Federbush mode decomposition of ϕ42

The procedure for computing the kernel for the quantum mechanical harmonic oscillator by path integrals using Fourier series to diagonalize the action (originally proposed by Feynman and Hibbs), can be shown to be inconsistent; it leads to a kernel which converges to 0. A rigorous computation starting directly from the Trotter formula is demonstrated. This leads to the correct kernel and throws light upon the convergence mentioned.