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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7
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2answers
1k views

Conceptual interpretation of the left- and right-handed spinor representations of the Lorentz group

I understand mathematically that the Lorentz group's Lie algrebra $\mathfrak{so(3,1)}$ (given by eqns. (33.11)-(33.13) in Srednicki's QFT book) is isomorphic to $\mathfrak{su(2) \times su(2)}$ (given ...
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Some subtleties in quantizing a fermi field

Consider the quantization conditions for a complex Fermi field $\Psi=\Phi_1+i\Phi_2$: $$\{\Psi(x),\Psi(y)\}=\{\Psi^\dagger(x)\Psi^\dagger(y)\}=0,~~~~ \{\Psi^\dagger(x),\Psi(y)\}=\delta(x-y)$$ Compare ...
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1answer
503 views

Tensor product representation of $SO(3)$ in the Hilbert space of particle with spin $S$

For a particle with a spin $S$, the rotation operator is given by $$ e^{iJ_i\theta/\hbar} $$ where $J_i$ is the component of the total angular momentum along the direction of the rotation axis. The ...
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1answer
62 views

How do I extract the groundstate of a quadratic field theory?

I have successfully solved a field theory quadratic in fermonic creation and annihilation operators via Bogolyubov transformation to a diagonal field theory. I now want to extract the groundstate of ...
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1answer
69 views

Anti-commutator relation of supercharges

Reading mutiple references on SUSY (e.g. Baer and Tata's Weak Scale SUSY and A SUSY Primer by S.P. Martin, arXiv:hep-ph/9709356), there seems to be different anti-commutation relation conventions for ...
5
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1answer
653 views

What is the spin of the Kalb-Ramond field?

In bosonic string theory the massless states of the closed string are given by a rank 2 tensor, which is divided into its three irreducible spherical tensors: symmetric traceless, antisymmetric and ...
5
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1answer
409 views

$SU(N)$ Yang-Mills Theory

Yang-Mills theory is based on the gauge group $G$ which we take to be $SU(N)$. Consider an example; $$\mathcal{L}=-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}-\sum_{j=1}^N\bar{\psi}_j(i\gamma^\mu D_\mu-m)\...
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2answers
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Why are usually 4x4 gamma matrices used? [duplicate]

As far as I understand gamma matrices are a representation of the Dirac algebra and there is a representation of the Lorentz group that can be expressed as $$S^{\mu \nu} = \frac{1}{4} \left[ \gamma^\...
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1answer
229 views

What is the $10$ in the $\mathbf{4}\otimes\mathbf{4}$ tensor product of $SO(6)$?

This is the question 22.D in Howard Georgi's Lie Algebras book, I thought about for a minute, but couldn't come up with a plausible answer. It's a fact that the SO(6) and SU(4) algebras are ...
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2answers
662 views

What is the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1) \subset su(2)$?

What is meant by the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1)\subset su(2)$? I have read it above eqn. (10) in this paper http://arxiv.org/abs/0812.3572 but have also heard it mentioned in ...
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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 ...
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379 views

Fierz identity for chiral fermions [closed]

First of all I define the convention I use. The matrices $\bar{\sigma}^\mu$ I will use are $\{ Id, \sigma^i \}$ where $\sigma^i$ are the Pauli matrices and $Id$ is the 2x2 identity matrix. I will use ...
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2answers
77 views

Transformation of $| JM\rangle$ under the group of rotations

I am following the Quantum Mechanics I, Galindo A., Pascual P. and in page 207 explaining the matrix representations of the Rotation Operators in the angular momentum it appears the next (obvious) ...
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Addition of $N$ spin halves

If I have two spin-halves, then \begin{align} \frac{1}{2} \otimes \frac{1}{2} = 0 \oplus 1. \end{align} If I have three spin-halves, then \begin{align} \frac{1}{2} \otimes \frac{1}{2} \otimes \frac{...
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2answers
947 views

Deriving Pauli Matrices

How does one derive using, say, the operator formula for reflections $$ R(r) = (I - 2nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) +...
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1answer
83 views

