We know that two SU(2) fundamentals have multiplication decompositions, such that 2⊗2=1⊕3.
In particular, the 3 has a vector representation of SO(3). While the 1 is the trivial representation of SU(2).
I hope to see the precise SO(3) rotation from the two SU(2) fundamental rotations.
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So let us first write two SU(2) fundamental objects in terms of an SO(3) object. In particular, we can consider the following three:
|1,1⟩=(10)(10)=|↑↑⟩,
|1,0⟩=1√2((10)(01)+(01)(10))=1√2(|↑↓⟩+|↓↑⟩),
|1,−1⟩=(01)(01)=|↓↓⟩.
where the |↑⟩ and ↓⟩ are in SU(2) fundamentals. And we shothand
|↑↑⟩≡|↑⟩|↑⟩ and so on.
question: How do we rotate between |1,1⟩, |1,0⟩, |1,−1⟩, via two SU(2) rotations acting on two SU(2) fundamentals? Namely, that is, construct an SO(3) rotation inside the two SU(2) fundamental rotations?
The SU(2) has three generators, parametrized by mx,my,mz; how do we write down the generic SO(3) rotations from two SU(2) rotations?
Let us consider an example, an SU(2) rotation U acting on the SU(2) fundamental (10) give rise to
U(10)=(cos(θ2)+imzsin(θ2)(imx−my)sin(θ2)(imx+my)sin(θ2)cos(θ2)−imzsin(θ2))(10)=(cos(θ2)+imzsin(θ2)(imx+my)sin(θ2))≡cos(θ2)+imzsin(θ2)(10)+(imx+my)sin(θ2)(01)
In other words, the SU(2) rotation U (with the |→m|2=1) rotates SU(2) fundamentals. Two SU(2) rotations act as
UU|1,1⟩=U(10)U(10)=(cos(θ2)+imzsin(θ2)(imx+my)sin(θ2))(cos(θ2)+imzsin(θ2)(imx+my)sin(θ2))
Hint: Naively, we like to construct
L±=Lx±iLy,
such that L± is an operator out of two SU(2) rotations acting on two SU(2) fundamentals, such that it raises/lowers between |1,1⟩, |1,0⟩, |1,−1⟩.
question 2: Is it plausible that two SU(2) are impossible to perform such SO(3) rotations, but we need two U(2) rotations?
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user annie marie heart