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6
votes
1answer
336 views

Vector and Spinor Representation in Ramond-Neveu-Schwarz Superstring Theory

I am learning Ramnond-Neveu-Schwarz Superstring theory (RNS theory). I often find the following notation, especially in the closed string spectrum etc.: $$\mathbf{8}_s,\mathbf{8}_v $$ And it is ...
0
votes
1answer
225 views

Angular Momentum Addition Theorem

If I have, for example a particle with $s = 3/2$ and $\ell = 2$, what are the allowed values of $j$? I'm slightly confused because I know that $j = \ell + s$, so surely there is only one allowed ...
3
votes
2answers
610 views

Quantization of orbital angular momentum

Probably a very simple question, but I can't find the answer on the Internet. I know nearly to nothing about quantum mechanics, but in statistical physics I'm confronted with the idea that the orbital ...
1
vote
0answers
82 views

References for Understanding Minahan's N=4 SCFT review

This is about the same paper as this thread: Some questions about chapter I.1 (by Minahan) of the "Review of AdS/CFT Integrability" but it was never answered. I have some different ...
8
votes
0answers
499 views

Decomposing a Tensor Product of $SU(3)$ Representations in Irreps

Can somebody explain in a simple way why, talking about representations, $3\otimes3=3\oplus6$, $3\otimes\bar{3}=1\oplus8$ and $3\otimes3\otimes3=1\oplus8\oplus8\oplus10$? Here $3$ and $\bar{3}$ are ...
2
votes
3answers
123 views

Please explain this statement about Lorentz transformations

I'm reading Sternberg's Group Theory and Physics. I have a question about chapter 1.2 Homeomorphisms. Background: A Lorentz Metric is defined as $||{\bf x}||^2=x_0^2-x_1^2-x_2^2-x_3^2$ And a ...
1
vote
2answers
125 views

Identity as a trivial reducible representation

In particle physics, I was taught that a representation of a group is a function $r: group \rightarrow matrices\,(n\times n)$ such that $r(g_1)r(g_2)=r(g_1g_2)$ and $r(e)=I_{n\times n}$. Then, that a ...
6
votes
1answer
157 views

Are group representations possible when the solution space is not a vector space?

As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector ...
1
vote
0answers
1k views

How is multiplicity given by 2S+1?

Suppose there are two electrons in an atom with $s_1 = \frac{1}{2}$, $l_1 = 1$ and $s_2 = \frac{1}{2}$, $l_2 = 1$. Hence the total $S$ (of the atom) may be +1 or 0. And total $L$ is either $+2$, $+1$ ...
7
votes
2answers
341 views

Why does $\mathcal L = -\frac14 F^{\mu\nu} F_{\mu\nu}$ imply Photons are massless?

The Lagrangian $\mathcal L = -\frac14 F^{\mu\nu} F_{\mu\nu}$ with $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ results in the four-potential's equation of motion $$ \underbrace{\partial^\mu ...
3
votes
3answers
369 views

Quantum mechanical angular momentum and spin formalism/notation

I am currently stuck on the following notation: $\frac{1}{2}\otimes\frac{1}{2} = 0 \text{ (antisym) } \oplus 1 \text{ (sym) }$ No matter what I tried, I couldn't derive the identity. I am sure that ...
1
vote
1answer
149 views

Action of the Lorentz group on scalar fields

The Lorentz groups act on the scalar fields as: $\phi'(x)=\phi(\Lambda^{-1} x)$ The conditions for an action of a group on a set are that the identity does nothing and that $(g_1g_2)s=g_1(g_2s)$. ...
8
votes
1answer
199 views

Why do we classify states under covering groups instead of the group itself?

Why do we always classify states under covering group representations instead of the group itself? For example see the following picture I lifted from 'Symmetry in physics' by Gross So in the first ...
6
votes
1answer
222 views

Equivalent Representations of Clifford Algebra

I'm reviewing David Tong's excellent QFT lecture notes here and am a little confused by something he writes on page 94. We've considered the standard chiral representation of the Clifford Algebra, ...
5
votes
2answers
470 views

Number of Components of a Spinor

I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral ...
1
vote
2answers
315 views

high spin atoms SU(2) representation

I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry. Why not $SU(N)$?
7
votes
5answers
633 views

The role of representation theory in QM/QFT?

