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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23
votes
11answers
4k 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 ...
10
votes
1answer
1k views

How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$ ?

This question is based on problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell (I'm reading this for fun- it isn't a homework problem.) Show, by explicit calculation, that ...
8
votes
2answers
544 views

Lie bracket for Lie algebra of $SO(n,m)$

How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by $$[J_{ab},J_{cd}] ~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$ ...
31
votes
8answers
2k views

Is there something similar to Noether's theorem for discrete symmetries?

Noether's theorem states that, for every continuous symmetry of a system, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any ...
18
votes
5answers
677 views

Tensor Operators

Motivation. I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. ...
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 ...
5
votes
2answers
1k views

Definition of Casimir operator and its properties

I'm not sure which is the exact definition of a Casimir operator. In some texts it is defined as the product of generators of the form: $$X^2=\sum X_iX^i$$ But in other parts it is defined as an ...
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 ...
21
votes
5answers
3k views

What is the usefulness of the Wigner-Eckart theorem?

I am doing some self-study in between undergrad and grad school and I came across the beastly Wigner-Eckart theorem in Sakurai's Modern Quantum Mechanics. I was wondering if someone could tell me why ...
7
votes
2answers
595 views

How does non-Abelian gauge symmetry imply the quantization of the corresponding charges?

I read an unjustified treatment in a book, saying that in QED charge an not quantized by the gauge symmetry principle (which totally clear for me: Q the generator of $U(1)$ can be anything in ...
3
votes
1answer
357 views

Why helicity is proportional to the spin of particle and has two values?

How can it be shown without using the little group formalism? Let's have the Wigner's classification for the irreducible represetation of the Poincare group. For the massless case the eigenvalues of ...
6
votes
1answer
845 views

Why does photon have only two possible eigenvalues of helicity?

Photon is a spin-1 particle. Were it massive, its spin projected along some direction would be either 1, -1, or 0. But photons can only be in an eigenstate of $S_z$ with eigenvalue $\pm 1$ (z as the ...
12
votes
3answers
360 views

How are symmetries precisely defined?

How are symmetries precisely defined? In basic physics courses it is usual to see arguments on symmetry to derive some equations. This, however, is done in a kind of sloppy way: "we are calculating ...
8
votes
3answers
755 views

Group Theory in General Relativity

In Special Relativity, the Lorentz Group is the set of matrices that preserve the metric, i.e. $\Lambda \eta \Lambda^T=\eta$. Is there any equivalent in General Relativity, like: $\Lambda g ...
4
votes
1answer
199 views

Linearizing Quantum Operators

I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. $$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$ The right hand side ...
3
votes
2answers
109 views

Unitrary groups and infinitesimal transformations - Schwingers way of deriving Lie groups

In Schwinger's source theory book, he suggests if $G_a$ are the hermitian generators of the Unitary group, then we have an infinitesimal transformation is given by : $$ G = \sum_{a=1}^n ...
21
votes
4answers
2k views

Could the Periodic Table have been done using group theory?

These three questions are phrased as alternative-history questions, but my real intent is to understand better how well different modeling approaches fit the phenomena they are used to describe; see 1 ...
13
votes
2answers
670 views

Is the G2 Lie algebra useful for anything?

Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional E8 comes up in string theory. But G2? I've always wondered about that one. ...
7
votes
4answers
2k views

How to prove that proper orthochronous Lorentz transformations form a group?

Proper orthochronous Loentz transform are Lorentz transforms that satisfy the conditions (sign convention of Minkowskian metric $+---$) $$\det \Lambda=+1, \qquad \Lambda^0{}_0 \geq +1.$$ How to prove ...
9
votes
2answers
309 views

How to prove $(\gamma^\mu)^\dagger=\gamma^0\gamma^\mu\gamma^0$?

Studying the basics of spin-$\frac{1}{2}$ QFT, I encountered the gamma matrices. One important property is $(\gamma^5)^\dagger=\gamma^5$, the hermicity of $\gamma^5$. After some searching, I stumbled ...
7
votes
1answer
211 views

Double connectivity of $SO(3)$ group manifold

Is there any physical significance of the fact that the group manifold (parameter space) of $SO(3)$ is doubly connected? EDIT 1: Let me clarify my question. It was too vague. There exists two ...
6
votes
1answer
246 views

Fundamental Representation of $SU(3)$ is a complex representation

Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule: $\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$. Now ...
4
votes
2answers
167 views

