Coherent plane waves are idealized mathematical constructs used in classical physics to simplify certain calculations and analyses. In reality, physical systems are rarely described by perfectly coherent plane waves. Quantum fields, on the other hand, do not exhibit the same coherence properties as classical waves, and the reasons for this can be understood from the principles of quantum mechanics.
Quantum fields, such as the electromagnetic field or the Higgs field, are described by quantum field theory, which combines quantum mechanics and special relativity. In quantum field theory, fields are represented by operators that satisfy specific commutation relations, such as the canonical commutation relations. These commutation relations imply that certain pairs of observables, such as position and momentum or energy and time, cannot be simultaneously measured precisely.
The uncertainty principle of quantum mechanics states that there is an inherent limit to the precision with which certain pairs of observables can be simultaneously known. This implies that the position and momentum (or energy and time) of a quantum field cannot be precisely defined at the same time. Consequently, the amplitude of a quantum field cannot be perfectly uniform across all space and time.
Additionally, quantum fields can exhibit fluctuations or "quantum noise" even in their ground state. These fluctuations arise from the fundamental nature of quantum mechanics and cannot be eliminated. The presence of these fluctuations introduces randomness and uncertainty into the amplitudes of quantum fields.
In summary, the coherence properties of classical waves, including coherent plane waves, are not applicable to quantum fields due to the inherent uncertainty and fluctuations imposed by quantum mechanics. Quantum fields are subject to the principles of quantum mechanics, which introduce uncertainty and limit the precise definition of observables such as position and time.