Physics:Quantum configuration space
configuration space in classical field theory, the configuration space of the field is an infinite-dimensional space. In quantum field theory, it is expected that the Hilbert space is also the L^2 space on the configuration space of the field, which is infinite dimensional, with respect to some Borel measure naturally defined. However, it is often hard to define a concrete Borel measure on the classical configuration space, since the integral theory on infinite dimensional space is involved. Thus the intuitive expectation should be modified, and the concept of quantum configuration space should be introduced as a suitable enlargement of the classical configuration space so that an infinite dimensional measure, often a cylindrical measure, can be well defined on it. While in the classical theory we can restrict ourselves to suitably smooth fields, in quantum field theory we are forced to allow distributional field configurations. In fact, in quantum field theory physically interesting measures are concentrated on distributional configurations. That physically interesting measures are concentrated on distributional fields is the reason why in quantum theory fields arise as operator-valued distributions. The example of a scalar field can be found in the references
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