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.

Filter by
Sorted by
Tagged with
52
votes
1answer
3k views

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)...
55
votes
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 ...
17
votes
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 ...
17
votes
2answers
6k 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 = ...
61
votes
20answers
31k 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 ...
25
votes
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
votes
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 ...
19
votes
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
votes
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?
28
votes
4answers
11k 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?
27
votes
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 ...
17
votes
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}} = \...
12
votes
1answer
367 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 ...
15
votes
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 ...
4
votes
1answer
250 views

Expansion coefficients of an arbitrary state in the Hilbert space of one-particle states

I was going through my notes on the unitary irreducible representations of the Poincare group and the subsequent construction of one particle states and I stumbled across the following steps in the ...
23
votes
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 ...
4
votes
1answer
266 views

Why is orbital angular momentum quantized according to $I= \hbar \sqrt{\ell(\ell+1)}$?

I simply have no idea how this result is found $$I=\hbar \sqrt{\ell(\ell+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that $I_z=\hbar m_j$. ...
11
votes
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, ...
8
votes
1answer
1k views

How to get result $3 \otimes 3 = 6 \oplus \bar{3}$ for $SU(3)$ irreducible representations?

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that $$ 3\otimes 3 =6 \oplus \bar{3}~? $$ I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...
4
votes
1answer
1k views

How does $SU(2)$ group enters quantum mechanics?

What is the reason that $SU(2)$ group enters quantum mechanics in the context of rotation but not $SO(3)$? What really rotates and which space it rotates? It cannot be the physical electron that ...
20
votes
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 $...
11
votes
2answers
462 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 & ...
4
votes
1answer
684 views

Construction of interaction Lagrangian invariant under isospin $SU(2)$ transformations

The problem considered here is, as the title says, to construct a Lagrangian that is invariant with respect to $SU(2)$ transformations. I will present first the context and the reason of my ...
2
votes
3answers
312 views

Solving the Lie algebra of generators: path from algebra to matrix representation

Given the Lie algebra, what is the systematic way to construct the matrix representation of the generators of the desired dimension? I ask this question here because it is the physicists for whom ...
3
votes
1answer
1k views

Matrix order in Dirac equations

The trace of matrix is always sum of its eigen values , which can be seen if $\hat{U}$ transforms the matrix $\alpha_i$ into it's diagonal form . $$ \begin{pmatrix} A_1 & 0 & \cdots & 0 \...
23
votes
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 ...
11
votes
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^\...
9
votes
2answers
651 views

Does the $\bf{1+3}$ representation of $SU(2)$ also represent $SU(2)\times SU(2)$?

I'm a bit confused about this following issue concerning representations of $SU(2)$. Denote by 1 the 1-dimensional representation of the group $SU(2)$ (=the spin 0). Similarly, denote by 2 and 3 the ...
5
votes
3answers
523 views

Symmetry in terms of matrices

When we encounter a problem in physics which can be expressed in terms of matrices or tensors, why do we decompose the tensor in terms of its symmetric and antisymmetric or trace components? What is ...
29
votes
4answers
8k 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 ...
20
votes
6answers
4k 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} \...
34
votes
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 ...
23
votes
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 ...
5
votes
2answers
1k views

Covariant derivative applied to a vector vs. applied to a matrix?

I know there are (say) two different definitions/representations of the covariant derivative: one is the covariant derivative applied to a vector $F$, which reads as $$DF=\partial F+iAF$$ (adapted ...
13
votes
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 ...
11
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 ...
11
votes
2answers
3k views

What is the relationship between symmetry and degeneracy in quantum mechanics?

Let me remind you about the following classical examples in quantum mechanics. Example 1. Bound states in 1-dim potential V(x). Let $V(x)$ be a symmetric potential i.e. $$V(x) = V(-x)$$ Let ...
10
votes
2answers
466 views

Does the lagrangian contain all the information about the representations of the fields in QFT?

Given the Lagrangian density of a theory, are the representations on which the various fields transform uniquely determined? For example, given the Lagrangian for a real scalar field $$ \mathscr{L} = ...
7
votes
4answers
2k views

How do you find a particular representation for Grassmann numbers?

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
12
votes
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 ...
5
votes
1answer
2k 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 ...
2
votes
2answers
393 views

Representations of Lie algebras in physics

Why is an invariant vector subspace sometimes called a representation? For example in Lie algebras, say su(3), the subspace characterized by the highest weight (1,0) is an irreducible representation ...
7
votes
1answer
390 views

From irreducible representations of the Lorentz algebra to irreducible representations of the Lorentz group

My lecture notes state that we need to classify all finite-dimensional irreducible representations of the proper, orthochronous Lorentz group in order to formulate a QFT for particles with non-zero ...
7
votes
3answers
4k views

Is the adjoint representation of $SU(2)$ the same as the triplet representation?

Is the triplet representation of $SU(2)$ the same as its adjoint representation? Where the convention for the adjoint representation used is the one used in particle physics, where the structure ...
6
votes
3answers
2k views

Proof that $(1/2,1/2)$ Lorentz group representation is a 4-vector

Taken from Quantum Field Theory in a Nutshell by Zee, problem II.3.1: Show by explicit computation that $(\frac{1}{2},\frac{1}{2})$ is indeed the Lorentz vector. This has been asked here: How do I ...
6
votes
1answer
901 views

On finite-dimensional unitary representations of non-compact Lie groups

In this thread Lorentz transformations for spinors, V. Moretti made a claim as follows: "it is possible to prove that no non-trivial finite-dimensional unitary representation exists for a non-compact ...
5
votes
6answers
585 views

Queries about rotational groups $\mathrm{SO}(3)$ and $\mathrm{SU}(2)$ in QM

In a QM text I am using (Sakurai 2nd edition 'Modern Quantum Mechanics'), he describes two rotation groups, namely the $\mathrm{SO}(3)$ rotation group and $\mathrm{SU}(2)$ rotation group (unitary ...
1
vote
1answer
247 views

How to find the remaining subgroup after some Higgs field gets a VEV?

Say we have a group $G$ and a set of Higgs fields in a representation $R$ of $G$. One of the Higgs fields in $R$ gets a VEV, how can I determine the remaining subgroup after this symmetry breaking? ...
2
votes
2answers
129 views

What does matrices act on different spaces mean in QFT?

I have a Dirac kinetic term in a Lagrangian. $$ i\bar{\psi}\gamma^\mu D_\mu\psi = i\bar{\psi}\gamma^\mu\partial_\mu\psi + g\bar{\psi}\gamma^\mu\psi A^a_\mu T^a,$$ However, I usually heard that ...
4
votes
1answer
424 views

Does $GL(N,\mathbb{R})$ own spinor representation? Which group is its covering group? (Kaku's QFT textbook)

In Kaku's QFT textbook page 54, there is a saying: $GL(N)$ does not have any finite-dimensional spinorial representation. This implicates that $GL(N)$ owns infinite-dimensional spinorial ...