GENERAL ARTICLE Swinging in Imaginary Time

104 RESONANCE February 2010

GENERAL ARTICLE

Swinging in Imaginary Time

More on the Not-So-Simple Pendulum

Cihan Saclioglu

Keywords

Pendulum, Jacobi elliptic func-

tions, tunneling, WKB, instan-

tons.

Cihan Saclioglu teaches

physics at Sabanci

University, Istanbul,

Turkey. His research

interests include solutions

of Yang–Mills, Einstein

and Seiberg–Witten

equations and group

theoretical aspects of

string theory.

1When the exact quantum me-

chanical calculation of the tun-

neling probability for an arbitrary

potential U(x) is impracticable,

the Wentzel–Kramers–Brillouin

approach, invented indepen-

dently by all three authors in the

same year as the Schrödinger

equation, provides an approxi-

mate answer.

When the small angle approximation is not made,the exact solution of the simple pendulum is aJacobian elliptic function with one real and oneimaginary period. Far from being a physicallymeaningless mathematical curiosity, the secondperiod represents the imaginary time the pendu-lum takes to swing upwards and tunnel throughthe potential barrier in the semi-classical WKBapproximation1 in quantum mechanics. The tun-neling here provides a simple illustration of simi-lar phenomena in Yang{Mills theories describingweak and strong interactions.

1. Introduction

Consider a point mass m at the end of a rigid masslessstick of length l. The acceleration due to gravity is gand the oscillatory motion is con¯ned to a vertical plane.Denoting the angular displacement of the stick from thevertical by Á and taking the gravitational potential tobe zero at the bottom, the constant total energy E isgiven by

E =1

2ml2

µdÁ

dt

¶2

+mgl(1¡ cos Á)

= mgl(1¡ cosÁ0); (1)

where Á0 is the maximum angle. Isolating (dÁ=dt)2 andtaking its square root gives Á as an implicit function ofthe time t through

t =

sl

2g

Z Á

0

dÁ

(cos Á¡ cosÁ0)1=2

105RESONANCE February 2010

GENERAL ARTICLE

=

sl

4g

Z Á

0

dÁ¡sin2(Á0=2)¡ sin2(Á=2)

¢1=2 : (2)

Following the treatment in Landau{Lifshitz [1], we de-¯ne sin » = sin(Á=2)= sin(Á0=2), in terms of which theperiod T becomes

T = 2

sl

g

Z Á0

0


¢1=2

= 4

sl

g

Z ¼2

0

d»

(1¡ k2 sin2 »)1=2: (3)

This can be used to de¯ne K(k), the complete ellipticintegral of the ¯rst kind, via

T ´ 4

sl

gK(k) : (4)

When k2 = (sin(Á0=2))2 can be neglected, one recovers

the period 2¼pl=g of the simple small angle pendulum.

Note that the standard elliptic function notation K(k)is a bit misleading: K is really a function NOT of k,but of k2. Hence for k2 ¿ 1, we can consistently keepk as a small non-zero quantity while neglecting k2. Thesubstitution y = sin » shows that (3) can also be writtenas

K(k) =

Z ¼2

0

d»

(1¡ k2 sin2 »)1=2

=

Z 1

0

dy

((1¡ y2)(1¡ k2y2))1=2: (5)


GENERAL ARTICLE

This is an appropriate

point to mention an

essential difference

between trigonometric

and elliptic functions:

Considered as

functions of the

complex variable

z = x + iy, the former

are bounded and have

a real period along the

x-axis, but blow up

exponentially as y

goes to positive or

negative infinity, while

the latter have an

additional imaginary

period along the y-

axis.

2. Elliptic Functions and Integrals

Let us convert the de¯nite integral (5) into an inde¯niteone by replacing the upper limit 1 by y. Also introducingthe dimensionless time variable x = t

pg=l, this yields

x =

Z y

0

dy0

((1¡ y02)(1¡ k2y02))1=2: (6)

The right-hand side of (6) is known as the Legendre ellip-tic integral of the ¯rst kind and is denoted by F (arcsiny; k). The inverse of F , implicitly de¯ned by (6), is theJacobian elliptic function y = snx [2]. Assembling allthis, the exact time dependence of the angle is found tobe

Á(t) = 2 arcsin

µ

sin

µrg

lt

¶

sinÁ0

2

¶

; (7)

which becomes the familiar

Á(t) = Á0 sin

µrg

lt

¶

(8)

in the limit sin Á0 ¼ Á0.

