Yes, it is possible to have a purely quantum field without any particles. In quantum field theory (QFT), fields are fundamental entities that pervade all of space and time. According to QFT, particles can be understood as excitations or quanta of these underlying fields.
Fields in QFT are described by mathematical equations, such as the Klein-Gordon equation for scalar fields or the Dirac equation for fermionic fields. These equations govern the behavior of the fields and allow for the quantization of the field, resulting in particles.
However, the vacuum state of a quantum field also exists, which represents the lowest energy state of the field with no particles present. This vacuum state is not completely devoid of activity. Instead, it is a state with fluctuations and virtual particles that continually appear and disappear due to the inherent uncertainty and dynamics of quantum mechanics.
These vacuum fluctuations can have observable effects, such as the Lamb shift or the Casimir effect, and they are an essential part of the quantum field. The concept of a particle is often associated with excitations or localized disturbances in the field, but the field itself can exist without any such excitations, resulting in a state with no particles present.
Therefore, a purely quantum field without any particles, in the sense of localized excitations, can still exhibit fluctuations and properties associated with the underlying quantum nature of the field.