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. ...

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3
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1answer
206 views

Is time reversal operator not a representation of Lorentz group?

I'm puzzled why every book says that time reversal operator is a representation of full Lorentz group. Because of physical consideration, time reversal is an antilinear operator. While the definition ...
4
votes
1answer
225 views

Is General Relativity based on a Symmetry?

In short: Is there any kind of symmetry one can start with to derive general relativity (GR)? Longer: Einstein had the opinion that GR was the generalisation of special relativity, because instead of ...
2
votes
0answers
33 views

Measure of interaction of two quarks and Casimir operators [closed]

Let's have two quarks, which refers to representations of $r_{1}$ and $r_{2}$ of color symmetry group. They create bounded state which refers to the representation $r$. There is a statement that ...
3
votes
1answer
178 views

What is the four-dimensional representation of the $SU(2)$ generators?

Recently, I have been learning about non-Abelian gauge field theory by myself. Thanks @ACuriousMind very much, as with his help, I have made some progress. I am trying to extend the Dirac field ...
3
votes
2answers
94 views

Unification of the electroweak theory

Can the electroweak theory be described by the spontaneous symmetry breaking of $SU(3)$ to $SU(2)\times U(1)$?
5
votes
1answer
108 views

Symmetries of AdS$_3$, $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$

Basically, I want to know how one can see the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry of AdS$_3$ explicitly. AdS$_3$ can be defined as hyperboloid in $\mathbb{R}^{2,2}$ as $$ ...
1
vote
0answers
42 views

Uncharged Gluon Representation

I'm currently learning for an exam and in an old exam the question was posed: If Gluons were uncharged, under which representation would they transform? The ...
3
votes
1answer
84 views

Two different transformation laws for Quantum Fields

I found a nice answer to a problem that was bothering me for quite a while in a lecture script (unfortunately in german). The first step of the answer, is what remains unclear to me. The script states ...
8
votes
1answer
191 views

Group representations as vectors and isomorphism between weights and matrix generators

This might be something basic, but it is unclear to me. So I am used to work with representations of groups as matrices. These matrices represent the structure of the Lie algebra by satisfying the ...
1
vote
0answers
41 views

What is the difference between the compact U(1) group and non-compact U(1) group? [duplicate]

Is compact $U(1)$ and non-compact $U(1)$ just two different representations for the same group or the same gauge theory? If not, what the difference of them? Or are there any properties that are ...
1
vote
1answer
55 views

Correspondence between one-parameter subgroups of $G$ and $T_eG$

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
3
votes
1answer
95 views

Permutations of two identical particles in two dimensions

In three spatial dimensions there are only two possible statistics: Bose-Einstein and Fermi-Dirac. This is the fact related with the statement that first homotopic group of 3-dimensional configuration ...
3
votes
1answer
152 views

Group theoretical reason that Gluons carry charge and anticharge

I was wondering how it is possible to see from the $SU(3)$ Gauge Theory alone that Gluons carry two charges colors: $g\overline{b}$ etc. Some background: The W-Bosons (pre-symmetry breaking) ...
2
votes
3answers
130 views

Generators of the Diffeomorphism Group

So what are the generators of a Diffeomorphism Group? For simplicity, let's consider $ Diff(R^2) $ (diffeomorphisms of the euclidean plane.) Diffeomorphisms are differentiable, invertible ...
3
votes
1answer
128 views

Dirac group representation

I am currently taking a representation theory class (from a physicist), and I am very confused about the Dirac groups' irreducible representations. First of all, all the Dirac matrices in the ...
2
votes
1answer
181 views

Group Theoretic definition of a particle

We intuitively have a sense of what a particle means in the conventional sense. But is it possible to have a group theoretical definition of a particle, I mean in terms of irreducible representations ...
2
votes
1answer
117 views

Hilbert space decomposition into irreps

I'm currently following a course in representation theory for physicists, and I'm rather confused about irreps, and how they relate to states in Hilbert spaces. First what I think I know: If a ...
2
votes
1answer
85 views

Symmetry transformation on Quantum Field

I stumbled upon this point several times, the latest beeing this question: Connection between conserved charge and the generator of a symmetry I want to understand, why Quantum fields transform under ...
3
votes
0answers
56 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 ...
2
votes
1answer
91 views

Invariant tensors of Symplectic and Exceptional groups.

We know that for special orthogonal groups $SO(N)$ there exists invariant tensors (invariant under the group action). These are $\delta_{ij}$ and the totally anti-symmetric $\epsilon_{m_1,m_2,...m_N}$ ...
2
votes
2answers
210 views

A whole lot of doubts on Lorentz representation

Can someone tell me in layman's language how the $(1/2,1/2)$ represents a vector field and $(0,1/2)$ or $(1/2,0)$ represents spinors and $(0,0)$ represents scalar field. Please don't be pedantic on ...
4
votes
1answer
165 views

Group theoretic way to find charges after SSB

I was wondering what is the group theoretic way to find the resulting charges of matter fields after a scalar field is given a vev. In the case of the EW symmetry breaking, one can directly read the ...
1
vote
1answer
93 views

The Euler-Poincare equation

Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone ...
14
votes
3answers
779 views

Can someone please qualitatively explain unitary group from a physics perspective?

Unitary Groups is the most mysterious thing for me when studying physics. All my physics endeavor ends when author starts talking about unitary groups. This is often the case because in a lot of the ...
1
vote
0answers
37 views

Lie theory and particle physics [duplicate]

I have recently been reading Intro to Lie algebras and representation theory by Humphreys, and when I am finished I am interested in reading about Lie groups and Lie algebras and their applications to ...
3
votes
2answers
110 views

What's the relationship between uncertainty principle and symplectic groups?

