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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Representation of The Poincare Group

I am currently trying to understand the representations of the conformal group. I am following the script by J.D Qualls. At page 29, the author finds the effect of $L_{\mu\nu}$ by "studying the ...
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Representation of $SO^{+}(3,1)$ for scalar fields

As far as i know, the generators of the representation of the group of the orthochronous Lorentz transformations $SO^{+}(3,1)$ can bewritten in the following form: $$J^{\mu \nu} = i(x^{\mu}\partial^{\...
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Figure of $SO(10)$ grand unified theory in $E_6$ from Wikipedia

In this figure from Wikipedia, it shows that the representations of particles of $SO(10)$ grand unified theory, their patterns of charges for particles in the $SO(10)$ model, rotated to show the ...
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Figure of $SO(10)$ grand unified theory from Wikipedia

In this Figure from Wikipedia, it shows that the representations of particles of $SO(10)$ grand unified theory, representations as numbers labeled in several axes. Descriptions: The patterns of weak ...
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Why is it useful to characterize relativistic particle states by this kind of wavefunction?

This question is a followup to my previous one "Why are momentum eigenstates in QFT plane waves? " as it made sense for me to ask this separately, in a self-contained manner. In QFT we have ...
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Are Lie algebras of groups unique?

Take for example $GL(2,\mathbb R)$, the group of $2\times2$ invertible matrices with real entries. By considering small variations from the identity, it is clear that one needs four parameters to ...
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Why are momentum eigenstates in QFT plane waves?

In Weinberg's The Quantum Theory of Fields, Chapter 2, he works out the Wigner classification of the unitary representations of the Poincare group. In particular, one finds that the possible one-...
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What is the motivation for the $\mathrm{Spin}(n)$ group to be the double cover of $SO(n)$? [duplicate]

The $\mathrm{Spin}(n)$ group is defined to be the double cover of $\mathrm{SO}(n)$. In the case of $n > 2$, this agrees with the universal cover. However, for $n=2$, the physically relevant group ...
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Meaning of the symbol: top clockwise arrow $\curvearrowright$ (html code ↷)

In Tachikawa's book on ${\cal N}=2$ Supersymmetric dynamics for pedestrians (pg 72 https://arxiv.org/abs/1312.2684), he uses a symbol ↷. $F$ is the flavor symmetry group and $G$ is the gauge group. $F,...
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What is the inverse map of $R_{jk}(U)=\frac{1}{2}{\rm Tr}(\sigma_j U \sigma_kU^\dagger)$?

Given a $2\times 2$ unitary, unimodular matrix $U\in {\rm SU}(2)$, the (elements of the) corresponding $3\times 3$ rotation matrix $R\in {\rm SO}(3)$ can be obtained from the map $$R_{jk}(U)=\frac{1}{...
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How is $j=1/2$ representation, $U(R(\theta,\hat{\bf n}))=e^{i{\sigma}\cdot{\hat {\bf n}}\theta/2}$, is a projective representation of ${\rm SO}(3)$?

A projective unitary representation of ${\rm SO(3)}$ satisfies $$U(R_1)U(R_2)=e^{i\phi(R_1,R_2)}U(R_1R_2)\tag{1}$$ where $R_1,R_2\in {\rm SO(3)}$. How to show that the $j=1/2$ representation, $U(R(\...
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Lorentz Transformation of massive Spin-1 fields

First, let me give a summary of the relevant background: In the QFT book by Schwartz, in Chapter 8.2.2, we derive a Lagrangian for a massive Spin-1 field: $$ \mathcal{L}=\frac{1}{2} A_{\mu} \square A_{...
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Finite vs infinite group representations

I'm reading about the representation theory of the first chapter in Woit's book. I have a small confusion between finite/infinite sets $M$ and actions finite/infinite. As usual, the action of a group $...
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Are Dirac/Weyl/Majorana fermions exclusive?

I think we can be pretty sure that fermions exist. We have several ways to describe them (Dirac, Weyl, Majorana, maybe someone I'm missing?), with different equations and number of components. My ...
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Massive Spin-1 Lagrangian: Removal of Spin-0 degree of freedom

I am currently reading Schwartz on QFT and the Standard Model and arrived now at Chapter 8.2.2, where he derives a Lagrangian for a massive Spin-1 field. The final Lagrangian looks like this: $$ \...
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How to compute the tensor product $[0,1] \times [2,k]$ in $SP(4)$?

