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22
votes
3answers
948 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 ...
21
votes
5answers
127 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 ...
15
votes
3answers
446 views

Idea of Covering Group

$SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence? I heard that this fact is related to the description of bosons and fermions. But how does it ...
13
votes
1answer
1k views

Mathematically, what is color charge?

A similar question was asked here, but the answer didn't address the following, at least not in a way that I could understand. Electric charge is simple - it's just a real scalar quantity. Ignoring ...
12
votes
1answer
401 views

What really are superselection sectors and what are they used for?

When reading the term superselection sector, I always wrongly thought this must have something to do with supersymmetry ... DON'T laugh at me ... ;-) But now I have read in this answer, that for ...
11
votes
2answers
724 views

Can any rank tensor be decomposed into symmetric and anti-symmetric parts?

I know that rank 2 tensors can be decomposed as such. But I would like to know if this is possible for any rank tensors?
11
votes
2answers
2k views

Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?

The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
11
votes
1answer
168 views

Majorana-like representation for mixed symmetric states?

Is there a generalization of the Majorana representation of pure symmetric $n$-qubit states to mixed states (made of pure symmetric $n$-qubit)? By Majorana representation I mean the decomposition of ...
10
votes
2answers
196 views

When are there enough Casimirs?

I know that a Casimir for a Lie algebra $\mathfrak{g}$ is a central element of the universal enveloping algebra. For example in $\mathfrak{so}(3)$ the generators are the angular momentum operators ...
10
votes
2answers
203 views

What's the relationship between $SL(2,\mathbb{C})$, $SU(2)\times SU(2)$ and $SO(1,3)$?

I'm a beginner of QFT. Ref. 1 states that [...] The Lorentz group $SO(1,3)$ is then essentially $SU(2)\times SU(2)$. But how is it possible, because $SU(2)\times SU(2)$ is a compact Lie group ...
10
votes
1answer
534 views

Dimension of Dirac $\gamma$ matrices

While studying the Dirac equation, I came across this enigmatic passage on p. 551 in From Classical to Quantum Mechanics by G. Esposito, G. Marmo, G. Sudarshan regarding the $\gamma$ matrices: ...
10
votes
2answers
335 views

$\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = ...
9
votes
1answer
212 views

Why does the eightfold way work?

Last year I attended an introductory particle physics course, in which the Eigthfold Way for classifying hadrons has been discussed. The main idea consists in grouping hadrons in multiplets (i.e ...
9
votes
1answer
111 views

The difference between $\mathcal{N}=2$ short multiplets and BPS states

I have some questions about the construction of $\mathcal{N}=2$ supermultiplets for chiral matter. I know that the supermultiplet should not include spin one states since they are always in the ...
9
votes
1answer
282 views

Introduction to spinors in physics, and their relation to representations

First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a ...
8
votes
3answers
517 views

Why does spin have a discrete spectrum?

Why is it that unlike other quantum properties such as momentum and velocity, which usually are given through (probabilistic) continuous values, spin has a (probabilistic) discrete spectrum?
8
votes
2answers
408 views

What does “the ${\bf N}$ of a group” mean?

In the context of group theory (in my case, applications to physics), I frequently come across the phrase "the ${\bf N}$ of a group", for example "a ${\bf 24}$ of $SU(5)$" or "the ${\bf 1}$ of ...
8
votes
4answers
242 views

What does “carry a representation” mean (in SUSY algebra)?

I come from a maths background and am struggling with some of the more physical texts on SUSY. In particular they claim that the fermionic generators $Q_A^i$ carry a representation of the Lorentz ...
8
votes
1answer
456 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 ...
7
votes
1answer
351 views

Boosts are non-unitary!

The boost transformations are not unitary unlike rotations, the boost generators are not Hermitian. When this induces transformations in the Hilbert space, will those transformation be unitary? I ...
7
votes
1answer
191 views

Can a spinor be defined as any quantity which transforms linearly under Lorentz transformations?

Recently I’ve come across a few papers from China (e.g. Xiang-Yao Wu et al., arXiv:1212.4028v1 14 Dec 2012) that make the following statement: ...any quantity which transforms linearly under ...
7
votes
4answers
2k views

What is the difference between a spinor and a vector or a tensor?

Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?
7
votes
2answers
99 views

Definition of a spinor and applications to GR

I understand the construction of the Clifford algebra $C(r,s)$ and in turn the corresponding $Pin$ and $Spin$ groups. I would like first to clarify that $Spin(r,s)^e$ is the universal covering group ...
7
votes
2answers
125 views

Why the lowest order of matrices in Dirac equation are 4x4 matrices?

Why the lowest order of matrices in Dirac equation (Relativistic Quantums) are 4x4 matrices (and can not be 2x2 matrices)? How to prove it?
6
votes
2answers
177 views

Tensor decomposition under $\mathrm{SU(3)}$

In Georgi's book (page 143), he calculates the tensor components of $3\otimes 8$ under the $\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning ...
6
votes
1answer
139 views

Representations of the Poincare group

Which type of states carry the irreducible unitary representations of the Poincare group? Multi-particle states or Single-particle states?
6
votes
2answers
143 views

Why aren't the spin-3/2 fields in the (3/2,0)+(0,3/2) representation?

