Kroeger H - Fractal Geometry in Quantum Mechanics

8/13/2019 Kroeger H - Fractal Geometry in Quantum Mechanics

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FRACTAL GEOMETRY IN

QUANTUM MECHANICS, FIELD THEORY

AND SPIN SYSTEMS

H. KROG GER

DeHpartement de Physique, UniversiteH Laval, QueHbec, QueH., Canada G1K 7P4

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

H. KroKger/Physics Reports 323 (2000) 81}181 81


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E-mail address:[email protected] (H. KroKger)

Physics Reports 323 (2000) 81}181

Fractal geometry in quantum mechanics, "eld theoryand spin systems

H. KroKger

De&partement de Physique, Universite& Laval, Que&bec, Que&., Canada G1K 7P4

Received May 1999; editor: J. Bagger

Contents

1. Introduction to fractal geometry 84

2. Fractal geometry in quantum mechanics 87

2.1. Brownian motion versus motion in

quantum mechanics 87

2.2. Path integral quantization and fractal

geometry of quantum paths 90

2.3. Numerical simulations for di!erent

potentials 94

2.4. Can we measure experimentally the

geometry of quantum mechanicalpropagation? 100

3. Quantum physics on fractal space}time 108

3.1. SchroKdinger and Dirac equation viewed

from statistical mechanics 108

3.2. The principle of scale invariance 109

3.3. Quantum physics on Cantor sets 109

4. Fractal geometry and quantum "eld

theory 110

4.1. Self-similarity and renormalization group

equation 110

4.2. Self-similarity and scale dependence.

Hadron structure functions in QCD 113

4.3. Geometry of propagation of a relativistic

particle 117

4.4. Non-local order parameters for

con"nement in lattice gauge theory 134

4.5. Fractal geometry and critical behavior of

lattice "eld theories 147

5. Fractal geometry and quantum gravity 152

5.1. Random surfaces and quantization of

gravity 152

5.2. Fractal structure 1545.3. Numerical results from lattice simulations.

Gravity coupled to matter: Ising model and

3-state Potts model 160

6. Fractal geometry and spin systems 161

6.1. Critical behavior of spin systems as

a function of non-integer dimension of

space}time: Ising model 162

6.2. Geometry of critical clusters: Ising model 169

6.3. Fractal geometry of topological excitations:

X> spin model 174

7. Concluding discussion and outlook 175

References 177

0370-1573/00/$- see front matter 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 5 1 - 4


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Abstract

The goal of this article is to review the role of fractal geometry in quantum physics. There are two aspects:

(a) The geometry of underlying space (space}time in relativistic systems) is fractal and one studies the

dynamics of the quantum system. Example: percolation. (b) The underlying space}time is regular, and fractal

geometry which shows up in particular observables is generated by the dynamics of the quantum system.Example: Brownian motion (imaginary time quantum mechanics), zig-zag paths of propagation in quantum

mechanics (Feynman's path integral). Historically, the "rst example of fractal geometry in quantum

mechanics was invoked by Feynman and Hibbs describing the self-similarity (fractal behavior) of paths

occurring in the path integral. We discuss the geometry of such paths. We present analytical as well as

numerical results, yielding Hausdor!dimension d"2. Velocity-dependent interactions (propagation in

a solid, Brueckner's theory of nuclear matter) allow for d(2. Next, we consider quantum "eld theory. We

discuss the relation of self-similarity, the renormalization group equation, scaling laws and critical behavior,

also violation of scale invariance, like logarithmic scaling corrections in hadron structure functions. We

discuss the fractal geometry of paths of the path integral in "eld theory. We present numerical results for the

length of propagation and fractal dimension for the free fermion propagator which is relevant for the

geometry of quark propagation in QCD. Then we look at order parameters for the con"nement phase inQCD. The fractal dimension of closed monopole current loops is such an order parameter. We discuss

properties of a fractal Wilson loop. We look at critical phenomena, in particular at critical exponents and its

relation to non-integer dimension of space}time by use of an underlying fractal geometry with the purpose to

determine lower or upper critical dimensions. As an example we consider the ;(1) model of lattice gauge

theory. As another topic we discuss fractal geometry and Hausdor!dimension of quantum gravity and also

for gravity coupled to matter, like to the Ising model or to the 3-state Potts model. Finally, we study the role

that fractal geometry plays in spin physics, in particular for the purpose to describe critical clusters. 2000

Elsevier Science B.V. All rights reserved.

