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.

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 (any more than, say, irrational numbers defy logic), it is simply expressed in terms of what mathematicians call imaginary numbers.

Origins

Mathematically, imaginary time $\scriptstyle \tau$ may be obtained from real time $\scriptstyle t$ via a Wick rotation by $\scriptstyle \pi /2$ in the complex plane: $\scriptstyle \tau \ =\ it$ , where $i^{2}$ Is defined to be $-1$ , and $i$ is known as the imaginary unit.

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^[1]

In fact, the names "real" and "imaginary" for numbers are just a historical accident, much like the names "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^[2]

In cosmology

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 $d$ in relativistic spacetime is given by the usual formula but with time negated:

d^{2}=x^{2}+y^{2}+z^{2}-t^{2}

where $x$ , $y$ and $z$ are distances along each spatial axis and $t$ is a period of time or "distance" along the time axis.

Mathematically this is equivalent to writing

d^{2}=x^{2}+y^{2}+z^{2}+(it)^{2}

In this context, $i$ 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 $\tau$ , and the equation rewritten in normalised form:

d^{2}=x^{2}+y^{2}+z^{2}+\tau ^{2}

Similarly its four vector may then be written as

(x_{0},x_{1},x_{2},x_{3})

where distances are represented as $x_{n}$ , $c$ is the velocity of light and $x_{0}=ict$ .

In physical cosmology, imaginary time may be incorporated into certain models of the universe which are solutions to the equations of general relativity. In particular, imaginary time can help to smooth out gravitational singularities, where known physical laws break down, to remove the singularity and avoid such breakdowns (see Hartle–Hawking state). The Big Bang, for example, appears as a singularity in ordinary time but, when modelled with imaginary time, the singularity can be removed and the Big Bang functions like any other point in four-dimensional spacetime. Any boundary to spacetime is a form of singularity, where the smooth nature of spacetime breaks down.^[3] With all such singularities removed from the Universe it thus can have no boundary and Stephen Hawking has speculated that "the boundary condition to the Universe is that it has no boundary".^[4]

However the unproven nature of the relationship between actual physical time and imaginary time incorporated into such models has raised criticisms.^[5] Penrose has noted that there needs to be a transition from the Riemannian metric (often referred to as "Euclidean" in this context) with imaginary time at the Big Bang to a Lorenzian metric with real time for the evolving Universe. Also, modern observations suggest that the Universe is open and will never shrink back to a Big Crunch. If this proves true then the end-of-time boundary still remains.^[3]

In quantum statistical mechanics

The equations of the quantum field can be obtained by taking the Fourier transform of the equations of statistical mechanics. Since the Fourier transform of a function typically shows up as its inverse, the point particles of statistical mechanics become, under a Fourier transform, the infinitely extended harmonic oscillators of quantum field theory.^[6] The Green's function of an inhomogeneous linear differential operator, defined on a domain with specified initial conditions or boundary conditions, is its impulse response, and mathematically we define the point particles of statistical mechanics as Dirac delta functions, which is to say impulses. ^[7] At finite temperature $T$ , the Green's functions are periodic in imaginary time with a period of $\scriptstyle 2\beta \ =\ 2/T$ . Therefore, their Fourier transforms contain only a discrete set of frequencies called Matsubara frequencies.

The connection between statistical mechanics and quantum field theory is also seen in the transition amplitude $\scriptstyle \langle F\mid e^{-itH}\mid I\rangle$ between an initial state $I$ and a final state $F$ , where $H$ is the Hamiltonian of that system. If we compare this with the partition function $\scriptstyle Z\ =\ \operatorname {Tr} \ e^{-\beta H}$ we see that to get the partition function from the transition amplitudes we can replace $\scriptstyle t\,=\,\beta /i$ , set $F = I = n$ and sum over $n$ . This way we don't have to do twice the work by evaluating both the statistical properties and the transition amplitudes.

Finally by using a Wick rotation one can show that the Euclidean quantum field theory in (D + 1)-dimensional spacetime is nothing but quantum statistical mechanics in D-dimensional space.

References

Notes

^ Hawking (2001), p.59.
^ Coxeter, H.S.M.; The Real Projective Plane, 3rd Edn, Springer 1993, p. 210 (footnote).
^ ^a ^b Roger Penrose. The Road to Reality. Vintage. 2005. pp.769-772.
^ Hawking 2001, p.85.
^ Robert J. Deltete & Reed A. Guy; "Emerging from Imaginary Time", Synthese, Vol. 108, No. 2 (Aug., 1996), pp. 185-203.
^ Uwe-Jens Wiese, "Quantum Field Theory", Institute for Theoretical Physics, University of Bern, 21 August 2007, page 63.
^ Andy Royston; "Notes on the Dirac Delta and Green Functions", 23 November 2008.

Bibliography

Hawking, Stephen W. (1998). A Brief History of Time (Tenth Anniversary Commemorative ed.). Bantam Books. p. 157. ISBN 978-0-553-10953-5.
Hawking, Stephen W. (2001). The Universe in a Nutshell. United States & Canada: Bantam Books. pp. 58–61, 63, 82–85, 90–94, 99, 196. ISBN 0-553-80202-X.

External links

The Beginning of Time — Lecture by Stephen Hawking which discusses imaginary time.
Stephen Hawking's Universe: Strange Stuff Explained — PBS site on imaginary time.

[1] Hawking (2001), p.59.

[2] Coxeter, H.S.M.; The Real Projective Plane, 3rd Edn, Springer 1993, p. 210 (footnote).

[Penrose-3] Roger Penrose. The Road to Reality. Vintage. 2005. pp.769-772.

[4] Hawking 2001, p.85.

[5] Robert J. Deltete & Reed A. Guy; "Emerging from Imaginary Time", Synthese, Vol. 108, No. 2 (Aug., 1996), pp. 185-203.

[6] Uwe-Jens Wiese, "Quantum Field Theory", Institute for Theoretical Physics, University of Bern, 21 August 2007, page 63.

[7] Andy Royston; "Notes on the Dirac Delta and Green Functions", 23 November 2008.

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