We suggest a closed form expression for the path integral of quantum transition amplitudes. We introduce a quantum action with renormalized parameters. We present numerical results for the V ∼ x 4 potential. The renormalized action is relevant for quantum chaos and quantum instantons.

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We suggest a closed form expression for the path integral of quantum transition amplitudes to construct a quantum action. Based on this we propose rigorous definitions of both, quantum instantons and quantum chaos. As an example we compute the quantum instanton of the double well potential.

We review our new method, which might be the most direct and efficient way for approaching the continuum physics from Hamiltonian lattice gauge theory. It consists of solving the eigenvalue equation with a truncation scheme preserving the continuum limit. The efficiency has been confirmed by the observations of the scaling behaviors for the long wavelength vacuum wave functions and mass gaps in (2+1)-dimensional models and (1+1)dimensional σ model even at very low truncation orders. Most of these results show rapid convergence to the available Monte Carlo data, ensuring the reliability of our method.