PACS: 03.65.!w; 03.70.#k; 04.60.!m; 05.45.Df

Keywords: Fractal geometry; Quantum mechanics; Quantum "eld theory; Quantum gravity; Spin systems

H. Kro(ger/Physics Reports 323 (2000) 81}181 83


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nine and replacing each piece of straight line of curve (b). This continues ad in"nitum. The curvesare self-similar, i.e., parts of each curve are identical (apart from the scale change) to the previouscurve. Now consider the length. Let us denote byathe length of the straight line part of the originalgenerator which has length l"4a. After scale reduction the straight line part has lengtha/3, a(1/3),2, a(1/3)and the reduced generator has lengthl"4a/3, l"4a(1/3),2, l"4a(1/3).In the limit nPR this goes to zero. The lengths of the curves are: "4a, "4a(4/3),

"4a(4/3),2, "4a(4/3) in the nth step. In the limit nPR the length goes to in"nity bya power law.

As another example let us assume we want to measure the length of the coastal line of England.One takes a yardstick, representing a straight line of a given length. Let denote the ratio of theyardstick length to a"xed unit length. The one walks around the coastline and measures the lengthof the coast using the particular yardstick (starting a new step where the previous step leaves o!).The number of steps multiplied with the yardstick length gives a value () for the coastal length.Then one repeats the same procedure with a smaller yardstick say. Doing this for many values of yields a function versus. Let us assume there is a power law

()& . (1.1)

This looks very much like the critical behavior of a macroscopic observable at the critical point,e.g., magnetization of a ferromagnet when temperature approaches the critical temperature. In thatcasewould be called a critical exponent. One observes for a wide range of scales that the lengthof the British coast obeys such a power law. The fractal dimension d

is de"ned by

"d!d

, (1.2)

where the topological dimension d"1 for the curve. For the coastline of England one "nds

experimentally d+1.25. For Koch's curve, one can choose as the ratio of the straight line parts

corresponding to nth and 0th step, "(1/3). This satis"es Eq. (1.1), yielding "(log 4!log 3)/log 3 and thusd

"log 4/log 3"1.262 .

Why and where does fractal geometry play a role in quantum physics? In the case of non-relativistic quantum mechanics the most obvious example is the geometry of propagation (i.e., ofa typical path) of a massive particle. Under very general assumptions this is a curve of fractaldimensiond

"2. The underlying physical reason is Heisenberg's uncertainty principle, which by

itself is closely related to the commutation relation of position and momentum. A heuristicargument, leading tod

"2 goes like this: writing the uncertainty relation

x p& , (1.3)

and putting

p"m(x/t) , (1.4)

we obtain

(x)&t . (1.5)



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When we consider the propagation between two given points in space and time, where the timeinterval is broken into N subintervals of lengtht, we "nd for the length of the path

"N x"

tx&

x&x . (1.6)

