This fifth paper of the foundational series presents new foundations for quantum mechanics and quantum field theory in terms of coherent spaces and clarifies some aspects of the thermal interpretation from an abstract mathematical point of view.

In a nutshell, coherent spaces (more technical details are here and here) are a novel mathematical concept that can be characterize as a nonlinear version of Hilbert spaces. To get the axioms of a coherent space from those of a Hilbert space, the vector space axioms are dropped while the notion of inner product and its properties is kept and called a coherent product. In a quantum mechanical context, the coherent space is the classical phase space of the system, whose points label the coherent states of its quantum space. For example the spinning electron and higher spin particles can be described in this context, including the half-integrality condition for the spin which is a result of the coherence condition. Another example are Klauder spaces that have the Fock space as their quantum space. Starting with a specific coherent space, a new one can be created for example by taking derivatives or linear combinations of the original coherent states.

Symmetries of a coherent space are its invertible coherent maps. Coherent maps can be quantized by applying a quantization map which produces a coherent operator on the quantum space of the coherent space. The quantization map promotes symmetries to q-observables and infinitesimal symmetries to quantum symmetries. By this relation, complicated computations in quantum mechanics can be reduced to more tractable computations on the coherent space itself. For exactly solvable systems, by means of coherent states techniques can be reduced to solving the dynamics of the classical system. If a problem has a dynamical symmetry group such that the (discrete or continuous) spectrum of all elements of its Lie algebra L is exactly computable, then the spectrum of the system can be found exactly. Such systems are called integrable. The coherent quantization of Kähler manifolds is equivalent to traditional geometric quantization of Kähler manifolds. But in the coherent setting, quantization is not restricted to finite dimensional manifolds, which is important for quantum field theory.

Quantities of a classical or quantum mechanical system can be modeled as L*-algebras, the corresponding uncertain values are defined via the general uncertainty principle. In this context, the Ehrenfest equations for quantities are nicely derived. The Dirac-Frenkel approach applied to the quantum space of a coherent space gives classical dynamics on the coherent space. Well known methods in numerical quantum mechanics, such as Hartree-Fock calculations for example, can be seen as instances of the coherent Dirac-Frenkel approach. The semi-quantal equations obtained from the coherent Dirac-Frenkel variation are chaotic under high resolution. In the thermal interpretation, this chaoticity is used to explain the probabilistic behavior in quantum mechanics.

In field theory, the quantities (or smeared fields) are obtained by integrating the fields which are distributions with a test function. To do quantum field theory in the coherent setting, infinite-dimensional coherent spaces are needed such that issues like taking limits and defining appropriate measures is more involved than in the finite dimensional case. Quantum field theory in spacetime can be obtained by using causal coherent manifolds.

This for now last paper of the foundational series is my personal favorite. The unification of classical and quantum mechanics in the single mathematical framework of coherent spaces looks really intriguing and some kind of beautiful to me. As it is partly a summary of the mathematical underpinnings of this new framework, it nicely motivates one to look up the details in the related papers where they are given. Putting down the abstract mathematica underpinnings of the thermal interpretation does probably not make me agree with its rather drastic change of notions concerning microscopic physics. Nevertheless I some kind of like the mathematical derivations and discussions given here as for example the derivation of the Ehrenfest equations in the context of Lie algebras. The coherent variation principle is a nicely explained seems to have many interesting applications, even though I am not sure if I agree with the notion that the intrinsic quantum uncertainty is explainable by the chaoticity of the semi-quantal equations. The discussion of putting quantum field theory in the coherent context is in intriguing outlook of is still to come.