Lecture
The Wightman theorem — a theorem of axiomatic quantum field theory. It reveals the inadequacy of describing a field by means of an operator in Hilbert space. To describe a field, a generalized operator-valued function must be used instead. It was proved by A. Wightman
Formulation
In quantum field theory, it follows from the axioms of relativistic and translational invariance, locality, and spectrality that the description of a field at a definite point of space-time cannot have the meaning of an operator in Hilbert space other than a numerical constant. A field can only be described by a generalized operator-valued function.
An American theoretical physicist. Born in Rochester. Graduated from Yale University (1942). Since 1949 he has worked at Princeton (professor since 1960).
His scientific work is devoted to quantum field theory, elementary particle physics, and nuclear physics. He studied the universality of the Fermi interaction, and the formation of mesoatoms as a result of the capture of a meson into an outer orbit with subsequent loss of energy either through radiation or through the Auger effect. In quantum field theory he is known for the theorem named after him (the Wightman theorem). He introduced the concept of superselection.
The axioms of relativistic and translational invariance of quantum theory mean the invariance of transformations of the scalar products of four-dimensional vectors with respect to the inhomogeneous Lorentz group, and the invariance of the mean value of an observable quantity with respect to proper Poincaré transformations.
The principle of locality of relativistic quantum theory means that measurements of field components at space-time points separated by a spacelike interval are independent of one another
The principle of spectrality of quantum theory means that only representations of the universal covering group of the Poincaré group with positive energy are realized in the space of state vectors .
This theorem can be applied in areas where quantum fields are studied, such as quantum electrodynamics or quantum chromodynamics. In these areas researchers use mathematical models to describe the interaction of elementary particles and fields, and the Wightman theorem helps to refine these models and make them more precise.
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