Abstract
The essence of the path integral method in quantum physics can be expressed in terms of two relations between unitary propagators, describing perturbations of the underlying system. They inherit the causal structure of the theory and its invariance properties under variations of the action. These relations determine a dynamical algebra of bounded operators which encodes all properties of the corresponding quantum theory. This novel approach is applied to non-relativistic particles, where quantum mechanics emerges from it. The method works also in interacting quantum field theories and sheds new light on the foundations of quantum physics.
Highlights
Path integrals [3, 4] are a standard tool in the theoretical description of quantum systems
One can show [2] that the algebra AL0 is equipped with a norm which promotes it to a C*-algebra, i.e. an algebra which can be realized as a norm closed subalgebra of the algebra of bounded operators on some Hilbert space. Before showing that this algebra contains the entire information incorporated in the conventional quantum mechanical setting, let us emphasize that our scheme of constructing dynamical algebras covers a large set of systems of physical interest
In the present letter we have established a relation between the path integral approach to quantum physics and the framework of dynamical algebras, established in [1, 2]
Summary
Path integrals [3, 4] are a standard tool in the theoretical description of quantum systems. Starting from a classical theory, describing paths (orbits) of the underlying system in configuration space and an action, governing its dynamics, they provide formulas for the propagators, i.e. the time ordered scattering operators in the resulting quantum theory These formulas yield useful algorithms for the treatment of concrete problems. It is gratifying that the essence of the path integral formalism can be replaced by simple algebraic relations without having to dive into the mathematical subtleties of functional integrals on infinite dimensional configuration spaces. This reformulation sheds new light on the foundations of quantum physics. It is the aim of the present letter to clarify its relation to the path integral formalism
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