The beginning of the most potent formula in physics begins with a thin S,
which stands for an integral, a type of sum. A second S, which stands for
the action quantity, appears farther on. These two twin Ss work together to
make an equation that is possibly the best future predictor ever
created.

The Feynman route integral is the name of the oracular formula. It
accurately predicts the behavior of every quantum system, including an
electron, a light ray, and even a black hole, as far as physicists can
determine. Many physicists consider the path integral to be a direct window
into the essence of reality because it has achieved so many
achievements.

Renate Loll, a theoretical physicist at Radboud University in the
Netherlands, remarked, "It's how the world actually is.

However, despite being featured in hundreds of scientific articles, the
equation is more of a philosophical statement than a precise formula. It
implies that all conceivable outcomes are combined to create our reality.
However, it does not specifically instruct researchers on how to do the sum.
In order to design and compute the integral for various quantum systems,
scientists have spent decades creating a library of approximation
approaches.

The approximations are accurate enough that daring scientists like Loll are
now seeking the ideal path integral, which combines every imaginable shape
of space and time to yield a universe that resembles our own. But they
encounter much uncertainty about which possibilities should be included in
the total in their attempt to demonstrate that reality is, in fact, the sum
of all potential realities.

Every Path Leads to One

Erwin Schrödinger developed an equation outlining how the wavelike states
of particles change from moment to moment in 1926, which is when quantum
mechanics truly took off. Paul Dirac developed a different interpretation of
the quantum world during the following ten years. His concept founded on the
old idea that things go in the direction that requires the "least action" to
get from A to B, or, more broadly, the way that requires the least amount of
time and effort. Later, Richard Feynman came upon Dirac's work and developed
the concept, revealing the path integral in 1948.

The double-slit experiment, the classic quantum mechanics demonstration,
puts the philosophy's core on full display.

A barrier with two slits is fired at by physicists, who then watch the
particles settle on a wall beyond the barrier. If the particles were
bullets, each slit would have a group of them. Instead, particles fall in
stripes down the rear wall. The experiment implies that the wave indicating
the particle's potential locations is what passes through the slits. Two
wavefronts that are developing interfere with one another, creating a
succession of peaks where the particle might ultimately be found.

Because it suggests that both of the particle's potential pathways through
the barrier have a physical existence, the interference pattern is an
extremely odd outcome.

Even in the absence of any obstructions or slits, this is how particles are
assumed to behave by the path integral. Consider first slicing the barrier
in thirds. On the distant wall, the interference pattern will change to
reflect the new potential path. Now continue to create slits until the
barrier is made entirely of them. Finally, place all-slit "barriers" to
enclose the remaining area. In certain ways, a particle sent into this area
follows every path through every slit to the far wall, even strange paths
with looping diversions. And somehow, when added up correctly, all of those
possibilities equal what you would anticipate if there were no obstacles: a
solitary brilliant spot on the opposite wall.

Many physicists take this extreme interpretation of quantum behavior
seriously. University of Montreal physicist Richard MacKenzie remarked, "I
think it entirely real.

Nevertheless, how can an endless number of wavy routes combine to form a
single straight line? Feynman's method is, broadly speaking, taking each
possible path, calculating its action (the time and energy needed to travel
the way), and then using that information to derive a number called an
amplitude that indicates how likely it is for a particle to follow that
path. The total amplitude for a particle traveling from point A to point B
is then obtained by adding all of the amplitudes together; it is an integral
of all pathways.

Since each path's amplitude is the same, swerving courses first appear to
be equally likely as straight ones. But most importantly, amplitudes are
complex quantities. Complex numbers behave like arrows whereas real numbers
mark points on a line. For various pathways, the arrows point in various
directions. And the total of two arrows going in opposite directions is
zero.

The end result is that for a particle moving through space, the amplitudes
of all nearly straight pathways point in the same direction and reinforce
one another. However, as the amplitudes of winding pathways point in all
directions, these paths compete with one another. The single classical path
of least action is shown to arise from an infinite number of quantum choices
by leaving just the straight-line path behind.

Feynman demonstrated how Schrödinger's equation and his route integral are
comparable. Feynman's approach has the advantage of offering a more logical
guide for interacting with the quantum world: Compile all of the
options.

Combined Ripples Total

The concept of particles as quantum field excitations, or things that fill
space with values at every point, quickly spread among physicists. A field
can ripple in different ways here and there, similar to how a particle might
migrate from one location to another via distinct routes.

Fortunately, quantum fields may also be represented by the route integral.
Gerald Dunne, a particle physicist at the University of Connecticut, said:
"It's apparent what to do." "You sum over all configurations of your fields
rather than over all pathways." You choose the field's initial and final
configurations, then take into account any conceivable histories that
connect them.

