Quantum Calculus: Fund. Theorem & Plank Time?

[duffbeerforme]

Hi, first post. I'm not a physics buff at all and this is probably an easy question to answer.
when looking at the fundamental theorem of calculus you take the limit as say t goes to zero (t being time). But does quantum physics say that t is not continuous.. something like a smallest time step such as plank time?.. and would this change calculus when dealing with real world problems?

thanks

 

[lbrits]

Well, t need not be time, and as far as calculus is concerned, it can be anything. In quantum mechanics time and space are usually treated as being continuous variables, and it is usually only when you throw gravity (or field theory) in the mix that you need to ask questions about discreteness of time.

Aside: In quantum mechanics, things are sometimes discrete and sometimes not. It's not fundamental to quantum mechanics, but rather the spaces on which the quantities are defined. For example, angles live on a compact space, and so angular momentum gets quantized. On the other hand, distances are unbounded and so these aren't quantized.

Back to your question. I don't know if this would really change real-world problems. The definition of a limit says that you never actually have to take t=0, but just "as close as you need to". That's sort of the heart of the delta-epsilon definition of the limit.

 

[denian]

for any help

I can provide some insights on this topic. Quantum calculus is a relatively new field that combines principles from quantum mechanics and calculus. The fundamental theorem of calculus, which states that the integral of a function can be calculated by finding the antiderivative of that function, has been a cornerstone of traditional calculus. However, in the realm of quantum mechanics, time is not considered a continuous variable, but rather it is quantized into discrete units called Planck time.

This means that in quantum calculus, the concept of taking the limit as time goes to zero does not apply. Instead, we must consider the smallest possible unit of time, the Planck time, and how it affects the behavior of particles at that scale. This can have significant implications when dealing with real-world problems, especially in the realm of quantum mechanics where the behavior of particles can be drastically different at the quantum level.

In short, the existence of Planck time does change the way we approach calculus in the context of quantum mechanics. It requires us to consider the discrete nature of time and how it affects the behavior of particles at the quantum level. However, this does not mean that traditional calculus is no longer applicable in the quantum realm. It simply means that we must adapt our mathematical tools to better understand the behavior of particles at this scale. I hope this helps to answer your question.