TY - GEN
T1 - Why Nature Made a Choice of Clifford and not Grassmann Coordinate?
AU - Borstnik, N.S. Mankoc
AU - Nielsen, Holger Frits Bech
PY - 2018/12/1
Y1 - 2018/12/1
N2 - This is a discussion on fermion fields, the internal degrees of freedom of which are described by either the Grassmann or the Clifford anticommuting "coordinates". We prove that both fields can be second quantized so that their creation and annihilation operators fulfill the requirements of the commutation relations for fermion fields. However, while the internal spins determined by the generators of the Lorentz group of the Clifford objects Sab and Sab (in the spin-charge-family theory Sab determine the spin degrees of freedom and Sab the family degrees of freedom) are half integer, the internal spin determined by Sab (expressible with Sab + Sab) is integer. Nature "made" obviously the choice of the Clifford algebra, at least in the so far observed part of our universe. We discuss here the quantization-first and second-of the fields, the internal degrees of freedom of which are functions of the Grassmann coordinatesa and their conjugate momenta, as well as of the fields, the internal degrees of freedom of which are functions of the Clifford a. Inspiration comes from the spin-charge-family theory ([1,2,9,3], and the references therein), in which the action for fermions in d-dimensional space isequal to R ddx E 1 2 (̄ ap0a ) + h:c:, with p0a = fap0+ 1 2E fp; Efag-, p0= p-1 2Sab!ab-1 2 S ab! ab. We write the basic states as products of those either Grassmann or Clifford objects, which allow second quantization for fermion fields, and look for the action and solutions for free fields also in the Grassmann case in order to understand why the Clifford algebra "wins in the competition" for the physical (observable) degrees of freedom.
AB - This is a discussion on fermion fields, the internal degrees of freedom of which are described by either the Grassmann or the Clifford anticommuting "coordinates". We prove that both fields can be second quantized so that their creation and annihilation operators fulfill the requirements of the commutation relations for fermion fields. However, while the internal spins determined by the generators of the Lorentz group of the Clifford objects Sab and Sab (in the spin-charge-family theory Sab determine the spin degrees of freedom and Sab the family degrees of freedom) are half integer, the internal spin determined by Sab (expressible with Sab + Sab) is integer. Nature "made" obviously the choice of the Clifford algebra, at least in the so far observed part of our universe. We discuss here the quantization-first and second-of the fields, the internal degrees of freedom of which are functions of the Grassmann coordinatesa and their conjugate momenta, as well as of the fields, the internal degrees of freedom of which are functions of the Clifford a. Inspiration comes from the spin-charge-family theory ([1,2,9,3], and the references therein), in which the action for fermions in d-dimensional space isequal to R ddx E 1 2 (̄ ap0a ) + h:c:, with p0a = fap0+ 1 2E fp; Efag-, p0= p-1 2Sab!ab-1 2 S ab! ab. We write the basic states as products of those either Grassmann or Clifford objects, which allow second quantization for fermion fields, and look for the action and solutions for free fields also in the Grassmann case in order to understand why the Clifford algebra "wins in the competition" for the physical (observable) degrees of freedom.
UR - http://bsm.fmf.uni-lj.si/bled2018bsm/index.html
M3 - Conference article
SN - 1580-4992
VL - 19
SP - 175
EP - 215
JO - Blejske Delavnice iz Fizike
JF - Blejske Delavnice iz Fizike
IS - 2
ER -