, where the little circles mark out the measured elevations on your route. Now if you were to add up all of those elevations from the beginning to the end and then just divide the total by the number of measurements or points on the chart (fifty in this case), you would have just computed an approximate value of the path integral of the “parameter of elevation” along this particular path. As you might have guessed already, the path integral in this case is just a measure of the average elevation along your bike route; and that pretty much is what simple path integrals do—they compute the average of some parameter along a specific path. However, we have to keep in mind that this average (equal to 27 meters in ) is not an exact value for the path integral, but only an approximation. But we can always get closer to the exact value by tracking the elevations more frequently. Say instead of noting the elevation every 100 meters, the GPS is set to mark it at every meter; we would then have 5,000 points (instead of 50), one for each meter of path, and they will be closer together. The resulting average will give a much better approximation for the path integral—but it is still not quite exact. So how do we get the exact value? This is where calculus comes in. Let’s imagine taking this to the limit of a continuum—suppose we have a magic GPS that measures the elevation at every infinitesimal point on the path, continuously. In that limit, the average we get becomes the exact path integral, which corresponds to the exact average elevation during this bike trip. You might wonder, “Since there are an infinite number of points on that path, wouldn’t the sum or integral be infinite as well?” But remember we are also dividing by the total number of intervals, and there are infinitely many of them as well, which compensates for it, so as a result even in the continuum limit the path integral remains a finite number not very different from the first approximate value of 27 meters we got from the chart in .
, is that we are now rating the level of happiness as well as sadness. On this scale, 0 means neither sad nor happy, 10 would be the happiest one could ever feel, and –10 is the absolute nadir of sadness and depression. Now suppose we have a mood-meter (like the GPS in the bike ride) to track a person’s mood at regular intervals and mark them up on a chart. In , we see an example of such tracking done daily for someone for fifty consecutive days. If this particular person seems to be a moody sort with strong fluctuations of mental state daily, that’s just to make the point and is not necessarily the typical scenario. The chart is pretty self-explanatory: Each dot on the chart corresponds to a value between +10 and –10 as measured along the vertical axis on the left. The thin line connecting those dots makes it easier to track the changes. As you can see, this is very much like the elevation chart in that we drew earlier, and just as we did there, if we average over all the points along this path, we find an approximation for the path integral of happiness/sadness over those fifty days for this particular person. As with the example of the bike ride, we could in principle take the continuum limit by tracking the person’s mood at every instant and get an exact average or path integral of happiness/sadness over that period of time. But notice, and this is important, although the person’s mood and state of mind fluctuates between all the way up at +8 and down close to –8, the average (or path integral) of happiness/sadness is just about 1.5, as indicated by the dotted line in . Let us call such path integrals over any period of time the “net-joy” value for that person for that particular period.
. What remains more or less constant is the net sum or the path integral of happiness + sadness over the entirety of each of those possible paths. In the end, we all end up charting one unique path as we go through life and make our choices, the path that we call our life story or biography. But for most people the ups and downs in life mostly average out regardless of what that path is. Perhaps this simple observation can help us relax somewhat, as we all inevitably ponder the eternal “what if”s of life and are sometimes tormented by visions of the paths not taken. Because no matter what path we might have taken, the sum of happiness and sadness would probably average out to be close to what we have right now. If we chart our lives to experience extremes of sensation, it is almost inevitably negated by the opposite extremes. Things that make us deliriously happy will inevitably make us feel like shooting ourselves at other times.
, which can now be interpreted as just different possible paths in our lives. Everyone’s life has two certain events: birth and death. But there are infinite possible paths that connect those two fixed points of destiny of each life. Feynman’s path integral point of view is that everything in the universe, including you and me, can be viewed as traversing all of those infinitely many possible paths or “histories” from one fixed point (birth) in time and space to the other (death)—sort of like being simultaneously on all possible world lines of life. Of course, the obvious question here is, if we really are following infinite possible paths of destiny in life, how is it that we are aware of only one? Well, as mentioned a few times before, in quantum mechanics, everything in the universe is described by waves of probability. But waves interfere with each other constructively or destructively, as described in ; so that if two waves have their crests lined up, then they add up to interfere constructively to result in waves with a bigger crest, but if the crest of one lines up with the trough of another, they interfere destructively to effectively cancel each other out. So essentially what happens in this path integral worldview of quantum mechanics is that we add up the waves of probability of all the infinite possible paths one could take. When we do that, the probability waves for practically all the paths (except in the vicinity of the one we actually follow in life) undergo destructive interference with one another, canceling each other out. Only close to the actual path we follow in life do the probability amplitudes add up constructively, resulting in the true reality in which we live out our lives. The general idea is sketched in . The vision this evokes has a hard-to-accept dizzying madness about it and yet has a poetic beauty to it at the same time.
To keep it simple and closer to the ideas in physics, we will assume that the time and place of the starting and ending points are truly fixed and unchanging. If we allow for flexibility in the time and place of birth and death, that just opens up more possible paths, which does not change our general conclusions here in any way.