1
vote
0answers
21 views

(A,B)-Representation of Lorentz Group: Coefficient functions of fields

I have a question regarding the construction of general causal fields in Weinberg's book on quantum field theory. In his conventions a field that transforms according to the irreducible (A,B) ...
2
votes
1answer
198 views

Lorentz group representations in QFT: what's the vector space?

In QFT, a representation of the Lorentz group is specified as follows: $$ U^\dagger(\Lambda)\phi(x) U(\Lambda)= R(\Lambda)~\phi(\Lambda^{-1}x) $$ Where $\Lambda$ is an element of the Lorentz group, ...
1
vote
0answers
92 views

I want to decompose a tensor product using Littlewood-Richardson rule, How do I find the component of this in each irreducible space?

Let me set up the notation I am using. $(abc,de)$ denotes the standard Young tableau where the first row is $abc$ and the second row is $de$. Each young tableau corresponds to the young symmetriser, ...
3
votes
1answer
96 views

From Symmetry Group to Physics Equations

To the extent that I know: There are symmetry groups like the rotation groups SO(3), the Groups of Poincare Transformations,... If the physics of a system has a symmetry group G, then it can be ...
1
vote
0answers
33 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 ...
8
votes
2answers
266 views

What's a lepto-diquark?

This questions refers to Slansky's Group theory for unified model building, page 106 of chapter 7. He assigns the weight $(1)(01)$, which is stepwise projected from $E_6$ to $SU(2)\times SU(3)$, to a ...
10
votes
1answer
102 views

Triality and charge

I have a few questions about triality for the representations of $SU(3)$. (I have seen the wikipedia page, but it does not make the connection with physics.) What is triality, how can you compute ...
3
votes
1answer
142 views

Reducibility of tensor products of Lorentz group representations

Consider the statement: (34.29 in Srednicki's QFT text) $$\tag{34.29} (2,1)\otimes(1,2)\otimes(2,2)~=~(1,1)\oplus\ldots$$ Where of course, $(a,b)$ label representations of Lorentz group in the usual ...
4
votes
2answers
108 views

What guarantees the existence of unitary operators implementing Lorentz Transformations?

This should be a very basic question. In introductory QFT books, often one of the first things we see is the following claim: for every Lorentz transformation $\Lambda$, we can associate an unitary ...
5
votes
1answer
51 views

How can one (formally) determine the particle content of a free field theory?

Here's my question: Suppose I'm given a free field theory, where my fields are functions $\phi:\mathbb{R}^4 \rightarrow V$, and the equations of motion are a system of linear Lorentz-invariant ...
5
votes
2answers
242 views

The Fifth Gamma Matrix

This is regarding $\gamma^5$, the fifth gamma matrix in quantum field theory. I know its defining properties, namely, $$\gamma^5= -i\gamma^0 \gamma^1 \gamma^2 \gamma^3 $$ with ...
6
votes
1answer
168 views

Representations of the Poincare group

Which type of states carry the irreducible unitary representations of the Poincare group? Multi-particle states or Single-particle states?
7
votes
1answer
398 views

Boosts are non-unitary!

The boost transformations are not unitary unlike rotations, the boost generators are not Hermitian. When this induces transformations in the Hilbert space, will those transformation be unitary? I ...
9
votes
1answer
141 views

The difference between $\mathcal{N}=2$ short multiplets and BPS states

I have some questions about the construction of $\mathcal{N}=2$ supermultiplets for chiral matter. I know that the supermultiplet should not include spin one states since they are always in the ...
2
votes
2answers
264 views

Why does a Lorentz scalar field transform as $U^{-1}(\Lambda)\phi(x)U(\Lambda) = \phi(\Lambda^{-1}x)$?

This problem is from Srednicki page 19. Why $U^{-1}(\Lambda)\phi(x)U(\Lambda) = \phi(\Lambda^{-1}x)$? Can anyone derive this? $\phi$ is a scalar and $\Lambda$ Lorentz transformation.
2
votes
1answer
118 views

How to find a particular representation for the gamma matrices?

I asked this question as a subquestion in another thread, but got the answer below and thought it deserved a thread of its own. Two well-known representation of the gamma matrices are the Weyl and ...
2
votes
3answers
177 views

How do you find a particular representation for Grassmann numbers?

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
3
votes
0answers
51 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
votes
1answer
103 views

Is it possible to Vectorialize Quantum Field Theories?

If I take the rules for classical electrodynamics in the covariant formulation (the closest to QFT), I have a tensor that describes the field, $F_{\mu\nu}$. Now we know that we can take some of the ...
-1
votes
1answer
119 views

What is a supermultiplet?

In Quantum field theory by Lewis H. Ryder, a supermultiplet is mentioned with no explanation as to what one is.
4
votes
1answer
222 views

Some questions about the paper, “AdS description of induced higher spin gauge theory”

I am referring to this paper. I guess that in this paper one is trying to relate the massless spin $s$ gauge fields in $AdS_4$ to conformal spin $s$ theory on $S^3$. So am I right that the ...
2
votes
1answer
216 views

Can one prove the full spin-statistics theorem from the spin 0, 1/2 and 1 cases?

Using second quantization for scalar field, spinor field and vector fields, we can get commutation and anticommutation relations for the birth and destruction operators of the fields, which leads us ...
6
votes
2answers
768 views

Irreducible Representations Of Lorentz Group

In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as ...
4
votes
1answer
314 views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
12
votes
1answer
483 views

What really are superselection sectors and what are they used for?

When reading the term superselection sector, I always wrongly thought this must have something to do with supersymmetry ... DON'T laugh at me ... ;-) But now I have read in this answer, that for ...
5
votes
1answer
283 views

Labelling representations using isospin and hypercharge

Can someone explain how isospin and hypercharge can be used to label representations? What is the meaning of the term singlet, doublet etc in this context? In particular how can I use it to label ...
3
votes
0answers
110 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 ...
4
votes
2answers
733 views

Why do many people say vector fields describe spin-1 particle but omit the spin-0 part?

We know a vector field is a $(\frac{1}{2},\frac{1}{2})$ representation of Lorentz group, which should describe both spin-1 and spin-0 particles. However many of the articles(mostly lecture notes) I've ...
8
votes
1answer
519 views

Representations of Lorentz Group

I'd be grateful if someone could check that my exposition here is correct, and then venture an answer to the question at the end! $SO(3)$ has a fundamental representation (spin-1), and tensor product ...
3
votes
0answers
125 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 ...
5
votes
1answer
228 views

About 2+1 dimensional superconformal algebra

I would like to get some help in interpreting the main equation of the superconformal algebra (in $2+1$ dimenions) as stated in equation 3.27 on page 18 of this paper. I am familiar with supersymmetry ...
6
votes
2answers
166 views

Why aren't the spin-3/2 fields in the (3/2,0)+(0,3/2) representation?

Why is it that spin-$\frac 32$ fields are usually described to be in the $(\frac 12, \frac 12)\otimes[(\frac 12,0)\oplus(0,\frac 12)]$ representation (Rarita-Schwinger) rather than the $(\frac ...
4
votes
1answer
339 views

Gauge invariance and the form of the Rarita-Schwinger action

in Weinberg Vol. I section 5.9 (in particular p. 251 and surrounding discussion), it is explained that the smallest-dimension field operator for a massless particle of spin-1 takes the form of a field ...
4
votes
1answer
173 views

Massive excitations in Conformal Quantum Field Theory

Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of ...