In the context of quantum mechanics (QM) and quantum field theory (QFT), the concept of energy conservation is more subtle than in classical physics. The conservation of energy is intimately related to time translation symmetry, and in quantum theories, the notion of energy conservation is tied to the conservation of the Hamiltonian operator.
In non-relativistic quantum mechanics, the Hamiltonian operator represents the total energy of a system. If the Hamiltonian operator commutes with the time evolution operator, which is the case for systems with time translation symmetry, then the expectation value of the Hamiltonian (i.e., the average energy) remains constant over time. This can be seen as a form of energy conservation.
However, in relativistic quantum field theory, such as quantum electrodynamics (QED), the situation is more complicated. The Hamiltonian operator is not a conserved quantity due to interactions and the creation and annihilation of particles. In this context, energy is not conserved locally in the same way as in classical physics.
Nevertheless, energy can still be conserved in a broader sense. In QFT, a quantity known as the total energy-momentum tensor is conserved globally. This conservation law accounts for the exchange of energy and momentum between different parts of a system, including particles and fields.
It's important to note that the non-conservation of energy in certain scenarios does not imply energy coming out of nothing in the sense of spontaneous creation from nothingness. The fluctuations and transformations of energy in quantum systems are governed by the principles of quantum mechanics, which involve complex processes and interactions. Energy can be transferred, transformed, and redistributed within a system, but it does not arise from a complete absence of everything.
In cosmology, the total energy of the universe is a topic of ongoing investigation and debate. The concept of energy conservation in the context of the entire universe is complex due to the expansion of space, the nature of dark energy, and other factors. The question of whether the total energy of the universe is conserved remains an active area of research.