Feynman’s sum over paths [1]
Overview
This approach, also known as ‘sum over histories’, describes an averaging of all the paths a particle can possibly take from its starting point to its destination. According to Richard Feynman, particles must be viewed as travelling from one location to another along every possible path and along all possible paths between them. He then assigned to each possible path or history a particular probability, and then used this idea to make predictions. It works spectacularly well to predict the future, says Stephen Hawking , so we presume it works to retrodict the past as well. [1A]
For example if you shine light on a barrier with two slits in it with a photographic plate behind to see which electrons make it through which slit, the experiment will confirm that each electron that makes it through to the screen actually goes through both slits. Not only that but the same electron must be passing simultaneously through both slits!
Feynman took this one step further by postulating that in travelling from the source to a given point on the screen, each individual electron actually traverses every possible trajectory simultaneously. “It goes in a nice orderly way through the left slit. It simultaneously goes in a nice orderly way though he right slit. It heads toward the left slit, but suddenly changes course and heads through the right. It meanders back and forth, finally passing through the left slit. It goes on a long journey to the Andromeda galaxy before turning back and passing through the left slit on its way to the screen. And on and on it goes. The electron, according to Feynman, simultaneously ‘sniffs’ out every possible path connecting its starting location with its final destination”, and you add up the probabilities of all conceivable paths.[2] The probability that the electron (or photon - any particle) arrives at any chosen point is therefore built up from the combined effect of every possible way of getting there, rather like working out the probabilities of a person who enters a subway station from one entrance or turnstile exiting via another specific location when numerous routes and choices are available[3].
Left:: The double slit experiment revisited. Depicted here are a few of the infinity of trajectories for a single electron travelling from a source to a phosphorescent screen. Note that one electron actually passes through both slits[4]
Gavin Hesketh likens this to electrons making multiple copies of themselves when they collide and exchange photons, maybe one, two or four, maybe more. [5] "For every possible path a particle might travel from A to B, and every possible number of photons it might fire off or absorb, there is a copy. All of these copies come back together at the end, so that when we next make a measurement of each electron, we find just one particle again. But in between measurements, when we are not looking at them, particles seem to split into many copies, and explore all possible ways to behave". And this is not an analogy: it is a description of real mathematics comprising the "path integral" version of quantum mechanics, so called because that's what the mathematics does: it adds up every possible path a particle might take, and every possible interaction a particle might have.
Particles don't exchange just one or two photons, but every possible combination, and "only by adding up all these different possibilities, all the different paths and interactions, and treating them as if they all happened at the same time, only then do we get a realistic description of what happens when electrons collide".
Hesketh does goes on to draw an analogy about his journey to work. He could go by bus or train and his colleagues wouldn't know which option he exercised, and in real life it was obviously just one. "But quantum mechanics is different. An electron me really would split into two copies and take both the bus and the train. The electron me that takes the bus would also split many many more times, as thousands of copies of that bus spread out in all directions taking every possible route across town. All of these different routes converge on my office, and only by accounting for them all can we correctly account for how an electron behaves".
For high energy particle physics, Feynman's path integral approach is part of the phenomenon known as "quantum field theory". And experiment has shown that here there are no so-called loopholes or 'hidden variables'. "Particles do exist in these weird quantum states where every possibility exists at the same time, sometimes cancelling each other out. In practice, these weird states generally exist for just a tiny fraction of a second, and usually over a very small distance.... Not only is the universe random, but it does everything - all at once".
[1] Greene (2000), 108-112.
[1A] Stephen Hawking, Brief Answers to the Big Questions, John Miller, London, 2018 at 54.
[2] Greene (2000), 112; Hawking, A brief History of Time, op cit, 56-59.
[3] Cited in Greene (2000), 111.
[4] Source: Fig 4.10, Greene (2000), 110. For a much better animated graphic, see the links on the page Energy, matter and light: waves and particles at the same time
[5] Gavin Hesketh, The Particle Zoo - The search for the fundamantal nature of reality, Quercus, Hachette, London, 2016, 30-35.
