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Introductions to Density Functional Theory (DFT) usually discuss the Hohenberg–Kohn theorems which prove that there exist universal functionals of density that can be used to determine ground state properties of a system. This has extensions for degenerate ground states, or incorporating spin, or magnetic interactions, or even time dependent systems and excited states.

But all these seem to arrive at the proof by discussing a system in terms of a fixed number of particles, and the wavefunction in terms of the particle positions. This means DFT has its basis in non-relativistic fixed-particle-number quantum mechanics.

Is it possible for density functional theory to also be applied to a quantum field theory, such as QED or QCD?

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I suspect this isn't quite what you're looking for, but it's too long to share in a comment:

Relativistic corrections can be added with augmentation methods. Usually though, relativistic corrections arise in the core of (typically heavy) atoms and may therefore be incorporated with small adjustments to the pseudopotentials in canonical DFT. This is discussed in Martin's Electronic Structure book.

As for a non-constant number of particles (electrons usually), there do exist methods for non-canonical ensemble sampling. See, for example, Mermin's approach to finite-temperature DFT.

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