This is an appropriate point to mention an essentialdi®erence between trigonometric and elliptic functions:Considered as functions of the complex variable z =x + iy, the former are bounded and have a real periodalong the x-axis, but blow up exponentially as y goes topositive or negative in¯nity, while the latter have an ad-ditional imaginary period along the y-axis. We will haveto state this and a few other properties of elliptic func-tions without proof, as the required complex analysis isa bit involved. We refer the interested reader to Math-ews and Walker [2], whose treatment we mostly follow.For snx, we have already met the real period K(k) in(4); the imaginary period 2K 0 is de¯ned via


GENERAL ARTICLE

Box 1. Elliptic Functions

A function f(z) is said to be doubly periodic in the complex z-plane with periods !1 and!2 if f(z + n!1 + m!2) = f(z) for any pair of integers n;m. Double periodicity alongtwo independent directions requires the ratio !1=!2 to have an imaginary part. Thus thefunction maps a single parallelogram cell ¦ de¯ned by !1 and !2 to the entire z-plane. If thesingularities of f(z) consist of poles, the resulting meromorphic doubly-periodic function iscalled an elliptic function. By Liouville's theorem, f(z) must have poles in ¦ if it is not tobe a constant. Since the parallel sides of the parallelogram @¦ bounding ¦ give equal andopposite contributions, and f(z) is meromorphic,

H@¦f(z)dz = 0 by Cauchy's theorem. This

means the residues of the poles in ¦ must add up to zero. The simplest way to get vanishingtotal residue is either to choose two simple poles with opposite residues, or one double pole in¦. These correspond to the Jacobi and the Weierstrass elliptic functions, respectively. Thelatter, denoted by }, is easier to represent explicitly (in the form of an in¯nite double sumover all lattice points), but our present problem of a planar pendulum involves the Jacobielliptic functions (interestingly, the Weierstrass ones appear in the treatment of the sphericalpendulum!). The lattice is generated by !1 = K and !2 = iK(k0) that are de¯ned in (5)and (9). The double periodicity properties follow by integrating (6) in the complex y-planealong di®erent contours: sn(2K ¡ y) = sny is proven by choosing a path that goes from 0to y to 1 along the real axis, encircles 1 in a clockwise sense and comes back to y on a pathgoing left just below the real axis. Using sn(¡y) = ¡sny which follows from (6), we havesn(2K+y) = ¡sny. Shifting the argument again by another 2K, we get sn(4K+y) = +sny.Extending the contour from 1 to 1=k, clockwise encircling 1=k and coming back to y on apath just below the real axis yields sn(y + 2iK0) = sny on account of (9) and the fact thatfor 1 < x < 1=k,

p1¡ x2 = §i

px2 ¡ 1.

Historically, the study of elliptic functions originated around the end of the seventeenthcentury in the study of arc lengths of ellipses and a curve called Bernoulli's lemniscate,leading to the integral

R ®0

dx/p

1¡ x4. Count Fagnano presented an addition theorem for

this integral in 1750. Euler extended the theorem to integrals withp

1 + ax2 + bx4 inthe denominator. Legendre's studies of more general integrands such as E + F=

pQ(x),

where E and F are rational functions and Q is a cubic or quartic polynomial culminatedin his 1832 Treatise on Elliptic functions and Euler integrals. In 1828 Abel and Jacobialmost simultaneously de¯ned new functions by inverting such integrals and showed thatthey were doubly-periodic. The terminology was also changed: these functions are nowcalled elliptic, while Legendre's elliptic functions are referred to as elliptic integrals. Jacobi'selliptic functions can also be expressed as the ratios of quadratic combinations of Jacobi £-functions that are quasiperiodic rather than periodic in the complex plane. The so-calledRamanujan £-function is a further generalization of Jacobi £-functions, but it is not due toRamanujan himself. Ramanujan introduced what he called \mock £-functions", which arenowadays regarded as a class of `modular forms', de¯ned by their transformation propertiesunder modular transformations of the complex plane. Such a transformation takes z to(az + b)=(cz + d), where a; b; c; d are integers and ab ¡ cd = 1. For more on Ramanujan'sremarkable work in this area we refer the reader to the book Ramanujan's lost notebook,George E Andrews and Bruce C Berndt, Part I, Springer, New York, 2005.


GENERAL ARTICLE

Figure1.Therealand imagi-

nary 'paths'aremostclearly

understood by referring to

the potential. The solid line

represents the real time

oscillations, while the dot-

ted one corresponds to

imaginary time tunneling.