I need help understanding the role of representation theory in QM/QFT. My understanding of representation theory in this context is as follows: there are physical symmetries of the system we are ...
3
votes
2answers
493 views

Lorentz transformations in Dirac equation

Let's denote a spinor $\xi$. If $(\theta ,\phi)$ are the parameters of a rotation and pure Lorentz transformation, then how $\xi$ could be written as $$\xi ~\rightarrow~ \exp\left(\ i ...
2
votes
1answer
171 views

Taylor series for unitary operator in Weinberg

On page 54 of Weinberg's QFT I, he says that an element $T(\theta)$ of a connected Lie group can be represented by a unitary operator $U(T(\theta))$ acting on the physical Hilbert space. Near the ...
5
votes
4answers
580 views

Calculating the commutator of Pauli-Lubanski operator and generators of Lorentz group

The Pauli-Lubanski operator is defined as $${W^\alpha } = \frac{1}{2}{\varepsilon ^{\alpha \beta \mu \nu }}{P_\beta}{M_{\mu \nu }},\qquad ({\varepsilon ^{0123}} = + 1,\;{\varepsilon _{0123}} = - ...
1
vote
2answers
470 views

Tensor product decomposition of SU(2)

I have a rather trivial question. I am looking for the decomposition of $1/2\otimes 1/2\otimes 1/2$. It should give, $0,1/2$ and $3/2$. I thought one must get as the overall dimension of this space 8, ...
5
votes
2answers
709 views

Particle as a representation of the Lorentz group

In QFT one may refer to a particle as a representation of the Lorentz group (LG). More accurately - every particle is a quantum of some field $\phi(x)$ that belongs to some representation of the LG. I ...
9
votes
2answers
548 views

Schwinger representation of operators for n-particle 2-mode symmetric states

A bosonic (i.e. permutation-symmetric) state of $n$ particles in $2$ modes can be written as a homogenous polynomial in the creation operators, that is $$\left(c_0 \hat{a}^{\dagger n} + c_1 ...
3
votes
1answer
479 views

Does a spin-2 particle really return to its previous state after 180° rotation?

It is often claimed that spin-2 particles return to their previous state after $\pi$ rotation, just like spin-1/2 particles return after $4\pi$ rotation. But my calculation suggests otherwise. Let z ...
4
votes
2answers
203 views

Are there any known potentially useful nontrivial irreducible representations of the Lorentz Group $O(3,1)$ of dimension bigger than 4? Examples?

Are there any known potentially useful, nontrivial, irreducible representations of the Lorentz Group $O(3,1)$ of dimension more than $4$? Examples? A $5$-dimensional representation? EDIT: Is there ...
2
votes
1answer
148 views

How do representations of an isometry group correspond to degrees of freedom/entropy in a system?

To put the question into context: I am currently writing my bachelors thesis on de Sitter space, specifically, $dS_4$. I am trying to show that while the horizon entropy is finite, the isometry group ...
21
votes
4answers
2k views

Could the Periodic Table have been done using group theory?

These three questions are phrased as alternative-history questions, but my real intent is to understand better how well different modeling approaches fit the phenomena they are used to describe; see 1 ...
12
votes
1answer
541 views

Is this a quaternion representation of the equations of motion of General Relativity?

In The Quaternion Group and Modern Physics by P.R. Girard, the quaternion form of the general relativistic equation of motion is derived from $du'/ds = (d a / d s ) u {a_c}^* + a u ( d {a_c}^* / ...
12
votes
2answers
3k views

Adding 3 electron spins

I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers ...
3
votes
2answers
303 views

Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients

An identity exists for CG coefficients: $$\langle j_1 m_1 j_2 m_2 |J M \rangle = (-1)^{j_1+j_2-J} \langle j_2 m_2 j_1 m_1|J M\rangle,$$ But why is there a phase factor $(-1)^{j_1+j_2-J}$? It seems ...
1
vote
1answer
872 views

Yukawa Coupling of a Scalar $SU(2)$ Triplet to a Left-Handed Fermionic $SU(2)$ Doublet

Suppose we have a field theory with a single complex scalar field $\phi$ and a single Dirac Fermion $\psi$, both massless. Let us write $\psi _L=\frac{1}{2}(1-\gamma ^5)\psi$. Then, the Yukawa ...
1
vote
1answer
456 views

Wigner-Eckart projection theorem

I'm following the proof of Wigner-Eckart projection theorem which states that: $$\langle \bf{A} \rangle ~=~ \frac{\langle \bf{A} \cdot \bf{J} \rangle}{\langle {\bf{J}}^2 \rangle} \langle \bf{J} ...
5
votes
2answers
648 views

Is this a quaternion Lorentz Boost?

