What is the right time for path integrals

Aoyama, H, Harano, T

The complex-time formalism is developed in the framework of the path-integral formalism, to be used for analysis of the quantum tunneling phenomena. We show that subleading complex-time saddle-points do not account for the right WKB result. Instead, we develop a reduction formula, which enables us to construct Green functions from simple components of the potential, for which saddle-point method is applicable. This method leads us to the valid WKB result, which incorporates imaginary-time instantons and bounces, as well as the real-time boundary conditions.

Mark S. Swanson

David Bailin, A. Love

There are two widely used approaches to quantum field theory. The first is based on field operators and the canonical quantisation of these field operators. The second approach involves path integrals over classical fields. This chapter develops the idea of path integrals (or functional integrals) in a very intuitive way without any attempt at mathematical precision or rigour. It exploits the analogy between vectors and functions, and between matrices and differential operators on functions. Since very few path integrals can be performed exactly, the chapter concentrates on Gaussian path integrals. These are important in their own right, but much more so because they can be used in approximation schemes when the exact path integral is intractable.

R. J. Rivers

Mikoto Matsuda, Takehisa Fujita

The path integral formulation can reproduce the right energy spectrum of the harmonic oscillator potential, but it cannot resolve the Coulomb potential problem. This is because the path integral cannot properly take into account the boundary condition, which is due to the presence of the scattering states in the Coulomb potential system. On the other hand, the Sommerfeld quantization can reproduce the right energy spectrum of both harmonic oscillator and Coulomb potential cases since the boundary condition is effectively taken into account in this semiclassical treatment. The basic difference between the two schemes should be that no constraint is imposed on the wave function in the path integral while the Sommerfeld quantization rule is derived by requiring that the state vector should be a single-valued function. The limitation of the semiclassical method is also clarified in terms of the square well and $δ(x)$ function potential models.