5
votes
3answers
134 views

Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the ...
15
votes
2answers
326 views

Why are only linear representations of the Lorentz group considered as fundamental quantum fields?

As described in many Q&As around here, fundamental quantum fields are expressed as irreducible representations of the Lorentz group. This argument is entirely clear - we live in a ...
8
votes
2answers
120 views

Why gauge $SU(N)$ and not $SO(N)$?

When building models people typically gauge $SU(N)$ but rarely try to gauge $SO(N)$ (the only example I know about is $SO(10)$, but even that isn't quite $SO(10)$ but actually its double cover). At ...
2
votes
0answers
28 views

How does the choice of a particular vacuum in a field theory problem decide the number of Goldstone bosons?

How does the field expansion method (by this I mean expanding your fields about a chosen VEV and plugging into a given potential so that the masses of the fields are given by the coefficients in ...
1
vote
0answers
13 views

How does the choice of a basis decide how many Goldstone bosons there are under spontaneous symmetry breaking?

I have a question about how the basis you choose in a field theory problem semmingly decides how many Goldstone bosons you get after spontaneous symmetry breaking. For SU(2), if you choose the 3 Pauli ...
8
votes
2answers
265 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 ...
3
votes
1answer
86 views

How to construct an invariant Lagrangian under a Lie group $G$ generally?

How to construct an invariant Lagrangian under a Lie group $G$ generally? For example, if we have $SO(5)$'s generators which are constructed by some operators, then the question is that: is it ...
10
votes
1answer
96 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 ...
1
vote
1answer
47 views

Read-off particle from (projected) Dynkin labels

In the review of Slanksy "Group theory for unified model building" in chapter 6: How do one relate the projected Dynkin diagrams from for example $\overline{5}+10$ of $su(5)$ to the corresponding ...
4
votes
2answers
97 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 ...
2
votes
1answer
47 views

Unit determinant for relevant symmetry groups in QFT

When treating QFT we want our theory to be invariant under different symmetry groups, for example, the Standard Model is a non-abelian gauge theory with the symmetry group $U(1)×SU(2)×SU(3)$. ...
9
votes
0answers
151 views

Differential geometry of Lie groups

In Weinberg's Classical Solutions of Quantum Field Theory, he states whilst introducing homotopy that groups, such as $SU(2)$, may be endowed with the structure of a smooth manifold after which they ...
8
votes
1answer
144 views

Assumptions of the Coleman-Mandula Theorem

In the original paper All Possible Symmetries of the S-Matrix, by S. Coleman and J. Mandula, they prove their famous 'no go' theorem regarding the possible extensions of Poincaré symmetry. The ...
6
votes
1answer
119 views

Question about the Noether charge algebra

I'm reading these notes - page 8 and 9 - and I'm a bit confused. If we consider a field $\phi$ (which can be either bosonic or fermionic) transforming as: \begin{equation} \phi(x) \rightarrow \phi(x) ...
1
vote
1answer
121 views

Question on derivation of Ward identity

I'm currently reading these notes about the Ward identity (pages 259 - 261). I will repeat some of the steps to make the question self-contained. Let us consider a local transformation on the field ...
9
votes
2answers
167 views

From representations to field theories

The one-particle states as well as the fields in quantum field theory are regarded as representations of Poincare group, e.g. scalar, spinor, and vector representations. Is there any systematical ...
9
votes
2answers
269 views

In what sense is the renormalization group equation a group?

The renormalization group equation is given by: \begin{equation} \left[\mu \frac{\partial}{\partial \mu} + \beta \frac{\partial}{\partial g} + m \gamma_{m^2} \frac{\partial}{\partial m} - n \gamma_d ...
4
votes
0answers
79 views

Unitary gauge for non-abelian case

I'm reading Chapter 19 of Mandle and Shaw's Quantum field theory. In the first section it is explained that one can go with a $SU(2)$ followed by a $U(1)$ transformation from ...
6
votes
1answer
164 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
2answers
360 views

$(\frac{1}{2},\frac{1}{2})$ representation of $SU(2)\otimes SU(2)$

The representation $(\frac{1}{2},\frac{1}{2})$ of the Lorentz group correspond to a four- vector or a spin-one object. Right? Does it imply that any four-vector is identical to a spin-one object or ...
7
votes
1answer
394 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 ...
3
votes
1answer
196 views

Scalar field transformation and generators

When we do a transformation (norm preserving one) for a given quantity, from what I have understood it seems like there is a representation of the group element for each quantity depending how they ...
2
votes
0answers
62 views

Quantum Field Theory and Lie Theory [duplicate]

I am reading Vol.1 of "The Quantum Theory Of Fields" by S. Weinberg. However I have come to a halt when connected Lie groups were introduced. I have solid knowledge in elementary group theory and ...
2
votes
2answers
104 views

A question about relativistic spin operator

The question comes from Ryder's Quantum Field Theory, 2nd edition. The author was looking for relativistic spin operator. It was concluded that it cannot be $J^2:=\mathrm{J} \cdot \mathrm{J}$, where ...
11
votes
2answers
499 views

What does a $SU(2)$ doublet really mean?

