Quantum field theory without divergences

It is shown that loop divergences emerging in the Green functions in quantum field theory originate from correspondence of the Green functions to unmeasurable (and hence unphysical) quantities. This is because no physical quantity can be measured in a point, but in a region, the size of which is constrained by the resolution of measuring equipment. The incorporation of the resolution into the definition of quantum fields $\varphi (x)\to {\varphi }^{(A)}(x)$ and appropriate change of Feynman rules results in finite values of the Green functions. The Euclidean ${\varphi }^4$-field theory is taken as an example.

Figure 1

Feynman diagrams for the Green functions $G^{(2)}$ and $G^{(4)}$ for the resolution-dependent fields.

Figure 2

Scale-decay factors for the two-point and four-point Green functions. The bottom curve is the graph of (15) as a function of $A^2$; the top curve is the graph of (20) divided by ${\lambda }^2/(256{\pi }^6)$ as a function of $A^2$. $m=s^2=1$ is set for both curves.