The Helmholtz Hamiltonian system
We look at the nonlinear Hamiltonian system
where T is a temperature parameter. It preserves the Helmholtz free energy Hamiltonian as well as the total probability of a complex valued wave on a simplicial complex G. The internal energy is the Heisenberg energy , where is the Green function giving the potential energy between linked simplices x and y. The entropy is the entropy of the probability distribution of the quantum state which is . As the energy definition extends to the Grothendieck ring, the Helmholtz system can be run on every element of this ring.
Here is an evolution of the system in the case when the complex is the Whitney complex of the icosahedron which has 12+30+20=72 cells. It is a bit of a strange system. It is a differential equation in and has 2 integrals of motions at least. We have run it with . For T=0, it would be the Schroedinger evolution in which the Hamiltonian is the inverse of the Laplacian, as this is the internal potential theoretic energy of the geometry. Numerically, we have integrated using Runge-Kutta and checked numerically that the energy and total probability are both preserved very well:
For , the Helmholtz free energy of a probability measure p on a simplicial complex G is , with , where U is the internal energy and S is the entropy of . Denote by a minimal measure and the minimal free energy. (We have seen that is discontinuous already for .) We could now start with a wave for which the probability amplitude is a minimum.
Zero temperature: Schroedinger evolution
For T=0, we get a linear quantum evolution , where is an initial probability distribution. It is a Hamiltonian evolution which preserves the and . The linear system
describes the evolution of the quantum state on the simplicial complex . It defines an evolution of probabilities . The inverse g of the connection Laplacian behaves similarly as a Laplacian itself as g(x,y)=0 if the maximal simplices containing x and y do not intersect.
Infinite temperature: Independent oscillators
The pure entropy case when is dynamically quite uninteresting as each cell in the cellular complex evolves independently from each other. The system is then
which means for . This means that the probability amplitude does not change under the evolution. Each cell oscillates with a frequency which depends on only.
Energy and probability conservation
To the nonlinear Hamiltonian belongs the nonlinear Hamiltonian system , where and . We call it the Helmholtz system. For an eigenstate, . For , we have which gives . What is if . The gradient is . The equations now become complex close to a nonlinear-Schroedinger equation. We get .
Proof of invariance
The justification that and are integrals is obvious and no different than any other Hamiltonian system like the nonlinear Schrödinger equation: if is differentiable then is Hamiltonian: and . Now so that so that by the chain rule . To the invariance of the norm: is purely imaginary. The expression is conjugate. The sum given by the product rule is therefore zero.
Related with known models
When evolving the Ginzburg Landau equation with complex t, purely imaginary t lead to a non-linear Schrödinger equation which features solitons. The nonlinear Schrödinger equation is , where is the probability amplitude of the complex wave and T is a real parameter. Its Hamiltonian is which is . Nonlinear effects can lead to localized waves. It would be interesting to know whether this happens in the Helmholtz case too. As satisfies , one can argue that the quartic nonlinear Schrödinger equation approximates for small $\psi$ the Helmholtz case, suggesting so that for small , the Helmholtz system behaves similar as the nonlinear Schrödinger system.
The integrability question
The most obvious question is whether the Helmholtz system is integrable. Despite the fact that integrability is rare in Hamiltonian dynamics, it is surprising how many important systems are integrable and how much structure is hidden if integrability happens: a firework of mathematics awaits then. Even the simplest systems like the Toda lattice have a staggering amount of interesting mathematics related to it: Operator theory, scattering theory, algebraic geometry.
Is there a chance that the Helmholtz system is integrable for any simplicial complex?
More fundamentally, is there an isospectral deformation of the connection Laplacian which leads asymptotically to the Helmholtz system
Since the differential geometry defined by the exterior derivative d of a simplicial complex has a nice integrable structure given by a Lax deformation of d (leading to an isospectral deformation of the Dirac operator ), one can ask whether It is one of the most exciting topics in partial differential equations.r there is a similar deformation of the Helmholtz Laplacian:
|Is there an isospectral deformation of the connection Laplacian.|
The story is different as the connection Laplacian is not of the form , its spectrum is not symmetric and unlike the Hodge Laplacian is not partitioned.
The soliton question
An other question, which is exciting is in the context of nonlinear quantum mechanics. Linear quantum mechanics does not really feature particles. In classical quantum mechanics this is a conundrum. Klaus Hepp told us as undergraduate students in a classical QM course that we have to answer the “particle-wave” dichotomy problem on our own. Indeed, quantum mechanics alone does not answer that. One needs a quantum field theory for that. But that theory is orders of magnitudes more difficult. Nonlinear Hamiltonian dynamics shows that solitons have particle-like features. It does not solve the quantum field theory problem but it has the advantage that it is relevant and measurable. Solitons can be seen and measured. The emergence of rogue waves for example in the ocean is still unexplained. At any moment of time, one believes that there exist about 10 rogue waves in the ocean. Obviously, these nonlinear waves can not be explained by linear partial differential equations. Related to the integrability question is
|Is there a chance that the Helmholtz system features solitons and so particle like structures?|
[Update May 5, 2017: there is an interesting article in the Swiss SRF (the Swiss TV) on Monster waves. The trouble is that theory does not allow their existence (one can explain only 15 meter waves), not 26 meter waves as measured since 1995.
[Added April 13, 2017] The entropy functional is natural because it is made of multiplicative components. This, and the remarkable multiplicative character of energy (at least in the case of a constant wave) makes the combination of geometric internal energy and entropy natural. We originally had hoped that the energy functional is multiplicative too but that is not the case. Only if is constant, the multiplicative property holds. But entropy S is universally multiplicative. Any way, even if U had been multiplicative, the combination U-TS would not have been multiplicative. So, except except referring again to the special properties of energy and entropy, there is not yet much evidence, why the system should be interesting. But in order to understand the difference between the super symmetric Hodge Laplacian and the not-super symmetric connection Laplacian, it might be useful to study and compare the corresponding Schrödinger evolutions. Also interesting would be to look at nonlinear Schrödinger evolutions with entropy nonlinearity and Hodge Laplacian.
Assume the internal energy is some fundamental Hamiltonian describing properties of space and the addition incorporates a heat bath taking into account all kind of other forces and interactions, then one could assume that in a cosmological setting when the geometry is small (any free evolution of the Dirac operator of a geometry always leads to a fast initial inflation), the effect is that the temperature T is large. As the temperature decreases, the free energy decreases and dynamically could lead to some interesting geometries.
Finally, speaking about Hamiltonians, if one looks at the notion of energy not as a geometric notion of space but as an entity which has physical relevance, there is the question whether there is a possibility to measure it.