arXiv:2412.16183v1 Announce Type: new
Abstract: Two distinct energy-momentum tensors of the theory of weak gravity and spinor quantum mechanics are analyzed with respect to their four-divergence and expectation values of energy. The first energy-momentum tensor is obtained by a straightforward generalization of the symmetric energy-momentum tensor of a free Dirac field, and the second is derived by the second Noether theorem. We find that the four-divergences of both tensors are not equal. Particularly, the tensor derived by the generalization procedure does not match the four-divergence of the canonical energy-momentum tensor. As a result, both tensors predict distinct values for the energy of the Dirac field. The energy-momentum tensor of the non-extended theory with the correct expression for four-divergence obtained by the second Noether theorem is asymmetric. This contradicts the requirements of general relativity. To rectify this situation, the Lagrangian of the theory is extended with the Lagrangian of the free electromagnetic field on curved spacetime. Then, the symmetric energy-momentum tensor of quantum electrodynamics with the required four-divergence is obtained by the second Noether theorem. Moreover, the energy-momentum tensor appears in the interaction Lagrangian term of the extended theory. In addition, we show that the Lagrangian density of the extended theory can be recast into the Lagrangian density of a flat spacetime theory, contrary to the statement made for the non-extended theory.
Conclusion:
In this study, two different energy-momentum tensors in the theory of weak gravity and spinor quantum mechanics are analyzed. It is found that the four-divergences of both tensors are not equal, leading to distinct energy predictions for the Dirac field. Moreover, the energy-momentum tensor derived by the generalization procedure does not match the canonical energy-momentum tensor, and the one derived by the second Noether theorem is asymmetric, contradicting the requirements of general relativity.
To address this discrepancy, the Lagrangian of the theory is extended with the Lagrangian of the free electromagnetic field on curved spacetime. This extension allows the symmetric energy-momentum tensor of quantum electrodynamics, with the correct four-divergence, to be obtained through the second Noether theorem. Furthermore, the energy-momentum tensor appears in the interaction Lagrangian term of the extended theory.
The study also demonstrates that the Lagrangian density of the extended theory can be recast into the Lagrangian density of a flat spacetime theory, contrary to the previous statement made for the non-extended theory.
Future Roadmap:
- Investigate further the implications of the distinct energy predictions for the Dirac field based on the different energy-momentum tensors.
- Explore the consequences of the asymmetry in the energy-momentum tensor on general relativity and its compatibility with other theories.
- Further examine the extended theory with the Lagrangian of the free electromagnetic field on curved spacetime, and investigate its implications and predictions.
- Evaluate the significance of the appearance of the energy-momentum tensor in the interaction Lagrangian term, and study its effects on other quantum mechanical systems.
- Compare and contrast the recasting of the Lagrangian density from an extended theory into that of a flat spacetime theory, and analyze any implications or limitations of this recasting.
- Consider the possible modification or refinement of the existing theory to reconcile the discrepancies and address the contradictions with general relativity.
Challenges:
- Understanding the underlying reasons for the unequal four-divergences of the energy-momentum tensors and the implications on energy predictions.
- Exploring the consequences of the asymmetric energy-momentum tensor on the compatibility of the theory with general relativity.
- Investigating the extended theory and analyzing its predictions, particularly in relation to other quantum mechanical systems.
- Determining the significance and effects of the appearance of the energy-momentum tensor in the interaction Lagrangian term.
- Thoroughly examining the recasting of the Lagrangian density and its potential implications and limitations.
- Developing modifications or refinements to the theory to resolve the discrepancies and ensure consistency with general relativity.
Opportunities:
- Advancing knowledge and understanding in the theory of weak gravity and spinor quantum mechanics.
- Contributing to the field of quantum electrodynamics and its interaction with curved spacetime.
- Exploring potential connections between the extended theory and other areas of physics.
- Engaging in interdisciplinary research to bridge the gaps between different theories.
- Promoting further discussions and collaborations among physicists to address the challenges and opportunities in this field.
This study highlights the discrepancies and contradictions in the energy-momentum tensors of weak gravity and spinor quantum mechanics. It presents a roadmap for future research, outlining the challenges that need to be overcome and the opportunities for advancing knowledge and understanding in this field.