Classical Interactions in dx,dt→0 Contradicting Special Relativity?
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2025
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| _version_ | 1866901227502043136 |
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| author | Ruggeri, Francesco R. |
| author_facet | Ruggeri, Francesco R. |
| contents | <p dir="ltr"> In a previous note (1), we argued that special relativity introduces two sets of dx, dt intervals, with only one, dx1,dt1, linked to classical Newtonian mechanics. We argued that the second set, dx2,dt2, is associated with 2-slit interference patterns etc. In the case of a photon, the two sets merge. The first set is the Newtonian dx1 = v dt1 which allows for dx ,dt→0. The second set is dx2=constant/p and dt2=constant/E.</p> <p dir="ltr"> The point we wish to make here is that if E and p, Newtonian variables in the nonrelativistic limit, are really associated with fixed length dt2 and dx2 intervals, then this contradicts the notion of interactions occurring in dx1, dt1→0, unless dx2, dt2 fit inside dx, dt, which are being made arbitrarily small. This can only hold as an approximation, with dx1, dt1 taken as tiny compared to system lengths . Newtonian mechanics is associated with changes in E and p, but these are said to occur in dx1,dt1→0, whereas these variables are linked with bigger intervals, dx2,dt2. </p> <p dir="ltr">If, however, dx2,dt2 are not small compared to characteristic lengths (times) such as those found in quantum bound problems, then there is a contradiction between dx2,dt2 and dx1,dt1 and the view of interactions occurring in dx1,dt1 need not hold as it doesn’t in quantum bound problems. </p> <p dir="ltr"> The problem which arises is that E and p are defined in terms of v in special relativity and v is automatically linked with dx1,dt1 for a particle with rest mass, mo, but at the same time are closely associated with fixed length dx2,dt2. If E and p are interaction variables and are linked with dx2,dt2 (fixed), then it seems that interactions which affect E and p may occur over dx2, dt2 (as they do in 2-slit interference). Nevertheless, the notion of v does not disappear and must be associated with dx1,dt1. Thus, it seems that in a bound problem, for which time is removed, various p values are linked with interactions in dx2= constant/p, but p is linked with a velocity which is associated with dx1 and dt1. This is fine if one does not consider a force as changing v at a given x,t. Instead, there may exist various p values linked with dx2. This suggests a statistical averaging process which would allow dx1,dt1 to apply to averaged variables, whereas the averaging process itself utilizes probabilities linked with dx2, dt2. </p> <p dir="ltr"> The question then becomes: What should be averaged and what probability distribution should be used? In a classical bound problem in a time-independent framework, one has a constant E, but KE(x), v(x) and p(x) at each x. We have argued that v(x) must be linked with dx1 and dt1. v(x) follows from both p(x) and KE(x), except in the latter it is vrms(x), thus which should be used to create an average? If KE(x) is used, then KEave(x)+V(x)= E. If p(x) is used, then dpave(x)/dt = -dV/dx. The interaction variable is p, but it is linked with an interval dx2=constant/p so we use the probability function exp(ipx) to represent minimal information in space with fixed constant/p intervals. The problem with dpave/dt is that:</p> <p><strong> </strong></p> <p dir="ltr">dpave/dt = dpave(x)/dx dx/dt where dx/dt = v which must be calculated using dx1,dt1 mixing </p> <p><strong> </strong></p> <p dir="ltr">dx2,dt2 and dx1,dt1 intervals which one does not want with an exp(ipx) averaging scheme. Instead, one averages over KE and uses vrms. </p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15161286 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Classical Interactions in dx,dt→0 Contradicting Special Relativity? Ruggeri, Francesco R. <p dir="ltr"> In a previous note (1), we argued that special relativity introduces two sets of dx, dt intervals, with only one, dx1,dt1, linked to classical Newtonian mechanics. We argued that the second set, dx2,dt2, is associated with 2-slit interference patterns etc. In the case of a photon, the two sets merge. The first set is the Newtonian dx1 = v dt1 which allows for dx ,dt→0. The second set is dx2=constant/p and dt2=constant/E.</p> <p dir="ltr"> The point we wish to make here is that if E and p, Newtonian variables in the nonrelativistic limit, are really associated with fixed length dt2 and dx2 intervals, then this contradicts the notion of interactions occurring in dx1, dt1→0, unless dx2, dt2 fit inside dx, dt, which are being made arbitrarily small. This can only hold as an approximation, with dx1, dt1 taken as tiny compared to system lengths . Newtonian mechanics is associated with changes in E and p, but these are said to occur in dx1,dt1→0, whereas these variables are linked with bigger intervals, dx2,dt2. </p> <p dir="ltr">If, however, dx2,dt2 are not small compared to characteristic lengths (times) such as those found in quantum bound problems, then there is a contradiction between dx2,dt2 and dx1,dt1 and the view of interactions occurring in dx1,dt1 need not hold as it doesn’t in quantum bound problems. </p> <p dir="ltr"> The problem which arises is that E and p are defined in terms of v in special relativity and v is automatically linked with dx1,dt1 for a particle with rest mass, mo, but at the same time are closely associated with fixed length dx2,dt2. If E and p are interaction variables and are linked with dx2,dt2 (fixed), then it seems that interactions which affect E and p may occur over dx2, dt2 (as they do in 2-slit interference). Nevertheless, the notion of v does not disappear and must be associated with dx1,dt1. Thus, it seems that in a bound problem, for which time is removed, various p values are linked with interactions in dx2= constant/p, but p is linked with a velocity which is associated with dx1 and dt1. This is fine if one does not consider a force as changing v at a given x,t. Instead, there may exist various p values linked with dx2. This suggests a statistical averaging process which would allow dx1,dt1 to apply to averaged variables, whereas the averaging process itself utilizes probabilities linked with dx2, dt2. </p> <p dir="ltr"> The question then becomes: What should be averaged and what probability distribution should be used? In a classical bound problem in a time-independent framework, one has a constant E, but KE(x), v(x) and p(x) at each x. We have argued that v(x) must be linked with dx1 and dt1. v(x) follows from both p(x) and KE(x), except in the latter it is vrms(x), thus which should be used to create an average? If KE(x) is used, then KEave(x)+V(x)= E. If p(x) is used, then dpave(x)/dt = -dV/dx. The interaction variable is p, but it is linked with an interval dx2=constant/p so we use the probability function exp(ipx) to represent minimal information in space with fixed constant/p intervals. The problem with dpave/dt is that:</p> <p><strong> </strong></p> <p dir="ltr">dpave/dt = dpave(x)/dx dx/dt where dx/dt = v which must be calculated using dx1,dt1 mixing </p> <p><strong> </strong></p> <p dir="ltr">dx2,dt2 and dx1,dt1 intervals which one does not want with an exp(ipx) averaging scheme. Instead, one averages over KE and uses vrms. </p> <p> </p> |
| title | Classical Interactions in dx,dt→0 Contradicting Special Relativity? |
| url | https://doi.org/10.5281/zenodo.15161286 |