LettingtP0, and comparing with Eqs. (1.1) and (1.2), implies d"2. This can be made more

rigorous. The result is also supported by numerical simulations.Another example for the usefulness of fractal geometry in quantum mechanics is percolation, in

particular, the SchroKdinger equation has been studied on a Sierpinski gasket.If fractal geometry plays a role in non-relativistic quantum mechanics, one expects it to play

a role in quantum "eld theory, too. For example, one can ask the question: What is the geometry ofpropagation for a relativistic quantum particle? Because particle number #uctuates in relativisticphysics, the answer is not as simple as in the non-relativistic case. In fact, there is no unambiguousanswer. Quantum"eld theories describe many-body systems in nature. Those often display phasetransitions. At the critical point, we know that scaling laws describe the singular behavior ofmacroscopic observables. This behavior is related to the mathematical property of self-similarity,which is inherent in the renormalization group equation. On the other hand, self-similarity isa characteristic feature of a fractal. Thus it is not surprising that fractal geometry plays a role forcritical phenomena. E.g., the geometry of critical clusters in certain spin systems (Ising clusters) canbe described by fractal geometry. Fractal geometry of random walks and random surfaces has beenstudied in two-dimensional models of statistical mechanics at the critical point.

In the context of lattice gauge theory, where the Wilson loop plays the role of an order parameterdistinguishing a con"ned from a decon"ned phase, it has been suggested that fractal geometryplays a role at the decon"nement phase transition. New order parameters of gauge theories inconnection with fractal geometry have been suggested by Polikarpov and co-workers. The fractal

dimension of closed monopole current loops is such an order parameter. For example, themonopole current loop becomes a fractal object at the transition point. The #ux lines of the Abelianmonopole current in the con"ning phase are found to have a fractal dimension D

'1 for ;(1) as

well as for Abelian projected"nite temperatureS;(2), but haveD"1 in the non-con"ning phase.

Those "ndings by Polikarpov and co-workers strongly suggest that con"nement of quarks andgluons has something to do with fractal geometry. Another area where fractal geometry has turnedout to be a useful tool, is the determination of upper or lower critical dimensions. Using latticeswhich are fractal, it has been possible to simulate physical systems in non-integer space dimensionsand to study non-perturbatively critical behavior as a function of such non-integer dimension. Thishas been done for the ;(1) model of lattice gauge theory and for the Ising spin model. For such

purpose fractal geometry is a nice and useful tool, complementary to the perturbative technique of expansion.

Finally, in quantum gravity, the dynamics of the system is determined by curved space}time. It istempting to ask: Does curved space}time exhibit any particular features in terms of fractalgeometry? The answer is yes! We discuss, e.g., the relation between the Hausdor!dimension d

and

the critical exponent, related byd"1/"4 in pure gravity. This establishes that the Hausdor!

dimension can play the role of a critical exponent. In particular, workers have studied theHausdor!dimension of topology of 2-Dquantum gravity as well as gravity coupled to matter, liketo the Ising model or to the 3-state Potts model. The "nding that the Hausdor!dimension in

86 H. Kro(ger/Physics Reports 323 (2000) 81}181


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Fig. 2. Random motion in a 2-D square lattice, from a computer simulation. Figure taken from Ref. [149].

quantum gravity plays the role of a critical exponent (critical exponents determine the universalityclass of the theory) suggests that the Hausdor!dimension might play such a role also in other "eldtheories. The problem is to "nd suitable observables.

Many of the questions asked and techniques used in the context of fractal geometry andquantum"eld theory have been"rstly investigated for spin systems. In particular, the determina-tion of lower or upper critical dimensions, critical behavior as a function of non-integer dimension

of the embedding lattice and the geometry of critical clusters have been investigated "rst for spinsystems. We review those topics, paying attention to the Ising model and the X> model.

In this review, we will present concepts and applications of fractal geometry in quantum systems.We will try to do this in a pedagogical manner. We hope to give an account of the developments ofrecent years.

