Lattice Regularization and Measurement Tensors
July 28, 2023
Lattice regularization makes a continuous field calculable by placing its degrees of freedom on discrete sites and links. Operators then become transformations of measurable tensor structure rather than distant formal abstractions.
Brian Greenforest connects that computational picture across scalar fields, vector fields, fermions, spinors, spacetime amplitudes, and momentum-space energy representations.
Turn Fields Into Tensor Operations
A scalar assigns one value to a site. Vector and tensor fields add direction and multilinear structure, while fermionic descriptions require spinor components and their transformation rules.
Tensor products combine systems; contractions connect matching indices; traces summarize invariant structure. Those operations let a finite lattice represent interactions that approach the continuous theory as spacing decreases.
Move Between Spacetime and Momentum
Fourier transforms reorganize a field from position and time into frequency, momentum, and energy variables. The physics stays connected while different regularities become visible.
Field theorists, numerical researchers, and hardware designers can make those operators executable, inspect the tensor flow, and build tools that reveal the geometry inside the calculation.
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Lattice regularization. Operators are just continuous version of measurement tensors. Spacetime probability amplitudes of your favorite fields and the field directions, if these are vector fields, or more complex tensors of tensors, if these are fermions, or normal scalars, like Higgs. Fun what tensor products, contraction, and traces can do for you! And keep the mystery that Fourier transform can always flip it into the energy phase space representation! Even for spinors. 😎📚🤓
