Quantum Ostrogradsky theorem
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Springer
Received: July 8, 2020 Accepted: August 9, 2020 Published: September 4, 2020
Hayato Motohashia,1 and Teruaki Suyamab a
Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan b Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
E-mail: [email protected], [email protected] Abstract: The Ostrogradsky theorem states that any classical Lagrangian that contains time derivatives higher than the first order and is nondegenerate with respect to the highestorder 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 nondegenerate theories, and prove that Hamiltonian is still unbounded at the level of quantum field theory. Keywords: Cosmology of Theories beyond the SM, Differential and Algebraic Geometry ArXiv ePrint: 2001.02483
1
Present address: Division of Liberal Arts, Kogakuin University, 2665-1 Nakano-machi, Hachioji, Tokyo, 192-0015, Japan.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)032
JHEP09(2020)032
Quantum Ostrogradsky theorem
Contents 1
2 Ostrogradsky theorem 2.1 Classical analysis 2.2 Quantum analysis
2 2 2
3 Theory with third-order EOM 3.1 Classical analysis 3.2 Quantum analysis
3 4 4
4 Quantum field theory
6
5 General theories
7
6 Conclusion
8
1
Introduction
The Ostrogradsky theorem states that any Lagrangian which contains more than first-order time derivatives and is non-degenerate with respect to the highest-order derivatives leads to a classical Hamiltonian which is not bounded due to its linear dependence on the canonical momenta [1, 2]. It implies that there is in general no stable configuration, which is known as the Ostrogradsky instability. Recently, a generalization of the Ostrogradsky theorem has been investigated extensively. It was proved in [3] that even though a higher-derivative Lagrangian is degenerate with respect to the highest-order derivatives and hence avoids the original Ostrogradsky theorem, a nondegeneracy with respect to the next-highest order derivatives still makes the Hamiltonian unbounded. Furthermore, analysis on more general Lagrangian involving multiple kinds of variables with arbitrary higher derivatives was established recently [4–7]. These generalizations of the original Ostrogradsky theorem are powerful and provide severe restriction for constructing consistent higher-derivative theories. However, the unboundedness of the classical Hamiltonian does not always mean that the corresponding quantum Hamiltonian is also unbounded since the canonical momentum cannot be varied independently of its conjugate variable due to the uncertainty principle. For instance, the Hamiltonian o
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