What do we really mean when we say that the neutron and proton wavefunctions together form an $SU(2)$ doublet? What is the significance of this? What does this transformation really doing to the ...
2
votes
0answers
72 views

Interesting identity on $SU(3)$

In arXiv:hep-ph/1307.5414 Grabovsky use an interesting identity which is not derived in the paper: ...
2
votes
2answers
260 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.
1
vote
3answers
316 views

Difference between $SU(2)$ and $SU(2)$ gauge transformations?

I hear this jargon all the time, so what is the difference? (Of course this is nothing special to $SU(2)$, but rather I just took it as an example)
6
votes
1answer
96 views

In which field theories with fermions do string- and fivebrane structures not come up?

A year ago, username @Greg Graviton asked in a thread here about the Spin group as covering of the spatial rotations. A subquestion was: What other groups, even larger than SU(2) are there that ...
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 ...
9
votes
2answers
404 views

How to prove $(\gamma^\mu)^\dagger=\gamma^0\gamma^\mu\gamma^0$?

Studying the basics of spin-$\frac{1}{2}$ QFT, I encountered the gamma matrices. One important property is $(\gamma^5)^\dagger=\gamma^5$, the hermicity of $\gamma^5$. After some searching, I stumbled ...
6
votes
1answer
219 views

A Puzzle about $SO(3)$

Lie algebra of nonabelian group is $[T^a,T^b]=if^{abc}T^c$. For $SO(3)$ case, is the representation $T^a_{ij}=-i\epsilon^{aij}$ fundamental or adjoint? The fundamental representation is defined as ...
5
votes
2answers
507 views

Why does adjoint representation matter in some field theories?

Recently I am reading a paper about monopoles. In several cases, it seems that writing fields in adjoint representation of the gauge group makes a difference. Once it leads to different group after ...
3
votes
2answers
172 views

Under which representation of U(1) transform electron and photon gauge field?

I know that under $SU(2) \times SU(2)$, the left-handed electron transforms under $ ( \frac{1}{2},0 ) $ representation and the vector gauge field $A_\mu$ under $ ( \frac{1}{2},\frac{1}{2}) $. Since ...
3
votes
0answers
145 views

Fields with SO(3) diagonal subgroup symmetry

I read about a Higgs field $\vec{\phi}=\frac{1}{2}a\hat{r}\cdot \vec{\sigma}$ (in the context of 't Hooft-Polyakov monopole) with SO(3) diagonal subgroup symmetry consisting of simultaneous and equal ...
2
votes
1answer
317 views

Noether First and Second Theorem

I have this question related to the the Noether's Theorems. I want to know a rigorous enough enunciation of this theorem, the context is Classical Field Theory without fancy geometrical structures ...
3
votes
1answer
442 views

Why helicity is proportional to the spin of particle and has two values?

How can it be shown without using the little group formalism? Let's have the Wigner's classification for the irreducible represetation of the Poincare group. For the massless case the eigenvalues of ...
4
votes
1answer
257 views

Lorentz group and classification of fields by their transformation under Lorentz transformations

Let's have Lorentz group with generators of 3-rotations, $\hat {R}_{i}$, and Lorentz boosts, $\hat {L}_{i}$. By introducing operators $\hat {J}_{i} = \frac{1}{2}\left(\hat {R}_{i} + i\hat ...
2
votes
2answers
129 views

Lorentz homogeneous group and observables

For generators of the Lorentz group we have the following algebra: $$ [\hat {R}_{i}, \hat {R}_{j} ] = -\varepsilon_{ijk}\hat {R}_{k}, \quad [\hat {R}_{i}, \hat {L}_{j} ] = -\varepsilon_{ijk}\hat ...
4
votes
1answer
255 views

What exactly is the connection between gauge transformations and symmetry groups?

For a given gauge transformation, say, the electromagnetic field, where observable quantities aren't affected by transformations of the form $$\mathbf{A}' = \mathbf{A} + \nabla \chi,$$ $$\phi' = \phi ...
4
votes
2answers
354 views

How to directly calculate the infinitesimal generator of SU(2)

We commonly investigate the properties of SU(2) on the basis of SO(3). However, I want to directly calculte the infinitesimal generator of SU(2) according to the definition $$X_{i}=\frac{\partial ...
5
votes
2answers
1k views

Definition of Casimir operator and its properties

I'm not sure which is the exact definition of a Casimir operator. In some texts it is defined as the product of generators of the form: $$X^2=\sum X_iX^i$$ But in other parts it is defined as an ...
4
votes
1answer
303 views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
6
votes
0answers
103 views

Is the search for a Simple-group-based Electro-Weak theory over?

Just wondering: We know that, in its current form of the $SU(2)_L\times U(1)$, the electroweak theroy rides a wave of huge success. However, is it not possible that the correct simple group ...
5
votes
0answers
102 views

Finding symmetry of a part of an equation, given the group transformation property of another part

I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
3
votes
1answer
79 views

Isometry group from information about the center of the group

I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
18
votes
5answers
969 views

Elegant approaches to quantum field theory

I have been reading Quantum Mechanics: A Modern Development by L. Ballentine. I like the way everything is deduced starting from symmetry principles. I was wondering if anyone familiar with the book ...
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
507 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 ...
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 ...