In this paper the authors give in eq.(A.4) the tensor product $[0,a] \otimes [0,b]$, with $[a,b]$ the Dynkin labels for irreps of $SP(4)$. How can one compute the tensor product $[0,1] \otimes [2,k]$, ...
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Little Group of massive particles in moving frame

My understanding of the little group till now was, that we take some standard-momentum and define the little Group as the subgroup of the Poincaré group that leaves this standard-momentum invariant. E....
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Equivalent representations of Poincare group

This question is regarding the paperA wavelet transform for the Poincaré group We have the Poincare group action as $$U(a, \Lambda)\psi(x, t)=\Psi(\Lambda^{-1}(x-a))$$ Then the author defines another ...
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Why use the double cover instead of the universal cover of $SO(2)$?

In QM, we are generally looking for projective representations of the underlying symmetry group (because of Wigner's theorem). So generally, we should be looking for representations of central ...
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In what meaningful way can we talk about the generators of the super-Poincaré algebra being “spinors” or “4-vectors” because of their indices?

It is often presented without much justification that the generators of the super-Poincaré algebra carry indices that imply they are elements of a representation space of the Poincaré algebra. In ...
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Tensor product of Spinorial Left/Right Lorentz Representation

I searched in various textbook (Zee Group and QFT, Weinberg 1-3, Cornwell, Maggiore) and all the similar question on this site but I didn’t find a final answer. My question is simple: What is the ...
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Completeness relations in Group Theory [migrated]

How should I approch the following? A group $G$, order $n$ , has $R^{(\mu)}$ as a irreducible matrix representation of dimension $l_{\mu}$. We have the following ortogonality relation: \begin{equation}...
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Matrix Representation of Four Dimensional Lorentz Transformation

According to Peskin and Schroeder page no.39, Just to see that we have this right, let us look at one particular representation (which we will simply pull out of a hat). Consider 4 x 4 matrices $$ ({\...
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Do spinors form a vector space?

In contradiction to a number of other authors (sample ref below), Gerrit Coddens, at France’s prestigious Ecole Poytechnique, asserts that: 2.2 Preliminary caveat: Spinors do not build a vector space ...
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Majorana fermions in Euclidean and Minkowski signatures - contradiction with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf Somehow I find some inconsistency with his Tables of Euclidean and ...
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The 'mutual' and the 'self' in terms of the 'conjugacy' of Euclidean and Minkowski Weyl fermions

Euclidean and Minkowski fermions are shown in the Table of Wikipedia. (see the bottom https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions) My question is that what does the conjugacy mean ...
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Why there are 2l+1 values of m for any given l where l is the azimuthal quantum number and m is the magnetic quantum number? [duplicate]

whats the reason that for any given value of l(azimuthal quantum number),there are 2l+1 values of m(magnetic quantum number)?why is it true,is it an experimental fact or does it have any mathematical ...
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Rank 3 tensor decomposition

I know the decomposition of general rank two tensors $X_{ab}$ \begin{align} X_{ab} &= X_{[ab]} + \frac{1}{n}\delta_{ab}\delta^{cd}X_{cd} + \left(X_{(ab)}-\frac{1}{n} \delta_{ab}\delta^{cd}X_{cd}\...
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Irreducible representations of $SO(3)$

Let $\pi_l:so(3)\rightarrow gl(v)$ be the Lie Algebra representation of $so(3)$, with spin $l$ half-integer. Let $F_3$ be one of the canonical basis of the Lie Algebra, in particular $$ F_3 = \begin{...
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Operator acting on tensor product of states

Working with $SU(2)$, from Clebsch-Gordan coefficient tables I have for example: $$|j=3/2, m=3/2\rangle \;= |1, 1\rangle \otimes |1/2, 1/2\rangle $$ How can I write $\hat{O}|j=3/2, m=3/2\rangle$ in ...
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How to find the Pauli matrices for $d>3$ spatial dimensions?

I understand what pauli matrices $\sigma_x, \sigma_y, \sigma_z$ do mechanically, but the fact they work still seems magical (not in a good way). It seems like a coincidence that we have 3 spatial ...
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Equivalence of Dirac matrix representations

I'm currently having fun with the Dirac matrices $\gamma^\mu$ which appear in the relativistic Fermion field equation (aka the Dirac equation). They need to satisfy the anticommutation relation $$ \{\...
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Why is the Lorentz boost a non-unitary operator in the single electron theory?