Lie group Homomorphism $SU(2) \to SO(3)$

The Lie algebra of $ \mathfrak{so(3)} $ and $ \mathfrak{su(2)} $ are respectively $$ [L_i,L_j] = i\epsilon_{ij}^{\;\;k}L_k $$ $$ [\frac{\sigma_i}{2},\frac{\sigma_j}{2}] = ...
6
votes
2answers
259 views

$(\frac{1}{2},\frac{1}{2})$ representation of $SU(2)\otimes SU(2)$

The representation $(\frac{1}{2},\frac{1}{2})$ of the Lorentz group correspond to a four- vector or a spin-one object. Right? Does it imply that any four-vector is identical to a spin-one object or ...
4
votes
2answers
417 views

Number of Components of a Spinor

I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral ...
7
votes
3answers
280 views

Why is the Symmetry Group for the Electroweak force SU(2)xU(1) and not U(2)

Let me first say that I'm a layman who's trying to understand group theory and gauge theory, so excuse me if my question doesn't make sense. Before symmetry breaking, the Electroweak force has 4 ...
6
votes
2answers
123 views

Galilean, SE(3), Poincare groups - Central Extension

After having learnt that the Galilean (with its central extension) with an unitary operator $$ U = \sum_{i=1}^3\Big(\delta\theta_iL_i + \delta x_iP_i + \delta\lambda_iG_i +dtH\Big) + ...
4
votes
2answers
329 views

Is there a 1-1 correspondence between symmetry and group theory?

The professor in my class of mathematical physics introduces the definition of groups and said that group theory is the mathematics of symmetry. He gave also some examples of groups such as the set ...
3
votes
0answers
127 views

Fields with SO(3) diagonal subgroup symmetry

I read about a Higgs field $\vec{\phi}=\frac{1}{2}a\hat{r}\cdot \vec{\sigma}$ (in the context of 't Hooft-Polyakov monopole) with SO(3) diagonal subgroup symmetry consisting of simultaneous and equal ...
4
votes
1answer
205 views

Is the symmetry group of two spin 1/2 particles $SU(2) \times SU(2)$ or $SU(4)$?

This is a simple question. Please forgive me, as I am a lowly experimentalist. Suppose we have two free spin 1/2 particles, i.e. a 4-fold degenerate system. What is the set of symmetry operations ...
3
votes
1answer
59 views

Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is ...
2
votes
0answers
63 views

Is the algebra of a differential equation invariant under transformation?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
2
votes
1answer
113 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 ...
2
votes
3answers
506 views

Textbook on group theory to be able to start QFT

I am very enthusiastic about learning QFT. How much group theory would I need to master? Please could you recommend me a textbook on group theory, which would help me to start QFT?
1
vote
0answers
71 views

Matrix separability preservation under conjugation?

Someone know any paper about matrix separability preservation under conjugation? A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words: $C(P_{1} ...
1
vote
1answer
96 views

What is an isoscalar factor?

I need to find a definition for "the isoscalar factors of 3j-symbols for the restriction $SO(n)\supset SO(n-1)$...denoted by brackets with a composite subscript $(n: n-1)$..." They are given as: $$ ...
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 ...
8
votes
0answers
453 views

What is the difference between the properties of Electron spin and Photon polarization/helicity?

What is the difference between a photon's polarization/helicity and an electrons spin half? I know that the photon is spin 1 but isn't its polarization analogous to spin half? This question stems ...
6
votes
2answers
338 views

Infinitesimal Lorentz transformation is antisymmetric

The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that ...
5
votes
2answers
449 views

Do Lorentz Boosts in the same direction form a group?

I know that two consecutive Lorentz Boosts in different directions produce a rotation and therefore Lorentz Boosts don't form a group. But, my intuition tells me that, Lorentz Boosts in the same ...
2
votes
1answer
164 views

What exactly means “is a singlet under $SU(N)$” [duplicate]

I don't get a grip of what that exactly means. What IS an abstract singlet, doublet,... under $SU(N)$ or other groups?
2
votes
2answers
348 views

Two ways to form SU(2) singlets?

I am trying to reconcile the two ways of forming SU(2) singlets out of a pair of doublets. Method (1): If $v=\begin{pmatrix}v^1\\ v^2\end{pmatrix}$ and $w=\begin{pmatrix}w^1\\ w^2\end{pmatrix}$ are ...
2
votes
1answer
94 views

Charge of a field under the action of a group

What does it mean for a field (say, $\phi$) to have a charge (say, $Q$) under the action of a group (say, $U(1)$)?
-1
votes
1answer
133 views

Symmetry groups [closed]

I am quite new to this subject. I am just repeating in a few words, what I have learned so far: There are 4 fundamental forces of nature: strong, weak, electromagnetism and gravity. Physicists are ...