7.4 Complex projective space

Concept links · terms present in this machine draft; source roles are unverified: vector space · linear map · eigenvalue · group action

There is a diferent possible approach to characterizing states of a quantum system with $H = C^{2}$ . Multiplication of vectors in $H$ by a non-zero complex number does not change eigenvectors, eigenvalues or expectation values, so arguably has no physical efect. Thus what is physically relevant is the quotient space $(C^{2} - 0) / C^{*}$ , which is constructed by taking all non-zero elements of $C^{2}$ and identifying those related by multiplication by a non-zero complex number.

For some insight into this construction, consider first the analog for real numbers, where $(R^{2} - 0) / R^{*}$ can be thought of as the space of all lines in the plane going through the origin.

Figure 7.1: The real projective line $RP^{1}$

One sees that each such line hits the unit circle in two opposite points, so this set could be parametrized by a semi-circle, identifying the points at the two ends. This space is given the name $RP^{1}$ and called the “real projective $line^{s}$ . In higher dimensions, the space of lines through the origin in $R^{n}$ is called $RP^{n - 1}$ and can be thought of as the unit sphere in $R^{n}$ , with opposite points identified (recall from section 6.2.3 that $S O (3)$ can be identified with $RP^{3})$ .

What we are interested in is the complex analog $C P^{1}$ , which is quite a bit harder to visualize since in real terms it is a space of two dimensional planes through the origin of a four dimensional space. A standard way to choose coordinates on $\overset{ˉ}{CP}^{1}$ is to associate to the vector

(z _{2} z _{1}) \in C^{2}

the complex number $z_{1} / z_{2}$ . Overall multiplication by a complex number will drop out in this ratio, so one gets diferent values for the coordinate $z_{1} / z_{2}$ for each diferent coset element, and elements of $C P^{1}$ correspond to points on the complex plane. There is however one problem with this coordinate: the point

on the plane corresponding to

¹₀

does not have a well-defined value: as one approaches this point one moves of to infinity in the complex plane. In some sense the space $\overset{ˉ}{CP}^{1}$ is the complex plane, but with a “point at infinity” added.

$CP^{1}$ is better thought of not as a plane together with a point, but as a sphere (often called the “Riemann sphere”), with the relation to the plane and the point at infinity given by stereographic projection. Here one creates a oneto-one mapping by considering the lines that go from a point on the sphere to the north pole of the sphere. Such lines will intersect the plane in a point, and give a one-to-one mapping between points on the plane and points on the sphere, except for the north pole. Now, the north pole can be identified with the “point at infinity”, and thus the space $C P^{1}$ can be identified with the space $S^{2}$ . The picture looks like this

Figure 7.2: The complex projective line $C P^{1}$

and the equations relating coordinates $(x_{1}, x_{2}, x_{3})$ on the sphere and the complex coordinate $z_{1} / z_{2} = z = x + i y$ on the plane are given by

x = \frac{x _{1}}{1 - x _{3}}, y = \frac{x _{2}}{1 - x _{3}}

and

x_{1} = \frac{2 x}{x ^{2} + y ^{2} + 1}, x_{2} = \frac{2 y}{x ^{2} + y ^{2} + 1}, x_{3} = \frac{x ^{2} + y ^{2} - 1}{x ^{2} + y ^{2} + 1}

Figure 7.3: Complex-valued coordinates on $C P^{1} - {\infty}$ via stereographic projection.

Digression. For another point of view on $C P^{1}$ , one constructs the quotient of $C^{2}$ by complex scalars in two steps. Multiplication by a real scalar corresponds to a change in normalization of the state, and we will often use this freedom to work with normalized states, those satisfying

⟨ ψ ∣ ψ ⟩ = z_{1} \overline{z}_{1} + z_{2} \overline{z}_{2} = 1

Such normalized states are unit-length vectors in $C^{2}$ , which are given by points on the unit sphere $S^{3} \subset R^{4} = C^{2}$

With such normalized states, one still must quotient out the action of multiplication by a phase, identifying elements of $\overset{ˉ}{S}^{3}$ that difer by multiplication by $e^{i θ}$ . The set of these elements forms a new geometrical space, often written $S^{3} / U (1)$ . This structure is called a “fibering” of $S^{3}$ by circles (the action by phase multiplication traces out non-intersecting circles) and is known as the “Hopf fibration”. Try an internet search for various visualizations of the geometrical structure involved, a surprising decomposition of three dimensional space (identifying points at infinity to get $S^{3})$ into non-intersecting curves.

Acting on $H = C^{2}$ by linear maps

(z _{2} z _{1}) \to (α γ β δ) (z _{2} z _{1})

takes

z = \frac{z _{1}}{z _{2}} \to \frac{α z + β}{γ z + δ}

Such transformations are invertible if the determinant of the matrix is non-zero, and one can show that these give conformal (angle-preserving) transformations of the complex plane known as “M¨obius transformations”. In chapter 40 we will see that this group action appears in the theory of special relativity, where the action on the sphere can be understood as transformations acting on the space of light rays. When the matrix above is in $S U (2) (γ = - \overline{β}, δ = \overline{α}$ $α \overline{α} + β \overline{β} =$ 1), it can be shown that the corresponding transformation on the sphere is a rotation of the sphere in $R^{3}$ , providing another way to understand the nature of $S U (2) = S p in (3)$ as the double cover of the rotation group $S O (3)$

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 75、76、77、78、79、80、81、82、83、84、85、86、87、88

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：bfb0bda113ac1733dd883dd63c618e40cc1f08b1a13e7fa976c698e605889637

OCR 产物 SHA-256：bfb0bda113ac1733dd883dd63c618e40cc1f08b1a13e7fa976c698e605889637

数学物理阅读

书架

7.4 Complex projective space

7.4 Complex projective space

关系图谱