What is the meaning of equivalence classes in complex projective space?

In summary, the conversation discusses the concept of complex projective space and how it is defined as the set of planes in 4-dimensional space that pass through the origin. However, there is confusion about how this relates to the Grassmannian G(2,4) and the fact that two planes can intersect at a single point in 4-dimensional space. This leads to the realization that something is wrong with this way of thinking about complex projective space.
  • #1
Thorn
23
0
I've been thinking...and am starting to think that I don't understand complex projective space...So, it's defined as ( Cn+1 \{0,0} / C\{0} ). Now, I think this is just the set of planes in 4 space that pass through the origin... and one can consider how they would all intersect a 3 sphere and think of it as S3/U(1) where U(1) is the circle group... and the hopf function will take all these circles and map them to the 2 sphere isomorphically... but the problem I have is... just pick any 3 of the 4 basis vectors in C^2 and span two planes with them...essential you can just look at R^3 for this... and think of the plane spanned by XY and XZ...well they intersect at the whole X axis...which means there are elements that belong to both planes...but in the case of ( Cn+1 \{0,0} / C\{0} ), these planes are supposed to be equivalence classes...meaning it should divide the space into disjoint sets...and thus, you can't have an element in 2 equivalence classes...Unless, both these planes are actually in the same equivalence class, which is just mind blowing since you can find a bunch more planes that will intercet XY and XZ and before you know it, all of R^3 will belong to the same equivalence class...So, clearly something is wrong with this way of thinking of it...Anyone?
 
Mathematics news on Phys.org
  • #2
Thorn said:
I've been thinking...and am starting to think that I don't understand complex projective space...So, it's defined as ( Cn+1 \{0,0} / C\{0} ). Now, I think this is just the set of planes in 4 space that pass through the origin...

The set of 2-dimensional subspaces in R^4 is by definition the Grassmannian G(2,4). It has real dimension 2(4-2)=4, so it can not be the same as CP1, which has 2 real dimensions.
 
  • #3
Thorn said:
I've been thinking...and am starting to think that I don't understand complex projective space.
Heck, I've thought that for years!
 
  • #4
In four dimensions, two planes can intersect in a single point, not a line. Take the x-y and z-w planes, for example, which intersect only at the origin.
 

1. What is Complex Projective Space?

Complex Projective Space, denoted by ℂℙ^n, is a mathematical concept that extends the concept of Euclidean space to include points at infinity. It is a space in which each point represents a line through the origin in ℂ^n. It is a key concept in algebraic geometry and is used in many areas of mathematics, including algebraic topology, differential geometry, and complex analysis.

2. How is Complex Projective Space different from Euclidean space?

While Euclidean space is a finite space with a fixed number of dimensions, Complex Projective Space is an infinite space with infinitely many dimensions. In Euclidean space, parallel lines never meet, but in Complex Projective Space, every two lines intersect at a point. Additionally, in Complex Projective Space, all lines are considered equivalent, while in Euclidean space, lines can have different orientations and lengths.

3. How is Complex Projective Space used in physics?

Complex Projective Space is used in physics, particularly in quantum mechanics and string theory, to describe the symmetries of physical systems. These symmetries are represented by points in Complex Projective Space, and the transformations between these points correspond to the physical changes in the system. This concept is crucial in understanding the fundamental laws of nature and in predicting the behavior of physical systems.

4. What are some real-world applications of Complex Projective Space?

Complex Projective Space has many real-world applications, including in computer graphics and computer vision. It is used to project 3D images onto a 2D screen, allowing for a more efficient representation of 3D objects. It is also used in image recognition and pattern recognition to identify and classify objects in images. Additionally, it has applications in robotics, where it is used to represent the configuration space of robotic systems.

5. How is Complex Projective Space related to the concept of duality?

Complex Projective Space is closely related to the concept of duality in mathematics. In particular, the points in Complex Projective Space can be interpreted as lines in a dual space, and vice versa. This duality allows for a deeper understanding of projective geometry and is used in various areas of mathematics, including algebraic geometry and differential geometry.

Similar threads

B Invariant under rotation: Banal, obvious, or noteworthy?
    • General Math
Replies
2
Views
1K
A Explicit example of two complex numbers for double cover U(1) of SO(2) for a specified angle
    • General Math
Replies
2
Views
399
A How to visualise complex vector spaces of dimension 2 and above
    • Linear and Abstract Algebra
Replies
4
Views
2K
What is the concept of the extended complex plane/Riemann Sphere?
    • General Math
Replies
1
Views
1K
A Proving that the real projective plane is not a boundary
    • Topology and Analysis
2
Replies
38
Views
4K
A What is this unitary invariant?
    • Quantum Physics
Replies
3
Views
752
I What does the equation dΩ32 = dr2 + r2dΩ22 mean in terms of flat space?
    • Cosmology
Replies
6
Views
2K
Where (what) is an orbit? Where is Space?
    • STEM Educators and Teaching
Replies
3
Views
911
What is an Equivalence Relation and its Examples?
    • General Math
Replies
1
Views
2K
Why the projection of S on the xy plane cannot be used?
    • Calculus and Beyond Homework Help
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top