Negative mass does not necessarily imply negative energy in quantum theory or quantum field theory.
In classical physics, negative mass would indeed correspond to negative energy. However, in the context of quantum theory and quantum field theory, the situation is more nuanced.
In quantum field theory, particles and fields are described by quantum states and operators, and energy is associated with the positive-definite operator known as the Hamiltonian. The energy spectrum of a quantum system is bounded from below, meaning that the energy cannot take arbitrarily negative values.
While it is possible to introduce concepts such as "negative energy states" in certain formalisms, they are typically interpreted as states with lower energy relative to a reference energy level. These negative energy states can still have positive energy when compared to the overall energy spectrum of the system.
Moreover, the concept of negative mass itself is somewhat problematic in the context of quantum theory. The mass of a particle is typically associated with positive values, and a negative mass would introduce several theoretical difficulties and inconsistencies. For example, it could violate energy conditions, causality, and stability requirements.
It's worth noting that the concept of negative mass has been explored in certain speculative theories, such as exotic matter or the hypothetical concept of "phantom energy" in cosmology. However, these theories often involve non-standard assumptions and are not yet well-supported by experimental evidence.
In summary, while negative mass and negative energy can be conceptually introduced in some contexts, they are not inherent or commonplace in quantum theory or quantum field theory.