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Surfaceology is an emerging field within condensed matter physics and mathematics, notable for its approach to calculating scattering amplitudes in quantum field theory. It hypothesizes a deep connection between geometry, topology, and particle physics.

Surfaceology may be able to replace Feynman diagrams, which translate into complex equations for describing particle interactions. Surfaceology yields the same result by in effect assembling large numbers of Feynman diagrams into a more compact representation. Surfaceology does not make use of supersymmetry and can describe both supersymmetric and nonsupersymmetric particles.

Background

Surfaceology is one of a host of theories that attempt to replace conventional notions of spacetime with more fundamental concepts.

Alternatives include string theory, branes,

Standard model

The standard model of physics (connections, curvature, spinors, the Dirac operator, quantization), is based in part on symmetries: properties that do not change when an object is subjected to space-time translations, such as a 90-degree rotation. Each particle has other internal symmetries, such as electric charge. The many decades of unsuccessful attempts to merge general relativity theory with quantum mechanics have led some theorists to attempt to discard the notion of space-time in favor of potentially more fundamental concepts. Others cotinue to try to solve the puzzle with space-time intact. The traditional issue is that general relativity does not describe events happening at very short distances, while quantum mechanics fails at the long distances at which general relativity is unmatched. The standard model exploits strong parallels between highly precise experimental observation and unrelated mathematical insights. The classical notion of spacetime is based on the Riemannian geometry of spinors, which emerged long before its application to physics was established. In particular, it relies on the notion of a space, including spin, the principal bundle of spin-frames with spin-connection and Vielbein dynamical variables.

Unsolved mysteries

Conventional space-time physics cannot describe the beginning of the universe.

A full quantum gravity theory known as a “nonperturbative” theory would also explain black holes.

Definition and scope

Surfaceology involves using curve integrals to compute scattering amplitudes, which are crucial in understanding how particles interact at a quantum level. This method simplifies and potentially revolutionizes how physicists approach these calculations by focusing on the geometry of surfaces outside of traditional three-dimensional space and time.

History

In the late 1940s, Julian Schwinger, Sin-Itiro Tomonaga, and Richard Feynman won the 1965 Nobel Prize for their work on quantum electrodynamics. Feynman’s scheme was the most visual and dominated quantum physics.

In the early 2000s, Nima Arkani-Hamed began looking for solutions. In the mid-2000s, Ruth Britto, Freddy Cachazo, Bo Feng, and Edward Witten dis recursion relations, showing how to condense hundreds of Feynman diagrams to simple lines in specific situations.

In 2013, Arkani-Hamed and his student Jaroslav Trnka discovered the amplituhedron, a geometric object that describes the outcomes of certain particle interactions. However, the object did not apply to real-world particles. Arkani-Hamad showed that in special cases, the amplitude (measure of change) of an interaction could be derived without knowing how the particles moved in space-time. Arkani-Hamed’s team later found that associahedrons worked in a similar way.

In 2019, Arkani-Hamed recruited mathematicians Salvatori and Hadleigh Frost to help look for a geometrical means to computing all such amplitudes.

Carolina Figueiredo became interested in the subject after attending a talk by Arkani-Hamed. In the fall of 2022, Figueiredo discovered that collisions involving three types of subatomic particles produced the same debris. This led to the further discovery that the nominally independent theories describing those particles were essentially the same.

In September 2023 Arkai-Hamed's group published their findings, still unable to describe real particles.

• Quantum Geometry: Geometric objects can encode the outcomes of quantum particle collisions across different theoretical universes.
• Mathematical Innovations: scalar-scaffolded gluons and the combinatorial origins of Yang-Mills theory can be understood through surfaceology.

Quantum interactions

When two quantum particles collide they can merge, split, disappear, or undergo any combination in any order. Feynman diagrams describe these interactions by drawing lines representing the particles’ trajectories through space-time. Each diagram captures one possible event sequence and is accompanied by an equation for an amplitude, a number that represents the odds of that sequence taking place. The theory states that macro-scale objects can be described by accumulating sufficient amplitudes.

One feature that has not been explained is that combining the eqsuations behind a large number of interactions may produce terms that cancel out, leaving simple answers—notably, a value of 1.

Objects

The amplituhedron is a curved shape whose contours encode the number and orientation of particles involved in an interaction. Its volume gives the amplitude of that interaction. This volume equals the sum of the associated Feynman diagrams' amplitudes, which depict the ways the interaction could evolve, using the momenta of the particles that exist before and after the interaction, but without reference to spatiotemporal dynamics. However, the amplituhedron works only for particles that come with partner particles, i.e., supersymmetric particles.

The associahedron is another geometric object. It has flat sides, and its volume gives amplitudes for the particles of a simplified quantum theory. The particles in this theory carry a type of charge called “color” that is also carried by quarks and gluons. Its particles also lack supersymmetric partners. However, associahedrons produce amplitudes for only short event sequences.

The shapes can be defined by polynomials (equations that sum a series of terms) that correspond to curves on a surface. These surfaces form the heart of surfaceology.

To calculate the odds of e.g., two particles colliding to form three particles any Feynman diagram that shows two particle trajectories coming in and three coming out can be used. The lines are thickened to form a surface, and curves are drawn across the surface. This redescribes the moving particles as a static structure.

Each curve can be seen as a sequence of left and right turns. Enumerating the ways to break up this sequence into smaller sequences generates the correct polynomial. Using the polynomials, along with data from experiments, the amplitude for the five-particle interaction can be easily calculated.

