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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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144 views

Representation Theory of $SL(2,\mathbb R)$

The representation theory regarding the finite-dimensional representations of $SL(2,\mathbb C)$ is well-understood; namely, they all decompose into irreducibles $V_n$, $\dim(V) = n > 0$. ...
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332 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
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364 views

What type of fields are continuous spin representations?

Continuous spin representations (infinite dimensional representations of the Lorentz group) are pretty rarely discussed, and usually not in that much mathematical details. And usually it is done in a ...
4
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1answer
134 views

What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
4
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167 views

Infinite-dimensionality of unitary representations of non-compact simple Lie Groups

I have a question about the argument given in On finite-dimensional unitary representations of non-compact Lie groups. I have been looking for a good proof for this claim for a little while now. I ...
4
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1answer
295 views

Finite dimensional representations of Lorentz group

I am trying to understand the topic in the title but I found some difficulties. For example, I understand that $\left(\frac{1}{2},0\right)\otimes\left(\frac{1}{2},0\right)=\left(1,0\right)\oplus(0,0)...
4
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325 views

Understanding the Monster CFT

I try to understand what the Monster CFT and its possible connection to 3 dimensional gravity at ($c=24$) is about (see https://arxiv.org/abs/0706.3359) To my best understanding (and please correct ...
4
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118 views

Mathematical Rationale for Fermion and Boson Spin Representations

I am beginning with the statement that: All physical states occur as one dimensional representations of $\mathfrak{S}_n$; they are either bosonic or fermionic. Where a fermionic state of n identical ...
4
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268 views

Why the Universal Covering Space of a (Spacetime) Symmetry Group?

Potential philosophical issues notwithstanding, it is commonly said that the definition of an elementary particle is an irreducible, unitary representation of the Poincaré group (times a gauge group ...
4
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159 views

Characters of extra representations in the double group of O

Looking at the character table for $\overline{O}$ (double group of $O$) in a book, I noticed that two out of three of the additional irreps (with respect to the five irreps from $O$ itself) are ...
4
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87 views

Highest weight unitary representations of $psl(2|2)$

I'm having some trouble understanding how to extend representation theory from Lie algebras to super Lie algebras, in particular with $psl(2|2)$. Ultimately I'm interested in 2D quantum sigma models ...
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209 views

Subgroups of the Clifford Group

We recall the definition of a Clifford group (over $n$ qubits) is the set of unitary transformations: $$\{U: UPU^\dagger\in\mathcal{P}\}$$ where $\mathcal{P}$ denotes the corresponding Pauli group (...
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699 views

Deducing Young Tableaux from symmetries

I have a particular problem, the following. $T^{a_1 \dots a_p;b_1 \dots b_p}$ is a tensor with the following symmetries. 1) $a_i$'s and $b_i$'s are completely antisymmetric, ie restricted to either$...
3
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47 views

Is there an easy way to compute $\exp(-i\pi J_2) |jm\rangle = (-1)^{j-m} |j,-m\rangle$

Is there an algebraic way to compute $\exp(-i\pi J_2) |jm\rangle = (-1)^{j-m} |j,-m\rangle$. I know this is basically the Wigner $d$-matrix (which I can just look up), but how is it derived in this ...
3
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43 views

Weight $h$ as a function of the central charge $c$ in a Unitary Highest Weight Irrep of the Virasoro algebra

A highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by $(c,h)$, with $c$ the central charge and $h$ the "Witt weight" (the weight of the Witt algebra part of ...
3
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121 views

Dimension of gamma matrices in dimensional regularization

When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form $$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$ For instance, in $d$ ...
3
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95 views

Geometry of Affine Kac-Moody Algebras

One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric quantization, using the Kähler structure of various $G/H$ spaces. Can one perform a ...
3
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170 views

General Commutator for Spherical Tensors (reformulated)

Edit: I think what I was looking for wasn't very well understood, so I'm reformulating my question to make it clearer. Hope this helps. In the book of "Angular Momentum: An Illustrated Guide to ...
3
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82 views

What is the physical meaning of Lie congruence classes?

