EPR Paradox For the Uninitiated (1/3)

Quantum Mechanics is very odd. It is both an immensely practical and a deeply troubling theory. As such, its uses stretch from optimizing nanoelectronics to examining the nature of reality. In the next few posts we'll take a look at how this bizarre theory forced physicists to rethink one of their most treasured tenets, locality.

Locality is the idea that things can only be affected by their immediate surroundings. By the early 20th century special relativity had shown that nothing, including information, could travel faster than light, thus making the meaning of ''immediate surroundings'' a bit more precise: Systems could only affect one another if the space-time gap between them was time-like. Or, in English, two events can only have a cause and effect relationship if light can traveled from the location of the first event to that of the second in the time between their respective occurrences.

Locality According to Special Relativity

For example, imagine the sun just now blew up. The sun blowing up is an event which happens, like every other event, at some point in space and time. The picture below is a space-time diagram. Obviously it's not a very accurate space-time diagram since it's not four dimensional; never mind, we'll make do. Any event whose location and time are known can be represented by a point on such a diagram. As always we have freedom to choose our coordinate system however we like. Let's make the explosion of the sun the event which takes place at the origin, the vertex of the cones. Ignore the "Observer" label.

Now let's consider another event: Some guy on Earth looking up at the sky. The guy is a certain distance and a certain direction from the sun. In reality we would, of course, need three dimensions to characterize his spatial location relative to the sun but let's pretend we can do it with two. Then the guy's space-time position at the instant of the sun's explosion lies somewhere on the Douglas Adams-esquely named "hypersurface of the present" (hyper because it's really three dimensional but they drew it as two). As time marches on, the space-time position of the observation event simply travels up the time axis, with its space coordinates remaining more or less the same (forget that the Earth is spinning and assume the guy is lazy). Eventually, the space-time point which represents the observation event will fall within what is labeled the "future light cone". It is at this point that our poor fellow will see the sun blow up. The events within the future light cone may have been caused by the explosion of the sun while the events within the past light cone may have been the causes of that explosion. Anything outside of those regions can have no causal relationship to whatever happened at the origin, in this case, the sun blowing up. The former events, which lie within the two cones, are called time like while the latter are called space like. Hopefully now my definition of locality makes more sense. To restate it:

Locality is the idea that two events can only be causally related if their space-time separation is time like (i.e.they lie within each others light cones).

By the way, the light cones can be thought of as follows: light propagates outward from the event at the origin at a fixed speed in every direction. Thus, at the initial moment obviously the light hasn't had time to get anywhere. After some time the light will have traveled all the way to the outer surface of some sphere (represented as a circle in our two dimensional representation of space). These spheres (circles for us) are all stacked on top of each other for each moment of time, resulting in the cones that you see in the diagram.

Thus, the future light cone tell us how far light will have traveled from the origin in a given time. Anything outside of this cone couldn't possibly be caused by whatever happened at the origin; this is because there is no way for the information produced by the event at the origin to have reached it without having traveled faster than light, which is impossible. Similarly, the past light cone encloses all of the events which could have been the causes of the event which took place at the origin. The longer ago a causal event took place, the larger the space it can effect at the present moment.

Now that we've introduced locality, in the next post we'll see what it has to do with Quantum Mechanics.