Must a Hamiltonian be Hermitian?

A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data. One would think that a quantum theory based on a non-Hermitian Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is possible to use a previously unnoticed physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite. This construction is general and works for any PT-symmetric Hamiltonian. The dynamics is governed by unitary time evolution. This formulation does not conflict with the requirements of conventional quantum mechanics. There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.Comment: Revised version to appear in American Journal of Physic

Faster than Hermitian Quantum Mechanics

Given an initial quantum state |psi_I> and a final quantum state |psi_F> in a Hilbert space, there exist Hamiltonians H under which |psi_I> evolves into |psi_F>. Consider the following quantum brachistochrone problem: Subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time tau? For Hermitian Hamiltonians tau has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, tau can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from |psi_I> to |psi_F> can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.Comment: 4 page

Semiclassical analysis of a complex quartic Hamiltonian

It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\half p^2+\half\mu^2x^2-\lambda x^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half \mu^2x^2+igx^3-\lambda x^4, whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter \lambda.Comment: 4 page

Scalar Quantum Field Theory with Cubic Interaction

In this paper it is shown that an i phi^3 field theory is a physically acceptable field theory model (the spectrum is positive and the theory is unitary). The demonstration rests on the perturbative construction of a linear operator C, which is needed to define the Hilbert space inner product. The C operator is a new, time-independent observable in PT-symmetric quantum field theory.Comment: Corrected expressions in equations (20) and (21

Making sense of the divergent series for reconstructing a Hamiltonian from its eigenstates and eigenvalues

In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials this paper explains how to use the elementary procedure of Euler summation to sum these divergent series and thereby to make sense of the formal statement of the completeness of the formal sum that represents the reconstruction of the Hamiltonian.Comment: 5 pages, version to appear in American Journal of Physic

Entanglement Induced Phase Transitions

Starting from the canonical ensemble over the space of pure quantum states, we obtain an integral representation for the partition function. This is used to calculate the magnetisation of a system of N spin-1/2 particles. The results suggest the existence of a new type of first order phase transition that occurs at zero temperature in the absence of spin-spin interactions. The transition arises as a consequence of quantum entanglement. The effects of internal interactions are analysed and the behaviour of the magnetic susceptibility for a small number of interacting spins is determined.Comment: 4 pages, 2 figure

Complex Extension of Quantum Mechanics

It is shown that the standard formulation of quantum mechanics in terms of Hermitian Hamiltonians is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct to explain experimental data. One might expect that a quantum theory based on a non-Hermitian Hamiltonian would violate unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed physical symmetry C of the Hamiltonian. Using C, an inner product is constructed whose associated norm is positive definite. This construction is completely general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalisation of it.Comment: 4 Pages, Version to appear in PR

Age-related alterations in efferent medial olivocochlear-outer hair cell and primary auditory ribbon synapses in CBA/J mice

Copyright \ua9 2024 D\uf6rje, Shvachiy, K\ufcck, Outeiro, Strenzke, Beutner and Setz.Introduction: Hearing decline stands as the most prevalent single sensory deficit associated with the aging process. Giving compelling evidence suggesting a protective effect associated with the efferent auditory system, the goal of our study was to characterize the age-related changes in the number of efferent medial olivocochlear (MOC) synapses regulating outer hair cell (OHC) activity compared with the number of afferent inner hair cell ribbon synapses in CBA/J mice over their lifespan. Methods: Organs of Corti of 3-month-old CBA/J mice were compared with mice aged between 10 and 20 months, grouped at 2-month intervals. For each animal, one ear was used to characterize the synapses between the efferent MOC fibers and the outer hair cells (OHCs), while the contralateral ear was used to analyze the ribbon synapses between inner hair cells (IHCs) and type I afferent nerve fibers of spiral ganglion neurons (SGNs). Each cochlea was separated in apical, middle, and basal turns, respectively. Results: The first significant age-related decline in afferent IHC-SGN ribbon synapses was observed in the basal cochlear turn at 14 months, the middle turn at 16 months, and the apical turn at 18 months of age. In contrast, efferent MOC-OHC synapses in CBA/J mice exhibited a less pronounced loss due to aging which only became significant in the basal and middle turns of the cochlea by 20 months of age. Discussion: This study illustrates an age-related reduction on efferent MOC innervation of OHCs in CBA/J mice starting at 20 months of age. Our findings indicate that the morphological decline of efferent MOC-OHC synapses due to aging occurs notably later than the decline observed in afferent IHC-SGN ribbon synapses