- #1
"Don't panic!"
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This has been causing some confusion to me as the two concepts seem very similar, if not the same (especially when taking special relativity into account).
As far as I understand, even in classical physics (i.e. even before considering QFT and the like), one requires that interactions are local, i.e. the dynamics of a physical system are only affected by their immediate surroundings (an object can only exert a direct influence on another object, at a given instant in time, if they are in direct contact with one another - the interaction occurs at a single spatial point).
If I understand correctly, in classical mechanics this doesn't imply causality between events as propagation of information is not bounded by the speed of light. Thus, Newton's law of gravity [tex]\mathbf{F}=\frac{GMm}{\vert\mathbf{r}-\mathbf{r}'\vert^{2}}[/tex] is causal because it describes the influence of a body of mass ##M## on a body of mass ##m## - it unambiguously describes the effect on a body caused by another body. It is, however, non-local as it describes a direct influence on one body by another that are spatially separated by a finite distance ##\vert\mathbf{r}-\mathbf{r}'\vert## (i.e. they are not in direct contact).
The issue I struggle with is that when we introduce special relativity the two concepts no longer seem distinct as a space-like separation between two physical systems immediately implies that they are not in causal contact (causality is in some sense required by demanding locality)?!
Finally, is the reason why in textbooks it is explained that interactions are local if they occur at a single point in spacetime because locality is the requirement that two physical systems must be in direct contact in order to influence one another directly? (If they are at the same spatial location, but at different points in time, then they can not directly influence each other. Similarly, if they are at the same point in time, but they are located at different spatial points, then they can not directly influence one another. It is only when they are located at the same point (or infinitesimally close to one another) in space and time that they can directly influence one another)
As far as I understand, even in classical physics (i.e. even before considering QFT and the like), one requires that interactions are local, i.e. the dynamics of a physical system are only affected by their immediate surroundings (an object can only exert a direct influence on another object, at a given instant in time, if they are in direct contact with one another - the interaction occurs at a single spatial point).
If I understand correctly, in classical mechanics this doesn't imply causality between events as propagation of information is not bounded by the speed of light. Thus, Newton's law of gravity [tex]\mathbf{F}=\frac{GMm}{\vert\mathbf{r}-\mathbf{r}'\vert^{2}}[/tex] is causal because it describes the influence of a body of mass ##M## on a body of mass ##m## - it unambiguously describes the effect on a body caused by another body. It is, however, non-local as it describes a direct influence on one body by another that are spatially separated by a finite distance ##\vert\mathbf{r}-\mathbf{r}'\vert## (i.e. they are not in direct contact).
The issue I struggle with is that when we introduce special relativity the two concepts no longer seem distinct as a space-like separation between two physical systems immediately implies that they are not in causal contact (causality is in some sense required by demanding locality)?!
Finally, is the reason why in textbooks it is explained that interactions are local if they occur at a single point in spacetime because locality is the requirement that two physical systems must be in direct contact in order to influence one another directly? (If they are at the same spatial location, but at different points in time, then they can not directly influence each other. Similarly, if they are at the same point in time, but they are located at different spatial points, then they can not directly influence one another. It is only when they are located at the same point (or infinitesimally close to one another) in space and time that they can directly influence one another)