Operator-Valued Functions in Quantum Field Theory: Degrees of Freedom?

[plasmon]

Is it correct to express quantum field theory as "operator valued function" or "operator function" to spacetimepoints. Also, how value of field at each point act as a separate degrees of freedom.

 

[A. Neumaier]

Is it correct to express quantum field theory as "operator valued function" or "operator function" to spacetimepoints. Also, how value of field at each point act as a separate degrees of freedom.

A quantum field is an operator-valued distribution. In many cases (in the relativistic case always) it is too singular to be a function. The separate degrees of freedom are not at a space-time point but (depending on the representation used) either at each point in a 3-space defined by a fixed time, or at a point on a mass shell in momentum space.