The claim that this solves a physics mystery could be seen as a hype.
- Non-hermitian quantum mechanics is a trick used to model dissipative dynamics. However, in some cases this phenomenological model yields a system, which is unitary and conserves probabilities. Authors show that in these cases the resulting system can be dual to a unitary system defined in a curved space.
- The finding is that a non-hermitian quantum dynamics in the 1-D lattice considered allowing probability conservation with a modified inner product, can be formulated as a hermitian and thus unitary dynamics in a 2-D lattice of a 2-D hyperbolic space with a curved metric. Dimensional reduction is involved.
Non-hermitian Hamiltonians can indeed allow also real energy eigenstates and probability conserving time evolution (see this). By replacing complex conjugation representing T with a PT transformation in the inner product, the Hamiltonian becomes hermitian and defines unitary time evolution and has a real energy spectrum. This is the case now.
- In the general case, non-hermitian Hamiltonians give rise to non-conserved probabilities and can be used to describe dissipative systems phenomenologically without taking into account the environment.
- In this 1-D lattice model Schrödinger equation for an energy eigenstate of a non-Hermitian Hamilton is discretized in such a manner that second partial derivative associated with kinetic energy is replaced with a sum -tLΨn+1- -tRΨn-1 in which hopping amplitudes appear. For
L=- R, the equation reduces to the analog of ordinary Schödinger equation.
- At continuum limit this equation reduces to a dimensionally reduced Schrödinger equation at the 1-D boundary of a hyperbolic upper plane if the momentum component vertical to the boundary vanishes. For a non-vanishing vertical momentum one obtains additional potential term. The 2-D Schrödinger equation is probability conserving so that one can say that a 1-D system in which non-hermitian Hamiltonian allows real energy eigenstates and the absence of dissipation is equivalent to a 2-D system with a hyperbolic metric.
One could perhaps see the situation differently. The points of this lattice represent state basis for a discretized space of localized quantum state so that one can also talk about state basic of Hilbert space for which one has introduced metric, which is non-Hermitian in 1-D case but becomes hermitian if the inner product is defined by a Käher metric of the discretized 2-D hyperbolic plane. Hilbert space basis, with states labelled by a lattice of ordinary Hilbert space, would be in question and its points label elements of the state basis of a Hilbert space. The hopping amplitudes in 1-D lattice would be non-hermitian inner product for the neighboring basis elements and would be non-symmetric but for the 2-D extension of Hilbert space basis Hilbert state basis would have Kähler metric. The reason why I made this point is that I have worked with a different idea involving fermionic Fock space with a Kähler metric which would replace the unitary S-matrix as a description of quantum dynamics (see this, this, and this)
- TGD generalizes the Einstein' geometrization program from gravitation by including also the standard model gauge interactions by replacing space-time with a 4-surface in H=M4×CP2. Quantum TGD generalizes the problem to the level of the entire quantum theory by introducing the "world of classical worlds" (WCW) as the space of space-time surface in H satisfying holography required to general coordinate invariance in 4-D sense.
- One can ask whether Einstein's geometrization of physics program could be extended also to Hilbert space level in the sense that a curved Hilbert space with a Kähler metric would replace unitary S-matrix: its identification and construction has been a long standing problem. Kähler metric would characterize fermionic interactions at Hilbert space level. The motivation is that in the infinite-D case the Kähler metric is highly unique and could in the TGD framework completely fix the fermionic physics. In fact, Kähler metric for the WCW would do the same for the bosonic physic). For instance, the Kähler metric of WCW is expected to be essentially unique if it exists at all: already for loop spaces this is the case. Constant curvature metric for which all points of space are metrically equivalent is in question.
- The Kähler metric satisfies orthogonality conditions guaranteeing probability conservation are very similar to those associated with a unitary S-matrix and this is possible under rather weak additional conditions. This would work in the fermionic Fock space. This would mean giving up the dogma of the unitary S-matrix proposed first by Wheeler. The problem of S-matrix is that any unitary transformation of Hilbert space can define S-matrix. The replacement of the S-matrix with Kähler metric could in infinite-D context fix the physics completely.