Any weight $\lambda$ characterising a representation of $\mathfrak{su}(N)$ is an element of one of the $N$ congruence classes defined by (ref.1) $$ \lambda_1+2\lambda_2+\cdots+(N-1)\lambda_{N-1}\quad\...
3
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140 views

Why is a spinor not a tensor?

The title says it. why is a spinor not a tensor? I know the transformation rules for a spinor but I cant see why it is not a tensor?
3
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389 views

Matching Dirac/Majorana/Weyl Spinor Degrees of Freedom in Minkowski signature

Question: How do we match the real degrees of freedom (DOF) of Dirac/Majorana/Weyl Spinor in terms of their quantum numbers (spin, momentum, etc) in any dimensions [1+1 to 9+1] in Minkowski signature?...
3
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305 views

Representation Theory of $SL(2, \mathbb{C})$

I'm a PhD. in mathematics (working mainly in complex algebraic geometry), but I'm looking for a "convincing" answer concerning the various applications of representation theory of the group $SL(2, \...
3
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380 views

4-vector from a spinor

Currently reading Aitchison's book on SUSY, and on page 35 (section 2.2) he asks the reader to prove that $\bar{\Psi}\gamma^\mu\Psi=\psi^\dagger\sigma^\mu\psi+\chi^\dagger\bar{\sigma}^\mu\chi$ ...
3
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108 views

Thomas precession, Lie algebra of the Lorentz group and the conservation of energy

If you read this post Thomas Precession, you will see a very good answer by WetSavannaAnimal, on the subject of Thomas Precession, which I am currently working my through, in conjunction with some ...
3
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202 views

Young Tableau Projectors: Does the order of symmetric and anti-symmetric projectors matter?

Given a Young Tableau we find the irreducible basis of an arbitrary tensor by projecting, The projectors are usually defined as first symmetrise over the row entries and then anti-symmetrise over the ...
3
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93 views

$\mathcal{N}=4$ SUSY in $d=3$ versus $\mathcal{N}=2$ in $d=4$

Which is the field content of the hypermultiplet and the vector multiplet in $\mathcal{N}=4 \ d=3$ Supersymmmetry? Is it correct to state that $\mathcal{N}=4$ in $d=3$ has $8$ supercharges, (since ...
3
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121 views

State space of strings: Spin-1 particles in the conformal gauge?

I obviously have a problem with basics of group theory. consider an open string in flat spacetime. there are usually two common gauge to solve the classical problem and quantize the strings: ...
3
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196 views

Isospin and Hypercharge of the SU(2) bps monopole embedding

I am reading the paper Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups - Weinberg, Erick J . In appendix C of this paper the author states, that the solution ...
3
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153 views

Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius

Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained ...
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206 views

Some more questions on conformal spinors of $SO(n,2)$

This is somewhat of a continuation of my previous question. I had stated there that a conformal spinor ($V$) of $SO(n,2)$ can be created by taking a direct sum of two $SO(n-1,1)$ spinors $Q$ and $S$ ...
3
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1answer
111 views

Why particles with certain properties can't exist

This is inspired by a recent post on why a free electron can't absorb a photon, though my question below is about something considerably more general. The argument in the accepted answer goes (in ...
2
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55 views

What is the spinor representation of other groups beside $SO(p, q)$?

I am studying a lecture about superconformal algebras and it claims that there is a superconformal algebra in $d=5$ where supercharges belongs to spinor representation of $F_4$ (which is an ...
2
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1answer
49 views

Question regarding radial raising/lowering operator for isotropic harmonic oscillator

I understand the symmetry structure of the 3D isotropic harmonic oscillator $H = \frac{\mathbf{P}^2}{2\mu} + \frac{1}{2}m\omega^2\mathbf{X}^2$ as follows. The energy levels are $E_N = \hslash \omega (...
2
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1answer
70 views

Relationship between boundary states and primary states of a Kazama-Suzuki model

In [1] and [2] the authors claim that the boundary states (not just the Ishibashi states) of a Kazama-Suzuki model are labelled in the same way as the primary states of the model, so that the boundary ...
2
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1answer
71 views

