- 4295 - REALITY PHYSICS - I'm still working on it! Richard Feynman’s path integral is both a powerful prediction machine and a philosophy about how the world is. A particle’s straight-line path through space can be understood as the sum of all its possible paths.
---------- 4295 - REALITY PHYSICS - I'm still working on it!
- The most powerful
formula in physics starts with a slender “S”, the symbol for a sort of sum
known as an integral. Further along comes a second “S”, representing a quantity
known as “action”. Together, these twin S’s form the essence of an equation
that is arguably the most effective diviner of the future yet devised.
-
- The oracular
formula is known as the “Feynman path integral”. As far as physicists can tell,
it precisely predicts the behavior of any quantum system, an electron, a light
ray or even a black hole. The path integral has racked up so many successes
that many physicists believe it to be a direct window into the heart of
“reality”.
-
- But the equation,
although it graces the pages of thousands of physics publications, is more of a
philosophy than a rigorous recipe. It suggests that our reality is a sort of
blending, a sum, of all imaginable possibilities.
-
- But it does not
tell researchers exactly how to carry out the sum. So physicists have spent
decades developing an arsenal of approximation schemes for constructing and
computing the integral for different quantum systems.
-
- The ultimate path
integral is one that blends all conceivable shapes of space and time and
produces a universe shaped like ours as the net result. But in this quest to
show that reality is indeed the sum of all possible realities, they face deep
confusion about which possibilities should enter the sum.
-
- Quantum mechanics
really got off the ground in 1926 when Erwin Schrödinger devised an equation
describing how the wavelike states of particles evolve from moment to moment.
-
- The next decade,
Paul Dirac advanced an alternative vision of the quantum world. His was based
on the notion that things take the path of “least action” to get from A to B,
the route that takes the least time and energy. Richard Feynman later stumbled
upon Dirac’s work and fleshed out the idea, unveiling the path integral in
1948.
-
- In the “double
split experiment” physicists fire particles at a barrier with two slits in it
and observe where the particles land on a wall behind the barrier. If particles
were bullets, they’d form a cluster behind each slit. Instead, particles land
along the back wall in repeating stripes.
-
- The experiment
suggests that what moves through the slits is actually a wave representing the
particle’s possible locations. The two emerging wavefronts interfere with each
other, producing a series of peaks where the particle might end up being
detected.
-
- In the double-slit
experiment, a wave passes through both slits at once and interferes with itself
on the other side. The wave represents a particle’s possible locations. The interference pattern is a supremely
strange result because it implies that both of the particle’s possible paths
through the barrier have a physical reality.
-
- The path integral
assumes this is how particles behave even when there are no barriers or slits
around. First, imagine cutting a third slit in the barrier. The interference
pattern on the far wall will shift to reflect the new possible route. Now keep
cutting slits until the barrier is nothing but slits.
-
- Finally, fill in
the rest of space with all-slit “barriers.” A particle fired into this space
takes, in some sense, all routes through all slits to the far wall, even
bizarre routes with looping detours. And somehow, when summed correctly, all
those options add up to what you’d expect if there are no barriers: a single
bright spot on the far wall.
-
- But how can an
infinite number of curving paths add up to a single straight line? Feynman’s
scheme, roughly speaking, is to take each path, calculate its action (the time
and energy required to traverse the path), and from that get a number called an
“amplitude”, which tells you how likely a particle is to travel that path. Then
you sum up all the amplitudes to get the total amplitude for a particle going
from here to there, an integral of all paths.
-
- Swerving paths look
just as likely as straight ones, because the amplitude for any individual path
has the same size. Amplitudes are
complex numbers. While real numbers mark points on a line, complex numbers act
like arrows. The arrows point in different directions for different paths. And
two arrows pointing away from each other sum to zero.
-
- The upshot is that,
for a particle traveling through space, the amplitudes of more or less straight
paths all point essentially in the same direction, amplifying each other. But
the amplitudes of winding paths point every which way, so these paths work
against each other. Only the straight-line path remains, demonstrating how the
single classical path of least action emerges from unending quantum options.
-
- Feynman showed
that his path integral is equivalent to “Schrödinger’s equation”. The benefit
of Feynman’s method is a more intuitive prescription for how to deal with the
quantum world: Sum up all the possibilities.
-
- Physicists soon
came to understand particles as “excitations in quantum fields” entities that
fill space with values at every point. Where a particle might move from place
to place along different paths, a field might ripple here and there in
different ways.
-
- Feynman himself
leaned on the path integral to develop a quantum theory of the electromagnetic
field in 1949. Others would work out how to calculate actions and amplitudes
for fields representing other forces and particles. When modern physicists
predict the outcome of a collision at the Large Hadron Collider in Europe, the
path integral underlies many of their computations.
-
- Despite its
triumph in physics, the path integral makes mathematicians queasy. Even a
simple particle moving through space has infinitely many possible paths. Fields
are worse, with values that can change in infinitely many ways in infinitely
many places. Physicists have clever techniques for coping with the teetering
tower of infinities, but mathematicians argue that the integral was never
designed to operate in such an infinite environment.
-
- Yet it gets
results that are beyond dispute. Physicists have even managed to estimate the
path integral for the strong force, the extraordinarily complex interaction
that holds together particles in atomic nuclei.
-
- First, they made
time an imaginary number, a strange trick that turns amplitudes into real
numbers. Then they approximated the infinite space-time continuum as a finite
grid. Practitioners of this “lattice” quantum field theory approach can use the
path integral to calculate properties of protons and other particles that feel
the strong force, overcoming mathematics to get solid answers that match
experiments.
-
- The greatest
mystery in fundamental physics, however, sits beyond experimental reach.
Physicists wish to understand the quantum origin of the force of gravity. In
1915, Albert Einstein recast gravity as the result of curves in the fabric of
space and time. His theory revealed that the length of a measuring stick and
the tick of a clock change from place to place.
-
- Space-time is a
malleable field, in other words. Other fields have a quantum nature, so most
physicists expect that space-time should too, and that the path integral should
capture that behavior.
-
- The British
physicist Paul Dirac rejiggered quantum mechanics in 1933 in a way that
considers the whole history, or path, of a particle, rather than its
moment-to-moment evolution. The American physicist Richard Feynman took that
idea and ran with it, developing the path integral in 1948.
-
- Space-time might
conceivably split, for instance, severing one location from another. Or it
might become punctured by tubes, wormholes, that link locations together.
Einstein’s equations allow for such exotic shapes, but forbids changes that
would lead to them; rips or mergers would violate causality and raise time
travel paradoxes.
-
- Making time
“imaginary” effectively turns it into another dimension of space. In such a
timeless arena there’s no notion of causality for wormhole-ridden or ripped
universes to spoil. Hawking used this timeless, “Euclidean” path integral to
argue that time began at the Big Bang and to count the space-time building
blocks inside a black hole. Recently, researchers used the Euclidean approach
to argue that information leaks out of dying black holes.
-
- Where causes
strictly precede effects. Adding up a bunch of standard space-time shapes
(approximating each one as a quilt of tiny triangles) and get something like
our universe, the space-time equivalent of showing that particles move in
straight lines.
-
- In 2019,
researchers rigorously defined the full integral, not just an approximation for
two-dimensional universes, but using mathematical tools that further muddied
its physical meaning. Such work only deepens the impression, among physicists
and mathematicians alike, that the path integral holds power that’s waiting to
be harnessed. I'm still working on it.
-
-
December 30, 2023
REALITY PHYSICS - I'm still working on it! 4295
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