Representation of Symmetry group

Suppose the SE $\boldsymbol{H}\psi=E\psi$ describes a closed system and $G$ is a symmetry group of the system. Then any transformation in $G$ leaves the form of the SE invariant. It seems plausible to ...
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1answer
518 views

Raising and lowering operators for a composite isospin $SU(2)$ system

Consider pion states composed of $q \bar q$ pairs where $q \in \left\{u,d \right\}$ transforms under an $SU(2)$ isospin flavour symmetry. These bound states transform in the tensor product $R_1 \...
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825 views

What is the meaning of SU(2) triplet scalar field? [closed]

The following is an about a Left-Right Symmetric model. $SU(2)\otimes SU(2)$ $(2\otimes 2=3\oplus 1)$ will generate a triplet, which in Left-Right Symmetric model is $$\vec{\Delta}=\begin{pmatrix}\...
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1answer
904 views

What is the Lorentz group representation for a general spin?

Setup, as I understand things so far: One way to think about where the spin of a quantum field comes from is that it is a consequence of the ways that different types of fields transform under ...
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1answer
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How to write the Clebsch-Gordan decomposition in tensor notation

Let be $G$ a Lie Group and $\textbf{N}$ its complex representation. It is known that any state $|\ ab\ \rangle\in \textbf{N}\otimes\textbf{N} = \oplus_I\textbf{r}_I$ may be decomposed through the ...
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3answers
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Representation under which Pauli matrices transform

In Peskin and Schroeder's Quantum Field Theory, there is an identity of Pauli matrices which is connected to the Fierz identity, (equation 3.77) $$(\sigma^{\mu})_{\alpha\beta}(\sigma_\mu)_{\gamma\...
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2answers
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Lie groups with same algebra

I had a problem when considering symmetry breaking in an SO(4) gauge theory: $\mathcal{L} = \left| D_\mu\phi \right|^2$ where $D_\mu$ is the SO(4) covariant derivative. Then assuming there is some ...
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2answers
455 views

Bilinears in adjoint representation

Below are two statements from my notes and I am trying to verify them explicitly. In both cases the fields are assumed to transform under the fundamental representation of $O(N)$ - --'The kinetic ...
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1answer
255 views

Bifundamental representations [closed]

Can someone give me explicit examples (in matrix form) of bifundamental representations? Illustrative would be for instance: a) SU(3) x SU(2) b) SO(4) x U(1) c) E6 x U(1) but other you may have ready ...
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2answers
626 views

Does there exist finite dimensional irreducible rep. of Poincare group where translations act nontrivially?

I read several textbooks of QFT and find that there are two ways to classify the particles or fields. The first one is to study the irreducible representation of Lorentz group (or exactly the ...
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0answers
63 views

Example of a symmetry and the group with which it is modelled? [duplicate]

Could you please provide a specific example of a symmetry and the group with which it is modelled? I am beginner to study symmetry in physics, please answer with just an example. This question is ...
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1answer
419 views

Wigner-d matrices for higher (than 1/2) spins

I’ve been reading ¨Halzen, F., and A. D. Martin. Quarks and Leptons. New York: Wiley Text Books, January 1984. ISBN: 9780471887416¨, and I’d like some clarification of a concept, please: I’m ...
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2answers
536 views

Why representations instead of just groups?

This question is essentially asking for a clarification on what has already been said in this one. What I don't understand is why it is the representations that are important in Quantum Field Theory ...
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0answers
178 views

What's the relation between representation theory and mass / electric charge?

This is a follow-up on this answer, where ACuriousMind wrote Formally, both the mass and the charge classify certain irreducible representations of the Poincaré group and the circle group, ...
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316 views

Non-physical representations of double group

In group theory, to account for electron spin, double group is introduced. The key difference between an ordinary point group and a double group is an extra element $\bar{E}$ with the meaning of a $2\...
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1answer
127 views

Decomposition of the adjoint representation of a spontaneously broken compact group

Let be $G$ a compact group, symmetry of the theory I am working with. $G$ is broken into one subgroup $H$. I define the generators of G as $T_A = \{T_a,T_\hat{a}\}$, where the first are the unbroken ...
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1answer
568 views

Number of supercharges in ${\cal N}\ge 1$ SUSY in $d\ge4$ dimensions

I have a couple of questions concerning supercharges and superalgebras: In four-dimensions, the minimal spinor representations are Weyl spinors. These have 4 real degrees of freedom (dof). This means ...
7
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2answers
233 views

Poincare representations for interacting field theory

I was going through Rudolf Haag's memoir http://link.springer.com/article/10.1140%2Fepjh%2Fe2010-10032-4 and came across these lines: '..in quantum field theory (or for any system of interacting ...
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1answer
282 views

What is physically irreducible representation?

