ABSTRACT
The success of Schroedinger non-relativistic wave mechanics of 1927 , was due not only to the prediction of the spectra of the hydrogen atom (already derived in the matrix mechanics of Heisenberg https://www.w3.org/1998/Math/MathML"> e t a l https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6574.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ) but also on the fact that it could be extended immediately from the three dimensional space of one particle to the n-body problem in https://www.w3.org/1998/Math/MathML"> 3 n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6575.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> dimensional configuration space. While Dirac one particle relativistic wave equation appeared in 1928 , almost simultaneously with Schroedinger's work, its extension to n particles introduced n times whose interpretation was not clear. In this paper, with the help of a unit temporaloid four vector https://www.w3.org/1998/Math/MathML"> u μ , μ = 0,1 , 2,3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6576.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> we generalize the Dirac equation to n-particles but involving only one time. This equation for https://www.w3.org/1998/Math/MathML"> n = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6577.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> becomes again the standard Dirac equation while https://www.w3.org/1998/Math/MathML"> u μ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6578.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is proportional to the total energy-momentum four vector https://www.w3.org/1998/Math/MathML"> P μ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6579.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , which remains an integral of motion even in the presence of interactions. Our generalized n-body Dirac equation, while still invariant under the Poincaré group, in the frame of reference in which the total momentum https://www.w3.org/1998/Math/MathML"> P = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq6580.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> involves both positive and negative energies but, using a Foldy-Wouthuysen transformation generalized to many body problems, we can retain only the positive energy part and obtain the energy spectra of three quark systems by standard methods of non-relativistic quantum mechanics
