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

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Quantum field theory is characterized as an operator-valued distribution rather than a conventional function due to its singular nature, particularly in relativistic contexts. The concept of degrees of freedom is nuanced, as they do not correspond to individual spacetime points but are instead associated with points in a defined 3-space at a fixed time or on a mass shell in momentum space. This distinction highlights the complexity of how fields operate within quantum mechanics. Understanding these aspects is crucial for accurately describing the behavior of quantum fields. The discussion emphasizes the importance of precise terminology in quantum field theory.
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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.
 
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plasmon said:
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.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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