Fractional-time Schrödinger equation: Fractional dynamics on a comb

The physical relevance of the fractional time derivative in quantum mechanics is discussed. It is shown that the introduction of the fractional time Schrödinger equation (FTSE) in quantum mechanics by analogy with the fractional diffusion ∂∂t→∂α∂tα can lead to an essential deficiency in the quantum mechanical description, and needs special care. To shed light on this situation, a quantum comb model is introduced. It is shown that for α = 1/2, the FTSE is a particular case of the quantum comb model. This exact   example shows that the FTSE is insufficient to describe a quantum process, and the appearance of the fractional time derivative by a simple change ∂∂t→∂α∂tα in the Schrödinger equation leads to the loss of most of the information about quantum dynamics.

► FTSE is insufficient to describe a quantum process.
► A quantum comb is an exact example, where the fractional time derivative is naturally introduced.
► Fractional time derivative reflects an interaction with an additional degree of freedom.