Why is it that spin-$\frac 32$ fields are usually described to be in the $(\frac 12, \frac 12)\otimes[(\frac 12,0)\oplus(0,\frac 12)]$ representation (Rarita-Schwinger) rather than the $(\frac ...
5
votes
2answers
673 views

Irreducible Representations Of Lorentz Group

In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as ...
5
votes
2answers
598 views

Can one show that ${\gamma^5}^\dagger = \gamma^5$ directly from the anticommutation relations?

Is it possible to show that ${\gamma^5}^\dagger = \gamma^5$, where $$ \gamma^5 := i\gamma^0 \gamma^1 \gamma^2 \gamma^3,$$ using only the anticommutation relations between the $\gamma$ matrices, $$ ...
5
votes
2answers
156 views

Problem counting spin states

I can't figure out how many different spin states I can create with a four-electron system. I think I can create a spin-zero state, three spin-one states, and five spin-two states. That gives me nine ...
5
votes
1answer
253 views

Grassmann Variables Representation?

It might be a silly question, but I was never mathematically introduced to the topic. Is there a representation for Grassmann Variables using real field. For example, gamma matrices have a ...
5
votes
1answer
261 views

Intuitive understanding of the irreps like Wigner-D matrix?

Wikipedia defines Wigner D-matrix as an irreducible representation of groups SU(2) and SO(3). What is a good way to visualize this representation? Is there any physical system which can be kept in ...
5
votes
1answer
223 views

About 2+1 dimensional superconformal algebra

I would like to get some help in interpreting the main equation of the superconformal algebra (in $2+1$ dimenions) as stated in equation 3.27 on page 18 of this paper. I am familiar with supersymmetry ...
4
votes
1answer
217 views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
4
votes
2answers
635 views

Why do many people say vector fields describe spin-1 particle but omit the spin-0 part?

We know a vector field is a $(\frac{1}{2},\frac{1}{2})$ representation of Lorentz group, which should describe both spin-1 and spin-0 particles. However many of the articles(mostly lecture notes) I've ...
4
votes
1answer
160 views

Massive excitations in Conformal Quantum Field Theory

Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of ...
4
votes
2answers
126 views

Unitary representations of the diffeomorphism group in curved spacetime

In (special) relativistic quantum mechanics there is a standard argument that says that the (rigged) Hilbert space of states $H$ should be equipped with a projective unitary representation $U$ of the ...
4
votes
1answer
200 views

Some questions about the paper, “AdS description of induced higher spin gauge theory”

I am referring to this paper. I guess that in this paper one is trying to relate the massless spin $s$ gauge fields in $AdS_4$ to conformal spin $s$ theory on $S^3$. So am I right that the ...
4
votes
1answer
167 views

Eigenvalue of $L_z$

In section 4.3 of Griffths' "Introduction to Quantum Mechanics", just below Figure 4.6, the sentence begins Let $\hbar \ell$ be the eigenvalue of $L_z$ at this top rung... Why is this valid? ...
4
votes
1answer
114 views

Decomposition of representations of the Virasoro algebra under $sl(2)$

The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight ...
4
votes
1answer
177 views

Question about the Killing form in Georgi

I'm trying to understand the Killing form described on page 49 in the book by Howard Georgi. He starts by saying that one defines the inner product between two generators $T_a$ and $T_b$ in the ...
4
votes
2answers
293 views

Irreducible Representations of SO(n) tensors

My interest is purely in $\text{SO}(n)$ tensors and how one works out their irrep decomposition. For instance, for rank 2 tensors we simply split into an antisymmetric part, a traceless symmetric part ...
4
votes
1answer
337 views

Holstein-Primakoff and Dyson-Maleev representation

In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
4
votes
1answer
229 views

Labelling representations using isospin and hypercharge

Can someone explain how isospin and hypercharge, can be used to label representations? What is the meaning of the term singlet, doublet etc in this context? In particular how can I use it to label ...
4
votes
1answer
148 views

The use of Hall algebras in physics

I asked the same question in mo. I think maybe here there are more physics guys to help me. I once read a statement (not memorized precisely) that a certain physics quantity between two states of ...
4
votes
1answer
278 views

Gauge invariance and the form of the Rarita-Schwinger action

in Weinberg Vol. I section 5.9 (in particular p. 251 and surrounding discussion), it is explained that the smallest-dimension field operator for a massless particle of spin-1 takes the form of a field ...
4
votes
1answer
139 views

Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice

Happy holidays, everyone! The following is part question, part visual gallery, and part classical mechanics problem. Inspired by snow over the weekend I began simulating the vibrations of the ...
3
votes
2answers
145 views

Why is $U(\Lambda)^{-1} = U(\Lambda^{-1})$ for a unitary representation?

This is from the beginning of Srednicki's QFT textbook, where he writes (approximately): In QM we associate a unitary operator $U(\Lambda)$ to each proper orthochronous Lorentz transformation ...
3
votes
2answers
138 views

The Fifth Gamma Matrix

This is regarding $\gamma^5$, the fifth gamma matrix in quantum field theory. I know its defining properties, namely, $$\gamma^5= -i\gamma^0 \gamma^1 \gamma^2 \gamma^3 $$ with ...
3
votes
1answer
89 views

Even-branes in IIA and odd-branes in IIB

The R-R sector of IIA and IIB are respectively given as, $8_s \otimes 8_c = [1]\oplus [3] = 8_v \oplus 56_t$ $8_s \otimes 8_s = [0]\oplus [2] \oplus [4]_+ = 1 \oplus 28 \oplus 35_+$ Now looking at ...