diff --git a/docs/src/assets/spinspace/addition.png b/docs/src/assets/spinspace/addition.png deleted file mode 100644 index 942f2ba..0000000 Binary files a/docs/src/assets/spinspace/addition.png and /dev/null differ diff --git a/docs/src/assets/spinspace/addition07.png b/docs/src/assets/spinspace/addition07.png deleted file mode 100644 index 0ca18d8..0000000 Binary files a/docs/src/assets/spinspace/addition07.png and /dev/null differ diff --git a/docs/src/assets/spinspace/addition08.png b/docs/src/assets/spinspace/addition08.png deleted file mode 100644 index 966661c..0000000 Binary files a/docs/src/assets/spinspace/addition08.png and /dev/null differ diff --git a/docs/src/assets/spinspace/addition09.png b/docs/src/assets/spinspace/addition09.png deleted file mode 100644 index 47efb9e..0000000 Binary files a/docs/src/assets/spinspace/addition09.png and /dev/null differ diff --git a/docs/src/assets/spinspace/addition1.png b/docs/src/assets/spinspace/addition1.png new file mode 100644 index 0000000..e79fca4 Binary files /dev/null and b/docs/src/assets/spinspace/addition1.png differ diff --git a/docs/src/assets/spinspace/addition2.png b/docs/src/assets/spinspace/addition2.png new file mode 100644 index 0000000..0b64036 Binary files /dev/null and b/docs/src/assets/spinspace/addition2.png differ diff --git a/docs/src/assets/spinspace/addition3.png b/docs/src/assets/spinspace/addition3.png new file mode 100644 index 0000000..a550070 Binary files /dev/null and b/docs/src/assets/spinspace/addition3.png differ diff --git a/docs/src/assets/spinspace/addition4.png b/docs/src/assets/spinspace/addition4.png new file mode 100644 index 0000000..8cee88c Binary files /dev/null and b/docs/src/assets/spinspace/addition4.png differ diff --git a/docs/src/newsreport.md b/docs/src/newsreport.md index 8b53b1a..92469a0 100644 --- a/docs/src/newsreport.md +++ b/docs/src/newsreport.md @@ -157,22 +157,22 @@ gauge3 = float(π) The geometry of "spin-vector addition" is shown. The spin-vectors exist in a spin-space that is equipped with three operations: scalar multiplication, inner product and addition. The addition of spin-vectors κ and ω results in another spin-vector κ + ω in the spin-space, which has its own flagpole and flag plane. Taking κ and ω as null vectors in the sphere of future null directions, the flagpole of κ is represented by a point (complex number) and the null flag of κ is represented as a point sufficiently close to κ that is used to assign a direction tangent to the sphere at κ. -![addition](./assets/spinspace/addition.png) +![addition1](./assets/spinspace/addition1.png) The tails of the flagpoles of κ, ω and κ + ω are in a circle in the sphere of future null directions. The circumcircle of the triangle made by joining the tails of the three spin-vectors makes angles with the flagpoles and null planes. Meaning, the distance between κ and the center of the circle is equal to the distance between ω and the center. Also, the distance of the addition of κ and ω and the circle center is the same as the distance between κ and the center. For the circumcircle, we have three collinear points in the Argand complex plane. However, lines in the Argand plane become circles in sections of the three-dimensional sphere. The angle that the flagpoles of κ and ω make with the circle should be twice the argument of the inner product of the two spin-vectors (modulus 2π with a possible addition of π). -![addition07](./assets/spinspace/addition07.png) +![addition2](./assets/spinspace/addition2.png) ```julia w = (Complex(κ + ω) - Complex(κ)) / (Complex(ω) - Complex(κ)) @assert(imag(w) ≤ 0 || isapprox(imag(w), 0.0), "The flagpoles are not collinear: $(Complex(κ)), $(Complex(ω)), $(Complex(κ + ω))") ``` -![addition08](./assets/spinspace/addition08.png) +![addition3](./assets/spinspace/addition3.png) In an interesting way, the argument (phase) of the inner product of κ and ω is equal to half of the sum of the angles that the spin-vectors make with the circle, which is in turn equal to the angle that U and V make with each other minus π (also see the geometric descriptions of the inner product to construct U and V). In the case of spin-vector addition, the angles that the flag planes of κ, ω and κ + ω, each make with the circle are equal. But, be careful with determining the signs of the flag planes and the possible addition of π to the flag plane of κ + ω. For determining flag plane signs, see also Figure 1-21 in page 64 of Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984). -![addition09](./assets/spinspace/addition09.png) +![addition4](./assets/spinspace/addition4.png) For example, the Standard Model is formulated on 4-dimensional Minkowski spacetime, over which all fiber bundles can be trivialized and spinors have a simple explicit description. For the Symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant.