Path Integral
Sum Over Histories
Every Path, an Arrow
Quantum theory can be written as one rule. Ask how likely something is to get from here at one moment to there at a later one, a photon from a lamp to your eye, an electron from a wire to a screen. Then take every path it could take between the two, straight, bent or looping, and give each path a little arrow. Every arrow is the same length. What differs is the way it points: each path turns its arrow by its action, counted in a tiny unit called the reduced Planck constant: a little over six of those units make one full turn. For a ball, the action is the total of energy of motion minus energy of height along the path; for a charge, electric and magnetic fields add a share of their own. For light the turn is simpler still: one full turn for every wavelength of the light along the way.
Now lay all the arrows end to end, in any order, and draw one arrow from the start of the chain to its end. Its length, squared, gives the chance. That is the heart of it. The rest of this page is what follows.
Take light reflecting off a flat mirror into your eye. Every spot on the mirror is a route, and laying the arrows in the order of the spots makes the picture plain. Routes off the ends of the mirror are long, and from one spot to the next their length changes fast, so their arrows swing round and round and their chain curls into tight little circles that get nowhere. Routes near the middle, where light reflects at equal angles, barely change in length from spot to spot. Their arrows point nearly the same way, and their chain runs straight. Almost all of the final arrow comes from there. The ends are not idle, though. Scrape away the strips of silver near one end whose arrows point one way, and the circles no longer close: that end alone sends light off at an angle the whole mirror never would.
One Path, or Many
How wide that middle band is depends on how fast the arrows turn. For light, the wave sets it. Light’s waves are far shorter than any mirror, so only a thin band near the middle counts. Radio waves as long as the mirror is wide turn their arrows slowly from one end to the other, and then the whole mirror counts: the waves come off all of it, with no single route.
For matter, the action sets it. Bend a thrown ball’s path by the width of an atom, and its action changes by tens of trillions of units, so its arrows spin wildly. Only paths far closer than a proton’s width to the one where the action is flat keep step, and the ball seems to follow one exact arc. That is how the principle of least action comes out of quantum theory, rather than being put in.
For an electron in an atom, paths as far apart as the atom is wide differ in action by only a few units. Its arrows turn slowly, like the radio waves’ on the mirror, paths far apart keep step, and a whole wide spread of them counts. There is no single path left to follow: the electron is a cloud over the whole atom, a wave whose strength at each place sets the chance of finding it there.
No sharp line divides the two. Physicists keep pushing heavier things through the same test. They have sent clusters of more than 7,000 sodium atoms, heavier than most proteins, through a row of gratings made of laser light, and the clusters still came out in stripes. Their action is enormous, but that is not what counts. What counts is how much it differs between the paths through neighboring gaps of a grating: only a turn or so, so their arrows still add, as long as nothing records which gap each cluster took. Where, if anywhere, the many paths stop counting is not settled. Proposals that ordinary quantum rules break down for large enough objects exist, and no experiment has yet seen one.
Both Slits Count
Send photons one at a time at two slits, and the rule needs no extra story. For each spot on the screen, add the arrows of every path through the left slit and every path through the right. Where the two sets arrive pointing the same way, the arrow is long and photons often land. Where they point opposite ways, they cancel and photons never land. Stripes appear, built one photon at a time.
Put anything at one slit that keeps a record of the photon passing, and the paths through the two slits end in different states of that record. Arrows are only added for paths that end in the same state, so the two sets no longer meet. Each set’s arrow is squared on its own, and the two chances are added: one broad, smooth spread of light, and no stripes. Nothing needs to look at the record. It is enough that it could be read.
Arrows can be turned even where nothing pushes. Physicists sent electrons at a tiny ring magnet sealed in a superconducting shell, which kept all of its magnetic field inside. Paths through the ring’s hole and paths round its outside met again beyond it. No force reached the electrons anywhere on their way, yet the stripes shifted. The magnet’s share in a charge’s action does not vanish where its field does: it differed between the two sets of paths and turned their arrows by different amounts.
Rolling Through the Wall
A ball rolling toward a hill too high for it rolls back. A quantum particle sometimes turns up on the far side. The path sum gives a picture of how. Replace time in the rule with a fourth direction of space, a mathematical step called imaginary time. The turning arrows become plain weights that only shrink, by an amount set by motion plus height instead of motion minus height. The path that keeps the most weight then obeys the same law as a ball rolling on the hill turned upside down, into a valley.
In the upside-down world the particle can simply roll across: down into the valley and up the far side, arriving at rest at the same height it set out from. That roll has an action of its own, and the chance of tunneling falls off as that action grows. A hill twice as wide, or a heavier particle, makes the roll cost more and the odds far longer: doubling the cost squares the odds, so one chance in a thousand becomes one in a million. The same reckoning gives how fast a nucleus throws out an alpha particle and how often molecules flip between two shapes.
Summing Over Fields
For a field, the paths are whole histories of the field: every way it could rise and fall at every point of space, moment by moment. The rule is the same. Each history gets an arrow turned by its action, and the arrows are added. Where the forces are weak, the sum can be worked out as a series of corrections to particles flying free, without meeting, and each correction is drawn as one of Feynman’s diagrams: lines for particles, and points where they meet. Lines that start and end inside a diagram are steps in the sum, not particles thrown from one to another.
Where the force is strong, as among the quarks inside a proton, no series will do, and the sum is done by computer. Space and time become a grid of points, time is turned imaginary so the arrows become weights, and the machine draws field shapes at random, each as often as its weight says. That is how the proton’s mass has been worked out from the theory of quarks and gluons. The same kind of sum gave the hardest part of the muon’s magnetism to work out, the part due to quarks and gluons. The most precise measurement of that magnetism, finished in 2025, agrees. Working that part out from older collider experiments instead gives a different answer. A newer experiment of that kind agrees with the computer sum, and why the older ones came out different is still argued over.
Turned imaginary, the quantum sum takes exactly the form of the sums that describe a warm material in balance at a steady temperature, with the reduced Planck constant in the place of temperature. The spread that quantum theory leaves in a field’s values plays the part of the warm material’s jiggle.
Is the Sum Real?
Does a particle really take every path? The rule is not an approximation: it gives the same predictions as every other way of writing quantum theory, and they have passed every test. What it says is happening between here and there is another matter. Some read the paths as real histories that all take part. Others read the sum as bookkeeping for a wave, with no path taken at all. Others again keep one real path for each particle, steered by a wave that goes every way. No experiment has yet told these readings apart.
There is a puzzle on the mathematicians’ side too. Summing over every path in real time has never been made a true sum in their strict sense, except in special cases: it has been proved that the usual way of measuring how much of each path to count cannot work there. For a single particle in imaginary time, the sum becomes exactly the mathematics of a random walk, the jiggling of a speck of dust in water, and is perfectly sound; the paths that count are jagged everywhere, like the speck’s. That is one more reason the computer sums are done in imaginary time. For fields such as those of quarks and gluons, it is sound on a grid of points. As the grid is made ever finer, a sound meaning has been proved for some simpler fields in fewer dimensions. For the fields of the Standard Model, in our four dimensions of space and time, it is not settled. Whether the real-time version can be put on the same footing is not settled either. Physicists use it with confidence all the same, because its answers agree with other methods and with experiment wherever they can be checked.
