In quantum field theory, excitations in a quantum field are caused by the presence of energy. These excitations are often referred to as particles, although it's important to note that in the context of quantum field theory, particles are understood as quantized excitations of the underlying field, rather than classical objects.
The behavior of quantum fields and their excitations is described by mathematical equations, such as the equations of motion derived from the underlying theory. For example, the quantum field describing electromagnetic interactions is described by the equations of quantum electrodynamics (QED).
In these equations, the excitations of the field can be interpreted as the quantized "quanta" of the corresponding physical properties. For instance, in the case of the electromagnetic field, these excitations are referred to as photons, which are the quantized particles associated with electromagnetic radiation.
The presence of energy in the system can lead to the creation and annihilation of these excitations. For instance, when a high-energy collision occurs, it can result in the creation of new particles or excitations in the corresponding quantum field. Conversely, particles can also be annihilated, combining their energy with the field.
It's important to note that the behavior of quantum fields and their excitations is governed by the principles of quantum mechanics, which introduce probabilistic aspects. The probability of finding a particular excitation or particle at a given location or with a specific energy is described by mathematical quantities known as probability amplitudes, which are derived from the equations of the theory.
In summary, excitations in a quantum field are caused by the presence of energy, and their behavior is described by the mathematical equations of the corresponding quantum field theory. The presence of energy can lead to the creation and annihilation of these excitations, resulting in a complex interplay of particles and fields.