Can we apply 'Quantization' only from motion equation ?

[Klaus_Hoffmann]

Can we apply 'Quantization' only from motion equation ??

Supposing you have the equation of motion (in terms of momenta and position)

$$F(\dot p_{a} , q_{a})=0$$

then can you obtain the 'Quantum analogue' without the intervention of the Lagrangian ?

and another question could we regard the expression

$$\int \mathcal D[q(t)] e^{-\int_{a}^{b}dt \mathcal L (q, \dot q , t)}$$

as a 'Zeta function' of something evaluated at a point s=1 where

$$\int_{a}^{b}dt \mathcal L (q, \dot q , t)} =logM[q(t)]$$

being M another functional, so the problems involving Functional integration (if not all many of them) could be sovled by Zeta regularization.

 

[Demystifier]

For a solution of the first problem, see
http://xxx.lanl.gov/abs/quant-ph/0311159

 

[Entropia]

I cannot provide a definitive answer to these questions as they are highly theoretical and require a deep understanding of both quantum mechanics and mathematical concepts such as functional integration and zeta functions. However, I can offer some insights and considerations.

Firstly, it is important to note that quantization is a complex process that involves converting classical systems and equations into their quantum counterparts. It is not a simple matter of applying a single equation or concept, but rather a combination of various mathematical and physical principles.

Regarding the first question, it is possible to obtain a quantum analogue from the equation of motion without the use of a Lagrangian. This approach is known as the Hamiltonian quantization, where the Hamiltonian operator is used to generate the quantum equations of motion. However, the Lagrangian approach is more commonly used and has a more intuitive connection to classical mechanics.

In regards to the second question, it is possible to view the expression as a zeta function evaluated at s=1, but it is not clear what the underlying quantity or function would be. Zeta regularization has been used in various areas of physics, but its applicability to functional integration in the context of quantization is still an open question.

In summary, while it may be possible to apply quantization solely from the motion equation, it is not a straightforward process and may not provide a complete understanding of the quantum system. Further research and analysis are needed to fully understand the role of zeta regularization in functional integration and its potential applications in quantization.