K 0 =

Z 1=k

1

dy

((y2 ¡ 1)(1¡ k2y2))1=2: (9)

The double-periodicity property means that

sn(z + 4K) = sn(z + 2iK 0) = snz : (10)

It can be shown that K 0(k) = K(k0) if k02 = 1¡ k2 bygoing over to the variable z =

p(1¡ k2y2)=(1¡ k2) in

(9). This `duality relation' will be useful in relating thependulum's behavior near the top and bottom points.

3. The Physical Meaning of the Imaginary Pe-riod

We will now show that the imaginary period K 0 coin-cides with the interval of imaginary time the pendulumspends while `swinging' from +Á0 to ¡Á0 not via thebottom point Á = 0 as in its normal motion, but bycontinuing beyond +Á0 to the top point Á = ¼ and allthe way down in the same direction to Á = 2¼¡Á0, whichis of course the same position in space as ¡Á0. Duringthis `motion', the total energy mgl(1¡ cosÁ0) is belowthe potential energy mgl(1¡ cos Á), making the kineticenergy 1

2ml2(dÁ=dt)2 negative; hence the time t may be

regarded as imaginary. Although we started with a fa-miliar classical mechanics problem, this is clearly remi-niscent of quantum tunneling, which raises the questionof what an ab initio WKB-like treatment of the problemwithout referring to elliptic functions would reveal.


GENERAL ARTICLE

Speci¯cally, we wish to calculate the imaginary time T 0

that the mass takes to cover the above described path.To do this, all we have to do is to change the limits inthe integral (2), giving

T 0=

Z 2¼¡Á0

Á0

dÁ

(dÁ=dt)= 2

sl

g

Z ¼

Á0


¢1=2 ;

(11)

where in the last equation we have used the symmetryof the potential around Á = ¼. In the new integrationrange ¼ ¸ Á ¸ Á0, sin » = sin(Á=2)= sin(Á0=2) ¸ 1.Setting y = sin » as before gives

T 0 = 2i

sl

g

Z 1=k

1

dyp

(y2 ¡ 1)(1¡ k2y2)= 2i

sl

gK 0(k) :

(12)

We note that i =p¡1 and the y-direction period K 0(k)

given in (9) naturally appear! The second, purely imag-inary period of the Jacobi elliptic function indeed coin-cides with the WKB tunneling time.

4. Instantons, Small Oscillations, Duality

The duality relationK 0(k) = K(k0) = K(p

1¡ k2) men-tioned earlier allows us to identify and interrelate someinteresting limiting cases in the pendulum's behavior.Our problem is non-linear, and it happens to share someimportant aspects of the also non-linear Yang{Mills ¯eldtheory. In the k2 ¼ 0 small-amplitude limit, both ourpendulum and Yang{Mills ¯elds can be treated pertur-batively, meaning just a few lowest powers in the ¯eldamplitudes (Á in our case) need be considered. An ex-ample in the pendulum problem is the ¯rst anharmonicterm used in [3]. However, in both cases, the imagi-nary time behavior of the small or even zero amplitudelimit contains rich non-perturbative phenomena relatedto the real time perturbative regime via duality. For

In both cases, the

imaginary time

behavior of the

small or even zero

amplitude limit

contains rich non-

perturbative

phenomena

related to the real

time perturbative

regime via duality.


GENERAL ARTICLE

Box 2. Yang{Mills Fields and their Instantons

According to the `Standard Model' of fundamental particle physics, strong, electromagneticand weak interactions are mediated by gauge ¯elds. The familiar electric and magnetic¯elds interact only with charges and currents but not directly with themselves, resulting ina linear theory. The Yang{Mills gauge ¯elds used for describing strong and weak interactions,however, have self-interactions, and are fundamentally non-linear. The pendulum consideredhere is a surprisingly useful toy model for illustrating novel features (such as instantons)of Yang{Mills ¯elds that arise from non-linearity. For our purposes, it will be su±cientto consider the simplest Yang{Mills gauge ¯elds as electric and magnetic ¯elds with anadditional internal symmetry index a that runs from 1 to 3; thus we now have Ea and Ba

