Can one falsify the state of the universe?

We can never find out everything about the state of the universe, but we can find out a great deal. In particular we can easily falsify the proposition that the state of the universe is some particular thing blindly assumed!

For the sake of simplicity, let us assume that the universe is in a pure state ψ. Let us also assume that we have somehow fixed a coordinate system (something like GPS). (We can naturally express doubts about whether this is possible, but physicists make plenty of apparently plausible assumptions that can be questioned.)

In the corresponding Hilbert space there are operators M(x) (more precisely, operator-valued distributions) which represent the mass density at the point x. For x in our solar system we know, for example, very well that we can measure M(x) fairly precisely and reproducibly, either directly or by extrapolation, and thus obtain thoroughly precise statistical information about it. Let ρ(x,m) be the appropriate probability density.

Quantum mechanics says that for sufficiently well-behaved functions f(m) the formula

ψ^*f(M(x))ψ = ∫ f(m) ρ(x,m) dm       (*)

holds with a precision also predicted for any chosen confidence level.

If one blindly chooses ψ (for example with a random generator, which produces unit vectors in the Hilbert space) we can test (*) with the same statistical predictions that are used to test the theory. Randomly chosen ψ will fail this test with an extremely high confidence level.

The state of the universe is thus fairly strongly constrained by the set of all facts about the universe known to us. If one extrapolates, using the same arguments that one uses to argue that the state of an n-particle system is uniquely determined, we can plausibly argue that the state of the universe is uniquely determined, even if we can never identify it precisely.

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Arnold Neumaier (Arnold.Neumaier@univie.ac.at)

A theoretical physics FAQ