Discrete spacetime means discrete momentum ?

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discrete spacetime means discrete momentum ??

the question is using De Broglie's Wavelength [tex] \lambda = h|p|^{-1} [/tex]

then in case space is discrete would mean that there is a minimum possible wavelength in nature , for example [tex] \lambda = k l_{p} [/tex] for Planck's lenght this would mean that the maximum allowed momentum modulus would be [tex] h(k l_{p})^{-1} [/tex] for 'k' a constant , then since momentum is finite and spacetime is also finite there wouldn't be any IR or UV divergences but does this make sense
 

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Yes. Lattice models are often used in calculations for high energy physics or in condensed matter when considering lattices. However, note that discrete spacetime may not be that simple --- if we take GR seriously and treat the spacetime as part of the system, it will be in a quantum superposition of states, and it may not be necessarily the case that physical properties are still as manifestly discretised.
 

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