In order to create a quantum theory of the electromagnetic field in 1949,
Feynman himself relied on the route integral. Others would figure out how to
determine the actions and amplitudes of fields that reflect different forces
and particles. The path integral is the basis of many of the calculations
that contemporary physicists use to forecast the outcome of a collision at
the Large Hadron Collider in Europe. Even a coffee mug with an equation
illustrating how the activity of the known quantum fields may be utilized to
compute the route integral is available in the gift store.

It is crucial to quantum physics, according to Dunne.

Mathematicians are wary of the path integral despite its success in
physics. The number of alternative pathways for a single particle traveling
through space is unlimited. Fields are worse because their values can vary
in an endless number of ways and locations. Mathematicians contend that the
integral was never intended to function in such an endless setting, in
contrast to physicists who have developed sophisticated strategies for
dealing with the teetering tower of infinities.

Yen Chin Ong, a theoretical physicist at Yangzhou University in China with
a background in mathematics, compared it to "black magic." Mathematicians
find it difficult to work with situations where the situation is
unclear.

However, it produces outcomes that are incontestable. The incredibly
intricate connection that binds particles in atomic nuclei together, the
strong force, has even allowed physicists to estimate its path integral.
They accomplished this using two primary techniques. The first unusual
approach they used to convert amplitudes into real numbers was to transform
time into a fictitious quantity. The infinite space-time continuum was then
roughly represented by a limited grid. The path integral may be used by
practitioners of this "lattice" quantum field theory technique to compute
the characteristics of protons and other particles that experience the
strong force, overcoming shaky mathematics to obtain reliable results that
are consistent with observations.

For a particle physicist like me, Dunne said, "it's the proof that the
thing works."

What is the sum of Space-Time?

But the biggest puzzle in fundamental physics is out of reach for
experiments. Scientists are interested in learning how the quantum nature of
gravity came to be. Albert Einstein reformulated gravity as the product of
curved space-time in 1915. His idea demonstrated that space-time is a
flexible field, or that a measuring stick's length and a clock's tick vary
depending on the location. The majority of scientists anticipate that since
other fields exhibit a quantum aspect, space-time should as well, and that
the path integral should represent this behavior.

Feynman's guiding principle is unmistakable: Physicists should sum over all
potential space-time configurations. But what precisely is feasible when we
take into account the structure of space and time?

For example, space-time might divide, separating one region from another.
Or it could be pierced by wormholes, tunnels connecting different places.
Such strange forms are permitted by Einstein's equations, but rips or
mergers are forbidden since they would break causality and create paradoxes
related to time travel. Tossing Swiss-cheese space-times into the
"gravitational path integral" is a controversial topic among physicists
since no one knows if space-time and gravity may participate in more risky
behavior at the quantum level.

One group believes that everything is input. For example, Stephen Hawking
promoted a route integral that allows for rips, wormholes, doughnuts, and
other ludicrous "topological" transformations between different geometries
of space. To speed up the calculations, he relied on the imaginary-number
trick. Making time fictitious effectively transforms it into an additional
spatial dimension. There is no concept of causality in such a timeless
environment, therefore wormhole-filled or torn universes cannot ruin it. In
order to prove that time began at the Big Bang and to count the pieces of
space-time that make up a black hole, Hawking employed this timeless,
"Euclidean" route integral. Recently, researchers asserted that information
escapes from dying black holes using the Euclidean method.

According to Durham University quantum gravity theorist Simon Ross, this
"seems to be the richer point of view to take." We don't entirely comprehend
certain stunning characteristics of the gravitational path integral, which
is defined to include all topologies.

However, the more insightful viewpoint has a cost. Some physicists object
to eliminating a reality-supporting component like time. According to Loll,
the Euclidean route integral "is basically entirely unphysical."

Her group makes an effort to keep time on the route, placing it in the
familiar and beloved space-time, where causes absolutely come before
consequences. Loll has discovered indications that the method could be
successful after spending years researching strategies to approximate this
considerably more difficult route integral. In one publication, for example,
she and her coworkers piled together a number of common space-time
configurations, roughly representing each one as a patchwork of small
triangles, and obtained something resembling our universe, demonstrating
that particles flow in straight lines in space.

Others are developing the timeless route integral for gravity and
space-time, taking into account all topological changes. For two-dimensional
universes, the complete integral, not just an approximation, was properly
determined in 2019 but with the use of mathematical techniques that further
obscured its physical significance. Such research only strengthens the
notion that the route integral possesses potential that is just waiting to
be realized, which is shared by both physicists and mathematicians. Ong
remarked, "Perhaps we still need to refine route integrals, but basically I
think it's simply a matter of time."