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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156 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 $I_z=\hbar m_j$. ...
1
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1answer
85 views

Total angular momentum operator

How do the eigenfunctions of the total angular momentum operator analytically look like? I mean the operator is given by $J = L+S$ so the eigenfunctions have to be tensor-product states, right? Can ...
3
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3answers
429 views

Why does the electron spin with a particular tilt?

I found this image for the classical description of the electron spin at hyperphysics Can you explain why the axis of rotation makes an angle of 60° with the z-axis and how this particular ...
1
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1answer
51 views

Why do we restrict the maximal supercharge to 32?

Many supersymmetry textbook state that the maximal supersymmetry in any dimension has 32 hermitian supercharges. (Actually for lowest number of supersymmetry $N=1$ the highest dimension is $D=11$) I ...
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1answer
137 views

How do states in Hilbert Space act like irreducible representations?

I am reading Georgi's book on group theory and I came across this sentence..." Hilbert space of any parity invariant system can be decomposed into states that behave like irreducible representations". ...
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1answer
99 views

Interpretation of vector mesons in QCD

It is well-known that scalar mesons are interpreted as pseudogoldstone bosons which is connected with spontaneous broken $SU(3) \times SU(3)$ symmetry to $SU(3) \times SU(3) / SU(3)_{chiral}$. Is ...
2
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1answer
65 views

Decomposition of the gravitino into helicity $\pm \frac{3}{2}$ and $\pm \frac{1}{2}$ components

I'm reading this book on string theory. When they decompose two dimensional gravitino (formula 7.16) $$ \chi_\alpha = \frac{1}{2}\rho^\beta \rho_\alpha \chi_\beta + \frac{1}{2}\rho_\alpha \rho^\gamma \...
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1answer
81 views
3
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1answer
145 views

Is an electron technically a set of two particles?

The electron - described as a four-spinor in the Dirac equation - transforms according to the $(1/2,0)\oplus(0,1/2)$ representation of the Lorentz group, so it is actually a direct sum of a left- and ...
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1answer
377 views

How do you add angular momentum of three or more particles in quantum mechanics?

I'm trying to find some information on how to add the angular momentum of three or more particles, but all the sources I look at deal with only two. In this case I understand that if the angular ...
2
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1answer
137 views

Embedding of particles into fields

For the classification of particles (Wigner 1939), we look for unitary representations of the Poincaré/Lorentz group. There are are only infinite-dimensional (non-trivial) unitary representations! To ...
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2answers
236 views

Explicit Spinor Representations in $SO(3)$ and $SO(4)$

The beautiful physical interpretation of the decomposition of the elasticity stress tensor into it's irreducible representations (mentioned in context below) has inspired the following question on ...
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0answers
82 views

Relation between representations/classifications

Generally a quantum system can be characterized in the following way: its states form a representation space for every symmetry group of that system. The representation has to be unitary (or anti-...
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1answer
82 views

Tensor operators and transformation of $O^s_{\ell}|j,m,\alpha\rangle$

In H. Georgi's Lie Algebras in Particle Physics one defines a tensor operator transforming under the spin-$s$ representation of $SU(2)$ as the set of operators $O^s_{\ell}$ (for $\ell=-s...s$) such ...
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3answers
937 views

Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $

I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
2
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0answers
64 views

manipulations with SU(N) Nekrasov partition function

Think of a Young tableau $R$ as collection of rows $y_1 \geq ... \geq y_d > y_{d+1}=0$ and all others zero, with $\ell(Y):= \sum_j y_j$ and for a box $s=(i,j)\in R$ we have $a_Y(s):=y_i-j$ and $l_Y(...
8
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1answer
173 views

How do you build a Lagrangian in particle/nuclear physics? (A specific example)

I know that the terms in the Lagrangian needs to be scalars (with respect to Lorentz symmetry etc.). Also I know that [see C. G. Tully (EPP) p. 85] in general, for $\psi$ in the fundamental ...
2
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1answer
100 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 ...
0
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1answer
178 views

Why do particles have spins such as $1/2$, $3/2$, $5/2$? [duplicate]

What does it mean to have 'half' spin? I have looked on Wikipedia and a few youtube videos on spin but they don't explain what it means to have $1/2$ spin. I am 18 and only starting to learning about ...
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1answer
55 views

What is the point of defining the lie algebra of the proper Lorentz group in a “covariant” way?

In Muller-Kirsten's book Introduction to Supersymmetry, the author first defines the proper Lorentz group's lie algebra basis in the standard manner - antisymmetric matrices consisting of $0$s and $\...
3
votes
2answers
274 views

Why is the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group realized as the vector space of complex $2\times 2$ matrices?

Why can we write an arbitrary object $v_{a \dot{b} }$ our transformations in this basis act on as $$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix} 1&0 \\ 0&1 \...
3
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2answers
141 views

Identifying irreps of $SU(2)$

How does one verify that, the representations of $SU(2)$ corresponding to $j=1/2$ or $j=1$ is irreducible? I think showing the irreducibility (taking the representative matrices into a block-diagonal ...
2
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2answers
101 views

Why can tensors be broken up into parts?

I have found these notes: http://www.physics.usu.edu/Wheeler/QuantumMechanics/QMWignerEckartTheorem.pdf Which state on page two that a matrix (M) can be broken up into rotationally independent pieces ...
0
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1answer
105 views

Why are Majorana fields usually used to introduce gravity in the Rarita-Schwinger Lagrangian?