instead of the electric ¯eld E and the magnetic ¯eld B. Just as the electromagnetic ¯eldscan be obtained from the potentials A and © via E = ¡ _A¡ r© and B = r£A (c, thespeed of light is taken as unity here), the Yang{Mills ¯elds are obtained from potentials(Aa(r; t);©a(r; t)), albeit in a more complicated and non-linear way. The potentials are theanalogues of the `coordinate' Á(t) and the `electric ¯elds' Ea the analogues of the `velocity'_Á. The ¯eld energy density is proportional to Ea ¢ Ea + Ba ¢ Ba, where, pursuing theanalogy, the ¯rst and second terms can be identi¯ed as `kinetic' and `potential' energies,respectively (the Einstein convention of summing over the repeated index a is understood).When one switches to imaginary time to investigate tunneling, Ea ! iEa, and the energydensity becomes ¡Ea ¢Ea +Ba ¢Ba. This allows non-vanishing ¯eld con¯gurations obeyingEa = §Ba with zero energy. Such ¯elds are said to be self-dual or anti-self-dual, and theequation essentially says (square root of the kinetic energy) = § (square root of the potentialenergy), just as in (15). In particular, there are (anti) self-dual ¯elds that vanish over allspace at t = ¡1, become non-zero for later times and vanish again at t = +1. Although the¯elds are zero at t = §1, the potentials need not be, and the ¯eld that interpolates betweentwo topologically di®erent potentials for vanishing ¯elds in the in¯nite past and the in¯nitefuture is called a Yang{Mills instanton. There are in¯nitely many topologically distinct`vacua' with vanishing ¯elds, just like the pendulum's minima at Á = 2¼n;n = 0;§1;§2; :::.A similar situation obtains for an electron in a periodic lattice potential with in¯nitely manydi®erent minima; because the electron can tunnel from one minimum to another, its groundstate is a superposition of all the minima. This is an example of a Bloch wave.

the Yang{Mills case, we refer the reader to Huang [4].In our problem, we start with the `perturbative limit'k2 = 0.

4.1 The k2 = 0 Case

Since K(k2 = 0) = ¼=2, we are in the small angle regimewhere T = 2¼

pl=g. Equation (9) then gives

T 0 = 2i

sl

g

Z 1

1

dyp

(y2 ¡ 1)= 2i

sl

gK 0(0) : (15)


GENERAL ARTICLE

The real oscillation

period of a pendulum

starting at the top

unstable equilibrium

point =, is also

infinite.

This is an example of

a topological

instanton, a

semiclassical solution

that tunnels in

imaginary time. It

starts from t = – at

the original ground

state = 0, and ends

up at the topologically

distinct one = 2

at t = +.

The integral is elementary, resulting inK 0(0) = 2arccosh(1) = 1. This is of course the degeneration of an el-liptic function to a trigonometric one as the imaginaryperiod goes to in¯nity. The duality relation then showsthat K(1), the real oscillation period of a pendulumstarting at the top unstable equilibrium point Á = ¼,is also in¯nite. The reason is simply that the motioncannot start without an initial perturbation, howeversmall.

Let us next examine the actual function Á(t) in imagi-nary time, setting the energy E (and thus also k) strictlyto zero so that the pendulum starts at Á = 0 in the in-¯nite past. After canceling out overall constants, thezero-energy condition becomes

¡( _Á)2 + (2g=l)(1¡ cos Á) = 0 ; (14)

or, using half angles and taking square roots,

_Á = §2g

lsin

Á

2: (15)

Let us choose the plus sign. The solution will be a func-tion of t¡ t0, with t0 an arbitrary constant which we setto zero, resulting in

Á(t) = 4 arctan(exp!t) : (16)

This is an example of a topological instanton [5], a semi-classical solution that tunnels in imaginary time. Itstarts from t = ¡1 at the original ground state Á = 0,and ends up at the topologically distinct one Á = 2¼at t = +1. The tunneling could also go in the op-posite direction. This corresponds to taking the mi-nus sign in (15) and might be called an anti-instanton.Thus the in¯nite period T 0 = 2i

pl=gK 0(0) found in

(13) can be viewed as the tunneling time of the instan-ton, which is de¯ned as a classical solution interpolat-ing in imaginary time between two ground states. The


GENERAL ARTICLE

The instanton

solution dominates

the imaginary time

Feynman path

integral for the

problem.

The ground state

of unquantized

Yang–Mills theory

is approximated by

a Bloch wave-like

superposition of

precisely such

topologically

inequivalent vacua

connected by

instantons.

ground state of unquantized Yang{Mills theory is ap-proximated by a Bloch wave-like [4] superposition of pre-cisely such topologically inequivalent vacua connectedby instantons, although the topological inequivalence ofYang{Mills vacua is not as intuitively graspable as ourelementary example.

