Imaginary time


The representation of time by real numbers has been successful, particularly in classical mechanics (CM). Another possibility, imaginary time, is the analytic continuation of real time from the real numbers R into the complex number plane C. This is the focus of this chapter. Imaginary time has found significant application in fundamental areas of quantum physics such as high energy particle physics and quantum cosmology.

We need to distinguish between imaginary time and complex time. A complex number я can be written in the form я = u + iv, where u and v are real and i is the square root of-1. Therefore C, the set of complex numbers, requires two real parameters to label all of its elements. When time is asserted to be complex, that then implies that time is being parametrized by two real dimensions. We now refer back to Chapter 12, where we discussed the consequences of time having two or more dimensions. The general conclusion there was that such a possibility leads to unstable ultrahyperbolic solutions to differential equations [Tegmark, 1997]. Another problem with complex numbers is that they are not an ordered set, so they are not consistent with the ordering property of time that we would naturally expect.

Clearly, complex time is problematical and we are going to avoid it. When we refer to imaginary time we mean a complex number of the form я = it, where т is real. This form of complex number has a single real parameter, т, that has the required ordering we want for time. We shall use the term complex time for a point in the complex plane that is parametrized by two independent real variables. Imaginary time is one dimensional, complex time is two dimensional. If we need to think of a parametrized curve (a function of a single real variable) in the complex plane not necessarily along the imaginary axis, we shall refer to it as complex path time. The set of complex numbers on a complex path is ordered by the values of the real parameter involved.

Images of Time. First Edition. George Jaroszkiewicz.

© George Jaroszkiewicz 2016. Published in 2016 by Oxford University Press.

Application to wave mechanics 227

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