What is the general form of the rotation matrix in SU(2) space?

In summary: The ##\hat{n}## and ##\varphi## are simply the components of the right hand vector ##\vec{\varphi}## in the coordinate system defined by the origin and the vector ##\hat{n}##.
  • #1
Splinter1
4
0
Hi. I know that the [itex]\sigma [/itex] matrices are the generators of the rotations in su(2) space. They satisfy
[tex] [\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k[/tex] It is conventional therefore to take [itex]J_i=\frac{1}{2}\sigma_i[/itex] such that [itex][J_i,J_j]=i\epsilon_{ijk}\sigma_k [/itex]. Isn't there a problem by taking these [itex] J_i [/itex] since [itex] \det J_i \neq 1[/itex]? (since we are talking about the special unitary group.)
Also, how does one arrive at the general form of the rotation matrix [itex]e^{i\bf{\sigma} \theta\cdot \bf{\hat{n}}/2}[/itex]? the factor of 1/2 obviously comes from the definition of J above. Where does the [itex] \hat n [/itex] come from?
 
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  • #2
Splinter1 said:
Hi. I know that the [itex]\sigma [/itex] matrices are the generators of the rotations in su(2) space. They satisfy
[tex] [\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k[/tex] It is conventional therefore to take [itex]J_i=\frac{1}{2}\sigma_i[/itex] such that [itex][J_i,J_j]=i\epsilon_{ijk}\sigma_k [/itex]. Isn't there a problem by taking these [itex] J_i [/itex] since [itex] \det J_i \neq 1[/itex]? (since we are talking about the special unitary group.)

No there is no problem here, su(2) is the Lie algebra of the group SU(2) and is a linear vector space. The determinant of the generators is not relevant (you will also notice that the determinants of the ##\sigma##s is -1.

Also, how does one arrive at the general form of the rotation matrix [itex]e^{i\bf{\sigma} \theta\cdot \bf{\hat{n}}/2}[/itex]? the factor of 1/2 obviously comes from the definition of J above. Where does the [itex] \hat n [/itex] come from?

Any SU(2) matrix can be written as an exponentiation of an element of the Lie group su(2). The ##\theta## and the ##\hat n## simply parametrise the three-dimensional Lie algebra su(2).
 
  • #3
Orodruin said:
No there is no problem here, su(2) is the Lie algebra of the group SU(2) and is a linear vector space. The determinant of the generators is not relevant (you will also notice that the determinants of the ##\sigma##s is -1.
I see. Thank you
Orodruin said:
Any SU(2) matrix can be written as an exponentiation of an element of the Lie group su(2). The ##\theta## and the ##\hat n## simply parametrise the three-dimensional Lie algebra su(2).
Ok. And [itex] \hat n [/itex] parametrizes it since it can include any (normalized) 3d vector, thus describing any linear combination, and we can choose it to be normalized since a multiplicative factor in the exponent doesn't matter. Is that correct?
 
  • #4
The general spin-1/2 rotation matrix reads
$$D(\vec{\varphi})=\exp(-\mathrm{i} \vec{\sigma} \cdot \vec{\varphi}/2),$$
where the magnitude of ##\vec{\varphi}## denotes the rotation angle and its direction ##\hat{n}=\vec{\varphi}/|\vec{\varphi}|## in the sense of the right-hand rule.

Since
$$\mathrm{det} D=\exp[\mathrm{Tr}(\ln D)]=\exp(-\mathrm{i} \mathrm{Tr} \vec{\varphi} \cdot \vec{\sigma}/2) \stackrel{!}{=} 1,$$
it follows that
$$\mathrm{Tr} \sigma_j=0, \quad j \in \{1,2,3 ,\}.$$
Further ##D^{\dagger}=D^{-1}## implies that
$$\sigma_j^{\dagger}=\sigma_j, \quad j \in \{1,2,3 \}.$$
Thus the generators of spin-1/2 rotations are the Hermitean traceless ##\mathbb{C}^{2 \times 2}## matrices, which build a vector space and together with the commutator a Lie algebra.
 
