Imaginary time

Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?...the words real and imaginary are picturesque relics of an age when the nature of complex numbers was not properly understood. Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories. Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary root i. Imaginary time is not imaginary in the sense that it is unreal or made-up (any more than, say, irrational numbers defy logic), it is simply expressed in terms of what mathematicians call imaginary numbers. Mathematically, imaginary time τ {displaystyle scriptstyle au } may be obtained from real time t {displaystyle scriptstyle t} via a Wick rotation by π / 2 {displaystyle scriptstyle pi /2} in the complex plane: τ   =   i t {displaystyle scriptstyle au = it} , where i 2 {displaystyle i^{2}} Is defined to be − 1 {displaystyle -1} , and is known as the imaginary unit. Stephen Hawking popularized the concept of imaginary time in his book The Universe in a Nutshell. In fact, the names 'real' and 'imaginary' for numbers are just a historical accident, much like the names 'rational' and 'irrational': In the Minkowski spacetime model adopted by the theory of relativity, spacetime is represented as a four-dimensional surface or manifold. Its four-dimensional equivalent of a distance in three-dimensional space is called an interval. Assuming that a specific time period is represented as a real number in the same way as a distance in space, an interval d {displaystyle d} in relativistic spacetime is given by the usual formula but with time negated: where x {displaystyle x} , y {displaystyle y} and z {displaystyle z} are distances along each spatial axis and t {displaystyle t} is a period of time or 'distance' along the time axis.