Observables labelling one-particle states in Quantum Field Theory

I'm studying introductory QFT using the first volume of Weinberg's series, and i'm having problems in understanding how single particle states of the free theory are labelled, i.e. what observables ...
2
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100 views

Finite conformal transformations of fields from infinitesimal

I know that in conformal field theories conformal group acts not by pushforwards but (e.g. for scalar field $\phi$ with conformal dimension $\Delta$) $$ \phi(x) \mapsto \phi'(x') = \left| \frac{\...
2
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1answer
39 views

Wicks contractions of stress-energy tensor and plane partitions

I am working out the number of wick contraction of a number $n$ of stress-energy tensor in 4D CFT. The strategy is as follows: For 1 stress energy tensor $T_{\alpha\beta}$, you have only one ...
2
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62 views

Understanding the idea behind the super-Poincaré algebra

On the Super-poincaré algebra wiki page (https://en.m.wikipedia.org/wiki/Super-Poincaré_algebra), it says: "If Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry ...
2
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101 views

Transformations of contravariant and covariant tensor operators

I've been able to convince myself that a set of contravariant tensor operators $\hat{O}^{x}$ for $x=1,2,...,n$ respond to a small transformation $\hat{A}$ as, \begin{equation} [\hat{A},\hat{O}^{x}]=-(\...
2
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1answer
51 views

Classical mechanics in coadjoint orbits

We know that coadjoint orbits are symplectic manifolds, and they can be used to find unitary representations of Lie groups and stuff, and it's also related to quantization. However, is it true that ...
2
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67 views

Young tableaus for $SO(n)$

I know how to use young tableaus to find irreducible representations and their dimensions of $SU(n)$. Are there similar rules for $SO(n)$?
2
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0answers
71 views

Are there supersymmetry algebras with higher spinor representations?

The super-Poincare algebra contains supersymmetry generators $Q^I$ which satisfy fermionic anticommutation relations. By the higher-dimensional analogue of the spin-statistics theorem, they must ...
2
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1answer
29 views

For what angles (and why) does the equation for finite rotation fail to work?

I am learning rotations and group theory/representations and my lecturer's note mentioned that: "The group is considered connected, but not simply connected [...] As a result, the formula for a ...
2
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0answers
44 views

Existence of an electric/magnetic dipole of a molecule

I have a molecule with a symmetry group $S_3=D_3$. I have to determine if it has a non-zero electric and/or magnetic dipole moment using the representation theory. I'm using the book of Jones "Groups, ...
2
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0answers
32 views

Interpretation of operators applied to systems with higher symmetry

Let me make an example from solid state physics to show what I mean: We can show that for a 2D hexagonal lattice the resistivity is isotropic by noticing that the resistivity tensor is symmetric ...
2
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0answers
157 views

What is the spin of an operator in QFT?

Operators in quantum field theory with $n$ Lorentz indices that are symmetrized and traceless are referred to as spin-$n$ operators. For example, a spin two operator would be \begin{equation} \bar{\...
2
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94 views

Rotation representations for spin higher than 1/2

I was reading through Feynman's Lectures Vol. III when I stumbled upon Eqs. 5.38 which describe the transformation amplitudes for a Stern-Gerlach experiment rotated about the y-Axis for an arbitrary ...
2
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0answers
147 views

Implications of Poincare symmetry: spin and mass

Is it correct for me to say that the symmetry of a quantum system with respect to the Poincare group leads to the concept of mass and spin? The postulates of the special theory of relativity demand ...
2
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0answers
154 views

Formal definition of gauge field and spinors in QFT

I am trying to pin down what spaces a spinor and gluon gauge field exactly occupy. I know that the spinor is a quantity $\psi_{i\alpha f}(\vec x, t)$ where $i$ is a color index in the fundamental ...
2
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142 views

Why isn't weak hypercharge quantized to integer values?

Page 527 of Srednicki's QFT book lists the matter content of the Standard Model as left-handed Weyl fields in three copies of the representation $(1, 2, −1/2)⊕(1, 1, +1)⊕(3, 2, +1/6)⊕(\bar{3̄}, 1, −...