The product G ( R → → , R → → ″ ; N − − n ) G ( R → → ″ , R → → ′ ; N ) ) expresses the weight factor for a chain which starts at R ′ , passes through R ″ in n steps, and ends at R after N steps. The integration over all possible midpoints R ″ gives back the statistical weight for a chain starting at R ′ , and terminating at R . It should now be clear that the path integral is simply a sum over all possible literal paths the polymer can form between two fixed endpoints.