Imaginary time

Imaginary time is a mathematical representation of time that appears in some approaches to special relativity and quantum mechanics. It finds uses 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 unit i. Imaginary time is not imaginary in the sense that it is unreal or made-up; it is simply expressed in terms of imaginary numbers.

In mathematics, the imaginary unit ${\displaystyle i}$ is ${\displaystyle {\sqrt {-1}}}$, such that ${\displaystyle i^{2}}$ is defined to be ${\displaystyle -1}$. A number which is a direct multiple of ${\displaystyle i}$ is known as an imaginary number.^{[1]: Chp 4} A number that is the sum of an imaginary number and a real number is known as a complex number.

In certain physical theories, periods of time are multiplied by ${\displaystyle i}$ in this way. Mathematically, an imaginary time period ${\textstyle \tau }$ may be obtained from real time ${\textstyle t}$ via a Wick rotation by ${\textstyle \pi /2}$ in the complex plane: ${\textstyle \tau =it}$.^[1]: 769

Stephen Hawking popularized the concept of imaginary time in his book The Universe in a Nutshell.

"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?"

— Stephen Hawking^[2]: 59

In fact, the terms "real" and "imaginary" for numbers are just a historical accident, much like the terms "rational" and "irrational":

"...the words real and imaginary are picturesque relics of an age when the nature of complex numbers was not properly understood."

— H.S.M. Coxeter^[3]

In the

in the same way as a distance in space, an interval ${\displaystyle d}$ in relativistic spacetime is given by the usual formula but with time negated: $${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}-t^{2}}$$ where ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ are distances along each spatial axis and ${\displaystyle t}$ is a period of time or "distance" along the time axis (Strictly, the time coordinate is ${\displaystyle (ct)^{2}}$ where ${\displaystyle c}$ is the

${\displaystyle c=1}$

Mathematically this is equivalent to writing $${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}+(it)^{2}}$$

In this context, ${\displaystyle i}$ may be either accepted as a feature of the relationship between space and real time, as above, or it may alternatively be incorporated into time itself, such that the value of time is itself an imaginary number, denoted by ${\displaystyle \tau }$. The equation may then be rewritten in normalised form: $${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}+\tau ^{2}}$$

Similarly its

$${\displaystyle (x_{0},x_{1},x_{2},x_{3})}$$

${\displaystyle x_{n}}$

${\displaystyle x_{0}=ict}$

${\displaystyle c}$

Hawking noted the utility of rotating time intervals into an imaginary metric in certain situations, in 1971.^[4]

In

• Euclidean quantum gravity
• Multiple time dimensions