The universe as a formal model

The deterministic dynamics of the universe is given in the Schrödinger picture by the von Neumann equation, and in the Heisenberg picture by the Heisenberg equation. The Heisenberg picture is more complete, because it allows us to model temporal correlations, and is more easily generalisable to the relativistic case.

In this discussion, however, we will assume a non-relativistic universe, and we will use the Schrödinger picture for simplicity's sake. The state of the universe will thus be modelled by a time-dependant ensemble.

In the thermal interpretation the universe as a whole is classically deterministic and has a classical Hamiltonian description (symplectic if one assumes the state of the universe to be pure, Poissonian in the general case.)

To save writing in working with Poisson brackets (particularly on paper - I have written down thousands of these brackets) I have invented the simple notation

f ⌉ g = {f,g}.

The symbol ⌉ is pronounced 'Lie'. Thus the Poisson bracket appears as a binary operation. We call this the Lie-product of f and g. It is bilinear in f and g and has the properties

f ⌉ f = 0

f ⌉ g = -g ⌉ f

f ⌉ gh = (f ⌉ g)h + g(f ⌉ h) (Leibnitz)

f ⌉ (g ⌉ h) = (f ⌉ g) ⌉ h + g ⌉ (f ⌉ h) (Jacobi)

and two analogous formulae obtained from the Jacobi identity by cyclic permutation of the arguments.

The variables in the thermal interpretation are elements f of a fixed algebra E of operators on a dense subspace of a universal Hilbert space. Typical variables are for example integrals:

f = ∫ dx^{3} a^{*}(x) c(x,∇x) a(x)

where a(x) is a quantum field and c(x,∇x) a differential operator, as well as linear combinations of products of such functions and generalisations thereof.

The deterministic differential equation for the value <f> = <f>_{t} of the variable f at the time t (in the following we suppress the time argument when it is not absolutely necessary) is then a classical Hamiltonian dynamics of the form

d/dt <f> = <H> ⌉ <f>, (*)

where H, the Hamiltonian of the universe, is a special self-adjoint variable, and (as one can easily check)

<f> ⌉ <g> := <iħ[f,g]>

is a classical Lie-Poisson bracket in the space of smooth functions

F(<f_{1}>,<f_{2}>,…,<f_{n}>) (**)

defined with arbitrary <f_{i}>.

In accordance with the quantum mechanics tradition, we assume that ensembles are 'normal' and may therefore be described by density matrices. Thus we describe the state of the universe at the time t by a density matrix ρ(t), a Hermitian, semi-definite trace class operator with trace 1. (It is impossible to say whether the state is pure, because the part of the universe accessible to us is small, and projection onto it inevitably destroys its purity.)

The universal density matrix ρ(t) determines the universal ensemble by the prescription

<f>_{t} := trρ(t)f

for the objective value of every variable f at time t. Of these, the ones which are observable in principle are those that vary sufficiently slowly in time and space; all others are 'hidden', that is, inaccessible to experiment.

All measurement values that can in principle be calculated from raw measurements have the form (**), with the exclusion of points where the calculation rule is non-differentiable. In any case the expressions on the right hand side of (*) are in general no longer directly measurable variables, but contain 'hidden' correlations. This makes the classical dynamics non-local and produces quantum effects.

With a bit of thought one discovers that (*) is nothing but the traditional von Neumann dynamics for the density matrix, only expressed in terms of the variables of the thermal interpretation. From this follows the complete consistency of the thermal interpretation with the formalism of quantum mechanics.

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