##SU(2, \mathbb C)## parametrization using Euler angles

In summary, the SU(2, ℂ) group, which represents the unitary transformations of two complex dimensions, can be parametrized using Euler angles. This parametrization involves expressing elements of SU(2) in terms of three angles—typically denoted as θ, φ, and ψ—allowing for a clear geometric interpretation of rotations in three-dimensional space. By using these angles, one can construct the corresponding unitary matrices that describe the transformations in a systematic way, facilitating applications in quantum mechanics and related fields.
  • #1
cianfa72
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TL;DR Summary
About the ##SU(2, \mathbb C)## parametrization using Euler angles.
Hi,
I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements

\begin{pmatrix}
e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\
ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}
\end{pmatrix}
where ##\theta, \psi## and ##\phi## are the three Euler angles. In particular ##\theta## runs in the closed interval ##[0, \pi]## whereas ##\psi## and ##\phi## in the range ##[0, 2\pi]## -- such a parametrization includes all and only ##SU(2 , \mathbb C)## group elements. That is actually a "closed box" in ##\mathbb R ^3## so to get a chart from it we need to exclude the box "boundary". This way we get a not global chart for ##SU(2)##. Since we know it is homeomorphic to ##\mathbb S^3## we can cover it with at least two charts.

Apart the above chart, which is one of the other charts for it ? Thanks.
 
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  • #2
Off topic: Can you quote a textbook that uses the notation ##SU(2,\mathbb C)##?
 
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  • #3
martinbn said:
Off topic: Can you quote a textbook that uses the notation ##SU(2,\mathbb C)##?
I think it is not a standard notation, see for instance this lecture.
 
  • #4
Btw the following map ##\varphi: \mathbb C^2 \rightarrow \mathbb C^4##
$$(a,b)\mapsto \begin{bmatrix}
a & -\overline b \\
b & \overline a
\end{bmatrix}$$
is injective and continuous (as map from ##\mathbb C^2## to ##\mathbb C^4##). The inverse map ##\varphi^{-1}## restricted from the image ##A=\varphi(\mathbb C^2)## to ##\mathbb C^2## should be the projection ##\pi|_A##. Hence it is continuous in the subspace topology from ##\mathbb C^4## the set ##A## is endowed with. Therefore ##\varphi## is homeomorphis with its image ##A## (in the subspace topology).

Is the above correct ?
 
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  • #5
Any comment ? Thanks.
 
  • #6
cianfa72 said:
TL;DR Summary: About the ##SU(2, \mathbb C)## parametrization using Euler angles.

Hi,
I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements

\begin{pmatrix}
e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\
ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}
\end{pmatrix}
where ##\theta, \psi## and ##\phi## are the three Euler angles. In particular ##\theta## runs in the closed interval ##[0, \pi]## whereas ##\psi## and ##\phi## in the range ##[0, 2\pi]## -- such a parametrization includes all and only ##SU(2 , \mathbb C)## group elements. That is actually a "closed box" in ##\mathbb R ^3## so to get a chart from it we need to exclude the box "boundary". This way we get a not global chart for ##SU(2)##. Since we know it is homeomorphic to ##\mathbb S^3## we can cover it with at least two charts.

Apart the above chart, which is one of the other charts for it ? Thanks.


Consider [itex]\mathbb{R} \to \mathbb{S} : x \mapsto e^{ix}[/itex]. We can obtain a two-chart atlas for [itex]\mathbb{S}[/itex] from this by taking the domains to be open invervals of width [itex]2\pi[/itex] whose images between them cover [itex]\mathbb{S}[/itex]; for example [itex](-\pi,\pi)[/itex] which covers everything except -1 and [itex](0, 2\pi)[/itex] which covers everything except 1.

Apply this logic to the above parametrization.
 
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  • #7
pasmith said:
for example [itex](-\pi,\pi)[/itex] which covers everything except -1 and [itex](0, 2\pi)[/itex] which covers everything except 1.
You mean -1 and 1 as complex numbers (i.e. the pairs (-1,0) and (1,0) under the identification ##\mathbb C \cong \mathbb R^2##).

