Elegant Shorthand: Einstein Summation and Notational Efficiency

It’s just indices! When Einstein introduced his summation convention, he eliminated pages of sigma symbols with a single rule: repeated index means sum over it. Write $v = v^\alpha e_\alpha$ instead of $v = v^0 e_0 + v^1 e_1 + v^2 e_2 + v^3 e_3$. The pattern encoded in the notation itself—one superscript, one subscript, automatic summation. Brilliant compression.

I did something similar with my diagrams. Before them, calculating quantum electrodynamics amplitudes meant wading through Hamiltonian perturbation theory—pages of integrals and operator products where you’d lose track of what physical process you were computing. My pictures made it obvious: one line for each particle, one vertex for each interaction, arrows showing time direction. The entire calculation visible at a glance. Right notation transforms impossible problems into trivial ones.

When Notation Shapes What’s Thinkable

Decomposing velocity into components along basis vectors seems natural now, but it required conceptual machinery. You’re separating a geometric object—the velocity vector itself—from its coordinate representation. The vector exists independent of coordinates; components and basis vectors are how we talk about it in a particular coordinate system. This distinction becomes mechanical in Einstein notation because the index placement tells you everything: superscripts for components (contravariant), subscripts for basis vectors (covariant). The notation enforces geometric thinking—you can’t accidentally add quantities that don’t transform properly because mismatched indices won’t contract correctly.

Language does this chunking automatically. We don’t spell out logical conjunction each time—we have the word “and.” Neural networks perform similar compression, mapping high-dimensional inputs through learned geometric transformations until Belgium and Netherlands become separable by simple planes. The network compresses geographic coordinates through plane heights, folded plane heights, confidence scores—each layer a more compact representation making classification trivial in the final space.

Information transmission is fundamentally about selection from possibilities. Einstein summation selects which operations to perform based on index patterns. See repeated indices? Sum automatically. Convention encodes the pattern, reducing cognitive load from remembering to write summation symbols to just checking index placement.

Christoffel Symbols and Hidden Structure

Here’s where notation elegance really matters. Christoffel symbols $\Gamma^\lambda_{\mu\nu}$ describe how basis vectors change in curved spacetime—64 components in four dimensions. Writing them all out would obscure their purpose completely, burying geometry under arithmetic. But the compact notation reveals structure: they’re not tensors themselves, they’re connection coefficients computed from metric derivatives. Most components vanish or relate through symmetries. The notation compresses this complexity into a single object whose manipulation becomes mechanical: match indices, sum, done. You can derive geodesic equations or compute curvature tensors without losing sight of the geometric meaning.

Do neural weight matrices hide similar redundancy? Training discovers that vast parameter spaces compress into lower-dimensional manifolds of solutions. We initialize hundreds of thousands of weights, but the effective degrees of freedom might be much smaller. Can we invent notation that exposes this compression, makes the redundancy obvious rather than hidden?

Diagrams for Thought

My principle remains: find the right formulation and problems solve themselves. QED became tractable once I invented diagram notation—not because calculations got easier, but because the physical processes became visible. Could neural architecture search benefit from better visual notation? We draw boxes and arrows, but do those diagrams reveal or obscure the geometric transformations networks perform?

When basis vectors twist through curved spacetime, Christoffel symbols track their rotation. When activation vectors twist through network layers, what notation captures their transformation? Einstein made tensor manipulation mechanical by encoding operations in index patterns. What conventions could make deep learning’s geometric intuitions similarly automatic—visible in the notation itself rather than buried in matrix multiplications?