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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1answer
62 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
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1answer
116 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 ...
2
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1answer
205 views

Group theoretical reason that Gluons carry color-charge and anti-colorcharge

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
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3answers
207 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
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1answer
201 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
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1answer
217 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
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1answer
170 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
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1answer
116 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
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0answers
66 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
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1answer
111 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 $\epsilon_{m_1,m_2,...m_N}$ ...
4
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2answers
239 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
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1answer
171 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
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1answer
118 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 ...
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3answers
953 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 ...
2
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0answers
40 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 ...
4
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2answers
143 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
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2answers
105 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
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1answer
78 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 ...
4
votes
2answers
220 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 ...
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5answers
536 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
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1answer
80 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
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1answer
70 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
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1answer
176 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 ...
3
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1answer
92 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 ...
5
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1answer
1k 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, ...
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1answer
118 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. ...
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2answers
220 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
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2answers
172 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
87 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 ...
7
votes
3answers
701 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 ...
20
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2answers
736 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 ...
9
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3answers
1k 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
102 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
293 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 ...
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3answers
369 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
1answer
62 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 ...
5
votes
0answers
106 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
259 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
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0answers
45 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 ...
1
vote
1answer
76 views

Irrep decomposition of direct product of stress tensors

I have stress tensors direct product of the form $T^{ab}(x)T^{cd}(y)$. I want to write this in terms of a tensor $I^{abcd}$ in the form. $T^{ab}(x)T^{cd}(y)= I^{abcd}$. This is like decomposing the ...
3
votes
1answer
139 views

Real representation is physically real?

In Peskin & Schroder, Introduction to Quantum Field Theory equation (15.82) states that $$ t^a_{\bar{r}} = -(t^a_{r})^* = (t^a_{r})^T $$ Why is the representation which satisfies $$ ...
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votes
2answers
194 views

Reasons for choosing $SU(3)$ as the color group vs. $SO(4)$

What are the reasons that $SU(3)$ is used for QCD? Why wouldn't the simpler & smaller group $SO(4)$ make a better candidate?
7
votes
4answers
536 views

Coadjoint orbits in physics

I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.
4
votes
1answer
74 views

What does “action of a gauge group on a particle” mean?

I have come across this phrase left handed fermions transform under $SU(3)\times{}SU(2)\times{}U(1)$ differently from the way right handed fermions do. I am just beginning to learn about how the ...
3
votes
1answer
249 views

Why is the projective symmetry group (PSG) called projective?

As discussed by Prof.Wen in the context of the quantum orders of spin liquids, PSG is defined as all the transformations that leave the mean-field ansatz invariant, IGG is the so-called invariant ...
10
votes
1answer
313 views

What's a lepto-diquark?

This questions refers to Slansky's Group theory for unified model building, page 106 of chapter 7. He assigns the weight $(1)(01)$, which is stepwise projected from $E_6$ to $SU(2)\times SU(3)$, to a ...
10
votes
3answers
429 views

The $U(1)$ charge of a representation

My question is about the reduction of a representation of a group $SU(5)$ to irreps of the subgroup $SU(3)\times SU(2) \times U(1)$. For example the weights of the 10 dimensional representation of ...
3
votes
1answer
203 views

How to construct an invariant Lagrangian under a Lie group $G$ generally?

How to construct an invariant Lagrangian under a Lie group $G$ generally? For example, if we have $SO(5)$'s generators which are constructed by some operators, then the question is that: is it ...
12
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1answer
212 views

Triality and charge

I have a few questions about triality for the representations of $SU(3)$. (I have seen the wikipedia page, but it does not make the connection with physics.) What is triality, how can you compute ...
5
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2answers
103 views

Possible mechanics based on the known symmetries in the nature (investigating rumor)

Somewhere I've heard about a relative new mathematical result regarding mechanics. Specifically, there is a list of the known symmetries of mechanics (both Newtonian and relativistic), i.e. different ...