# Quantum fluctuations

Quantum fluctuations are thought to seed inhogeneities arising during cosmic inflation, during which the initial metastable state of the inflaton field ‘slow-rolls’ down the potential to a more stable vacuum. Quantum fluctuations in the vacuum, as evinced in the Casimir effect, may also be the source of dark energy, or at least contribute to its observed value.

The nature of quantum fluctuations remains controversial. Whilst quantum theory yields probabilistic predictions of the statistics of apparently random individual experimental outcomes, the extrapolation of such statistical structures down to the microscopic scale in the absence of an experimental context is questionable. It conflicts with Bohr’s philosophy, according to which quantum phenomena can only be defined and described in a given experimental context – one of the reasons that the application of quantum theory to the universe as a whole has always been seen as questionable. It also impacts on the measurement problem: in Everett’s many worlds theory, there is no fluctuating quantum state, other than at the effective level defined by branching and decoherence, whereas in dynamical collapse theories such as the GRW theory (after Ghirardi, Rimini, and Weber) quantum mechanics is modified precisely by introducing a stochastic element to the dynamics.

The lasting appeal of the picture of quantum-mechanical fluctuations taking place at the microscopic level derives from the interpretation of the expectation values of dynamical quantities in terms of statistical averages. For example, the notion of ‘uncertainty’ in Heisenberg’s uncertainty relations may, following Bohr, be taken to mean lack of definability, but it may also be taken to mean *ignorance* – the conventional meaning of the term in ordinary probability theories. The time-energy uncertainty relation is often invoked in this context. However, this is the one uncertainty relation that is the most problematic, as there is no time operator in quantum theory, and where the uncertainty relations are well-defined, from a mathematical point of view the quantities involved are in all cases given as expectation values.

Whilst the energy density of a quantum field in the vacuum (and hence perhaps dark energy) is also given by an expectation value, the notion of ‘fluctuation’ is not. It is an open question as to whether the term has any real explanatory power at the microscopic level. There does, however, remain the use of fluctuations at the microscopic level, following exponential growth in the inflationary period, to explain structure formation in the very early universe. Whether this is consistent with the many worlds theory depends on whether superpositions of states showing small variations in (expectation values) of energy densities decohere. Their superposition may have perfect symmetry, but none of the decohering components need do.

### Links

Stanford Encyclopaedia of Philosophy: the uncertainty principle >