imaginary time

Certain quantum calculations (notably the calculation of path integrals as a way to find quantum mechanical probabilities) involve an algebraic manipulation of the following kind: Wherever the time coordinate t occurs, it is replaced by i·t, where i is the “imaginary unit”, a number defined to have the remarkable property i²=i·i=-1. At the end of the calculation, the substitution is reversed. The combination T=i·t is called imaginary time.

Most such calculations occur in particle physics, in the framework of special relativity, where there are rigorous mathematical proofs showing how the use of imaginary time leads to correct results.

Imaginary time has also been employed in some candidate theories for a theory of quantum gravity, notably in certain types of quantum cosmology. This, however, involves the flexible time of general relativity, and both the details of the imaginary time recipe and the more general question whether or not imaginary time can usefully be employed in this context in the first place are still unresolved, and the object of current research.

Some more information about path integrals and the role of imaginary time can be found in the spotlight text The sum over all possibilities, while imaginary time in quantum cosmology is briefly discussed in Searching for the quantum beginning of the universe.