When first introducing the gravitational interaction for a spin-3/2 Rarita-Schwinger field, Majorana fields are usually used (see for example here at chapter 4, or in Ramond, (6.4.112) ). Why is this?...
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2answers
237 views

If $v_{a \dot{b}}$ transforms like a four-vector, what does $v_{a}^{\dot{b}}$ describe?

The $( \frac{1}{2}, 0)$ representation of the Lorentz group acts on left-chiral spinors $\chi_a$, the $( 0,\frac{1}{2} )$ representation on right-chiral spinors $\chi^{\dot a}$. The $( \frac{1}{2}, \...
9
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1answer
203 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 $s$ ...
4
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0answers
138 views

Decomposing a representation under a subgroup [closed]

I am trying to understand what is the method for decomposing representations of a group under one of its subgroups. I already had a look in Slansky, but I could not extract a concrete set of rules/...
5
votes
1answer
319 views

Branching rules for $SU(3)$

How does one compute the branching rules for $SU(3)\to SU(2)\times U(1)$.? In particular, I do not know how to put the abelian charges. Take for example the adjoint $\mathbf{8}$ of $SU(3)$. I can ...
2
votes
1answer
113 views

Representations of subalgebra in the super virasoro algebra

In the Virasoro algebra, which is generated by $L_n$, one has the obvious subalgebra spanned by $L_{-1}$ ,$L_{1}$ and $L_{0}$ which is isomorphic to the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$. The ...
5
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0answers
356 views

How to calculate $3\otimes 3$ and $3\otimes 3\otimes 3$ in $SU(3)$? [closed]

EDIT: I have boiled my question down to How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? A derivation would be nice too. OP: I know that I can ...
2
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2answers
197 views

Representations of Lorentz group on Hilbert spaces

Question 1: Representation theory and group theory are quite new subjects for me and I'm having troubles in understanding why a representation of a group on Hilbert space act on operators as (here I ...
8
votes
2answers
660 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 ...
3
votes
2answers
382 views

Lorentz algebra and its generators

I'm reading Maggiore's book A Modern Introduction to Quantum Field Theory and I'm getting a bit confused when he writes about Lorentz algebra: $$K^i = J^{i0},$$ $$J^{i}=\frac{1}{2}\epsilon^{ijk}J^{...
2
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1answer
112 views

Unitary representations of $SO(3)$ and $so(3)$

According to my skript: Quantum mechanic states $ψ ∈ \mathcal H$ changes under a rotation $R ∈ \text{SO(3)}, \vec{x} \rightarrow R\vec{x}$ according to $ψ \rightarrow U(R)ψ$, whereas $U(R)$ is a ...
2
votes
2answers
266 views

Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
10
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2answers
501 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 ...
8
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2answers
293 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} = ...
0
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4answers
124 views

Conceptualization and modelling of spin

I'm trying to get a decent understanding of the Bell inequality, and so am trying to understand spin both conceptually and mathematically. When I picture spin, I imagine a sphere rotating about its ...
4
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1answer
226 views

Irreducible Representations from Cartan Matrix

I need some help understanding how you can construct the irreducible representations of an algebra from knowing its roots or its Cartan matrix, which I am being told you most certainly can. ...
0
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1answer
109 views

Eigenvalues of Angular Momentum in Quantum Mechanics

The eigenvalue equation of the $L^2$ operator is given by $$L^2f_l^m = \hbar ^2l(l+1)f_l^m$$ Side: So a determinate state for some observable $Q$ is a state where every measurement of $Q$ returns ...
2
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0answers
83 views

Difference between the 1/2 representation of $SU(2)$ and the (1/2,1) representation of $SU(2)\times SU(2)$? [closed]

What's the difference between the $j = 1/2$ representation of $SU(2)$ and the $(j,j') = ( 1/2 , 1 )$ representation of $SU(2)\times SU(2)$?
7
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2answers
227 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 ...
2
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1answer
91 views

Ambiguity in ordering of isospin states for Clebsch-Gordan coefficients

In studying isospin for nuclear physics, I am confused a bit by an ambiguity I found. If a process that goes from $K^- + p \rightarrow \Sigma^0+ \pi^0$, I can write the isospin for the left hand side ...
5
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1answer
694 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 ...
1
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1answer
75 views

Energy as charge with respect to time translations in QM

Consider a non relativistic quantum mechanical system with Hamiltonian $\mathcal{H}$, and denote the states by $\psi \equiv \psi(t) \equiv | \psi(t) \rangle$. From the Schrödinger equation we know ...
2
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1answer
105 views

Solution space of a differential equation with 3D rotational symmetry

We know that the space of solutions will be invariant under 3D rotations, but why can we say that the space of solutions will constitute a representation of the rotation group $SO(3)$? We know that a ...
0
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1answer
132 views

Derivation of the irreducible representations of SO(3)

Is there a way to derive the representations of $SO(3)$ without the usual method with the ladder operators which also gives the ones of $SU(2)$? The usual way to do these calculations is to start ...
1
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1answer
128 views

What do position and momentum representations represent in QM?

In QFT we classify field operators according to how they transform under a given symmetry, i.e. in their being a basis for some representation of the symmetry group of the Hamiltonian/Lagrangian. This ...
3
votes
2answers
373 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 $$DF=\partial F+iAF$$ (...
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30 views

What experiments don't confirm Poincare group-based representation of particle behavior

What experiments don't confirm Poincare group-based representation of particle behavior? Sources: http://en.wikipedia.org/wiki/Particle_physics_and_representation_theory http://en.wikipedia.org/...