An Approach to Building Quantum Field Theory Based on Non-Diophantine Arithmetics

Mark Burgin’s Contribution to the Foundation of Mathematics

“…As shown in [16], the arithmetic A L becomes the standard arithmetic Z in the formal limit L → ∞. However, as noted above, from the point of view of verificationism, the value of L should be finite.…”

“…and one can prove [16] the following: (e) The results of addition, subtraction, and multiplication in the arithmetic A L cannot be greater than L and less than −L.…”

“…Based on these considerations, Mark proposed in [16] the following option for the functions g and h. Let us take a natural number L as the boundary parameter of the non-Diophantine arithmetic A L . We build the non-Diophantine arithmetic A L , using functions g and h to establish a weak projectivity between A L and Z:…”

“…Typical divergences in QFT are similar to the divergences in quantum electrodynamics (QED). As shown in [16], after the integration over hyperspherical angular variables, the integrals of the Feynman diagrams describing the electron self-energy, the photon self-energy, and the electron-photon vertex are…”

“…We now consider how the integrals in Equation ( 1) should be treated in non-Diophantine mathematics (NDM). We will use the version of NDM where the functions g and h are described in the preceding section, and it has been noted that those functions have been proposed by Mark Burgin in [16].…”