Compactified Kaluza-Klein Theories

1. Definition and Mechanism:
   • In compactified Kaluza-Klein theories, the extra dimensions are curled up into very small sizes, typically below the scale of 10−1810^{-18}10−18 meters​​.
   • These extra dimensions form a compact manifold which must be a solution to the higher-dimensional Einstein field equations​​.
2. Challenges:
   • The combination of macroscopic four-dimensional spacetime with the compactified extra-dimensional space must be a solution of the higher-dimensional field equations, which is straightforward in simpler cases but complex in higher dimensions​​.
   • The process of compactification often requires additional fields or modifications to the Einstein equations, sacrificing some of the elegance of a purely geometrical theory​​.
   • There is a need for matter fields to achieve compactification, and in some theories, these fields must be added by hand, which departs from the original goal of explaining all forces and matter geometrically​​.
3. Importance and Developments:
   • Compactification has led to significant developments in higher-dimensional unified physics, including supergravity and superstring theory​​.
   • Mechanisms like spontaneous compactification, influenced by additional fields or higher-dimensional cosmological constants, have become standard approaches​​.

Non-Compactified Kaluza-Klein Theories

1. Definition:
   • Non-compactified Kaluza-Klein theories consider the extra dimensions as real and extended, not curled up​​.
   • These theories often still use the term “Kaluza-Klein,” even though the dimensions are not compactified, which is somewhat contradictory to the original definition​​.
2. Challenges:
   • There is a lack of discussion on conformal rescaling in non-compactified theories because the extra dimensions are considered physically real and potentially observable​​.
   • Ensuring that the physical implications of these extended dimensions align with observations and do not contradict known physics is a significant challenge​​.

Summary of Key Points

• Compactified theories require extra dimensions to be compactified, which involves additional fields and modifications to fit the higher-dimensional field equations. These theories have driven much of the progress in high-dimensional unification theories.
• Non-compactified theories maintain extended extra dimensions but face different challenges in ensuring these dimensions are consistent with physical observations and are less discussed in terms of conformal rescaling.

The document provides a broad overview of both approaches, comparing their advantages and limitations within the context of higher-dimensional unified gravity theories.

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