When I use bilbao crystallographic server recently, I noticed a notation called physically irreducible representation. Paper says it is a direct sum of two complex conjugate representations (if $\...
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1answer
3k views

Rotations in Bloch Sphere about an arbitrary axis

I am trying to understand the following statement. "Suppose a single qubit has a state represented by the Bloch vector $\vec{\lambda}$. Then the effect of the rotation $R_{\hat{n}}(\theta)$ on the ...
2
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1answer
500 views

How is the Lorentz group related to spin? [closed]

I've been reading about the agebra of the Lorentz group. It is given by, $$G\equiv SO(1,3) ~\cong~ SU(2)\times SU^*(2)$$ Now, representations of this group $G$ as labelled by $(j,j')$ where $j$ is ...
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1answer
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Matrix for Ladder Operators?

I found this website which shows how to derive the matrices for $L_{+}, L_{-}$ and while I understand the derivation of the equation for $<lm|L_{+}|lm'>$ and $<lm|L_{-}|lm'>$ I do not ...
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1answer
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Spin 3/2 matrices in terms of Pauli matrices

Let $\sigma_i (\frac{3}{2})$ be the three generators of the irreducible spin 3/2 representation of $SU(2)$ (see http://easyspin.org/documentation/spinoperators.html for their explicit forms). ...
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3answers
545 views

Is the local Lorentz transformation a general coordinate transformation?

There is a saying in Nakahara's Geometry, Topology and Physics P371 about principal bundles and associated vector bundles: In general relativity, the right action corresponds to the local Lorentz ...
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2answers
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Good reference on the parametrization of $SU(3)$ and $SU(N)$

For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation: $s_2=\begin{bmatrix} e^{i\alpha}\cos(\theta) & -e^{-i\beta}\sin(\theta) \\ e^{i\beta}\sin(\...
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0answers
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Representation of Lorentz Tranformation on Fields and Wigners theorem

I've been reading about symmetries and I haven't been able piece this information together. I've the Lorentz transformation $$x^\mu \mapsto x^{\rho} = \Lambda_\nu^\mu x^\nu$$ First off, arn't we ...
6
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1answer
320 views

Dirac Spinors as Representation of $SL(2,\mathbb{C})$ over Grassmann algebra

Recently, I've learned that the Clifford algebra can be regarded as the quantization of Grassmann algebra. This is shown from the following two papers by Berezin. 'Classical spin and Grassmann ...
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0answers
298 views

Writing Breit-Pauli spin-spin-coupling Hamiltonian as a sum of irreducible spin tensor operators

The spin-spin interaction part of the Breit-Pauli Hamiltonian for two electrons contains the term \begin{equation} 3(\mathbf S_1 \cdot \mathbf r)(\mathbf S_2 \cdot \mathbf r) - r^2 \mathbf S_1 \cdot \...
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2answers
712 views

Chiral symmetry breaking and confusion with terminology

I am confused about Chiral symmetry breaking and the terminology we use. First of all, I think the symmetry is started with taking quark masses zero and writing the Lagrangian as; $$ L=-\frac{1}{4}(F_{...
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1answer
154 views

Use Cartan subalgebra in spinor representation to find weights of vector representation

For $SO(2n)$ we can construct the lie algebra elements by using antisymmetric combinations of $\gamma_\mu$ which obey the Clifford algebra. Up to some prefactor the elements $ S_{\mu \nu} = \alpha [\...
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1answer
109 views

Implication of rotational symmetry on scattering matrix/ scattering cross-section [closed]

Coleman, in his notes, uses the statement "In NRQM, rotational invariance simplifies scattering problems." In general, I want to know how does it work? How does the rotational invariance help ...