The concept of creating a particle from a vacuum is closely related to quantum field theory and the phenomenon known as vacuum fluctuations. In this context, particles can spontaneously appear and disappear in a vacuum due to the inherent uncertainty of quantum mechanics.
The energy required to create a particle from a vacuum is not a well-defined concept. In quantum field theory, particles are described as excitations of their respective fields. These fields permeate all of space, even in what we typically consider a vacuum. Vacuum fluctuations allow for the temporary creation of particle-antiparticle pairs, which quickly annihilate each other and return the system to its original state.
The energy associated with the creation of a particle from a vacuum is uncertain and can vary depending on the specific circumstances. It is related to the energy-time uncertainty principle of quantum mechanics. According to this principle, energy conservation can be violated for a short period of time, as long as the violation lasts for a sufficiently short duration.
To give you a rough idea, the energy associated with these virtual particles is often described as "borrowed" from the vacuum, and it is generally proportional to the uncertainty in energy multiplied by the uncertainty in time. This uncertainty principle sets a fundamental limit on how precisely we can simultaneously measure energy and time.
In summary, the energy required to create a particle from a vacuum is not a well-defined quantity, as the concept of "creation" in this context is associated with the inherent uncertainties of quantum mechanics. The energy associated with such processes is transient and subject to the energy-time uncertainty principle.