Making sense of non-Hermitian Hamiltonians

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT ) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT -symmetric non-Hermitian quantum-mechanical Hamiltonians are H = p2 + ix 3 and H = p2 -x4 . Amazingly, the energy levels of these Hamiltonians are all real and positive! Does a PT -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. In terms of C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.

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