The relativistic energy of a single mass particle includes its rest mass and kinetic energy. If the kinetic energy of a mass particle is zero (or in a relatively static reference frame), or a system with kinetic energy is in a momentum center system, its total energy (including the kinetic energy inside the system) is related to its static mass or invariant mass, and the famous relationship is E=mc2.
Therefore, as long as the observer's reference frame has not changed, the conservation of energy to time in special relativity still holds, the energy of the whole system remains unchanged, and the energy measured by observers in different reference frames is different, but the energy measured by each observer will not change with time. Invariant mass is defined by the relationship between energy and momentum, which is the minimum value of system mass and energy that all observers can observe. Invariant mass will be conserved, and the values measured by all observers are the same.
In quantum mechanics, the energy of a quantum system is described by a self-adjoint operator called Hamiltonian, which acts on the Hilbert space (or wave function space) of the system. If Hamiltonian is a time-invariant operator, the measurement of its occurrence probability will not change with time as the system changes, so the expected value of energy will not change with time. The conservation of localized energy under quantum field theory can be obtained by combining energy momentum tensor operator with Nott theorem. Because there is no global time operator in quantum theory, the uncertain relationship between time and energy can only be established under certain conditions, which is different from the uncertain relationship between position and momentum as the basis of quantum mechanics (see uncertainty principle). The energy of each fixed time can be accurately measured and will not be affected by the uncertain relationship between time and energy, so even in quantum mechanics, the conservation of energy is a clearly defined concept.
Energy must obey the law of conservation of energy. According to this law, energy can only be transformed from one form to another, and it cannot be generated or destroyed out of thin air. Conservation of energy is a mathematical conclusion drawn from translation symmetry (translation invariance) of time (see Nott theorem).
According to the law of conservation of energy, inflow energy is equal to outflow energy plus internal energy change.
This law is a fairly basic principle in physics. According to the translation symmetry (translation invariance) of time, the laws (theorems) of physics hold at any time.
The law of conservation of energy is the characteristic of many physical laws. From a mathematical point of view, the conservation of energy is the result of Nott's theorem. If the physical system satisfies continuous symmetry in time translation, its energy (yoke physical quantity of time) is conserved. On the contrary, if a physical system is asymmetric in time shift, its energy is not conserved, but if this system exchanges energy with another system and the synthesized larger system does not change with time, the energy of this larger system will be conserved. Because any time-varying system can be placed in a larger time-invariant system, energy conservation can be realized by redefining energy appropriately. For the physical theory of flat space-time, because quantum mechanics allows non-conservation in a short time (such as positive and negative particle pairs), energy conservation is not observed in quantum mechanics, but the law of energy conservation will be transformed into the law of mass and energy conservation in special relativity.
The law of conservation of mass and energy means that in an isolated system, the sum of relativistic kinetic energy and static energy of all particles remains unchanged during the interaction. The law of conservation of mass and energy is a special form of law of conservation of energy.
In the special theory of relativity, the mass-energy formula E=mc2 describes the corresponding relationship between mass and energy. In classical mechanics, mass and energy are independent of each other, but in relativistic mechanics, energy and mass are the same expression of mechanical properties of objects. In the theory of relativity, mass is extended to a mass energy value. It turns out that in classical mechanics, independent conservation of mass and energy are combined into a unified law of conservation of mass and energy, which fully embodies the unity of matter and motion.