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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Matrix representation of the CAR for the fermionic degrees of freedom

The canonical anticommutation relations (CAR) for a fermionic degree of freedom can be written as follows: $$ a^2 = \left( a^{\dagger} \right) ^2 = 0, $$ $$ a a^{\dagger} + a^{\dagger} a = 1. $$ ...
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95 views

Spinor Understanding: QFT vs pure Representation Theory

I have some questions on terminology used in QM & QFT and (pure mathematical) representation theory treating the concept of "spinor". Let us focus on Dirac spinor as described in https://en....
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1answer
34 views

Symmetry of the scattering super-operator

Suppose we have an initial ensemble described by a density matrix $\rho$ and any given member of the ensemble scatters from one of some set of scattering matrices $\{S_g \equiv O_g S O_g^\dagger : g \...
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45 views

Representation of $SU(2)$, i.e., spin

Let \begin{equation} X= \begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}, \qquad Y= \begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix}, \qquad H= \begin{bmatrix} 1 & 0\\ 0 & -1\\ \end{...
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36 views

Conformal weight of a coset model, and a specific case

Given a coset model $(G\times SO(2d))/H$, what is the expression for its conformal weight (in terms of its central charge or, alternatively, in terms of the highest weights of irreducible ...
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46 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 ...
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92 views

Eigenspaces of the hydrogen atom as representations of $SO(3)$

When we computing the discrete spectrum of the hamiltonian of the hydrogen atom $$H=\Big(-\frac{\hbar^2}{2m} \Delta - \frac{e^2}{r} \large),$$ by some explicit computation we get that eigenspace $...
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74 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 ...
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2answers
192 views

Confusions with gluons. How many of them are there?

Gluons are bicolored objects. They are made out of one color and one anticolor. Therefore, there seems to be nine possible states $r\bar{r},r\bar{b},r\bar{g},b\bar{r},b\bar{b},b\bar{g},g\bar{r},g\bar{...
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149 views

Induced representation in Zee's Group Theory

I am trying to understand the topic of Induced representation of the euclidean Group E(2) in A. Zee's Group theory in a Nutshell in Chapter IV.i3. The Lie algebra of E(2) has three elements $P_1, P_2,...
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251 views

Anticommutator of spin-1 matrices

We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is $\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I$. Does a similar relation hold for the spin-1 representation?
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81 views

Which matrices represent unitary projective representations of ${\rm SO(3)}$?

I was reading this post which triggered the following question. The group ${\rm SO(3)}$ is real orthogonal. However, it is possible to consider representations of ${\rm SO(3)}$ on a complex vector ...
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1answer
101 views

Confusion about trace in the vertex term of Lagrangian

I was reading through Mariano Quirós's lecture notes titled "Finite Temperature Field Theory and Phase Transitions". In Sec. 1.2, the author is calculating the one-loop effective potential at $T=0$. ...
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2answers
84 views

Single sequence of angular momentum ladder in quantum mechanics? — Why there is only a

How do you prove that there is only one sequence of angular momentum eigenstates connected by the ladder operator, within the subspace where the squared modulus of the angular momentum has a given ...
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4answers
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Why do all fields in a QFT transform like *irreducible* representations of some group?

Emphasis is on the irreducible. I get what's special about them. But is there some principle that I'm missing, that says it can only be irreducible representations? Or is it just 'more beautiful' and ...
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Simple/elementary explanation for $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$? [duplicate]

I am preparing a talk on the Eightfold Way, and am attempting to explain the spectra of the light mesons/baryons via representation theory. It will be delivered to students who have never seen ...
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SUSY Loop diagrams from a categorical viewpoint

In the paper "A Prehistory of $n$-Categorical Physics" J. Baez and A. Lauda give an account of the use of category theory throughout physics. In section “Penrose (1971)” starting from page 25 they ...
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1answer
193 views

$z$ component of angular momentum under Lorentz transformation for massless particle

This question is related to this Helicity states. Suppose we have $k=[\omega,0,0,\omega]$. In Weinberg's book The Quantum Theory of Fields: Volume I he defines the state $|k,\sigma\rangle$ as an ...
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How does angular momentum get quantized? [duplicate]

We know that the magnitude and direction of angular momentum is quantized in quantum mechanics. We can explain the quantization with the help of quantum numbers. But actually who is responsible for ...
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1answer
80 views

Helicity states

On page 71 of Weinberg's book The Quantum Theory of Fields: Volume I, he defines the operators $$A=J_2+K_1$$and $$B=-J_1+K_2$$ where ${\mathbf{J }}=(J_1,J_2,J_3)$ are the rotation generators and ${\...
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70 views

Helicity under rotation

Suppose that the state $|p,\sigma\rangle$ (for a massless particle) has 3 momentum ${\bf p}=p_3$ (that is the momentum is in the $z$ direction) and that $J_3|p,\sigma\rangle=\sigma|p,\sigma\rangle$ ...
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There is a classification of matrix representations of Euclidean group $\mathrm{E}(n)$ (also called $\mathrm{ISO}(n)$)?

I'm wondering if someone has already done a classification of matrix representations of the Euclidean group $\mathrm{E}(n)$, more specifically I'm trying to classify $\mathrm{E}(2)$. I watched a ...
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Gamma traceless

I read this Under what conditions is a vector-spinor gamma trace free. And also read many papers about higher spin, but no one explains why irreducible spinor is gamma traceless spinor? Can anyone ...
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352 views

Why are physicists so interested in irreps if in their non-block-diagonal form they mix all components of a vector?