2. Fractal geometry in quantum mechanics

2.1. Brownian motionversus motion in quantum mechanics

2.1.1. Brownian motion

The subject of this review is fractal geometry in quantum physics. In order to enter this subject letus consider a system described by classical statistical mechanics: Brownian motion. It describes themotion of a molecule in a liquid, undergoing collisions with other molecules. Each collision causesthe molecule to change its momentum. As a result one observes a sequence of erratic zig-zagmovements (see Figs. 2 and 3). It turns out that physics is well described by making the followingassumptions: The change of momentum of the molecule at a collision depends only upon the lastcollision. The change obeys a probabilistic law: It is a Gaussian stochastic process. Following

Itzykson and Drou!e [76] this can be nicely simulated numerically on a regular lattice inD spacedimensions (discretization x) and time (discretization t). After each time interval t, the moleculehops from one lattice site to a neighbor site chosen at random (each neighbor assigned equalprobability). Then one can ask for the probability P(x

, t

; x

, t

)50 that the molecule arrives at



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Fig. 3. A trail of a Brownian path in 2-D plane, from a computer simulation. Figure taken from Ref. [149].

site x

at time t

after having started from an initial position x

at time t

. This probability isdetermined by three conditions:

(i) Ift

equals t

it does not move

P(x

, t

; x

, t

)"

. (2.1)

(ii) The probability is normalized

P(x

, t

; x

, t

)"1 . (2.2)

(iii) At successive times, it can arrive only from neighbor sites

P(x, t#1; x

, t

)"1

2d

P(x, t; x

, t

) . (2.3)

Using the discretized (lattice) Laplace operator

f(x)"1

2d

[f(x#e)#f(x!e)!2f(x)] , (2.4)

the latter equation can be written as

P(x, t#1; x

, t

)!P(x, t; x

, t

)"

P(x, t; x

, t

) . (2.5)

In order to make the transition to the continuum limit, we generalize by allowing time steps a

andlattice steps of length a, by doing the substitution tPt/a

and xPx/a. Then p"P/a de"nes

a probability density. The continuum limit aP0 and aP0 exists and is well de"ned under the



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condition that the time scale and the spatial scale are related by aJa. Choosing in particular

a"(1/2d)a , (2.6)

the previous Eqs. (2.1), (2.2) and (2.5) yield for the probability density in the continuum limit

lim

p(x

, t

; x

, t

)"(x

!x

) , (2.7)

dx p(x, t; x, t)"1 , (2.8)

R

Rt!p(x, t; x, t)"0 . (2.9)

Eq. (2.9) is the di!usion equation (with unit di!usion constant). The solution is

p(x, t; x, t)"

1

[4(t!t)] exp!

(x!x

)

4(t!t) , (2.10)

which is essentially a Gaussian. From the solution Eq. (2.10) one obtains for t't

't

the

Kolmogorov equation

dx p(x, t; x, t) p(x, t; x, t)"p(x, t; x, t) . (2.11)This in turn allows to express the probability density p in terms of a multiple integral

p(x, t; x, t)"

dx

[4(t!t)]

1

[4(t!t)] exp!

1

4

(x!x

)

t!t , (2.12)

where x"x

and x

"x

. This gives a precise meaning to the path integral expression

p(x

, t

; x, t)"[dx(t)]exp!

1

4

dt x , (2.13)

where x(t)"x

and x(t

)"x

.

From Eq. (2.13) one can obtain a simple plausibility argument which shows that the averagepath of the Brownian motion is a fractal curve with fractal dimension d

"2. The main

contribution to the path integral comes from con"gurations where

dt x&O(1) or

(x)&t , (2.14)

where xmeans the increment of length of an average curve for a given time increment t. Supposethat a total time interval "n tis given. Then the length of an average curve, using Eq. (2.14), is

"n x"

tx"(x) . (2.15)

LettingtP0 hencexP0 and comparing with Eq. (1.2) yields d"2.



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2.1.2. Relation between Brownian motion and quantum mechanics

As has been shown by Nelson [112] and other authors (see, e.g., Ref. [149]), there is a closerelationship between Brownian motion and quantum mechanics. Brownian motion goes over intofree motion of a massive quantum mechanical particle, when replacing timet C itand the di!usioncoe$cientd

C /2m. Then one has the following correspondence between the di!usion equation

Eq. (2.9) and the SchroKdinger equation

R

Rtp"pC!

i

R

Rt"!

