Researchers present a version of quantum field theory built entirely from real numbers. Instead of using the usual imaginary unit i, they put a fixed 2×2 real matrix J wherever i normally appears. That matrix obeys J^2 = −1 and acts as a “real” replacement for complex numbers. The authors call the result real quantum field theory, or RQFT.

To show the idea works, the paper constructs the simplest quantum field: a real scalar field. The field is an operator on a real Kähler Fock space — a state space that describes many-particle states but is written with real entries. The field obeys the Klein–Gordon equation (the standard wave equation for a free scalar particle) and a modified quantization rule [Φ(t,x), Π(t,y)] = J δ^3(x−y). In formulas this shows up as mode expansions with factors like e^{−J(ωt−k·x)}. Creation and annihilation operators satisfy the usual commutation relations, but the complex phase factors are replaced by the matrix J. The explicit J used in the paper is the 2×2 real matrix [[0, −1], [1, 0]].

The authors build a parallel calculus around J. They introduce J-Fourier transforms and J-valued generalized functions, develop J-versions of Wightman functions and Feynman propagators, and state analogues of standard technical tools such as the Sokhotski–Plemelj formula and Cutkosky rules. The usual scattering operator (the S-matrix) in complex quantum field theory is unitary. In RQFT the corresponding real scattering operator is required to be both orthogonal and symplectic. Formally the Dyson series becomes S = T exp(J ∫ d^4x L_int(Φ)), with J replacing i.

Why does this matter? The paper argues that RQFT is an equivalent but conceptually different formulation of ordinary quantum field theory. For the scalar example the physical observables — the quantities you can measure such as scattering probabilities — agree with those of the usual complex formulation. One motivation is conceptual: the geometric language of gravity is naturally real, so expressing quantum fields in purely real terms could help connect particle physics with real differential geometry. The real viewpoint may also be useful for foundational questions about the role of complex numbers in quantum theory.