A model universe

In the thermal interpretation, the universe is described by three mathematical objects:

1. a fixed algebra E of operators on a dense subspace of a universal Hilbert space,

2. a self-adjoint universal Hamiltonian H in this algebra,

3. a normalised state ρ acted on by this algebra.

For the real universe, the algebra E of variables, together with the spacetime metric, is produced by the fields of the standard model, with the Hamiltonian being canonically derived from the associated forces. The state of the universe is however by and large unknown, as a knowledge of it in the context of the thermal interpretation would imply knowledge of all values of all fields and correlation functions in any order in all places and at all times. On the other hand, the states of many subsystems are fairly well understood, in particular those which have been explored experimentally.

Because of the as yet unsolved problems of quantum gravity and the difficulties of drawing conclusions about macroscopic detectors from the standard model, this fully realistic model is not well suited for the concrete analysis of a real measurement process.

To enable one to conceive of something more concrete than the abstract concept of 'universe', an example of a model universe will be provided that is simple enough to be easily specified and yet still able to reproduce important aspects of the real universe. As in the real universe, we will specify the algebra E of variables and the Hamiltonian H, but we will leave the condition ρ open and will observe at most the states imposed by it on subsystems (see the section “Physical systems and their measurement”) insofar as they are relevant to an experiment.

In addition we will consider a non-relativistic model universe whose matter is comprised of an undetermined number of elementary particles with mass m_{l}, spin s_{l} and charge eZ_{l}
(l=1,...L, Z_{l}∈ℕ) and electrons with mass
m_{0}, spin 1/2 und charge eZ_{0} (Z_{0}=-1).
The different particles are not distinguishable from one another. Associated basic variables have normal commutation relations, except for mass, charge and, electrons, Pauli matrices, the three dimensional positions coordinates x^{a} and the associated impulse p^{a}, one per particle.

Further, the model universe includes radiation, exclusively in the form of an undetermined number of photons, bosons with mass 0, spin 0 and charge 0. The photons have a frequency coming from an octave in the visible domain, and have three-dimensional wave vectors k with frequency ω=|k|. Associated basic variables are the radiation energy H_{rad} := ∫ dk |k| a^{*}(k)a(k)
and for every particle a, a radiation potential U^{a} := g ∫ dk a(k) exp(ik.x^{a}).
At the same time the usual commutation relations apply to the integrals over the spherical shell

K = {k | |k| in [1,2]}

a(k) are destruction operators and a^{*}(k) the adjoint creation operators. (Units are chosen so that c=ℏ=1 and the frequency of the lower end of the radiation spectrum is 1.)

The algebra of the variables is that of the basic variables and all Schwarz functions in the algebra of linear operators on the space

ℋ = ℋ_{matter} ⊗ ℋ_{rad},

where ℋ_{matter} is in the zero-charge sector of the charge operator

Q = ∑_{l=0}^{L} a_{l}^{*} Z_{l} a_{l}(x)

in the Fock space over the space of Schwarz functions in ℝ^{3} (with spin statistics appropriate to each case) and ℋ_{rad} is the Fock space over the space of Schwarz functions on K.

We will fix the Hamiltonian of the universe as

H = H_{matter} + H_{rad} + g ∑_{a} (U^{a}+(U^{a})^{*})

with a coupling constant g. H_{matter} is the matter Hamiltonian, given by the traditional formula with a Coulomb interaction.

The model universe is translation and rotation invariant, and scattering problems can be solved with the traditional formulae, without infrared or ultraviolet problems.

There are:

atoms, molecules, chemical reactions and molecular spectra,

no radioactivity or nuclear reactions,

no polarised light,

no gravitation,

no microscopic fields,

but the Coulomb interaction produces macroscopic electromagnetic fields.

Thus the majority of actual reality is qualitatively present, without the appearance of the typical difficulties of quantum field theory and gauge theory.

In particular the entire chemistry of the universe (with the exception of laser chemistry) is reproducible, as is the entirety of fluid mechanics, geometrical optics and practically the whole of solid state physics. A consequence of this is that all mechanical or hydraulic measurement devices and most optical and electric devices are modelable.

Particularly worth noting is that photographically recorded experiments with light sources and screens - like the double-slit experiment with light and photo detectors - as well as experiments with magnets - like the Stern-Gerlach experiment - can be performed in our model universe.

Of course this assumes that there are physicists who can conduct the experiments! To derive this from a microscopic model goes beyond the possibilities of modern physics, so that this can be accepted without proof. However physicists are essential to the execution of the trial; their role restricts itself to the preparation of the experiment and the subsequent scrutiny of the photographs. We will thus always observe physicists as a part of the irrelevant environment in which an experiment takes place. By this is in no way intended a belittling of the profession of experimental physics, whose members are, after all, responsible for all of our detailed knowledge about the universe.

In our simplified model of the universe the preparation of an experiment thus signifies no more than the assertion (supposition) that there could be a physical system in the model universe in an initial state that has the necessary features to comply with a complete description of the experiment. The task of the analysis is to explain the resulting observations.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