Explore the geometric aspects of quantum mechanics, including Hilbert spaces, projective spaces, real projective lines, planes, Möbius strip, Klein bottle, and more. Understand the differentiation between real and complex vector spaces, as well as the representation of states within one-dimensional subspaces. Dive into the mathematical intricacies of inner products, angles, rotations, and directions in quantum physics.
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
Vector Hilbert space Same state (state is not a vector) ? ? ~ ? ? States are represented by one-dimensional subspaces ? ? i.e. ray in Hilbert space Understand real projective spaces Understand complex projective spaces What the inner product means geometrically, how a complex vector space is different from a real one with double dimension, https://assumptionsofphysics.org/
Real projective line Set of all lines that pass through the origin (one dimensional-subspace, rays ) 0,0 ?,? ?[0,1] ? ?,1 = ? sin?,cos? ? 1 , ? 1 ,? 4 4 ? , ? 2 ; ,? ? 1,0 2 ? https://assumptionsofphysics.org/
Real projective plane Set of all lines that pass through the origin (one dimensional-subspace, rays ) ?[0,0,1] ? 0,1,0 ? ? ? 1,0,0 ? ? ?,?,1 = ?[sin?cos ?,sin?sin?,cos?] https://assumptionsofphysics.org/
Mbius strip Klein bottle Real projective plane https://assumptionsofphysics.org/
? 2? ?4 ?2 ?3 ??2 ??1 ?2 ?1 ?1 ?,? = ? ? cos???? ? ? = ? ? cos? Directions are bundled into planes. One angle within the plane (phase) and one angle across planes. Can only rotate planes onto planes. Each direction is independent. One angle defined between two vectors. Can rotate any direction onto any direction. https://assumptionsofphysics.org/
Real space Complex space Ray = real line that passes through the origin Ray = complex plane that passes through the origin ?,? = ? ? cos???? ? ? = ? ? cos? retained in the projective space NOT retained in the projective space https://assumptionsofphysics.org/
Spin 1/2 qubit ? = cos?/2 ?++ sin?/2???? = cos?/2? ??/2?++ sin?/2???/2|? |?+ ?|? ?+= ?+= ? = ? = 2/2 ?++ 2/2|? 2/2 ?++ ? 2/2|? 2/2 ?+ 2/2|? 2/2 ?+ ? 2/2|? |? |?+ |?+ |? |?+ |? spin is double cover https://assumptionsofphysics.org/ |? angle in vector space is half the angle in physical space
Vector space Projective space |?+ |?+ |?+ |? |?+ |? |? |? The physics is here! ?,? = ? ? cos????? ?,? ?,? ?,? ?,? = cos2??=1 + cos?? ? ? ? = https://assumptionsofphysics.org/ 2
Superposition of states probability distribution |?+ ?|? Superpositions are linear decompositions ? = ?+?++ ? ? ? = ????+ ???? |? diagonal force is a superposition of vertical and horizontal force |?+ |?+ |? |?+ |? Everything is a superposition of everything else ?+= 2/2 ?++ 2/2|? 2/2 ?++ 2/2|? |? https://assumptionsofphysics.org/ ?+=
Superposition is a property of ANY linear system https://en.wikipedia.org/wiki/Superposition_principle Note: linearity is a property of the VECTOR space, not of the projective space Quantum superposition is NOT a physical property! ?+= ?+= ? = ? = 2/2 ?++ 2/2|? 2/2 ?++ ? 2/2|? 2/2 ?+ 2/2|? 2/2 ?+ ? 2/2|? It is a property of the vector space representation https://assumptionsofphysics.org/ Coefficients are representation dependent
?2 ? ?12+ ?22? ? ?1?++ ?2? = ?1?++ ?2? phase shift that depends on both components |?+ ?2 vector space cos? = ?12+ ?22 Non linear map ? |? ? ?+ ? ?++ ? = ?+ ? ? = ?++ ??/ 2? = ??? Preserves the rays: colinear map = ? ? ?1?++ ?2? ? ? ? = ? ??1 ?++ ??2 ? ?(?),? ? = ? ? cos??? ? ??2 ? ??12+ ??22? |?| ?2 |?| ?12+ ?22? = ??( ? ) = ??1?++ ??2? phase of the inner product will change ? https://assumptionsofphysics.org/ = ??1?++ ??2?
The main difference in quantum mechanics is not the use of complex vector spaces, but the use of projective spaces A quantum state is not a vector in the Hilbert space, but a one-dimensional subspace, a complex plane (i.e. a ray ) For a spin 1/2 system, angles in Hilbert space are half the physical angles (half-sphere is stretched to a full sphere) Superposition (linearity) is a property of the vector space, not of the projective space, and therefore not fully physical https://assumptionsofphysics.org/
