Quantum fluctuations in the vacuum arise due to the inherent uncertainty and dynamics of quantum mechanics. According to Heisenberg's uncertainty principle, there is a fundamental limit to the precision with which certain pairs of physical quantities, such as position and momentum, can be known simultaneously.
In the vacuum, particles and antiparticles can spontaneously appear and annihilate in very short time intervals, obeying the principles of quantum mechanics. This is known as particle-antiparticle pair production. These virtual particles, which are not directly observable, contribute to the fluctuations in the vacuum.
Now, regarding the energy associated with these fluctuations, it's important to note that the concept of energy in quantum field theory is somewhat different from classical notions of energy. In quantum field theory, energy is not conserved locally in space and time due to the uncertainty principle.
The energy for the creation of these virtual particles comes from the inherent energy associated with the vacuum itself. According to quantum field theory, the vacuum is not an empty void but rather a state of lowest energy. This vacuum energy is sometimes referred to as the "zero-point energy." The uncertainty principle allows for temporary violations of energy conservation as long as the energy is restored within a very short time frame.
However, it's worth mentioning that the precise nature and origin of vacuum fluctuations, as well as the associated energy, are still subjects of ongoing research and debate in theoretical physics. Various approaches and mathematical frameworks, such as quantum field theory and quantum gravity, are being explored to better understand these phenomena.