According to the Feynman path integral recipe the tun-neling probability amplitude for the instanton is propor-tional to exp(iS=~), where S is the action for the entirepath. The latter can be calculated exactly in imaginarytime it. The result is a purely imaginary action, turningthe amplitude to exp(¡jSj=~). We start with

jSj = j

Z +1

¡1

L(t)dtj =

Z +1

¡1

f1

2ml2

µdÁ

dt

¶2

+mgl

(1¡ cos Á)gdt : (19)

Using (14), one can eliminate the _Á2 term in favor of thepotential energy, giving

jSj = 2mgl

Z +1

¡1

(1¡ cosÁ) dt : (20)

Replacing dt by dÁ=(dÁ=dt), using half-angle formulasand remembering that Á(¡1) = 0 and Á(+1) = 2¼,the integral becomes

jSj = 2mpgl3Z 2¼

0

sinÁ

2dÁ = 8m

pgl3 : (21)

Thus the instanton's contribution to the action isexp(¡8m

pgl3=~). It can be shown that the value of jSj

in (19) is a minimum (other than the trivial one S =0) and therefore the instanton solution dominates theimaginary time Feynman path integral for the problem.

4.2 The k = 1 Case

For completeness, let us also note that the resultK(0) =¼=2 for the small-angle simple pendulum impliesK(0) =


GENERAL ARTICLE

Box 3. The Action, WKB and Feynman's Path Integral

Let us limit ourselves to one coordinate x and the velocity _x. This is su±cient for discussingthe essential concepts, and our problem happens to be one-dimensional in any case. TheLagrangian is L = m _x2=2 ¡ U(x), where m is the mass and U is the potential energy of

a non-relativistic particle. The action S is given byR t0Ldt0 and ¯xing the initial and ¯nal

positions x(0) and x(t), one calculates an S(t) for each conceivable path connecting thesepoints. The classical path is found by extremizing S, which yields the equations of motion.In Feynman's path integral formulation of quantum mechanics, the quantity of interest is theprobability amplitude, denoted by < x(t); tjx(0); 0 >, which, by Feynman's recipe, equalsA§exp(iS[x(t)]=~). The sum is over all paths and A is a normalization constant. Each paththus weighs in with its phasor, i.e., a complex number of unit modulus. Since S is stationaryaround the classical path, the phasors for nearby paths are nearly parallel. They add upto give a huge contribution, while the randomly oriented phasors from non-classical pathscancel each other out. For macroscopic objects, the angle ®(S) ´ (Scl ¡ S)=~ represents abig deviation in the direction of the phasor from the classical; it will thus lead to destructiveinterference with other non-classical-path phasors. An elementary particle, on the otherhand, can follow non-classical paths for which ®(S) remains within a small angular range.

Of particular interest for our problem are tunneling paths where the momentum p(x) =p(E ¡ U(x))=2m is imaginary because U > E. When an exact quantum mechanical cal-

culation of the tunneling probability for an arbitrary U(x) is impracticable, the Wentzel{Kramers{Brillouin approach, invented independently by all three authors in the same yearas the SchrÄodinger equation, provides an approximate answer. The method is based onadding up the phase changes exp(ip(x)dx=~) as the approximate plane wave moves insteps of length dx. For a tunneling event with imaginary p(x) from x1 to x2, this givesa transmission amplitude proportional to exp(¡

R x2

x1

p(E ¡U(x))=2m)dx. The relation of

this expression to an also semi-classically evaluated Feynman recipe follows from puttingRLdt =

R(pdx=dt ¡ E)dt =

Rpdx ¡ E¿ , where ¿ is the transit time. Note that the semi-

classical approach involves taking E as constant during the passage and the ¯rst term is justthe WKB expression. A detailed calculation (R Shankar, Principles of quantum mechanics,second edition, Kluwer Academic/Plenum publishers, New York, 1994.) shows that the fac-tor < x2(t); tjx1(0); 0 > actually includes another factor of +E¿ in the exponential, leavingonly the WKB term.

K 0(1) = ¼=2 by the duality relation. Physically, thismeans that while the real time oscillations from Á0 = ¡¼to Á0 = +¼ take forever, the (imaginary) tunneling timearound the top point from ¡¼ + ² to ¼ ¡ ² is the sameas the usual period of a small-angle pendulum.

