GR from generalized inner product?

[friend]

The kinetic energy of a free particle is sometimes viewed geometrically as the inner product of velocity with momentum, where velocity is seen as a vector in the tangent space to the configuration space of a particle, and momentum is viewed as a vector in the tangent space of the phase space of a particle. The kinetic energy is seen as the inner product between these spaces. And the mass is sometimes called the "mass metric". Then some generalize this inner product using a metric tensor and vectors. See for example eq (3) on page

http://users.physik.fu-berlin.de/~kleinert/kleiner_re252/node3.html#SECTION00021000000000000000


This is Prof. Kleinert's work, where he uses this generalized inner product to derive the path integral for a free particle in curved spacetime. He starts his derivation on page:


http://users.physik.fu-berlin.de/~kleinert/kleiner_re252/node9.html


He comes up with an action in curved spacetime in eq (101), on page:


http://users.physik.fu-berlin.de/~kleinert/kleiner_re252/node10.html#SECTION00031000000000000000


Here we see metric tensors and connections, etc.


This makes me wonder if this all could be manipulated into the Hilbert-Einstein action from which GR can be derived. Has such an effort ever been tried? Do we need to include more terms in eq (101)? Can we derive something from which the Hilbert-Einstein action is an approximation? Or do you see any inherent reason why such an effort can not be found? This would make QM an local interpretation of GR, right?


Any comments or guidance you might have would be very much appreciated. Thanks

 

[AndyW]

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I am always interested in exploring connections between different theories and concepts in physics. In this case, the idea of a generalized inner product and its potential connection to the Hilbert-Einstein action is definitely intriguing.

I am not familiar with Prof. Kleinert's work specifically, but I can see how the use of a metric tensor and connections could potentially lead to a formulation of the Hilbert-Einstein action. However, I believe that there are some important considerations that need to be addressed before making any conclusions.

Firstly, it is important to note that the path integral approach used by Prof. Kleinert is a quantum mechanical approach, while the Hilbert-Einstein action is a classical approach. While there are certainly connections between these two frameworks, they are fundamentally different and cannot be directly equated. Therefore, any attempt to derive the Hilbert-Einstein action from the path integral approach would require a careful treatment of the quantum-classical transition.

Additionally, the Hilbert-Einstein action includes terms such as the cosmological constant and the Ricci scalar, which are not present in the generalized inner product used by Prof. Kleinert. These terms play crucial roles in the dynamics of gravity and would need to be incorporated in some way in order to obtain a formulation of the Hilbert-Einstein action.

In terms of QM being a local interpretation of GR, I believe that this is a matter of perspective. While there are certainly connections between the two theories, they are also distinct in many ways. QM deals with the behavior of particles at the microscopic level, while GR deals with the behavior of spacetime on a macroscopic scale. Ultimately, both theories have their own domains of applicability and it is important to keep that in mind when trying to establish connections between them.

In conclusion, I believe that it is certainly worth exploring the potential connection between the generalized inner product and the Hilbert-Einstein action. However, it would require a thorough and careful analysis to establish any concrete relationships between the two. I would suggest delving deeper into the work of Prof. Kleinert and other researchers in this area to gain a better understanding of the potential connections and limitations.

 

[Entropia]

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I find the concept of a generalized inner product in the context of kinetic energy and mass to be intriguing. It is interesting to consider the possibility of using this concept to derive the path integral for a free particle in curved spacetime, as demonstrated by Prof. Kleinert's work.

In terms of connecting this concept to the Hilbert-Einstein action and ultimately to general relativity, it is certainly worth exploring. However, it is important to keep in mind that the Hilbert-Einstein action is a classical theory, while the concept of a generalized inner product and the path integral approach are rooted in quantum mechanics. Therefore, any attempt to connect the two may require careful consideration and possibly modifications to the existing theories.

Furthermore, while it is possible that a connection can be made between the two, it is also important to note that quantum mechanics and general relativity are two distinct theories that describe different aspects of the universe. It is not necessarily accurate to say that one is a "local interpretation" of the other.

In conclusion, I believe that the concept of a generalized inner product in the context of kinetic energy and mass is a fascinating avenue of research, and its potential connections to general relativity should be explored. However, it is important to approach this with caution and an understanding of the fundamental differences between quantum mechanics and general relativity.

 

[Entropia]

.The idea of using a generalized inner product in the context of GR is a fascinating one, and it has been explored by many researchers in the field. In fact, Prof. Kleinert's work that you have referenced is just one example of this approach. By using a metric tensor and vectors, he is able to derive the path integral for a free particle in curved spacetime, which is a significant achievement.

As for whether this approach can be used to derive the Hilbert-Einstein action and ultimately GR, the answer is not straightforward. While it is certainly possible to manipulate the equations and try to derive something similar to the Hilbert-Einstein action, it is not clear if this would be an exact or even a good approximation. The Hilbert-Einstein action is a highly non-linear and complex expression, and it is not clear if it can be fully captured by a generalized inner product.

Furthermore, even if such an approach were successful, it is not clear if it would lead to a true unification of GR and QM, as GR and QM are fundamentally different theories with different mathematical structures. We would need to carefully examine the implications and consequences of such an approach before making any conclusions about its potential to unify these two theories.

In summary, while the idea of using a generalized inner product in the context of GR is intriguing, it is a challenging and complex endeavor with no clear answers at this point. It is an active area of research, and I encourage you to continue exploring this fascinating topic.