Linked Questions

16
votes
3answers
982 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 ...
13
votes
2answers
776 views

Can a spinor be defined as any quantity which transforms linearly under Lorentz transformations?

Recently I’ve come across a few papers from China (e.g. Xiang-Yao Wu et al., arXiv:1212.4028v1 14 Dec 2012) that make the following statement: ...any quantity which transforms linearly under ...
10
votes
1answer
890 views

What is the physical significance of the double connectivity of $\rm SO(3)$ group manifold?

Is there any physical significance of the fact that the group manifold (parameter space) of $SO(3)$ is doubly connected? There exists two equivalence classes of paths in the group manifold of SO(3) ...
9
votes
3answers
730 views

Understanding Group Theory in QFT

Recently it was asked what the reason for Pauli's Exclusion Principle, and the most well-recieved response looks like hieroglyphics to me: I think that while these "explanations" are all dancing ...
7
votes
1answer
646 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group? [duplicate]

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
6
votes
3answers
392 views

Why do we look at the representations of $SO(3)$ in QM?

I have a bit of an understanding issue why the representations of $SO(3)$ are so important for Quantum Mechanics. When looking at its Irreps one gets the Spin and Angular Momentum operators and thus ...
6
votes
1answer
2k 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 ...
5
votes
2answers
252 views

Do the spinor transformation matrices form a matrix representation of the corresponding Lorentz group?

Suppose $\Psi$ is a Dirac spinor, then let the transformation matrix $S$ be defined as usual: $\Psi'=S(\Lambda)\Psi$, where $\Lambda$ is the Lorentz transformation matrix. Then the questions is: for ...
5
votes
3answers
2k views

Complex numbers in quantum mechanics and in special relativity

Is there a physical relation between the use of complex numbers for the wavefunction in (non-relativistic) quantum mechanics and in special relativity (as formulated in the setting of Minkowski space)?...
5
votes
1answer
603 views

Representations of SO(3) and the classification of relativistic massive particles as in Weinberg's “The Quantum Theory of Fields”

I'm reading about the classification of relativistic massive particles in Weinberg's "The Quantum Theory of Fields", and I found something that doesn't convince me. In Chapter 2, paragraph 5, having ...
4
votes
2answers
413 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 ...
4
votes
1answer
1k views

How does $SU(2)$ group enters quantum mechanics?

What is the reason that $SU(2)$ group enters quantum mechanics in the context of rotation but not $SO(3)$? What really rotates and which space it rotates? It cannot be the physical electron that ...
4
votes
0answers
326 views

Why the Universal Covering Space of a (Spacetime) Symmetry Group?

Potential philosophical issues notwithstanding, it is commonly said that the definition of an elementary particle is an irreducible, unitary representation of the Poincaré group (times a gauge group ...
3
votes
1answer
130 views

Existence of spin-$\frac{1}{2}$ representation corresponds to $\text{SO}(3)$ having double cover?

I come across this article: https://skullsinthestars.com/2016/03/29/1975-neutrons-go-right-round-baby-right-round/ I quote here a part of this article: Spin 1/2 particles like the electron, ...
3
votes
1answer
327 views

Helicity quantization of massless particles

In Appendix B of QFT in a nutshell by Zee, a review of group theory is given. In the last paragraph of the appendix on page 533, he briefly discusses the helicity quantization of massless particles. ...

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