As far as I can understand what you said, If we define ##x =(\psi+\phi)/2## and ##y =(\psi - \phi)/2## then we obtain a two-chart atlas for ##SU(2)## starting from the parametrization in post#1. However to get that atlas we need a similar approach for ##\theta## that runs in the closed interval ##[0, \pi]##.
 
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  • #8
martinbn said:
Off topic: Can you quote a textbook that uses the notation ##SU(2,\mathbb C)##?
I try to use always the field. It is redundant in that case but a good habit in the general case, especially when groups with complex matrix entries are considered real manifolds. And it makes the mention of the characteristic unnecessary.
 
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  • #9
fresh_42 said:
I try to use always the field. It is redundant in that case but a good habit in the general case, especially when groups with complex matrix entries are considered real manifolds. And it makes the mention of the characteristic unnecessary.
But the point is the ##\mathbb C## is not the base field. If you view them as Lie groups then they are real groups (as in real manifolds) and the standard notation is ##SU(2)##. If you view them as algebraic groups they are algebraic varieties over the real numbers and the standard notation is ##SU(2,\mathbb C/\mathbb R)##. More generally ##SU(2, E/ F)## for a quadratic extension ## E/ F## are groups over the the field ##F##. Over the quadratic extension ##E## the group splits and is isomorphic to ##SL(2)##.
 
  • #10
martinbn said:
But the point is the ##\mathbb C## is not the base field.
Base field of what? ##SU(2)=SU(2,\mathbb{C})## has complex entries. If I consider it as a real manifold, I write ##SU_\mathbb{R}(2,\mathbb{C})## like I would indicate ##V_\mathbb{F}## as a vector space over ##\mathbb{F}## if there is a doubt about it. However, I try to avoid writing ##SU_\mathbb{R}(2,\mathbb{C}),## I prefer ##\mathfrak{su}_\mathbb{R}(2,\mathbb{C})## and let the group be what it is, complex. I generally do not like the fact that a) reals are most often automatically assumed to be the scalar field and b) that linear (in-)dependence almost never states the scalar field, which is sometimes a serious problem!
 
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  • #11
fresh_42 said:
Base field of what? ##SU(2)=SU(2,\mathbb{C})## has complex entries. If I consider it as a real manifold, I write ##SU_\mathbb{R}(2,\mathbb{C})## like I would indicate ##V_\mathbb{F}## as a vector space over ##\mathbb{F}## if there is a doubt about it. However, I try to avoid writing ##SU_\mathbb{R}(2,\mathbb{C}),## I prefer ##\mathfrak{su}_\mathbb{R}(2,\mathbb{C})## and let the group be what it is, complex. I generally do not like the fact that a) reals are most often automatically assumed to be the scalar field and b) that linear (in-)dependence almost never states the scalar field, which is sometimes a serious problem!
It has real dimension three. It cannot be complex since three is not even.
 
  • #12
martinbn said:
It has real dimension three. It cannot be complex since three is not even.
It is an algebraic group of matrices with complex entries like ##GL(n,\mathbb{C})## or ##SL(n,\mathbb{C})## are. I only use the same nomenclature that notes the field where the matrix entries are taken from.

Dimension kicks in if we consider them as Lie groups, i.e. consider the corresponding Lie algebra. And if we do so, it is even more important to distinguish the fields!
 
  • #13
fresh_42 said:
It is an algebraic group of matrices with complex entries like ##GL(n,\mathbb{C})## or ##SL(n,\mathbb{C})## are. I only use the same nomenclature that notes the field where the matrix entries are taken from.

Dimension kicks in if we consider them as Lie groups, i.e. consider the corresponding Lie algebra. And if we do so, it is even more important to distinguish the fields!
As algebraic groups the unitary groups are not complex. They become isomorphic to the special linear groups over an algebraicly closed field (already over the quadratic extension).
 
