An operator-valued distribution assigns an operator 
to each test function 
,
linearly and continuously in the weak sense. The operators act on a common dense
invariant domain 
in a Hilbert space. For every
,
the map 
is a scalar distribution. In a tempered realization the test functions belong to
the Schwartz space. The expression 
is symbolic. Smearing with 
defines the operator rather than requiring pointwise operator
values (Grangé and Werner 2006). Tensor components of an energy-momentum
tensor are examples of this framework.
Operator-Valued Distribution
See also
Distribution, Energy-Momentum Tensor, Hilbert Space, Quantum Field Theory, Schwartz SpaceExplore with Wolfram|Alpha
References
Grangé, P. and Werner, E. "Quantum Fields as Operator Valued Distributions and Causality." 4 Dec 2006. https://arxiv.org/abs/math-ph/0612011.Cite this as:
Weisstein, Eric W. "Operator-Valued Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Operator-ValuedDistribution.html