If given a su±ciently high initial velocity, our pendulumcan of course also complete full rotations in vertical cir-cles in a ¯nite amount of time (instead of swinging fromÁ0 = ¡¼ to Á0 = +¼ with the in¯nite period K 0(0)).


GENERAL ARTICLE

The period of the full rotational motion can be madearbitrarily small by increasing the initial energy, andtherefore cannot be identi¯ed with K or K 0 for any k =sin(Á0=2). Another way of achieving full rotations is ofcourse to drive the pendulum with an external motor.Interestingly, if the familiar value ! =

pg=l is chosen,

there is neither compression nor tension in the rod atthe topmost point.

5. Discussion: Tunneling and Imaginary Time inNature

Quantum tunneling as a microscopic phenomenon is ofcourse ubiquitous: The nitrogen atom tunnels back andforth across the equilateral triangle of hydrogen atomsin the ammonia (NH3) molecule about 2.4£1010 timesper second; alpha particles tunnel through the repulsiveCoulomb wall in nuclei and get out; a DC current °owsacross a thin insulating Josephson junction between twosuperconductors, to name a few familiar examples. It isnow even the basis of some high technology devices suchas tunnel diodes and scanning-tunneling microscopes.What we tried to show here is that a very familiar clas-sical mechanical system also reveals connections withquantum mechanics and non-perturbative phenomenain Yang{Mills theories in an exact mathematical treat-ment, but of course the actual probability that a macro-scopic pendulum will exhibit quantum tunneling is ofcourse fantastically small. For example, for m = 1 kgand l = 1 m, exp(¡8m

pgl3=~) is of order exp(¡1035)!

An appreciable probability is only possible if the argu-ment of the exponential is of order unity. In SI units, thisrequires l3=2 » (0:263 £ 10¡34)=m, which leaves a verynarrow window for simultaneously physically meaning-ful values of m and l. A length of 1 m leads to masses ofthe order of a billionth of an average atomic mass, whilea mass of 1 kg produces a length nearly at the Planckscale of 10¡34 m. On the other hand, if l = 1 nm, one¯nds m » 10¡22 kg. These last numbers are intriguingly


GENERAL ARTICLE

Suggested Reading

[1] L D Landau and E M Lifshitz, Mechanics, Butterworth and Heinemann,

Oxford, 2001.

[2] Jon Mathews and R L Walker, Mathematical methods of physics, Ben-

jamin, New York, 1965.

[3] H P Kaumidi and V Natarajan, The simple pendulum: not so simple after

all! Resonance,Vol.14, No.4,pp.357–366, 2009.

[4] K Huang, Quarks, leptons and gauge fields, World Scientific, 1982.

[5] R Rajaraman, Solitons and instantons, North-Holland, 1982.

[6] V Sazanova et al., Nature, Vol.431, pp.284–287, 2004.

[7] S W Hawking, A brief history of time, Bantam, New York 1988.

[8] A Vilenkin, Many worlds in one, Hill and Wang, 2006.

According to

Hawking and

Vilenkin, the universe

itself, the ultimate

uncontrolled

experiment and the

ultimate macroscopic

entity, may have

started in imaginary

time and switched to

our usual time ‘later’

(whatever ‘later’ may

mean in this setting)!

Address for Correspondence

Cihan Saclioglu

Faculty of Engineering and

Natural Sciences,

Sabanci University, 81474

Tuzla, Istanbul, Turkey.

Email:

[email protected]

within a few orders of magnitude of current nano- tech-nology applications [6] involving simple harmonic oscil-lators, but considering the weakness of gravity relativeto the competing forces at such small scales, it is veryunlikely that any evidence for the `imaginary time be-havior' of a simple pendulum will ever be seen in a con-trolled experiment.

According to Hawking [7] and Vilenkin [8], however,the universe itself, the ultimate uncontrolled experimentand the ultimate macroscopic entity, may have startedin imaginary time and switched to our usual time `later'(whatever `later' may mean in this setting)! In theVilenkin version, the imaginary time arises in connec-tion with the universe tunneling out of a state of zeroenergy just like the instanton here; indeed, the process isdescribed by something called the `de Sitter{Hawking{Moss instanton'. In the Hawking version, on the otherhand, imaginary time is principally used to smooth outthe Big Bang singularity. For more details and the de-bate about whether the approaches are equivalent, werefer the readers to Vilenkin's book.

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GENERAL ARTICLE Swinging in Imaginary Time

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