Questions tagged [representation-theory]

The systematic study of group representations, which describe abstract groups in terms of linear transformations of vector spaces, such that group elements or their generators are represented as matrices, reducing group-theoretic problems to linear-algebraic ones.

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61
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20answers
30k views

Comprehensive book on group theory for physicists?

I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that ...
53
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3answers
5k 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 ...
52
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1answer
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Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics: The universal cover: For instance, when representing the rotation group $\mathrm{SO}(3)...
36
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3answers
2k 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 ...
34
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2answers
2k views

Is there an elegant proof of the existence of Majorana spinors?

Almost all standard sources on the existence of Majorana spinors (e.g. Appendix B.1 to Polchinski's "String Theory", Vol. 2) do so in a way I consider inherently ugly: A priori, we are dealing with ...
29
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4answers
7k views

Irreducible tensors concept

This might be a little naive question, but I am having difficulty grasping the concept of irreducible tensors. Particularly, why do we decompose tensors into symmetric and anti-symmetric parts? I have ...
27
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2answers
5k 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 ...
27
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4answers
10k 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?
25
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4answers
5k 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: $$\...
24
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1answer
4k 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 ...
23
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2answers
3k 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 ...
23
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1answer
2k 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 ...
23
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2answers
1k 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 ...
22
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5answers
375 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 $|\phi\rangle=...
20
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6answers
4k views

Why is there this relationship between quaternions and Pauli matrices?

I've just started studying quantum mechanics, and I've come across this correlation between Pauli matrices ($\sigma_i$) and quaternions which I can't grasp: namely, that $i\sigma_1$, $i\sigma_2$ and $...
20
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6answers
3k views

Why is $\theta \over 2$ used for a Bloch sphere instead of $\theta$?

I'm a beginner in studying quantum info, and I'm a little confused about the representation of a qubit with a Bloch Sphere. Wikipedia says that we can use $$\lvert\Psi\rangle=\cos\frac{\theta}{2} \...
20
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2answers
3k 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?
19
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3answers
2k 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 $\mathrm{SU}(5)$" or "the ${\bf 1}$ ...
17
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3answers
1k 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?
17
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2answers
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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 = ...
17
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3answers
2k 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}} = \...
17
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2answers
1k views

Vector spaces for the irreducible representations of the Lorentz Group

EDIT: The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain ...
16
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2answers
583 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 $...
16
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2answers
1k views

Why are non-Abelian gauge theories Lorentz invariant quantum mechanically?

I seem to be missing something regarding why Yang-Mills theories are Lorentz invariant quantum mechanically. Start by considering QED. If we just study the physics of a massless $U(1)$ gauge field ...
15
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1answer
1k views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: $$[L_{\mu\nu},L_{\rho\sigma}]=i(\eta_{\mu\sigma}L_{\nu\rho}-\eta_{\mu\rho}L_{\nu\sigma}-\eta_{\nu\sigma}L_{\mu\rho}+\...
15
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3answers
3k views

Hypercharge for $U(1)$ in $SU(2)\times U(1)$ model

I understand that the fundamental representation of $U(1)$ amounts to a multiplication by a phase factor, e.g. EM. I thought that when it is extended to higher dimensional representations, it would ...
13
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4answers
2k views

Why Lie algebras if what we care about in physics are groups?

In physics, we want irreducible representations of the symmetry group of our system. However, one frequently sees representations of the corresponding Lie algebra being studied instead. Is it that in ...
13
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1answer
2k views

Boosts are non-unitary!

Unlike rotations, the boost transformations are non-unitary. Therefore, the boost generators are not Hermitian. When boosts induce transformations in the Hilbert space, will those transformation be ...
13
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5answers
2k views

Why isn't there a second baryon octet?

Let's temporarily ignore spin. If 3 denotes the standard representation of $SU(3)_F$, 1 the trivial rep, 8 the adjoint rep and 10 the symmetric cube, then it's well-known that $$ 3 \otimes 3 \otimes ...
13
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3answers
3k views

What are particle multiplets in the Standard Model?

