The instanton configuration in the SU(2)-gauge system with a Higgs doublet is constructed by using the new valley method. This method defines the configuration by an extension of the field equation and allows the exact conversion of the quasi-zero eigen-mode to a collective coordinate. It does not require ad hoc constraints used in the current constrained instanton method and provides a better mathematical formalism than the constrained instanton method. The resulting instanton, which we call “valley instanton”, is shown to have the desirable behaviors. The result of the numerical investigation is also presented.

Path-integral approach in imaginary and complex time has been proven successful in treating the tunneling phenomena in quantum mechanics and quantum field theories. Latest developments in this field, the proper valley method in imaginary time, its application to various quantum systems, complex time formalism, asympton theory for the large order analysis of the perturbation theory, are reviewed in a self-contained manner.

Microscopic tests of the exact results are performed in N=2 SU(2) supersymmetric QCD. We construct the multi-instanton solution in N=2 supersymmetric QCD and calculate the two-instanton contribution F 2 to the prepotential F explicitly. For N f = 1, 2, instanton calculus agrees with the prediction of the exact results, however, for N f = 3, we find a discrepancy between them.

Microscopic tests of the exact results are performed in N = 2 supersymmetric SU (2) QCD. We present the complete construction of the multi-instanton in N = 2 supersymmetric QCD. All the defining equations of the super instanton are reduced to the algebraic equations. Using this result, we calculate the two-instanton contribution F 2 to the prepotential F for the arbitrary N f theories. For N f = 0, 1, 2, instanton calculus agrees with the prediction of the exact results, however, for N f = 3, 4, we find discrepancies between them. We propose improved curves of the exact results for the massive N f = 3 and massless N

We study the path-integral formalism in the imaginary-time to show its validity in a case with a metastable ground state. The well-known method based on the bounce solution leads to the imaginary part of the energy even for a state that is only metastable and has a simple oscillating behavior instead of decaying. Although this has been argued to be the failure of the Euclidean formalism, we show that proper account of the global structure of the path-space leads to a valid expression for the energy spectrum, without the imaginary part. For this purpose we use the proper valley method to find a new type of instanton-like configuration, the "valley instantons". Although valley instantons are not the solutions of equation of motion, they have dominant contribution to the functional integration. A dilute-gas approximation for the valley instantons is shown to lead to the energy formula. This method extends the well-known imaginary-time formalism so that it can take into account the global behavior of the theory.