How are operator valued distributions defined?

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SUMMARY

Operator valued distributions are defined as continuous linear maps from the space of test functions to the space of bounded linear operators on a Hilbert space. This definition aligns with the concept of real-valued or complex-valued distributions, with the distinction that the values are linear operators. Key references for further understanding include the work of Streater & Wightman, which provides a precise definition and context for these distributions. Quantum fields exemplify operator valued distributions, particularly in their Fourier transform properties.

PREREQUISITES
  • Understanding of Hilbert spaces
  • Familiarity with continuous linear maps
  • Knowledge of quantum field theory
  • Basic concepts of Fourier transforms
NEXT STEPS
  • Study the definition of bounded linear operators in Hilbert spaces
  • Read "PCT, Spin and Statistics, and All That" by Streater & Wightman
  • Explore the properties of operator valued distributions in quantum field theory
  • Learn about the application of Fourier transforms in quantum mechanics
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Mathematicians, physicists, and students of quantum mechanics seeking to deepen their understanding of operator valued distributions and their applications in quantum field theory.

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How precisely are operator valued distributions defined?
 
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Isn't it defined exactly the same as a real-valued or complex-valued distribution, except that it takes values in the set of linear operators on a Hilbert space? See also posts #34 and #35 in this thread (option 2 only, since you convinced me option 1 doesn't work, as acknowledged in #43).

Hm, maybe I should have said "the set of bounded linear operators"...not sure about that.

There's probably an exact definition in Streater & Wightman. That's the first place I'd look for one.
 
jostpuur said:
How precisely are operator valued distributions defined?
An operator valued distribution is a continuous linear map from the space of test functions to the space of operators over a Hilbert space. Quantum fields are operator valued distributions which have a Fourier transform.
 

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