Quantizing a 4-derivative scalar field

Higher-derivative scalar field theories extend the familiar two-derivative kinetic term by introducing additional derivatives. In this talk, we will look at a 4-derivative scalar field and explore how some of its well-known pathologies arise. We’ll start with the equations of motion which give not only the usual plane-wave solutions but also growing modes. We’ll also investigate Ostrogradsky’s approach to higher-derivative systems and see why it leads to classical instabilities. Next, we’ll look at how to quantize the theory. We’ll see how ghost states emerge and what they imply for Green’s functions, Wightman functions, and the Källén–Lehmann spectral representation. Using this framework, we’ll discuss how negative-norm states show up in physical quantities such as inclusive rates. We’ll finish by discussing some proposed ways to deal with the issues we’ve encountered in higher-derivative scalar theories.