# Amplitude paths and Markov processes

• wofsy
In summary, mathematical models can be used to describe the state evolution of a free particle, but these paths may have physical meaning.
wofsy
In continuous Markov processes there is an idea of sample path. Starting at a point we follow the values of the process through time and find that almost surely the paths are continuous. In Brownian motion (Wiener process) the paths are crinkly curves, continuous paths of infinite variation.

The evolution of amplitudes seems to me to be a continuous Markov like process where complex amplitudes replace real probabilities. What are the sample paths for a free particle? Are they continuous? What is their variation? If these paths exist - why can't we define them as the paths of a quantum particle's state?

Your first paragraph is a discussion of mathematical concepts (Markov processes - brownian motion) which have physics applications, such as dust particles moving in air.

Your second paragraph seems to be a question of whether similar mathematical models can be used for quantum mechanical concepts. Personally I can't answer (I am a mathematician, not a physicist). However, it would be helpful if you could clarify your question, keeping in mind the difference between mathematics and physics.

mathman said:
Your first paragraph is a discussion of mathematical concepts (Markov processes - brownian motion) which have physics applications, such as dust particles moving in air.

Your second paragraph seems to be a question of whether similar mathematical models can be used for quantum mechanical concepts. Personally I can't answer (I am a mathematician, not a physicist). However, it would be helpful if you could clarify your question, keeping in mind the difference between mathematics and physics.

my question was two fold part mathematical part physical.

Mathematical: Is there a notion of state path for a complex continuous Markov proces such as the state evolution of a free particle?

Physical: If so, do these paths have any physical meaning?

Since I posted this question I spoke with a physicist friend who said that in QM there really are no state paths as in the path of a dust particle in a fluid.,

On the other hand, it seems that Bohm's hidden variable theory reproduces the state process as paths of actual particles. so it seems that these paths can be given physical meaning.

I think Feynman may have some formulation interms of paths but I really don't know.

A comment on your cautionary suggestion that I keep in mind the difference between mathematics and physics. Often mathematical ideas have physical corrolaries whether or not they are directly related to a currently known theory. Chern Simons invariants,the theory of connections and the idea of a Riemannian manifold come to mind.I was asking about a mathematical feature of a known theory which has even a better chance of having a physical corollary.

I also have a philosophical interest in the structure of QM as a theory but this is a physics thread so i won't go into it.

## 1. What is the difference between amplitude paths and Markov processes?

Amplitude paths are defined as a sequence of values that describe the change in amplitude over time in a system. On the other hand, Markov processes are stochastic processes where the next state of the system only depends on the current state, not on the previous states. In other words, amplitude paths describe the behavior of a system over time, whereas Markov processes describe the probabilistic nature of a system's transitions between states.

## 2. How are amplitude paths and Markov processes used in scientific research?

Amplitude paths and Markov processes are commonly used in many fields of science, such as physics, chemistry, biology, and economics. They are often used to model and analyze complex systems, such as financial markets, chemical reactions, and biological processes. By studying amplitude paths and Markov processes, scientists can gain a better understanding of the underlying dynamics and behavior of these systems.

## 3. How are amplitude paths and Markov processes related to each other?

Amplitude paths can be used to describe the behavior of a Markov process, as the amplitude values at each time point can be seen as the states of the system. Additionally, Markov processes can be used to model the transitions between amplitude paths, as the probabilities of transitioning between states can be described by Markov matrices.

## 4. Can amplitude paths and Markov processes be applied to real-world systems?

Yes, amplitude paths and Markov processes have been successfully applied to a wide range of real-world systems, including financial markets, climate patterns, and biological networks. By using these mathematical tools, scientists can gain insights into the behavior and characteristics of complex systems, which can then be used to make predictions and inform decision-making processes.

## 5. What are some limitations of using amplitude paths and Markov processes in research?

One limitation is that these mathematical models can be quite complex and challenging to interpret, especially when applied to highly nonlinear systems. Additionally, these models may not always accurately capture the full dynamics of a system, as they are based on assumptions and simplifications. Therefore, it is essential to carefully consider the limitations and assumptions of these methods when applying them to real-world systems.

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