Consider a group $\{G,\circ\}$, with elements $e,g_1,g_2,...$, represented by the matrices $\{D(e), D(g_1), D(g_2)...\}$. If all the matrices can be brought to block diagonal forms by a similarity ...
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1answer
36 views

How to make a triplet out of 2 doublets in the $SU(2)$ representation?

In Y.Grossman and Y.Nir "The Standard Model" book in chapter 4 (non abelian symmetrys) they present the law of whom we can have a triplet and singlet out of 2 doublets name them $\phi_a$ and $\phi_b$, ...
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249 views

Is Velocity Really a Vector?

In non-relativistic physics, physical quantities $Q$ are characterized by how they transform under a Galilean transformation $g \in \mathcal{G}$. $$ Q \rightarrow Q' = D[g]Q$$ where $D[g]$ is the ...
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33 views

Generators of 2D global conformal group in terms of differential operators?

I'm looking for a reference that lists generators of two dimensional global conformal group on a complex sphere in terms of differential operators that may act on quasi primary fields $\phi(z,\bar z)$....
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1answer
100 views

Why is there no state of total spin 0 for spin-1 and spin-2?

To my understanding, decomposing the tensor product of two particles with spins $s_1$ and $s_2$ works as follows: $$\mathcal{H_{s_1}}\otimes \mathcal{H_{s_2}}=\mathcal{H_{s_1+s_2}}\oplus\mathcal{H_{...
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4answers
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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 ...
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68 views

What does it mean to take the tensor product of two reps of the Lorentz group? [duplicate]

If I reduce the Lorentz group to the representation $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$, I can write left and right-handed Weyl spinors respectively as $\left( \frac{1}{2},0 \right)$ and $\left(...
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1answer
76 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 ...
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622 views

Position representation of spin states and spin operators

How can we represent a spin states $ \lvert S_x:+\rangle, \lvert S_y:+\rangle,\lvert S_z:+\rangle ,\lvert S_x:-\rangle, \lvert S_y:-\rangle $ and $\lvert S_z:-\rangle$ in position representation ...
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55 views

Spinor transformations as representations of $\mathrm{SL}(2, \mathbb{C})$

Background In the Weyl representation of the Dirac $\gamma$-matrices, the spinor transformations $S=e^{\frac{1}{2} \omega_{\alpha \beta} \Sigma^{\alpha \beta}} \in \,\mathrm{G}_{\mathrm{L}} \leq \...
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67 views

$3+3$ representation of $SO(4)$

In Zee's Group Theory in a Nutshell book, he says that the antisymmetric tensor $A^{ij}$ furnishes a 6 dimensional representation of $SO(4)$. He further argues that this 6 dimensional representation ...
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65 views

Representation of Poincaré group

Let's consider the most general Lorentz transformation: $x'^{\mu} = \Lambda^{\mu}_{\ \ \nu} x^{\nu} + a^{\mu}$. These transformations form the Poincaré group. The generators of translations of this ...
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149 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{\...
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Connection between $2n$ real fermions and $SO(2n)$

In section 11.4 of "Basic Concepts of String Theory" by Blumenhagen et al, they say: Consider a system of $2n$ two-dimensional real fermion (...) transforming as a vector of $SO(2n)$. I guess they ...
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2answers
46 views

Eigenvalues of quadratic Casimirs of simple Lie groups

I want to find a generic formula for calculating eigenvalue of quadratic casimirs of Lie groups, in terms of Dynkin labels. For a simple example if we take $SU(2)$, with $[R]$ indicating the highest ...
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1answer
126 views

Setting spinors and $SU(2)$ representations on the same patch

I am sorry for the naivety of this question, I am a mathematician and I am trying to put together different ideas. I am trying to understand the vocabulary of physics, in particular, I want to know: ...
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1answer
43 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 ...
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50 views

$SU(3)$ and flavor symmetry technical question

In the HW of a particle physics class I was asked about a global $SU(3)_G$ symmetry of $N$ complex scalar fields that transform as $\phi_i(3)$ with $i=1\dots N$, $i$ is the flavor index. The ...
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64 views

Representation of the Lorentz group using matrices of $SL(2,\mathbb{C})$

There is a correspondence between the Lorentz group and the group $SL(2,\mathbb{C})$. To each Lorentz transformation $\Lambda$ we can associate two matrices $\pm A(\Lambda) \in SL(2,\mathbb{C})$ such ...
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1answer
86 views

Representation and Lie algebra of $SO(3)$

Studyng the book Group Theory in Physics of Wu-Ki Tung, I have read: "... every representation of the [$SO(3)$] group is automatically a representation of the corresponding Lie algebra, (...) a ...
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39 views

Representation of the Lorentz group and correspondence with the $SL(2,\mathbb{C})$ group

We can find a correspondence between the restricted Lorentz group and the group $SL(2,\mathbb{C})$ if to each coordenate $x^{\mu}$ we associate a $2\times 2$ hermitian matrix $X$ given by $$X = x^{\mu}...
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39 views

Matrix representation in angular momentum basis

I'm trying to find a way to verify that the following expansion is valid for any potential, including noncentral ones, $$ \langle \textbf{k}' |V|\textbf{k}\rangle = \frac2\pi\sum_{lm} V_l (k', k) Y_{...
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2answers
172 views

What is a rotation group and how do we get its unitary representation?

The rotation group is ${\rm SO(3)}$. It is the group of $3\times 3$ orthogonal matrices $\{g(\theta)\}$ with unit determinant. So these are already defined in terms of $3\times 3$ matrices. But we use ...

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