  • #5
vanhees71 said:
The general spin-1/2 rotation matrix reads
$$D(\vec{\varphi})=\exp(-\mathrm{i} \vec{\sigma} \cdot \vec{\varphi}/2),$$
where the magnitude of ##\vec{\varphi}## denotes the rotation angle and its direction ##\hat{n}=\vec{\varphi}/|\vec{\varphi}|## in the sense of the right-hand rule.

Since
$$\mathrm{det} D=\exp[\mathrm{Tr}(\ln D)]=\exp(-\mathrm{i} \mathrm{Tr} \vec{\varphi} \cdot \vec{\sigma}/2) \stackrel{!}{=} 1,$$
it follows that
$$\mathrm{Tr} \sigma_j=0, \quad j \in \{1,2,3 ,\}.$$
Further ##D^{\dagger}=D^{-1}## implies that
$$\sigma_j^{\dagger}=\sigma_j, \quad j \in \{1,2,3 \}.$$
Thus the generators of spin-1/2 rotations are the Hermitean traceless ##\mathbb{C}^{2 \times 2}## matrices, which build a vector space and together with the commutator a Lie algebra.
Thank you. How do you see the geometrical meaning of the expression [itex] \exp(-\mathrm{i} \vec{\sigma} \cdot \vec{\varphi}/2[/itex]? i.e. how do you see that it means a rotation at an angle [itex] \varphi [/itex] about the axis [itex] \hat{\varphi} [/itex]? I thought about showing that the rotation matrix doesn't change the axis vector [itex] \hat{\varphi}=\hat{n} [/itex], as the axis should not rotate, but I'm not sure how to do this.
 
  • #6
You can map the three vectors to
##X=\vec{x} \cdot \vec{\sigma}.##
Then the rotation is given by
##X'=D(\vec{\varphi}) X D^{-1}(\vec{\varphi}).##
This is the socalled adjoint representation, which is the fundamental representation of rotations in terms of SO(3).
 
  • #7
vanhees71 said:
You can map the three vectors to
##X=\vec{x} \cdot \vec{\sigma}.##
Then the rotation is given by
##X'=D(\vec{\varphi}) X D^{-1}(\vec{\varphi}).##
This is the socalled adjoint representation, which is the fundamental representation of rotations in terms of SO(3).
But how [itex] \hat{n} [/itex] and [itex] \varphi [/itex] are interpreted as the axis and the angle of the rotation?
 
  • #8
Splinter1 said:
But how [itex] \hat{n} [/itex] and [itex] \varphi [/itex] are interpreted as the axis and the angle of the rotation?

Take an arbitrary vector and decompose it into a part parallel (or ant-parallel) to n, and a part orthogonal to n. The part parallel to n remains invariant (because it is along the axis of rotation), and the part orthogonal to n rotates by angle psi in a plane orthogonal to n.
 

1. What is SU(2) space?

SU(2) space, also known as the special unitary group, is a mathematical concept used to describe the rotation of vectors in three-dimensional space. It is a group of 2x2 unitary matrices with determinant 1, which can be used to represent rotations around any axis in three-dimensional space.

2. How are rotations represented in SU(2) space?

In SU(2) space, rotations are represented by unitary matrices with determinant 1. These matrices have four complex numbers, known as quaternions, as their elements. Each quaternion represents a rotation around a specific axis in three-dimensional space.

3. What is the significance of SU(2) space in physics?

SU(2) space plays a crucial role in various fields of physics, including quantum mechanics and particle physics. It is used to describe the symmetries of physical systems and is often used to represent the spin of particles. The properties of SU(2) space also have implications in the study of quantum entanglement and quantum computing.

4. How is SU(2) space related to the concept of angular momentum?

SU(2) space is closely related to angular momentum, as it is used to represent the rotations of particles in three-dimensional space. In quantum mechanics, the eigenvalues of the SU(2) operators correspond to the possible values of angular momentum for a particle. This is known as the correspondence between SU(2) space and angular momentum.

5. Can SU(2) space be extended to higher dimensions?

Yes, SU(2) space can be extended to higher dimensions, such as SU(3) for four-dimensional space. This allows for the representation of more complex symmetries and rotations in higher-dimensional spaces. However, SU(2) is the most commonly used group for representing rotations in three-dimensional space.

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