Action with Acceleration II: Euclidean Hamiltonian and Jordan Blocks

TL;DR

This paper investigates the Hamiltonian structure of Euclidean action with acceleration, focusing on pseudo-Hermitian cases and Jordan blocks, revealing unique propagator features and state space properties.

Contribution

It analyzes the state space and propagator of acceleration-based Euclidean actions, highlighting the role of dual vectors and Jordan block structures in non-pseudo-Hermitian cases.

Findings

Propagator calculation emphasizes dual state vectors' importance.

Hamiltonian can be a direct sum of Jordan blocks when not pseudo-Hermitian.

Distinct behavior in pseudo-Hermitian versus non-pseudo-Hermitian Hamiltonians.

Abstract

The Euclidean action with acceleration has been analyzed in [1], hereafter cited as reference I, for its Hamiltonian and path integral. In this paper, the state space of the Hamiltonian is analyzed for the case when it is pseudo-Hermitian (equivalent to a Hermitian Hamiltonian), as well as the case when it is inequivalent. The propagator is computed using both creation/destruction operators as well as the path integral. A state space calculation of the propagator shows the crucial role played by the dual state vectors that yields a result impossible to obtain from a Hermitian Hamiltonian acting on a Hilbert space. When it is not pseudo-Hermitian, the Hamiltonian is shown to be a direct sum of Jordan blocks.