7
votes
1answer
103 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
101 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
74 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 ...
8
votes
2answers
146 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 ...
8
votes
2answers
201 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
67 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
139 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?
6
votes
2answers
258 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
351 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
121 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
58 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
75 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 ...
10
votes
2answers
364 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
69 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
144 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
287 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
85 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
100 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
308 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
196 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
376 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 ...
2
votes
2answers
151 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
126 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
261 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
355 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
235 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
121 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
234 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
311 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
217 views

Different representations of the Lorentz algebra

I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
5
votes
0answers
98 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
74 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
897 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
146 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
456 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 ...
4
votes
2answers
417 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 ...
5
votes
4answers
497 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}} = - ...
4
votes
3answers
403 views

Must all symmetries have consequences?

Must all symmetries have consequences? We know that transnational invariance, for example, leads to momentum conservation, etc, cf. Noether's Theorem. Is it possible for a theory or a model to have ...
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)?
10
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 ...
2
votes
1answer
94 views

Charge of a field under the action of a group

What does it mean for a field (say, $\phi$) to have a charge (say, $Q$) under the action of a group (say, $U(1)$)?
3
votes
1answer
273 views

An odd relation with the epsilon/delta invariant tensors of SO(3)

The rotation group SO(3) can be viewed as the group that preserves our old friends the delta tensor $\delta^{ab}$ and $\epsilon^{abc}$ (the totally antisymmetric tensor). In equations, this says: ...
8
votes
2answers
178 views

Wilson Loops in Chern-Simons theory with non-compact gauge groups

VEVs of Wilson loops in Chern-Simons theory with compact gauge groups give us colored Jones, HOMFLY and Kauffman polynomials. I have not seen the computation for Wilson loops in Chern-Simons theory ...
17
votes
2answers
144 views

Can symmetry generators be used for quantization?

Take the Poincaré group for example. The conservation of rest-mass $m_0$ is generated by the invariance with respect to $p^2 = -\partial_\mu\partial^\mu$. Now if one simply claims The state where ...