has this same property. Since all reasonable observables (such as energy, charge, particle number, etc.) are built out of an even number of fermion fields, the commutation relation vanishes between any two observables at spacetime points outside the light cone. As we know from elementary quantum mechanics two simultaneously commuting observables can be measured simultaneously. We have therefore correctly implemented Lorentz invariance for the Dirac field, and preserved causality. More complicated field theories involving interactions (such as Yukawa theory, or quantum electrodynamics) can be analyzed too, by various perturbative and non-perturbative methods. Dirac fields are an important ingredient of the Standard Model. See also Dirac equation Spin–statistics theorem Spinor Composite Field Auxiliary Field References Peskin, M and Schroeder, D. (1995). An Introduction to Quantum Field Theory, Westview Press. (See pages 35–63.) Srednicki, Mark (2007). Quantum Field Theory, Cambridge University Press, . Weinberg, Steven (1995). The Quantum Theory of Fields, (3 volumes) Cambridge University Press. Quantum field theory Spinors