The particles of the standard model are often displayed in groupings known as multiplets. I know that this somehow relates to the underlying symmetries of the standard model, which can be viewed as ...
13
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2answers
680 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 ...
12
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2answers
3k views

What does it mean to transform as a scalar or vector?

I'm working through an introductory electrodynamics text (Griffiths), and I encountered a pair of questions asking me to show that: the divergence transforms as a scalar under rotations the ...
12
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2answers
3k 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 $P^{...
12
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1answer
355 views

Can Poincare representations be embedded in non-standard Lorentz representations?

My impression for how Poincare and Lorentz representations are linked in $3+1$ dimensions is: Assuming positive mass for simplicity, irreducible representations of the Poincare group are indexed by ...
12
votes
1answer
377 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 ...
12
votes
1answer
486 views

Representations of the Lorentz group in an arbitrary number of space-time dimensions$.$

Let $\mathrm{SO}(1,d-1)^\uparrow$ be the connected Lorentz group in $d$ dimensions. I am looking for a book/article where its finite-dimensional projective representations are studied in detail. ...
11
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2answers
2k views

In the quantum hamiltonian, why does kinetic energy turn into an operator while potential doesn't?

When we go from the classical many-body hamiltonian $$H = \sum_i \frac{\vec{p}_i^2}{2m_e} - \sum_{i,I} \frac{Z_I e^2 }{|\vec{r}_i - \vec{R}_I|} + \frac{1}{2}\sum_{i,j} \frac{ e^2 }{|\vec{r}_i - \vec{...
11
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3answers
1k views

Confusion about rotations of quantum states: $SO(3)$ versus $SU(2)$

I'm trying to understand the relationship between rotations in "real space" and in quantum state space. Let me explain with this example: Suppose I have a spin-1/2 particle, lets say an electron, ...
11
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2answers
1k views

Why are usually 4x4 gamma matrices used? [duplicate]

As far as I understand gamma matrices are a representation of the Dirac algebra and there is a representation of the Lorentz group that can be expressed as $$S^{\mu \nu} = \frac{1}{4} \left[ \gamma^\...
11
votes
2answers
449 views

What are the assumptions that $C$, $P$, and $T$ must satisfy?

I am not asking for a proof of the $CPT$ theorem. I am asking how the $CPT$ theorem can even be defined. As matrices in $O(1,3)$, $T$ and $P$ are just $$ T = \begin{pmatrix} -1 & 0 & 0 & ...
11
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1answer
305 views

Group notation $\otimes$ and $\oplus$ used for representations of quarks and mesons

I've been trying to figure out this statement from the PDG quark model summary (PDF). Following $\mathrm{SU}(3)$, the nine possible $q\bar{q}′$ combinations containing the light $u$, $d$, and $s$ ...
11
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2answers
2k 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 ...
11
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1answer
2k views

The conjugate representation in $\mathfrak{su}(2)$

Cheng & Li gives the following problem: Let $\psi_1$ and $\psi_2$ be the bases for the spin-1/2 representation of $\mathfrak{su}(2)$ and that for the diagonal operator $T_3$, \begin{align} ...
11
votes
1answer
379 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 ...
11
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2answers
5k 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 ...
10
votes
1answer
1k views

Why don't we construct a spin 1/4 spinor?

I am learning quantum field theory. I understand that the solution of Dirac equation has four states and each corresponds to a spinor. These four states are exactly the eigenstates of the spin ...
10
votes
3answers
3k 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 ...
10
votes
2answers
482 views

Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$?

The number of generators of Lie algebra $sl(2,\mathbb{C})$ is 6, and $sl(2,\mathbb{R})$ has 3 generators, Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$? Say \...
10
votes
1answer
338 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
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1answer
2k 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, $\...