2m"H . (2.16)

There is as well a correspondence between the path integral of the probability density of Brownianmotion, Eq. (2.13), and the path integral for the transition amplitude of quantum mechanics,

p(x

, t

; x, t)"[dx(t)]exp!

1

4

dt x Cx, t x, t"[dx(t)]expi

dtm

2x .

(2.17)

2.2. Path integral quantization and fractal geometry of quantum paths

2.2.1. Analytical results for free motion and the harmonic oscillator

While particles in classical mechanics follow smooth (di!erentiable) trajectories, the situation isdi!erent in quantum mechanics. As has been noted by Feynman and Hibbs [56], paths of a massiveparticle in quantum mechanics are non-di!erentiable, self-similar curves, i.e., zig-zag curves (seeFig. 4). Feynman and Hibbs have noticed in 1965 the property of (stochastic) self-similarity, whichplays an eminent role in many areas of modern physics. A decade later, Mandelbrot [104] has

introduced the concept of fractal geometry in nature. So what do we know about the fractalgeometry of quantum mechanical paths?

Due to the close relation between Brownian motion and quantum mechanics, one is tempted toguess: A typical path of a free massive particle in quantum mechanics is a fractal curve withd

"2. The plausibility argument employed for Brownian motion can be carried over directly to

the path integral of quantum mechanics, thus supporting the guess. Actually, Abbot and Wise [1]have shown that an average quantum mechanical path of free motion has fractal geometry withHausdor!dimension d

"2. This result is based on the de"nition of length of monitored paths.

Monitoring a path means to measure the position of a wave packet with some uncertainty inlocalization x at some discrete times t

, t

, t

,2, t

. Experimentally this can be done in the

following way: An electron is emitted from a source and passes through a sequence of screens eachof them carrying several holes (see Fig. 5). In order to determine by which hole the electron haspassed, one uses a source of light emitting photons placed behind each screen. The photon collideseventually with the electron. From the observation of the de#ected photon one can decide by whichhole the electron has passed. Suppose each hole has a sizex. In order to determine a fractaldimension from Eqs. (1.1) and (1.2) one needs to go to the limitx&P0. In order to localize theelectron with uncertainty x, the wave length of the photon "/p must obey (x. Thuseventually must go to zero. But that means, according to Heisenberg's uncertainty relationx p5that the momentum uncertainty pof the electron in the plane of the screen would go to



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Fig. 4. Typical paths of a quantum mechanical particle are highly irregular on a"ne scale. Although a mean velocity can

be de"ned, no mean-square velocity exists at any point. Paths are nondi!erentiable. Figure taken from Ref. [56].

in"nity! This can be interpreted such that the electron is not free but by the measurementundergoes interaction which in the limit xP0 becomes totally erratic.

This experimental idea of monitoring the paths underlies Abbot and Wise 's de"nition of lengthof quantum paths. They construct a wave packet being localized at t

"0 at x

"0 with

uncertainty in positionx. The wave packet is denoted by(x, t;x"0, t"0, x). They computethe evolution of the wave function for a time interval t and measure the lengthl by

l"dxx(x, t;x"0, t"0, x) . (2.18)AfterNmeasurements of position at time intervalst("Nt), the length of the average path is

l"Nl . (2.19)



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Fig. 5. Experiment to determine by which hole a particle (electron) has travelled between source and detector. A light

source is placed behind each screen. From the observation of light scattered by the electron one can determine the hole

which the electron has passed. Figure taken from Ref. [56].

Their calculation gives

lJ t/m x (2.20)

and

lJ/m x . (2.21)

According to Eqs. (1.1) and (1.2) this is equivalent to d"2. In summary, Abbot and Wise's result

d"2 for monitored paths can be interpreted such that the erratic paths are generated by

interaction via the position measurement.