This procedure works for all amplitudes, including lengthy event sequences. More complicated interactions translate into surfaces with holes for the curves to loop around, but do not break the procedure. The curves also correspond to faces of an associahedron, establishing that the associahedron and surfaceology are reflections of the same math.

The ultimate benefit of surfaceology is that each curve on the surface replaces the multitude of Feynman diagrams otherwise needed to characterize these interactions.

hidden zeros

One supersymmetric theory and another theory—trace phi cubed—have amplitudes that take the form of fractions and were the first for which Arkani-Hamed provided solutions. All their variables (particle momentums,...) are part of the denominator.

Singularities are collisions with small denominators and correspondingly high occurrence probabilities, implying that the fingerprints of any quantum theory. (Note that these singularities are unrelated to the singularities thought to sit at the heart of black holes.) Singularities led Arkani-Hamed to the amplituhedron and associahedron.

The quantum theories that describe real particles require variables in the numerator, too. For example, electrons have intrinsic angular momentum–spin–and terms that capture spin go in the numerator.

Figueiredo realized that the numerator could help find the geometric underpinnings of these real particle interactions. She looked for collisions in which the numerator of the fraction—instead of the denominator—is small. The overall value of these amplitudes approaches zero, so they represent collisions with minute probabilities.

Such “zeros” (low probability events) are more elusive than singularities. Their Feynman diagrams are difficult, and they are nearly impossible to observe experimentally. Figueiredo had learned from Arkani-Hamed’s group how to recast a trace phi cubed amplitude as the volume of an associahedron. So she adjusted the incoming and outgoing particles, looking for collisions that flattened the shape, collapsing its volume.

This led her to the zeros of trace phi cubed. She checked other theories, examining the same collisions but using pions—real particles with different rules. Pions have no known geometric theory, leading her to use Feynman diagrams. She showed that the pion theory outlawed the same collisions. The same collisions turned out to be unlikely under the Yang-Mills theory of gluons, completing the trifecta of electrons, pions, and gluons. Bourjaily validated the theory for collisions involving up to 14 particles.

The curves of trace phi cubed theory give an equation for an amplitude. The zeros make this amplitude very rigid; there’s only one part of the equation that can change while preserving the zeros, producing one of the three particles.

Initially, surfaceology applied only to collisions between bosons, which have integer amounts of spin. But other particles such as electrons are fermions, which have half-integer spin. A group led by Spradlin, Anastasia Volovich, and Marcos Skowronek worked out rules for curves that can accommodate certain "toy" fermions.

Shruti Paranjape, considering which quantum theories shared “hidden” zeros realized that in all of them it was possible to combine two amplitudes of one theory to make an amplitude of another theory, known as the double copy. She and her collaborators showed that theories that can be double-copied have the zeros Figueiredo had found.

The typical procedure is to draw only curves that don’t cross themselves. But if you include the

Self-intersecting curves produce amplitudes that describe interactions between strings rather than particles, possibly contributing to string theory, a candidate quantum gravity theory. Surfaceology might apply to gravitons, hypothetical particles that may produce the gravitational force. While working out how much each curve would contribute to a trace phi cubed amplitude,

The group came across curves that were unavoidable but that didn’t change the final answer. On surfaces that had holes, these curves circled around the holes infinitely. From the space-time perspective, these curves capture events beyond the trace phi cubed theory: colorless particles that could eventually describe gravitons.

Related theories

Holography

Holography is an alternative theory that seeks to capture the entirety of space-time by treating it as a higher dimension hologram of quantum particles moving around in one lower dimension. It would even explain the interiors of black holes,

In its current form, it shows how one dimension of space could emerge, and also depends on traditional quantum objects: some space, locality, and a clock, instead of those objects emerging as features of the theory.

Twistors

Twistors are a mathematically equivalent picture of space-time. Physical phenomena in space-time can be described in twistor space, or vice versa. Roger Penrose found them to dramatically simplify certain physical calculations. One unexplained aspect of twistor space is that certain particles can be either “right-handed” or “left-handed,” depending on whether follow their spin through space or not. However Twistor space best fits theories of purely right-handed or purely left-handed particles, rather than incorporating both types of particles and their interactions

Colored Yukawa theory

Scattering amplitudes for colored theories can be expressed as integrals over combinatorial objects constructed from surfaces decorated by kinematic data. The curve integral formalism includes theories with colored fermionic matter. A compact formula describes the all-loop, all-genus, all-multiplicity amplitude integrand of a colored Yukawa theory. The curve integral formalism manifests certain properties of the amplitudes. Non-trivial numerators can be merged into a single combinatorial object. -Loop integrated amplitudes can be computed in terms of a sum over combinatorial determinants.

Applications

Potential applications include understanding phenomena such as the behavior of surfaces (e.g., how a surface wrinkles depending on its curvature) and could lead to insights in areas like nanotechnology.

External links

• Musser, George (2015-11-03). Spooky Action at a Distance: The Phenomenon That Reimagines Space and Time--and What It Means for Black Holes, the Big Bang, and Theories of Everything. Macmillan. ISBN 978-0-374-29851-7.
• Musser, George (2023-11-09). Putting Ourselves Back in the Equation: Why Physicists Are Studying Human Consciousness and AI to Unravel the Mysteries of the Universe. Simon and Schuster. ISBN 978-0-86154-720-3.
• Musser, George (2017-05-16). "A Defense of the Reality of Time". Quanta Magazine. Retrieved 2024-12-01.
• Horgan, John (October 3, 2024). "The Beyond-Spacetime Meme". John Horgan (The Science Writer). Retrieved 2024-12-01.