Understanding Continuous Space, Spectra, and Planck Units in Quantum Spacetime

[denisv]

How are continuous space and continuous spectra of operators mathematically consistent with Planck units?

Shouldn't the quantum spacetime be a lattice (or what have you) of integer multiples of the Planck length?

 

[muppet]

Probably.
In ordinary QM treating spacetime as continuous has the distinct advantage that the wavefunction is a differentiable function. The level at which this is an approximation is really, really small, much smaller than the levels at which one uses ordinary QM. If you're probing those kind of distance scales you probably at least want to be using QFT anyway, if not BSM (beyond the standard model) physics; nobody claims that QM alone is a complete and accurate description of nature (or if they do, they're wrong).
Afraid I can't tell you much about anything more advanced than QM though (soon ... once I've done these exams :( ) Perhaps someone else on here can help. If not, try the dedicated BSM forum.

 

[Entropia]

I am happy to provide a response to this topic. The concept of continuous space and continuous spectra in quantum spacetime can be quite complex and requires a deep understanding of quantum mechanics and theoretical physics.

Firstly, let's start with the concept of continuous space. In classical physics, space is often described as a continuous entity, meaning that it can be divided into infinitely small units. However, in quantum mechanics, space is believed to be discrete and is described by the Planck length, which is the smallest possible length that can exist in the universe. This means that space is not continuous but rather made up of tiny discrete units.

Now, let's move on to continuous spectra. In quantum mechanics, physical quantities such as energy and momentum are described by operators, which have a corresponding spectrum of possible values. The spectrum of an operator is continuous if it can take on any value within a certain range, as opposed to discrete where it can only take on specific values. This continuous nature of spectra is consistent with the concept of continuous space, as it allows for a smooth and continuous distribution of energy and momentum within space.

So, how do Planck units fit into all of this? Planck units, such as the Planck length, Planck time, and Planck mass, are fundamental units of measurement that are derived from fundamental constants of nature. These units are believed to be the smallest possible units that can exist in the universe and are used to describe phenomena at the quantum level. Therefore, the concept of continuous space and continuous spectra is consistent with Planck units, as they both describe the discretization of physical quantities at the quantum level.

Now, to address your question about quantum spacetime being a lattice of integer multiples of the Planck length. While this idea has been proposed in some theories, it is not the only way to understand quantum spacetime. The concept of a lattice structure implies a discrete and rigid framework, whereas the idea of continuous space and continuous spectra allows for a more flexible and dynamic understanding of quantum spacetime. Ultimately, the exact nature of quantum spacetime is still a topic of ongoing research and debate among scientists.

In conclusion, the concepts of continuous space, continuous spectra, and Planck units are all mathematically consistent with each other in the framework of quantum mechanics. They provide a deeper understanding of the fundamental building blocks of the universe and how they interact at the quantum level. However, the exact nature of quantum spacetime and its relationship