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

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