let the spin begins

[AntonioLao]

The August paper of 1925 (introducing ½-integer quantum numbers for hydrogen, which yielded a bonus explanation of the 4686 angstroms fine structure in helium plus) penned jointly by two Dutch physicists, George Uhlenbeck and Samuel Goudsmit led to their discovery of electron spin in October. About the same duration, same ideas found in the August paper, Slater has also proposed them independently and earlier in January, the idea had occurred to Kronig after hearing Lande’s discussions on Pauli Exclusion Principle. The ultimate explanation of electron spin and the fine structure of energy levels, since the discovery of Zeeman Effect, were finally accomplished using Dirac equation. This equation uses a concept, already discovered and mathematically formalized afterward by Elie Cartan into a theory of spinors between 1913 and 1937.

Associating the word ‘spinors’ to the components of Dirac equation was introduced by Ehrenfest in a letter to van der Waerden questioning its similarity to tensors. Many physicists at this stage understood that relationships must exist between spinors, vectors, and tensors. Nevertheless, what are these relationships? In other words, what were the connections between the spinor group and the Lorentz group for rotations? The 3D spinors are composed of two components for 4 degrees of freedom, while the 3D vectors are composed of three components for 6 degrees of freedom. The 4D spinors are composed of four components for 8 degrees of freedom, while the 4D tensors are composed of 16 components for 32 degrees of freedom. So, what can be the connections? One possible connection could be the time axis. However, time unit vectors do not exist. How can we define these time axes?

[Guille]

Two questions, that when answered, will let me work on this questions and I will give you my mathematical-based answer:

What about 3d tensors?
and
What about 4d vectors?

Is the 4rth dimension refered here, time?

[AntonioLao]

3D tensors are usually called vectors. 4D tensors are used by both special relativity and general relativity.

[Guille]

3D tensors are usually called vectors. 4D tensors are used by both special relativity and general relativity.

And what about a 4d vector? Is it called tensor?

[Guille]

And, finally,

What is the number of components and degree of freedom of 3d Scalars and 4d Scalars?

[AntonioLao]

For 4D vectors, see site http://mathworld.wolfram.com/Four-Vector.html
all dimensional scalars have only one component.

[Guille]

For 4D vectors, see site http://mathworld.wolfram.com/Four-Vector.html

Thanks for the link.

all dimensional scalars have only one component.

And how many degrees of freedom?

[Guille]

3D tensors are usually called vectors.

Is this because a 3d tensor and a 3d vector have the same number of components and dregrees of freedom?

By the way, the link that you gave me (http://mathworld.wolfram.com/Four-Vector.html) lacks of one last piece of information that I need, to proceed inmy number theory investigation/researchs:

What is the degree of freedom of a 4d vector?

[AntonioLao]

GUILLE,

If the 4D object is a particle, it has 16 components and 8 degrees of freedom. If it is the continuous field, it has infinite components and dof's.

[Guille]

GUILLE,

If the 4D object is a particle, it has 16 components and 8 degrees of freedom. If it is the continuous field, it has infinite components and dof's.

What is then light? It has wave-particle duality, so it has both descriptions, doesn't it?