https://matpitka.blogspot.com/2022/06/a-duality-between-certain-non-hermitian.html

## Sunday, June 05, 2022

### A duality between certain non-hermitian systems in flat space with unitary systems in curved space

I learned about an interesting new  duality (see  the  popular article). The claim of the article "Curving the space by non-Hermicity" of Zhou et al (see this") is that dissipative dynamics, which allows a phenomenological description using non-hermitian  Hamiltonian, can in some cases be transformed to a non-dissipative dynamics by introducing non-flat Hilbert space Kähler metric.

The claim that this solves a physics mystery  could  be seen as a hype.

1.  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.
2.  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.

3. 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.
This is shown concretely for  what is known as the Hatano-Nelson model.
1. 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.
2. 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.
Authors interpret the 2-D hyperbolic  lattice as a curved lattice in 3-D  ordinary space so that Schrödinger equation is written in the induced metric, which is hyperbolic.  This is true for the considered  very special kind dissipative interaction with environment.

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)

1. 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.
2. 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.
3. 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.
For a summary of earlier postings see Latest progress in TGD.