My main reference on the subject is Sakurai's "Advanced Quantum Mechanics". Consider a single electron described by a bispinor that obeys the Dirac equation. The operator corresponding to a ...
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Difference between dual representation and antifundamental representation? And why is it interesting?

I'm confused about antifundamental and dual representations. Let $J_a$ be a set of generators, which satisfy $[J_a, J_b] = i f_{abc} J_c$, so that $e^{i\alpha J} \in \mathcal{SU}(n)$. For the complex ...
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Representation of Lorentz group and its signifiance

I started Quantum Field Theory, I have a hard time understanding how we use the decomposition of Lorentz group as the direct sum of two $SO(3)$. From what I understand, if we take a representation of ...
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Confusion about group representation terminology in physics

Background I see a lot of confusing group representation terminology in physics writing. Here is a typical example, taken from D. J. Griffiths' Introduction to Elementary Particles, talking about ...
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The exact role of representation theory in quantum theory; why projective representations [closed]

I am a mathematician trying to understand the role of representation theory in physics. There have been countless questions on this site like this, but they sadly don't seem to answer me. My ...
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Classifying irreducible subspaces for angular momentum according to symmetrization

I previously asked Proof that $L=S=0$ for filled electron subshells? which motivated me to look more deeply into the restrictions the Pauli exclusion principle places on multi-particle angular ...
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What is wrong with this proof that $h^\vee$ the dual Coxeter number is always 1?

(Not putting this in the math stack exchange because this is all about structure constants which are more familiar to physicists.) Say we have the basis of a Lie algebra $\mathfrak{g}$ with dimension $...
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What is the IRREP of the $p_z$ orbital in a $C_s$ point group?

What is the representation on the $p_z$ orbital in a $C_s$ symmetry group? The $C_s$ symmetric group has two irreducible no. of representations $A^{'}$ ---> $s, x, y, xy$ and $A^{''}$ ---> $...
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Excitations and representation categories

Apologies if this question is a little vague — I'm fuzzy on the details, which is why I ask. In quantum mechanics, we commonly encounter the idea that types of particle-like charges or excitations ...
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Is the representation of dimension 2 of $SU(2)$ also a representation of the Lorentz group?

Is the representation of dimension 2 of $SU(2)$ also a representation of dimension 2 of the part connected to the identity of the group $SO(3,1)$? To better explain my question: The representation of ...
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Spinors and spin group

It seems to me that spinors (pinors) are loosely defined as representations of the spin (pin) group $Spin(p,q)$ ($Pin(p,q)$), which double covers the spacetime symmetry group $SO(p,q)$ ($O(p,q)$). $\...
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Solution to Dirac equation

We take the solution of Dirac equation as 4 component wave function (Dirac Spinor). But how do we know that it can't be a square or rectangular matrix like 4x2 or 4x4 matrix?
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How to decompose $SO(3)$ tensors into irreducible parts?

From A. Zee's book Group Theory in a Nutshell, the author claimed to decompose a $SO(3)$ tensor space full of tensors of form $P^{ijk}$ into irreducible subspaces made of tensors of form $\tilde U, W, ...
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Since two dimensions is enough to faithfully represent $SU(2)$, how can there be irreducible representations of dimension $n>2$?

I am new to Lie goups and representation theory, and I am confused about irreducible representations of $SU(2)$ of dimension $n>2$. Let us take the more intuitive example of $SO(2)$. I seems ...
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Why am I getting this tensor rotation wrong?

Let $$\rho_\theta \equiv \rho(R_\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\tag{1}$$ be a representation of $SU(2)$, and consider the ...
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Dirac spinor in the chiral basis

In the chiral basis, the gamma matrices take the form $$ \gamma^0=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \quad \gamma^j=\begin{bmatrix}0 & -\sigma^j \\ \sigma^j & 0\end{bmatrix} $$...
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How to show that $\sigma^2\psi_L^*$ transforms as a right-handed spinor? (Peskin&Schroeder)

In Peskin & Schroeder, it is written that the quantity $\sigma^2\psi_L^*$ transforms as a right-handed spinor. What confuses me is that I only get the correct result when considering the following:...
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135 views

Positive Matrices Representation

Is it true that any positive matrix $\hat{H}$ can be rewritten in terms of pauli operators ($\sigma_0, \sigma_1, \sigma_2, \sigma_3$) as: \begin{align} \hat{H} = \sum_{i,j}c_{ij} \sigma_i \otimes \...

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