3 ms·

A Child’s Guide to Spinors (2016) [pdf]

62 points by bindidwodtj 6y ago

enugu 6y ago
Thanks for posting this! For someone introduced to spinors via Clifford algebras, with the applications to physics distant, this was very helpful to read. Some small points - The remarks on 'double cover' in Section 5 can be improved. 'Double cover' usually doesnt refer to the dimension of the vector space which the groups acts on, but rather the geometric fact there is map from SU(2) to SO(3) which takes 2 points to a single point. For example see, https://indico.cern.ch/event/243629/attachments/415251/576988/L2.pdf https://indico.cern.ch/event/243629/attachments/415251/57698... Writing SO(4) as SU(2)+SU(2) also doesnt make sense for this reason. The simplest example of double cover being the map z → z^2 for unit circle in complex plane or in terms of angles θ → 2θ. This also relates to the 'square root' and 'half' remark in the first paragraph of Introduction. The general definiton of 'covering space' in a topology text book can be useful to read. Also, I think there is a typo in RHS of Eqn 3.1.2. But again, this was really helpful to read for someone who learnt the formal definition long time ago and was out of touch with spinors.
deleted 6y ago
[deleted]
nabla9 6y ago
This is great. If you are not a physicist, know some math and really would like to have some knowledge about theoretical physics past analogies, metaphorical explanations and silly pictures, I strongly suggest The Road to Reality by Roger Penrose. You still have to invest the time but, not as much as getting a degree in physics.
zentiggr 6y ago
For someone who doesn't quite have the math background to truly operate at this level, this does seem far more approachable than most of the references to spinors that I've seen before. Looking forward to time to more than skim it and start delving into the relevant foundations.
flubert 6y ago
Do spinors get "easier" in geometric algebra?
- chr1 6y ago
  They arise very naturally, but unfortunately it still doesn't seem to give an intuitive geometric example of objects that are described by spinors. See https://www.sciencedirect.com/science/article/abs/pii/S0003491604002209 https://www.sciencedirect.com/science/article/abs/pii/S00034... (which can be accessed via sci-hub).
javajosh 6y ago
>Today we look upon electrons and other spin-1/2 particles as having an intrinsic angular momentum with no classical counterpart. This gave rise to the notion of such particles living in ‘‘spin space,’’ an abstract two-dimensional internal space that requires a description beyond that of scalars, vectors and tensors. Okay I admit it, this annoyed me. Electron spin doesn't require 2 dimensions to describe, its 1 dimensional, and even more, it requires 1 bit. Other spin values require slightly more space on a one dimensional line, but still. I like it better when maths people are meticulous.
- cuspycode 6y ago
  Terminology is not always unambiguous. The spin quantum number is discrete for electrons {-1/2,1/2}, but spin also means the physical entity that gives rise to the so-called intrinsic angular momentum, which is a vector.
  - javajosh 6y ago
    I thought he was pretty unambiguous there. If we wants to talk about the underlying wave function before collapse, fine, its path in 2D space. But the measured spin only requires 1 bit. Note that the spin is NOT a 2D vector, but a path, which itself can described by a sequence of vectors. Yes yes, Bell's Theorem, EPR, collapse, and all, but I still think it was sloppy.
    - n4r9 6y ago
      Honestly I'm not sure what your complaint is here. Or at least you seem to be very adamant that some particular interpretation/ontology is obviously the right one. In standard quantum theory, electron spin can be modeled by a 2 dimensional complex vector. It's also possible to model the spin of an isolated system using 3 dimensional real vector (the "Bloch sphere").
- chr1 6y ago
  It would have been 1 bit if all measuring devices in the universe were oriented along the same axis. If you prepare electron in a device with one orientation, then measure it again with a different device you need the vector.

62 points by bindidwodtj 6y ago

Thanks for posting this! For someone introduced to spinors via Clifford algebras, with the applications to physics distant, this was very helpful to read. Some small points - The remarks on 'double cover' in Section 5 can be improved. 'Double cover' usually doesnt refer to the dimension of the vector space which the groups acts on, but rather the geometric fact there is map from SU(2) to SO(3) which takes 2 points to a single point. For example see, https://indico.cern.ch/event/243629/attachments/415251/576988/L2.pdf https://indico.cern.ch/event/243629/attachments/415251/57698... Writing SO(4) as SU(2)+SU(2) also doesnt make sense for this reason. The simplest example of double cover being the map z → z^2 for unit circle in complex plane or in terms of angles θ → 2θ. This also relates to the 'square root' and 'half' remark in the first paragraph of Introduction. The general definiton of 'covering space' in a topology text book can be useful to read. Also, I think there is a typo in RHS of Eqn 3.1.2. But again, this was really helpful to read for someone who learnt the formal definition long time ago and was out of touch with spinors.

deleted 6y ago

[deleted]

nabla9 6y ago

This is great. If you are not a physicist, know some math and really would like to have some knowledge about theoretical physics past analogies, metaphorical explanations and silly pictures, I strongly suggest The Road to Reality by Roger Penrose. You still have to invest the time but, not as much as getting a degree in physics.

zentiggr 6y ago

For someone who doesn't quite have the math background to truly operate at this level, this does seem far more approachable than most of the references to spinors that I've seen before. Looking forward to time to more than skim it and start delving into the relevant foundations.

flubert 6y ago

Do spinors get "easier" in geometric algebra?

chr1 6y ago
They arise very naturally, but unfortunately it still doesn't seem to give an intuitive geometric example of objects that are described by spinors. See https://www.sciencedirect.com/science/article/abs/pii/S0003491604002209 https://www.sciencedirect.com/science/article/abs/pii/S00034... (which can be accessed via sci-hub).

javajosh 6y ago

>Today we look upon electrons and other spin-1/2 particles as having an intrinsic angular momentum with no classical counterpart. This gave rise to the notion of such particles living in ‘‘spin space,’’ an abstract two-dimensional internal space that requires a description beyond that of scalars, vectors and tensors. Okay I admit it, this annoyed me. Electron spin doesn't require 2 dimensions to describe, its 1 dimensional, and even more, it requires 1 bit. Other spin values require slightly more space on a one dimensional line, but still. I like it better when maths people are meticulous.

cuspycode 6y ago
Terminology is not always unambiguous. The spin quantum number is discrete for electrons {-1/2,1/2}, but spin also means the physical entity that gives rise to the so-called intrinsic angular momentum, which is a vector.
- javajosh 6y ago
  I thought he was pretty unambiguous there. If we wants to talk about the underlying wave function before collapse, fine, its path in 2D space. But the measured spin only requires 1 bit. Note that the spin is NOT a 2D vector, but a path, which itself can described by a sequence of vectors. Yes yes, Bell's Theorem, EPR, collapse, and all, but I still think it was sloppy.
  - n4r9 6y ago
    Honestly I'm not sure what your complaint is here. Or at least you seem to be very adamant that some particular interpretation/ontology is obviously the right one. In standard quantum theory, electron spin can be modeled by a 2 dimensional complex vector. It's also possible to model the spin of an isolated system using 3 dimensional real vector (the "Bloch sphere").
chr1 6y ago
It would have been 1 bit if all measuring devices in the universe were oriented along the same axis. If you prepare electron in a device with one orientation, then measure it again with a different device you need the vector.