Does a Functional Integral Really Need a Lagrangian?

Abstract

Path integral formulation of quantum mechanics (and also other equivalent formulations) depends on a Lagrangian and/or Hamiltonian function that is chosen to describe the underlying classical system. The arbitrariness presented in this choice leads to a phenomenon called Quantization ambiguity. For example both L₁ = ˙q² and L₂ = e^q˙ are suitable Lagrangians on a classical level (δL₁ = δL₂), but quantum mechanically they are diverse.

This paper presents a simple rearrangement of the path integral to a surface functional integral. It is shown that the surface functional integral formulation gives transition probability amplitude which is free of any Lagrangian/Hamiltonian and requires just the underlying classical equations of motion. A simple example examining the functionality of the proposed method is considered.