Negative Energy Density Excitations in Black Hole Evaporation

In the black hole evaporation process discovered theoretically by Dr. Hawking, in addition to "Hawking radiation," which is thermal radiation of matter propagating to spatial infinity, something called "negative energy density excitation" appears. As shown in Figure 1, the black hole absorbs this negative energy created outside the event horizon, causing its mass-energy to decrease, which is the cause of this evaporation.

In classical general relativity, the energy of matter was always non-negative, but in "quantum field theory," which incorporates the effects of quantum mechanics, negative energy density excitations can be created in various states, not just in black holes. For simplicity, let us consider a free scalar field in 1+1 dimensional spacetime below.

Here, if we let m be the mass of the scalar field particles, the angular frequency and wavenumber satisfy the following relationship.

In this case, the energy density is

given by, but the quantum energy density operator is defined by subtracting a constant from this Equation (2) so that this expectation value vanishes in the vacuum state |0〉. This adjustment of the constant is called "normal ordering," and in mathematical expressions, it is written by enclosing the operator with "::".

If we write the energy density operator defined by this normal ordering using particle creation and annihilation operators, it becomes

If we calculate the Hamiltonian, which is the total energy, we integrate this over the spatial coordinate x, so the "annihilation operator × annihilation operator" terms and "creation operator × creation operator" terms remaining in Equation (4) vanish. However, when considering energy density, the fact that these two terms do not vanish is the reason why negative energy density excitations appear.

For example, let us consider the following linear superposition state of the vacuum state |0〉 and a normalized two-particle state |2〉. Here, let ϵ be a real number with a very small absolute value.

When calculating the expectation value of the energy density operator in this state, the term for the expectation value in the vacuum state becomes zero, but in addition to the term for the expectation value in the two-particle state, an interference term remains. For example,

does not become zero. The interference term is obtained by multiplying the real part of this term by a coefficient of the magnitude of ϵ. And if ϵ is small, the term for the expectation value in the two-particle state becomes a minute quantity of the second order of ϵ, so ultimately, the interference term remains the largest in the expectation value of the energy density in the state of Equation (3). Then, by changing the value of x, it can be seen that the expectation value of negative energy density, which appears oscillating in the spatial direction, indeed emerges. Thus, in quantum field theory, negative energy density excitations are a common occurrence.

Similarly, it is known that negative energy density excitations appear in condensed matter systems such as quantum spin chains in states close to the ground state. Such negative energy density excitations also play an important role in Quantum Energy Teleportation (QET).