What is the basic idea behind lattice theory or computer-based QFT calculations?(adsbygoogle = window.adsbygoogle || []).push({});

For example, take a scalar field, and the functional path integral:

[tex]W[J(x)]=\int [d\phi(x)]e^{i\int \mathcal L \mbox{ }d^4x+i\int J(x)\phi(x)d^4x} [/tex]

W[J] is the starting point for all types of quantum-field-theoretic calculations.

Is this the quantity that's calculated by computers, summing over a grid of points instead of performing a path integral?

Usually when calculating this quantity by hand, J(x) is left arbitrary. For a computer calculation, don't you have to insert a specific J(x) for the computer to evaluate the path integral?

Do the computers evaluate this integral in position or momentum space? If it's in momentum space, it would be like this:

[tex]W[J(k)]=\int [d\phi(k)]e^{iS+\frac{i}{(2\pi)^4}\int J(k)\phi(-k)d^4k} [/tex]

where the action term S is a little more complicated to write for the interaction Lagrangian terms, but is still just a functional of [tex]\phi(k) [/tex]. I guess the lattice spacing would be '1/a' where 'a' is the spacing in position space?

But still, what is inserted for J(k)? For a scattering process, I guess it would have to be a delta function in momentum space in order to attach a real particle, and sum each of these delta functions for all particles (if the path integral is evaluated in position space instead, you would need some sort of delta function times [tex] e^{\pm ikx}[/tex] for each attached scattering particle of 4-momentum k)?

I've seen some impressive charts that come from computer calculations of QCD: just using the Lagrangian of quarks and gluons, the computer calculations predict confinement and asymptotic freedom, and all the values of hadron masses (the only input being the renormalized coupling strength at the Z-mass)!

I'm just wondering if I can apply these computer calculations to things besides QCD, instead of learning all these tricks like drawing Feynman diagrams and applying Feynman rules and various tricks to evaluate Feynman rules and diagrams like Feynman parameterization and stuff.

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# What is the basic idea behind lattice theory?

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