In quantum field theory (QFT), the wave-particle duality is indeed resolved in a sense that particles are described as excitations of quantum fields rather than discrete entities. However, the uncertainty principle still holds in QFT, albeit in a slightly different form.
The uncertainty principle, often associated with Heisenberg, is a fundamental principle in quantum mechanics that states that certain pairs of physical properties, such as position and momentum, cannot be simultaneously known to arbitrary precision. In QFT, this principle is manifested through the commutation relations between field operators.
In QFT, fields are described by operator-valued fields that satisfy specific commutation or anticommutation relations, depending on whether the field is a bosonic or fermionic field, respectively. These commutation or anticommutation relations give rise to the uncertainty principle.
For example, in the case of a scalar field, the field operator and its conjugate momentum operator satisfy the commutation relation:
[ϕ(x), π(y)] = iδ(x-y),
where ϕ(x) is the field operator, π(y) is the conjugate momentum operator, and δ(x-y) is the Dirac delta function. This commutation relation leads to the uncertainty principle between the field and its conjugate momentum.
The uncertainty principle in QFT is related to the inherent quantum fluctuations of the field. These fluctuations are present even in the vacuum state, giving rise to particle creation and annihilation processes. The uncertainty principle places a fundamental limit on the simultaneous knowledge of field observables and their conjugate momenta.
In summary, while the wave-particle duality is resolved in QFT by describing particles as excitations of quantum fields, the uncertainty principle remains a fundamental principle that is expressed through the commutation relations between field operators. The uncertainty principle persists as a consequence of the quantum nature of fields and their associated fluctuations.