The momentum space associated with “tachyonic particles” proves to be rather
intricate, departing very much from the ordinary dual to Minkowski space
directly parametrized by space-time translations of the Poincar’e group. In
fact, although described by the constants of motion (Noether invariants)
associated with space-time translations, they depend non-trivially on the
parameters of the rotation subgroup. However, once the momentum space is
parametrized by the Noether invariants, it behaves exactly as that of ordinary
particles. On the other hand, the evolution parameter is no longer the one
associated with time translation, whose Noether invariant, $P_o$, is now a
basic one. Evolution takes place in a spatial direction. These facts not only
make difficult the computation of the corresponding representation, but also
force us to a sound revision of several traditional ingredients related to
Cauchy hypersurface, scalar product and, of course, causality. After that, the
theory becomes consistent and could shed new light on some special physical
situations like inflation or traveling inside a black hole.
The conclusions of the text can be summarized as follows:
- The momentum space associated with “tachyonic particles” is complex and different from Minkowski space.
- The constants of motion associated with space-time translations depend non-trivially on the parameters of the rotation subgroup.
- Once the momentum space is parametrized by the Noether invariants, it behaves like that of ordinary particles.
- The evolution parameter is no longer associated with time translation, but with a spatial direction.
- These facts make the computation of the corresponding representation difficult and require a revision of traditional ingredients.
- After this revision, the theory becomes consistent and could provide new insights into inflation or traveling inside a black hole.
Future Roadmap
Looking ahead, there are both challenges and opportunities on the horizon in this field. Here is a possible roadmap for readers to consider:
1. Understanding Tachyonic Particles
One of the immediate challenges is to further explore the intricacies of momentum space associated with tachyonic particles. Researchers should focus on understanding the non-trivial dependence on parameters from the rotation subgroup and how it affects the behavior of these particles. This understanding is crucial for advancing the field.
2. Parametrizing Momentum Space
A key opportunity lies in finding an effective way to parametrize the momentum space using Noether invariants. This would help simplify computations and make the behavior of tachyonic particles more similar to that of ordinary particles. Researchers should investigate different methods and approaches to achieve this parametrization.
3. Revising Traditional Ingredients
To ensure consistency in the theory and overcome challenges related to Cauchy hypersurface, scalar product, and causality, a thorough revision of traditional ingredients is necessary. Researchers should critically examine these elements and propose alternative formulations that accommodate the unique characteristics of tachyonic particles. This revision may require interdisciplinary collaborations and new mathematical frameworks.
4. Special Physical Situations
Once the theory is consistent, there are promising opportunities to apply it to special physical situations, such as inflation or traveling inside a black hole. Researchers should explore these scenarios and investigate how the revised theory can shed new light on these phenomena. This could lead to breakthroughs in our understanding of the universe and open up avenues for further research.
Overall, the road ahead in studying tachyonic particles is challenging but full of potential. By addressing the complexity of momentum space, revising traditional ingredients, and exploring special physical situations, researchers can make significant advancements in this field and uncover new insights into fundamental physics.