# Mathematical Definition of Locality

What is the mathematically precise definition of principle of locality in physics for a continuous space-time in the sense that an object is only directly influenced by its immediate surroundings?

• The laws of physics should be formulated in terms of differential equations in partial derivatives, which is valid at each point of space-time. Oct 31 '14 at 9:21
• Hyperbolic differential equations... Oct 31 '14 at 10:06
• @ValterMoretti So all laws of physics must be formulated in terms of hyperbolic differential equations and we have satisfied the intuitive condition for locality in physics ? Please can you expand on this comment a little bit and make it into an answer :) Oct 31 '14 at 12:03
• Hyperbolicity (with respect to the metric of the spacetime) just assures that the velocity of propagation is bounded... Oct 31 '14 at 12:23
• @ValterMoretti Is the velocity of propagation being bounded precisely what we mean by locality in physics ? Oct 31 '14 at 13:06

In quantum field theory, the casuality is manifested as a requirement imposed on the Heisenberg fields $\varphi(x)$: they must be locally commutable:$$[\varphi(x),\varphi(y)]=0$$ when $x-y$ are spacelike. The aim is to make one measurement cannot affect the other which is outside the lightcone of the first one. The casuality can also give a condition for $S$ matrix: $$\frac{\delta}{\delta\varphi(x)}\left(\frac{\delta S}{\delta\varphi(y)}S^{\dagger}\right)=0$$ when $x-y$ is spacelike or $y^{0}>x^{0}$.

• Probable typo: "causality" instead of "casuality" Oct 31 '14 at 12:15

In Topology, there is considerable effort given to describing what makes two points "close" to each other. The answer is always to define a closeness function appropriate to the space, and then to apply the function to one fixed point and two test points; the function will determine which one of them is "closer" to the fixed point.

As you can see this is completely arbitrary but does allow the expression and solving of higher-order problems.

Let's take a boring Cartesian plane. The "closeness" function is (x^2 + y^2)^-1, where x and y are the differences in position on the two axes between our fixed reference point and the test point. (Someone make sure I didn't get that backwards... but should be the length of a straight line between the points)

That's all well and good, and for any point from our fixed reference point, you could describe a circle with the above on the Cartesian plane, of all points at least as close as the test point. Anything in that locus is said to be "in the neighborhood" of the reference point.

When you get to non-euclidian spaces, distance isn't so obvious and you have to work it out; the resulting function is a pretty good high level descriptor of the space itself. Take for example,

• A wrinkled up piece of lettuce. How could you travel from one point to the other? Would you be allowed to jump from one part to the other? Or remain inside the lettuce?

• A "taxicab topology" as in the one-way grid of Manhattan? It's not so straightforward to drive legally from one grid address to another.

• what about the surface of a sphere? Most of the time there is a direct shortest track, but at some special points there is no shortest path. Bring this further, to the surface of a doughnut, and you get the picture of using closeness to describe shapes in an arbitrary way.

• Ok, so dealing with topological spaces that have a metric, still there is no notion of immediate surroundings as there is no closest point. So, what you wrote is just another intuitive picture of the concept in topological terms. Oct 31 '14 at 12:56
• +1, just to have your perspective as a formal answer on this thread. Oct 18 '17 at 8:06