Tensors and spinors arise mathematically from the representation of the rotation group $SO(3)$ as a ball in 4D with all antipodal points on the surface identified. In this picture it is shown that there exist two classes of loops corresponding to two types of a continuous sequence of rotation both equivalent to no rotation at all; one which starts at the origin and stays within the ball's interior and can be shrunk own to a point and another which goes to the surface, back to the antipodal point and again to the origin. This can also be shrunk down to an arbitrarily small sized loop. In this argument there was no reference whatsoever to elementary particles but was an argument based only on the nature of 3D space. Why, then, is this property not one of every point in space rather than a property of elementary particles (matter)?
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You question is rather unclear. The loops or the 3-sphere as the universal cover of $\mathrm{SO}(3)$ have nothing to do with "points in space" because the fermionic representations are not linear representations of the rotations group, only projective ones. For more on this, see http://physics.stackexchange.com/q/96045/50583 and http://physics.stackexchange.com/q/203944/50583 – ACuriousMind Mar 07 '16 at 12:09
1 Answers
I think your question is why the picture of physics is one way or the other.
Your picture is a space-time in which the space-time points have properties such as spin (and perhaps therefore other quantum numbers like charge, mass, momentum, baryon number,...?). But this isn't the picture of Quantum Mechanics. In QM there is an object (eg: particle, table, or star) that you do transformations to. These transformations include rotations, boosts, space-time translations, and a variety of gauge transformations like isospin rotations. Particles are described and distinguished from each other according to how they transform under all these transformations. These transformations usually form a Lie Group with abstract generators like $J_x,J_y,J_z,P_x,P_y,P_z,E,Q,...$. The commutation relations between the generators make the group what it is mathematically. Each particle corresponds to a vector (ket) which is transformed by some representation of these abstract operators. There are various ways to represent the abstract generators and the space they operate on. The representation could be as matrices operating on vectors, or as derivatives acting on functions. For example, an electron corresponds to a 2-vector (spinor) that is rotated by a 2x2 matrix representation of $J_z$. You have proposed another representation of rotations as a manipulation (matrix multiply?) of "a ball in 4D with all antipodal points on the surface identified". Might this provide a new representation of the rotation group that is not isomorphic to one of the matrix or derivative reps we have now?
So, my long winded answer to your question, in the concepts of present QM you have just found another representation of rotations. If you want to develop a picture of a space-time with quantum numbers at each point, you have a novel and large task ahead to hook up with experimental knowledge.
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