# Variational principles for Probabilities.

• mhill
In summary, there is ongoing research and discussion about the application of concepts from General Relativity to Quantum Mechanics. Some propose that the Schrödinger equation can be derived from the geodesic equation in GR, known as geometric quantum mechanics. While the idea of considering the function P(r,t) as a minimal surface is intriguing, it may not directly apply to this scenario. This is an exciting area of research that has the potential to deepen our understanding of both theories.
mhill
I think this may sound odd and strange, the idea ocurred to me while i was watching a TV program about GR.

If Einstein and other proved that the equations of motion could be derived from Geodesic, or that point particles moved on Geodesic under no forces, could it be applied to QM

I mean , for example the function $$P(r,t)=|\Psi (r,t)|^{2}$$ is some kind of minimal surfaces, in the sense that it would minimize the integral taken over R-4

$$A(\Omega)= \int_{\Omega} |Gra(P(r,t))|^{2}dxdydzdt$$

Thank you for sharing your thoughts on this topic. It is not odd or strange at all to consider the application of concepts from General Relativity (GR) to Quantum Mechanics (QM). In fact, there has been a lot of research and discussion about this very topic in the scientific community.

To answer your question, yes, it is possible to apply the concept of geodesic motion to QM. In fact, some scientists have proposed that the Schrödinger equation, which describes the behavior of quantum particles, can be derived from the geodesic equation in GR. This idea, known as geometric quantum mechanics, suggests that the motion of quantum particles can be described as geodesic motion on a curved space-time.

In terms of your suggestion about the function P(r,t), it is an interesting idea to consider it as a minimal surface. However, it is important to note that the concept of minimal surfaces is more closely related to the concept of minimal action in classical mechanics, rather than geodesic motion in GR. Therefore, it may not directly apply to this scenario.

Overall, the idea of applying concepts from GR to QM is an ongoing and exciting area of research. It has the potential to deepen our understanding of both theories and potentially lead to new insights and discoveries. Thank you for bringing up this interesting topic.

I find your idea intriguing and worth exploring. Variational principles have been applied in various fields of physics, including classical mechanics and general relativity. Therefore, it is not unreasonable to consider applying them to quantum mechanics as well.

The concept of minimal surfaces in relation to probability functions is an interesting one. It is true that the probability function in quantum mechanics, represented by |\Psi (r,t)|^{2}, can be seen as a surface that minimizes the integral over R-4. This could potentially lead to a deeper understanding of the underlying principles governing quantum mechanics.

However, it is important to note that quantum mechanics is a highly complex and abstract theory, and any attempt to apply variational principles to it would require careful consideration and mathematical rigor. It is also worth mentioning that the concept of minimal surfaces has been studied extensively in the field of differential geometry, but its application to quantum mechanics may require new insights and developments.

In conclusion, while your idea is thought-provoking, it would require further research and analysis to determine its validity and potential implications. Nevertheless, exploring new ways to understand and describe quantum mechanics is a valuable pursuit in the scientific community.

## 1. What are variational principles for probabilities?

Variational principles for probabilities are mathematical principles that are used to optimize probability distributions. These principles are based on the concept of variational calculus, which involves finding the function that minimizes or maximizes a given functional. In this case, the functional is related to the probability distribution and the goal is to find the distribution that best fits the given data or constraints.

## 2. How are variational principles used in probability theory?

Variational principles are used in probability theory to find the most likely probability distribution that fits a given set of data or constraints. This is achieved by finding the functional that minimizes or maximizes the distribution's parameters, such as mean and variance, and then solving for the optimal solution using calculus.

## 3. What is the difference between variational principles and other optimization methods?

Variational principles differ from other optimization methods in that they are based on calculus and involve finding the optimal solution to a functional, rather than directly optimizing a given objective function. This allows for more flexibility in the optimization process and often leads to more accurate and efficient solutions.

## 4. What are some applications of variational principles for probabilities?

Variational principles for probabilities have a wide range of applications, including statistical inference, machine learning, and signal processing. They are also commonly used in physics and engineering to model and analyze complex systems with uncertain parameters.

## 5. Can variational principles be applied to any type of probability distribution?

Yes, variational principles can be applied to any type of probability distribution, as long as the functional being minimized or maximized can be expressed in terms of the distribution's parameters. This includes both discrete and continuous distributions, such as the binomial, normal, and exponential distributions.

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