of the $N$ congruence classes defined by (ref.1) $$ \lambda_1+2\lambda_2+\cdots+(N-1)\lambda_{N-1}\quad\...
3
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0answers
57 views

$\mathbb{Z}_6$ symmetry in the Standard Model

It's a well-known result that the spontaneous symmetry breaking of $SU(5)$ would lead not to the usual $G_{SM}=SU(3)\times SU(2)\times U(1)$, but to $G_{SM}/\mathbb{Z}_6$. However, it's also often ...
3
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0answers
199 views

Significance of Irreducible Representations

I am reading texts on QFT and they place a lot of emphasis on finding irreducible representations of the Lorentz/Poincare Group. I also recall some level of discussion from non-relativistic quantum ...
3
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0answers
511 views

Cartan Killing metric and Casimir operators

I'm a little confused about Casimir operators and Cartan-Killing metric. The Lorentz group is a semi-simple group and its Cartan-Killing metric is non-degenerate, say $g_{ab}$; it is invertible and ...
3
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0answers
314 views

Representation Theory of $SL(2, \mathbb{C})$

I'm a PhD. in mathematics (working mainly in complex algebraic geometry), but I'm looking for a "convincing" answer concerning the various applications of representation theory of the group $SL(2, \...
3
votes
1answer
319 views

Does a gauge group $G$ determine the Principal $G$-bundle?

I'm trying to understand the mathematical underpinnings of gauge theories in the language of principal $G$-bundles and associated vector bundles. Not long ago, I had assumed that the physical choice ...

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