The quaternion Lorentz boost $v'=hvh^*+ 1/2( (hhv)^*-(h^*h^*v)^*)$ where $h$ is $(\cosh(x),\sinh(x),0,0)$ was derived by substituting the hyperbolic sine and cosine for the sine and cosine in the ...
1
vote
1answer
112 views

One-Plaquette Action and SU(2)'s Irreducible Representations

I have a typical single-plaquette partition function for a gauge-field $$ Z=\int [d U_{\text{link}}] \exp[-\sum_{p} S_{p}(U,a)]$$ with $U$ as the product of the the $U$'s assigned to each link around ...
1
vote
0answers
43 views

How to obtain deconfined theory from an s-confined N=1 susy gauge theory?

Is there a systematic procedure for obtaining a deconfined theory from an s-confining theory (as defined in hep-th/9610139 for example)?
12
votes
1answer
1k views

How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$ ?

This question is based on problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell (I'm reading this for fun- it isn't a homework problem.) Show, by explicit calculation, that ...
5
votes
1answer
426 views

Wigner-Eckart theorem of SU(3)

I have just come across the Wigner-Eckart theorem and am not sure on how to apply it. How do I find the matrix elements of $\langle u|T_a|v\rangle$ in terms of tensor components and the Gell-Mann ...
3
votes
1answer
328 views

How do I find the tensor components of all weights of a representation of $SU(3)$, e.g. the six dimensional representation $(2,0)$?

How do I find the corresponding tensor component $v^{ij}$ of the six dimensional representation of $SU(3)$ with Dynkin label $(2,0)$?
4
votes
1answer
659 views

Introduction to Physical Content from Adjoint Representations

In particle Physics it's usual to write the physical content of a Theory in adjoint representations of the Gauge group. For example: $24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus ...
2
votes
1answer
228 views

How could $\textbf{S}^2$ not be a multiple of the identity?

I'm self-studying quantum mechanics with Sakurai's book (Modern Quantum Mechanics, 2nd edition) and came across the following in reference to the operator $\textbf{S}^2$: As will be shown in ...
3
votes
1answer
338 views

Angular Momentum Addition Theorem - Sanity Check

Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as: $j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2| $ (Using conventional notation) , but I'm a little unsure how to ...
5
votes
2answers
265 views

If the S-matrix has symmetry group G, must the fields be representations of G?

If the fields in QFT are representations of the Poincare group (or generally speaking the symmetry group of interest), then I think it's a straight forward consequence that the matrix elements and ...
1
vote
1answer
258 views

Irreducible tensor representations with “covariant” indices

As a follow-up of my question on the "most general" $\mathrm{SU}(2)$-symmetric interaction of two spin 1/2 particles, I ponder the following question: Consider an operator acting just on one particle ...
3
votes
1answer
317 views

Constructing the “most general” two-particle spin interaction with $SU(2)$ symmetry

Suppose I want to write down an interaction term for an action for spin 1/2 fermions that is $SU(2)$-symmetric. I start from the most naive general form of such an action: $$S_{int} ~=~ \int_{4321} ...
3
votes
2answers
297 views

A question from Weinberg QFT text

In page 71 Weinberg's QFT, $$A\Psi^{\theta }_{a,b} ~=~(a\cos{(\theta )}-b\sin{(\theta )})\Psi^{\theta }_{a,b}.$$ He says that massless particles represented by $\Psi ^{\theta }_{a,b}$ are not ...
12
votes
1answer
788 views

What is a general definition of the spin of a particle?

In quantum field theory, one defines a particle as a unitary irreducible representations of the Poincaré group. The study of these representations allows to define the mass and the spin of the ...
3
votes
1answer
954 views

General procedure for Clebsch-Gordan expansions

I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps ...
3
votes
3answers
770 views

Building the meson octet and singlet

I am very lost in this topic. I understand that there are $3\times 3$ possible combinations of a quark and an anti-quark, but why should one decide arbitrarily (that's how it appears to me) that one ...
1
vote
1answer
418 views

Decomposition of SU(N) adjoint representation under SU(2)

I am having trouble in demonstrating that under SU(2) transformations the adjoint representation of SU(N) transforms as one spin 1, 2(N-2) spin $\frac12$ and singlets. I am trying to demonstrate it ...
8
votes
1answer
536 views

Identification of the state of particle types with representations of Poincare group

In the second chapter of the first volume of his books on QFT, Weinberg writes in the last paragraph of page 63: In general, it may be possible by using suitable linear combinations of the ...