Intrinsicallydefined HigherDerivative Carrollian Scalar Field Theories without Ostrogradsky Instability
Core Concepts
Carrollian higherderivative scalar field theories intrinsically defined on a Carrollian manifold have richer dynamics and are more resistant to Ostrogradsky instabilities compared to their Lorentzian counterparts.
Abstract
The paper derives the most generic Carrollian higherderivative scalar field theory intrinsically defined on a Carrollian manifold. The key insights are:

The solutions to these Carrollian theories represent n massless particles propagating at different speeds, unlike Lorentzian theories where all massless particles move at the speed of light. This allows for interference patterns that are forbidden in Lorentzian theories.

Carrollian higherderivative theories are more resistant to Ostrogradsky instabilities compared to Lorentzian theories. The Ostrogradsky instability, which leads to an unbounded Hamiltonian, can be resolved in Carrollian theories by appropriately choosing the coupling constants, without the need for additional constraints on the phase space.

The most general Carrollian higherderivative scalar field Lagrangian cannot be derived as a limit of the Lorentzian counterpart, demonstrating that Carrollian theories are more general and should not be thought of only as limits of Lorentzian theories.
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Intrinsicallydefined higherderivative Carrollian scalar field theories without Ostrogradsky instability
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The Lagrangian for the Carrollian higherderivative scalar field theory is given by:
L = φ□g1□g2 · · · □gnφ
where □gk = gk∂2t + ∂i∂i, and gk are taken to be negative coupling constants.
The dispersion relation for the solutions is:
ω = ±1/√(gi) k
where i ∈ {1, 2, ..., n}.
Quotes
"The solutions to these Carrollian theories represent n massless particles propagating at different speeds, unlike Lorentzian theories where all massless particles move at the speed of light."
"Carrollian higherderivative theories are more resistant to Ostrogradsky instabilities compared to Lorentzian theories."
"The most general Carrollian higherderivative scalar field Lagrangian cannot be derived as a limit of the Lorentzian counterpart, demonstrating that Carrollian theories are more general and should not be thought of only as limits of Lorentzian theories."
Deeper Inquiries
How do the particles with different propagation speeds in Carrollian higherderivative scalar field theories interact with each other?
In Carrollian higherderivative scalar field theories, the solutions to the field equations represent multiple massless particles propagating at different speeds, as indicated by the distinct dispersion relations derived from the Lagrangian. The interaction between these particles is facilitated by the nontrivial dynamics allowed by the Carrollian structure, which permits interference patterns that are not possible in Lorentzian theories.
The interaction can be understood through the superposition principle, where the field configurations can be expressed as linear combinations of plane waves, each corresponding to a different propagation speed. This leads to the possibility of constructive and destructive interference among the waves, resulting in complex field dynamics. The coupling constants in the Lagrangian play a crucial role in determining the nature and strength of these interactions.
Moreover, since the Lagrangian is not required to be Carroll boost invariant, it allows for the presence of nonzero local energy flux, which can lead to interactions that are dependent on the relative speeds of the particles. This contrasts with Lorentzian theories, where all massless particles must propagate at the speed of light, limiting the interaction dynamics. Thus, the rich structure of Carrollian theories opens up new avenues for exploring particle interactions in a way that is fundamentally different from traditional field theories.
Can the results in this paper be extended to other field theories, such as Dirac fields, Maxwell fields, or gravity, on Carrollian manifolds?
Yes, the results presented in this paper can be extended to other field theories, including Dirac fields, Maxwell fields, and gravity, on Carrollian manifolds. The intrinsic formulation of higherderivative scalar field theories provides a framework that can be adapted to these other types of fields.
For Dirac fields, one can construct a Carrollian version of the Dirac equation that respects the Carrollian structure, allowing for the exploration of fermionic dynamics in this nonrelativistic limit. Similarly, for Maxwell fields, one can define a Carrollian electromagnetic theory that incorporates the unique features of Carrollian geometry, such as the degenerate metric and the associated gauge invariance.
In the context of gravity, the strong coupling limit of gravity theories can be analyzed within the Carrollian framework, potentially leading to new insights into gravitational dynamics in the absence of lightspeed constraints. The stability properties and the ability to avoid Ostrogradsky instabilities, as demonstrated for scalar fields, can also be investigated in these extended theories, providing a richer understanding of their dynamics.
Overall, the intrinsic approach to defining field theories on Carrollian manifolds allows for a broader application of the results, paving the way for future research in various areas of theoretical physics.
What are the potential applications of the rich dynamics and stability properties of Carrollian higherderivative scalar field theories in areas like condensed matter physics, cosmology, or black hole physics?
The rich dynamics and stability properties of Carrollian higherderivative scalar field theories have several potential applications across various fields, including condensed matter physics, cosmology, and black hole physics.
In condensed matter physics, the unique interference patterns and propagation characteristics of particles with different speeds can be leveraged to study phenomena such as waveparticle duality and quantum coherence in materials. The ability to manipulate the coupling constants in the Carrollian Lagrangian may lead to novel materials with tailored properties, such as enhanced conductivity or unique optical behaviors.
In cosmology, Carrollian theories can provide insights into the behavior of fields in the early universe, particularly in scenarios involving rapid expansion or inflation. The nontrivial dynamics of Carrollian fields may offer new mechanisms for generating perturbations in the cosmic microwave background or for understanding the nature of dark energy and dark matter.
In the context of black hole physics, the stability properties of Carrollian higherderivative theories can be crucial for exploring the dynamics of fields in the strong gravitational regime. The ability to avoid Ostrogradsky instabilities allows for a more robust treatment of quantum fields near black hole horizons, potentially leading to new insights into information paradoxes and the nature of spacetime singularities.
Overall, the applications of Carrollian higherderivative scalar field theories are vast and varied, with the potential to influence our understanding of fundamental physics and to inspire new experimental and theoretical investigations.