# Tagged Questions

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### What are differences between Spin(3,1), SL(2,C), SO(3,1) and SU(2) representations? Which one is correct exact representation for spinor fields? [duplicate]

I want to understand which group transformations exactly represent spinor fields. That is, do spinor fields transform under the Lorentz group $\mathrm{SO}(3,1)$ or under $\mathrm{Spin}(3,1)$? What ...
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### Rotating a complex number

Let us begin in a two-dimensional Euclidean plane. The vector is e.g. $\vec{V}(x,y)$ It is often useful – but in this case, it's just a mathematical trick that doesn't make the complex numbers ...
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### What does “action of a gauge group on a particle” mean?

I have come across this phrase left handed fermions transform under $SU(3)\times{}SU(2)\times{}U(1)$ differently from the way right handed fermions do. I am just beginning to learn about how the ...
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### The $U(1)$ charge of a representation

My question is about the reduction of a representation of a group $SU(5)$ to irreps of the subgroup $SU(3)\times SU(2) \times U(1)$. For example the weights of the 10 dimensional representation of ...
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### 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 ...
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### Coset space and transitiviy

I have a question regarding coset space or homogeneous space $SO(n+1)/SO(n)$ which is simply $S^n$. I need some intuition regarding this result. As everyone knows that for a simple case of ...
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### What is the Physical Significance of Tr(A) w.r.t. Matrix Representations in Group Theory

I've seen the post on mathoverflow.SE asking almost the same question, and I have indeed flipped through said answers, but most are in a more general context ie quantum mechanics and do not provide a ...
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### Where do $L_+$ and $L_-$ live, if not in $\mathfrak{so(3)}$?

This question is continuation to the previous post. The lie algebra of $\mathfrak{so(3)}$ is real Lie-algebra and hence, $L_{\pm} = L_1 \pm i L_2$ don't belong to $\mathfrak{so(3)}$. However, ...
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### How symmetry is related to the degeneracy?

I have several questions about symmetry in quantum mechanics. It is often said that the degeneracy is the dimension of irreducible representation. I can understand that if the Hamiltonian has a ...
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### Symmetries in physics

Can you explain me some of the mathematical details of such concept as symmetries? In physics, we have some manifold, and fields are functions on this manifold. On the one hand, we have symmetries of ...
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### Any examples of commensurable subgroups appearing in physics?

I am a mathematician. I am studying and working on Hecke pairs which I am going to give the related definitions in the following. But first let me explain what I am looking for to learn by asking this ...
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### $(\frac{1}{2},\frac{1}{2})$ 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 ...
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### Adjoint representation of the Lorentz group

Is it possible to construct an adjoint representation for the Lorentz group?
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### Matrix representation of a triplet state

The $SU(2)$ triplet state is typically given in the fundamental representation as a column vector, e.g. \vec{\Delta} = \left( \begin{array}{c} \delta^{++} \\ \delta^+ \\ ...
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### Double connectivity of $SO(3)$ group manifold

Is there any physical significance of the fact that the group manifold (parameter space) of $SO(3)$ is doubly connected? EDIT 1: Let me clarify my question. It was too vague. There exists two ...
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### 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 ...
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### Decomposition of Representation Multiplication

How can the multiplication of spinor representations (of $SO(8)$) $8_+ \otimes 8_-$ be decomposed into $8_v \oplus 56_v$? Where can I read more about the decomposition rule of different ...
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### Fundamental Representation of $SU(3)$ is a complex representation

Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule: $\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$. Now ...
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### Conformal group in two dimensions [closed]

How can one show in a group-theoretical way that each of SO(d,2) and SO(d+1,1) is isomorphic to two-copies of Virasoro algebra for d=2?
Let us define a 4D rotation by using two unit quaternions: $$\mathring{q}_l=\frac{a+ib+jc+kd}{\left|a+ib+jc+kd\right|}$$ and $$\mathring{q}_r=\frac{e+ib+jc+kd}{\left|e+ib+jc+kd\right|}.$$ They differ ...