What's the relationship between uncertainty principle and symplectic groups? Does the symplectic groups mathematically capture anything fundamental about uncertainty principle?
2
votes
2answers
102 views

Constructing SUSY algebra via index structure

Often in literature the SUSY algebra is simply given, but various books, for example Bailin and Love, goes through the trouble of showing how the SUSY commutation relations are the only possible ones ...
0
votes
1answer
72 views

Rotating a complex number

Let us begin in a two-dimensional Euclidean plane. The vector is e.g. $\vec{V}(x,y)$ It is often useful – but in this case, it's just a mathematical trick that doesn't make the complex numbers ...
3
votes
2answers
125 views

What is the axial transformation of a group, i.e. $SU(3)$?

The Gell-Mann matrices $\lambda^\alpha$ are the generators of $SU(3)$. Applying an SU(3) - transformation on the triple $q = ( u , d, s )$ of 4-spinors looks like this: $$ q \rightarrow q' = e^{i ...
9
votes
5answers
402 views

Seeking a quality plain-language description of the Wigner-Eckart theorem

I'm a third year physics undergrad with a very cursory knowledge of quantum mechanics and the formalism involved. For instance, I understand roughly how tensors work and what it means for a tensor to ...
0
votes
1answer
76 views

Notation in the book Symmetry by Hermann Weyl

I'm having troubles understanding a notation of the symmetry groups in a book "Symmetry" by Hermann Weyl. On the page 80 of the 1952 Princeton University Press edition of the book, Weyl lists the ...
2
votes
1answer
61 views

relating spinor and fundamental representation for $E_8$

While proving a very important relation which is satisfied both by $SO(32)$ AND $E_8$, which makes it possible to factorize the anomaly into two parts. The relation is ...
3
votes
1answer
167 views

What is the idea behind counting the number of excited states and the representation of a group ?

While reading Polchinski's Chapter 1, I encountered the following on page 24, "For example, the $(D-1)$ dimensional vector representation of $SO(D-1)$ breaks up into an invariant and a $(D-2)$-vector ...
2
votes
1answer
83 views

Finding the stabilizer group given a state

Consider general pure state $|\psi\rangle$ in some hilbert space $\mathcal{H}$ (which could be a tensor product of other Hilbert spaces) I would like to know whether there is a way to ...
4
votes
1answer
823 views

Lorentz group representations in QFT: what's the vector space?

In QFT, a representation of the Lorentz group is specified as follows: $$ U^\dagger(\Lambda)\phi(x) U(\Lambda)= R(\Lambda)~\phi(\Lambda^{-1}x) $$ Where $\Lambda$ is an element of the Lorentz group, ...
0
votes
1answer
93 views

Connection to spin 1/2 electron system?

In another Physics stack exchange thread here, Spin matrix for various spacetime fields I obtained the generator of rotations of the SO(2) rotation group for an infinitesimal rotation of 2D vectors. ...
4
votes
2answers
170 views

Traces in different representation

I am actually working with Green-Schwarz anomaly cancellation mechanism in which I have came across a strange formula which relates trace in the adjoint representation (Tr) to trace in fundamental ...
3
votes
2answers
155 views

Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3)$ isometry group?

As the title says, is it possible to have a Riemannian Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3) $ isometry group?
3
votes
2answers
77 views

What is different in representation?

I'm sorry if this is somewhat a dumb question. First: "Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear ...
5
votes
3answers
459 views

Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the ...
16
votes
2answers
633 views

Why are only linear representations of the Lorentz group considered as fundamental quantum fields?

As described in many Q&As around here, fundamental quantum fields are expressed as irreducible representations of the Lorentz group. This argument is entirely clear - we live in a ...
6
votes
3answers
569 views

Integrating the generator of the infinitesimal special conformal transformation

(c.f Di Francesco, Conformal Field Theory chapters 2 and 4). The expression for the full generator, $G_a$, of a transformation is $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta \omega_{a}} \partial_{\mu} ...
3
votes
0answers
87 views

Is general covariance a symmetry?

Is general covariance a symmetry? If it is ,what is its symmetry group and corresponding generator?
4
votes
2answers
192 views

Algebra, commutators and test functions

I am trying to make sense out of the algebra of the generators of the conformal group and I am running into some issues regarding how to calculate commutators. For instance, for translations of a ...
11
votes
3answers
270 views

Why gauge $SU(N)$ and not $SO(N)$?

When building models people typically gauge $SU(N)$ but rarely try to gauge $SO(N)$ (the only example I know about is $SO(10)$, but even that isn't quite $SO(10)$ but actually its double cover). At ...
3
votes
0answers
40 views

How does the choice of a particular vacuum in a field theory problem decide the number of Goldstone bosons?

How does the field expansion method (by this I mean expanding your fields about a chosen VEV and plugging into a given potential so that the masses of the fields are given by the coefficients in ...
1
vote
0answers
28 views

How does the choice of a basis decide how many Goldstone bosons there are under spontaneous symmetry breaking?

I have a question about how the basis you choose in a field theory problem semmingly decides how many Goldstone bosons you get after spontaneous symmetry breaking. For SU(2), if you choose the 3 Pauli ...
6
votes
0answers
87 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 ...
8
votes
1answer
193 views

Why $SU(3)$ and not $U(3)$?

Is there a good reason not to pick $U(3)$ as the colour group? Is there any experiment or intrinsic reason that would ruled out $U(3)$ as colour group instead?
2
votes
0answers
40 views

What is 'heterotic string compactification'?

I've read that some exceptional groups arises in the context of 'heterotic string compactification'. Could someone explain (to a person studying physics but who doesn't know string theory) what ...