Paul BaldufThe #paperOfTheDay is "Quantum Ostrogradsky theorem" from 2020.
Classical #physics is based on functions such as the Lagrangian, action, or Hamiltonian (=energy), which always depend on at most the first time derivative of the quantity (such as a field or a particle position) in question. The classical equations of motion -- the Hamilton canonical equations -- are always first order partial differential equations in the variables "position" and "momentum", and the time evolution is determined from giving the position and the velocity at an initial time.
Structurally, it would be easy to write down a Lagrangian that depends on second time derivative. Then, if one sets up first-order canonical equations, there is a second canonical momentum, and a second canonical position for each variable. This would be a bit unintuitive, namely it would allow infinitely many distinct future time evolutions if position and velocity of a particle are given, but why not? The problem is that the so-obtained Hamiltonian is unbounded from below, and such system has no ground state. This is Ostrogradsky's theorem: A classical system with higher order time derivatives is inconsistent.
The present article proves that the same problem persists in quantum theory if one follows the usual path of canonical quantization. This is perhaps not surprising, but also not trivial: For example, the 1/r potential of a hydrogen atom has no classical minimum, but it does have a finite energy ground state in #quantum mechanics. Such effect is absent for the Ostrogradsky case, where the potential decays linearly, so that quantum fluctuations have no chance of making it bounded. https://link.springer.com/article/10.1007/JHEP09(2020)032
Classical #physics is based on functions such as the Lagrangian, action, or Hamiltonian (=energy), which always depend on at most the first time derivative of the quantity (such as a field or a particle position) in question. The classical equations of motion -- the Hamilton canonical equations -- are always first order partial differential equations in the variables "position" and "momentum", and the time evolution is determined from giving the position and the velocity at an initial time.
Structurally, it would be easy to write down a Lagrangian that depends on second time derivative. Then, if one sets up first-order canonical equations, there is a second canonical momentum, and a second canonical position for each variable. This would be a bit unintuitive, namely it would allow infinitely many distinct future time evolutions if position and velocity of a particle are given, but why not? The problem is that the so-obtained Hamiltonian is unbounded from below, and such system has no ground state. This is Ostrogradsky's theorem: A classical system with higher order time derivatives is inconsistent.
The present article proves that the same problem persists in quantum theory if one follows the usual path of canonical quantization. This is perhaps not surprising, but also not trivial: For example, the 1/r potential of a hydrogen atom has no classical minimum, but it does have a finite energy ground state in #quantum mechanics. Such effect is absent for the Ostrogradsky case, where the potential decays linearly, so that quantum fluctuations have no chance of making it bounded. https://link.springer.com/article/10.1007/JHEP09(2020)032
Quantum Ostrogradsky theorem - Journal of High Energy Physics
The Ostrogradsky theorem states that any classical Lagrangian that contains time derivatives higher than the first order and is nondegenerate with respect to the highest-order derivatives leads to an unbounded Hamiltonian which linearly depends on the canonical momenta. Recently, the original theorem has been generalized to nondegeneracy with respect to non-highest-order derivatives. These theorems have been playing a central role in construction of sensible higher-derivative theories. We explore quantization of such non-degenerate theories, and prove that Hamiltonian is still unbounded at the level of quantum field theory.
Stephen Brooks 🦆