This approach, also known as ‘sum over histories’, describes an averaging of all the paths a particle can possibly take from its starting point to its destination. According to Richard Feynman, particles must be viewed as travelling from one location to another along every possible path and along all possible paths between them. He then assigned to each possible path or history a particular probability, and then used this idea to make predictions. It works spectacularly well to predict the future, says Stephen Hawking , so we presume it works to retrodict the past as well. [1A]
For example if you shine light on a barrier with two slits in it with a photographic plate behind to see which electrons make it through which slit, the experiment will confirm that each electron that makes it through to the screen actually goes through both slits. Not only that but the same electron must be passing simultaneously through both slits!
Feynman took this one step further by postulating that in travelling from the source to a given point on the screen, each individual electron actually traverses every possible trajectory simultaneously. “It goes in a nice orderly way through the left slit. It simultaneously goes in a nice orderly way though he right slit. It heads toward the left slit, but suddenly changes course and heads through the right. It meanders back and forth, finally passing through the left slit. It goes on a long journey to the Andromeda galaxy before turning back and passing through the left slit on its way to the screen. And on and on it goes. The electron, according to Feynman, simultaneously ‘sniffs’ out every possible path connecting its starting location with its final destination”, and you add up the probabilities of all conceivable paths.[2] The probability that the electron (or photon - any particle) arrives at any chosen point is therefore built up from the combined effect of every possible way of getting there, rather like working out the probabilities of a person who enters a subway station from one entrance or turnstile exiting via another specific location when numerous routes and choices are available[3].
Left:: The double slit experiment revisited. Depicted here are a few of the infinity of trajectories for a single electron travelling from a source to a phosphorescent screen. Note that one electron actually passes through both slits[4]
Gavin Hesketh likens this to electrons making multiple copies of themselves when they collide and exchange photons, maybe one, two or four, maybe more. [5] "For every possible path a particle might travel from A to B, and every possible number of photons it might fire off or absorb, there is a copy. All of these copies come back together at the end, so that when we next make a measurement of each electron, we find just one particle again. But in between measurements, when we are not looking at them, particles seem to split into many copies, and explore all possible ways to behave". And this is not an analogy: it is a description of real mathematics comprising the "path integral" version of quantum mechanics, so called because that's what the mathematics does: it adds up every possible path a particle might take, and every possible interaction a particle might have.
Particles don't exchange just one or two photons, but every possible combination, and "only by adding up all these different possibilities, all the different paths and interactions, and treating them as if they all happened at the same time, only then do we get a realistic description of what happens when electrons collide".
Hesketh does goes on to draw an analogy about his journey to work. He could go by bus or train and his colleagues wouldn't know which option he exercised, and in real life it was obviously just one. "But quantum mechanics is different. An electron me really would split into two copies and take both the bus and the train. The electron me that takes the bus would also split many many more times, as thousands of copies of that bus spread out in all directions taking every possible route across town. All of these different routes converge on my office, and only by accounting for them all can we correctly account for how an electron behaves".
For high energy particle physics, Feynman's path integral approach is part of the phenomenon known as "quantum field theory". And experiment has shown that here there are no so-called loopholes or 'hidden variables'. "Particles do exist in these weird quantum states where every possibility exists at the same time, sometimes cancelling each other out. In practice, these weird states generally exist for just a tiny fraction of a second, and usually over a very small distance.... Not only is the universe random, but it does everything - all at once".
[1] Greene (2000), 108-112.
[1A] Stephen Hawking, Brief Answers to the Big Questions, John Miller, London, 2018 at 54.
[2] Greene (2000), 112; Hawking, A brief History of Time, op cit, 56-59.
[3] Cited in Greene (2000), 111.
[4] Source: Fig 4.10, Greene (2000), 110. For a much better animated graphic, see the links on the page Energy, matter and light: waves and particles at the same time
[5] Gavin Hesketh, The Particle Zoo - The search for the fundamantal nature of reality, Quercus, Hachette, London, 2016, 30-35.