Lattice dimension and Planck length
In an earlier post, we have seen that the spacetime lattice that forms the backbone of the proposed local-realistic QM model, has dimensions that correspond to the Compton length and time. These quantities are claerly variable with the mass and thus particle-dependent. How can different particle ensembles coexist and evolve on the same lattice, then?
That is possible if all particles have a mass that is an integer fraction of a common multiple m0. If, say, the electron mass is me=m0/ke, that of the top quark mass is mt=m0/kt, that of the tau particle is mτ=m0/kτ, etc., with all the k's integers. Now, what is this common multiple m0?
Scientists have worked on the hypothesis of a discrete spacetime since long time. Most often, people prefer the concept of minimal length scale, that is the fundamental limit to the resolution of structures beyond which we cannot discover anything more. See the excellent paper of Sabine Hossenfelder, Minimal Length Scale for Quantum Gravity on this purpose. A promising candidate for such minimal length is the Planck length, that is defined as
where G is the gravitational constant, h-bar is the reduced Planck constant, and c is the speed of light in a vacuum, so that the Planck length is purely defined by means of fundamental physical constants. Its approximated numerical value is of 1.616 10-35 m.
Now, the mass for which the (mass-dependent, remember!) Compton length is of the same order of magnitude of the Planck length is the Planck mass, that is defined as
so, again, using only the three fundamental physics constants introduced above. If we equate m0 to mP, the smallest value of X (the dimension of the lattice) becomes the Planck length, in accordance to its probable role of minimal length scale as postulated by many. For what assumed above about the integers k's, a, say, electron will "see" a lattice dimension of
that is, an integer multiple of the minimal length scale. The same happens for other particles.
In other terms, while a light particle (large k) such as the electron will "jump" by relatively large amounts of time and space at each iteration of the model, more massive particles (small k) move by smaller time and position steps. So different particles can coexist on the same spatiotemporal lattice and even interact.
Again, this is plausible if all elementary particles have a mass that is exactly an integer fraction of the Planck mass. Unfortunately neither of the elementary particles' (nor the Planck mass itself) mass can be known with absolute precision, but only with some degree of approximation due to the experimental techniques adopted for the measurements. But, in a purely speculative approach, let assume that these integer factors k's exist and that they are all prime numbers (or, at least, they do not share common divisors) and let us check whether this is possible.
Let us take the the known fermion masses. Their least common multiple (LCM) equals their product and should be not greater than m0. By multiplying these masses, we obtain a number of the order of 1019 MeV/c2, or 10-11 kg, which is indeed smaller and not so far from the Planck mass. There is a "just" factor 1000 missing, which means that the factors k's are 1000 times larger than strictly necessary.
If we consider, however, bosons in addition to fermions, the product of single masses exceeds the Planck mass. In this case there should be common divisors between the factors k's (that cancel out when the LCM is calculated) for the assumptions to be true. Things become more complex if we treat composite particles such as baryons and mesons, at least those that are particularly stable such as the proton, as "elementary" in the sense that they have their own lattice steps. More thoughts on these aspects qill follow in future posts.
That is possible if all particles have a mass that is an integer fraction of a common multiple m0. If, say, the electron mass is me=m0/ke, that of the top quark mass is mt=m0/kt, that of the tau particle is mτ=m0/kτ, etc., with all the k's integers. Now, what is this common multiple m0?
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| Particles with different mass have different lattice steps that are nevertheless integer multiples of the same minimal length |
Scientists have worked on the hypothesis of a discrete spacetime since long time. Most often, people prefer the concept of minimal length scale, that is the fundamental limit to the resolution of structures beyond which we cannot discover anything more. See the excellent paper of Sabine Hossenfelder, Minimal Length Scale for Quantum Gravity on this purpose. A promising candidate for such minimal length is the Planck length, that is defined as
where G is the gravitational constant, h-bar is the reduced Planck constant, and c is the speed of light in a vacuum, so that the Planck length is purely defined by means of fundamental physical constants. Its approximated numerical value is of 1.616 10-35 m.
Now, the mass for which the (mass-dependent, remember!) Compton length is of the same order of magnitude of the Planck length is the Planck mass, that is defined as
so, again, using only the three fundamental physics constants introduced above. If we equate m0 to mP, the smallest value of X (the dimension of the lattice) becomes the Planck length, in accordance to its probable role of minimal length scale as postulated by many. For what assumed above about the integers k's, a, say, electron will "see" a lattice dimension of
that is, an integer multiple of the minimal length scale. The same happens for other particles.
In other terms, while a light particle (large k) such as the electron will "jump" by relatively large amounts of time and space at each iteration of the model, more massive particles (small k) move by smaller time and position steps. So different particles can coexist on the same spatiotemporal lattice and even interact.
Again, this is plausible if all elementary particles have a mass that is exactly an integer fraction of the Planck mass. Unfortunately neither of the elementary particles' (nor the Planck mass itself) mass can be known with absolute precision, but only with some degree of approximation due to the experimental techniques adopted for the measurements. But, in a purely speculative approach, let assume that these integer factors k's exist and that they are all prime numbers (or, at least, they do not share common divisors) and let us check whether this is possible.
Let us take the the known fermion masses. Their least common multiple (LCM) equals their product and should be not greater than m0. By multiplying these masses, we obtain a number of the order of 1019 MeV/c2, or 10-11 kg, which is indeed smaller and not so far from the Planck mass. There is a "just" factor 1000 missing, which means that the factors k's are 1000 times larger than strictly necessary.
If we consider, however, bosons in addition to fermions, the product of single masses exceeds the Planck mass. In this case there should be common divisors between the factors k's (that cancel out when the LCM is calculated) for the assumptions to be true. Things become more complex if we treat composite particles such as baryons and mesons, at least those that are particularly stable such as the proton, as "elementary" in the sense that they have their own lattice steps. More thoughts on these aspects qill follow in future posts.
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