  • #14
cianfa72 said:
You mean -1 and 1 as complex numbers (i.e. the pairs (-1,0) and (1,0) under the identification ##\mathbb C \cong \mathbb R^2##).

As far as I can understand what you said, If we define ##x =(\psi+\phi)/2## and ##y =(\psi - \phi)/2## then we obtain a two-chart atlas for ##SU(2)## starting from the parametrization in post#1. However to get that atlas we need a similar approach for ##\theta## that runs in the closed interval ##[0, \pi]##.

You are only dealing with a quarter period in the [itex]\theta[/itex] dependence; the end points are distinct. You need only 4 charts whose domains are [itex](\theta,\psi,\phi) \in [0,\pi] \times A \times B[/itex] where [itex]A[/itex] and [itex]B[/itex] are independently either [itex](-\pi,\pi)[/itex] or [itex](0, 2\pi)[/itex].
 
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  • #15
pasmith said:
You are only dealing with a quarter period in the [itex]\theta[/itex] dependence; the end points are distinct. You need only 4 charts whose domains are [itex](\theta,\psi,\phi) \in [0,\pi] \times A \times B[/itex] where [itex]A[/itex] and [itex]B[/itex] are independently either [itex](-\pi,\pi)[/itex] or [itex](0, 2\pi)[/itex].
Take for instance the chart ##\varphi## with domain ##[0,\pi] \times (-\pi, \pi) \times (- \pi, \pi)##. Is this domain open in ##\mathbb R^3## ?
 
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  • #16
cianfa72 said:
Take for instance the chart ##\varphi## with domain ##[0,\pi] \times (-\pi, \pi) \times (- \pi, \pi)##. Is this domain open in ##\mathbb R^3## ?
No. You have a closed interval for the first part.
 
  • #17
jbergman said:
No. You have a closed interval for the first part.
Exactly. Is there a way to "breakdown" that closed interval ##[0,\pi]## in open intervals to get overlapping charts for ##SU(2)## from open sets in ##\mathbb R^3## ?
 
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FAQ: ##SU(2, \mathbb C)## parametrization using Euler angles

What is the significance of the SU(2, C) group in quantum mechanics?

The SU(2, C) group is significant in quantum mechanics as it represents the symmetry group of quantum states for spin-1/2 particles, such as electrons. It is fundamental in the description of quantum spin and is closely related to the concept of quantum entanglement and the representation of quantum gates in quantum computing.

How are Euler angles used to parameterize SU(2, C)?

Euler angles are used to parameterize SU(2, C) by expressing its elements in terms of three angles: θ (theta), φ (phi), and ψ (psi). This parameterization allows us to represent any element of SU(2, C) as a rotation in three-dimensional space, facilitating the understanding of the group's structure and its applications in quantum mechanics and other areas of physics.

What is the general form of an SU(2, C) matrix using Euler angles?

The general form of an SU(2, C) matrix using Euler angles (θ, φ, ψ) can be expressed as:

U(θ, φ, ψ) = cos(θ/2) -e^{-iφ}sin(θ/2) e^{iψ}sin(θ/2) cos(θ/2)

This matrix represents a unitary transformation that preserves the inner product in a two-dimensional complex vector space.

What are the ranges of the Euler angles in the SU(2, C) parameterization?

The ranges of the Euler angles in the SU(2, C) parameterization are typically defined as follows: θ ranges from 0 to π, while φ and ψ both range from 0 to 2π. These ranges ensure that each point on the corresponding manifold is uniquely represented, while also accounting for the periodic nature of the angles.

Can the SU(2, C) parameterization be used in quantum computing?

Yes, the SU(2, C) parameterization is extensively used in quantum computing, particularly in the design and implementation of quantum gates. Quantum gates can be represented as SU(2, C) matrices, enabling the manipulation of qubit states through rotations in Hilbert space, which is essential for quantum algorithms and error correction methods.

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