Thus it is natural to ask the question: What is the geometry and in particular the Hausdor!dimension of an unmonitored quantum mechanical path? There is indication that d"2 holds

also. Feynman and Hibbs [56] have shown in 1965 that unmonitored quantum paths arenon-di!erentiable, stochastically self-similar curves. They have done a calculation which almostproves d

"2. Their calculation includes the presence of any local potential. Moreover, their

calculation shows the close connection with Heisenberg's uncertainty principle. So let us recall herethe basic steps of Feynman and Hibbs' calculation. They consider the Hamiltonian

H"!

2m #


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where < denotes a local potential. The quantum mechanical transition element from a statex

,t"0 to a statex

, t" is given by

x

, t"x

,t"0"xexp!i

Hx

"

[dx(t)]exp

i

S[x(t)]

. (2.23)

The expression on the r.h.s. is the path integral, i.e., the sum over paths x(t) which start at x

at

t"0 and arrive atx

at t". The paths are `weightedaby a phase factor exp[iS[x(t)]/], where

S is the classical action corresponding to the above Hamiltonian for a given path,

S"

dtm

2x !


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Then in the limitt,P0, Eqs. (2.29) and (2.30) imply

(x!x

)"!

im

1 ,

(x)J

m

t . (2.31)

This is Feynman and Hibbs'important result on the scaling relation between a time incrementtand the corresponding average length increment of a typical quantum path. Further one can prove

(x)"2

x . (2.32)

Feynman and Hibbs scaling relation Eq. (2.31) and Eq. (2.32) imply

xJt . (2.33)

Then de"ning the length in analogy to Eq. (2.19), one has

"Nx"

txJ

x , (2.34)

and thus d"2. In summary, Feynman and Hibbs calculation yields d

"2 for unmonitored

quantum paths, in the presence of an arbitrary local potential. Moreover, it shows the close relationwith Heisenberg's uncertainty relation.

2.3. Numerical simulations for diwerent potentials

2.3.1. Free motion

There is further indication that d"2 is the correct result for unmonitored paths. This comes

from numerical simulations. First let us specify the quantum mechanical length de"nition. Weconsider a particle to travel fromx

att

tox

att

. We divide the time interval into Nslices such

that "Nt. The length of the classical trajectory is given by

"

x!x

, (2.35)

wherex"x(t

),x

"x(t

)"x

andx

"x(t

)"x

. According to Feynman and Hibbs [56] the

corresponding quantity in quantum mechanics is given by the transition element Eq. (2.25), wherethe action is given by Eq. (2.26) and the observable F[x(] is given by the classical length, Eq. (2.35),thus

"

x!x

"dx2dx

x!x

exp

i

S[x]

. (2.36)



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One should note that this is not an expectation value in the usual sense. This expression is anaverage of the observable length involving N time slices where average means summing over allpaths with weight factors given by the exponential action. Feynman and Hibbs refer to transitionelements as `weighted averagesa. The weighting function in quantum mechanics is a complexfunction. Thus the transition element is in general complex.

KroKger et al. [91] have computed this expression numerically, however, for imaginary time. The

latter has been done for the technical reason to be able to compute the path integral by MonteCarlo simulation. Physically it means to do the Euclidean path integral. For a given initial positionx

and"nal positionx

and a "xed time interval t

!t

""Nt the length

has been

computed from Eq. (2.36) and also the average increment of lengthx has been determinedfrom the path integral. In the continuum limit (tP0,xP0) one expects critical behavior,

&(t)&(x) . (2.37)

For consistency it is useful to compute also the weighted average for the observable of lengthsquared

S

"

(x!x

)

. (2.38)

This can be computed analytically

S

"D t

2m2m

D

(x!x

)

!2#

D

m . (2.39)

From the analytic expression one obtains the critical behavior

S

&(t)&(x) , (2.40)

with exponents B"0, C"0.Numerical results for motion inD"3 dimensions are shown in Fig. 6. For the observableS

one"nds consistency with the exact result Eq. (2.39). The extracted critical exponentsB"0.00037$0.00025 andC"!0.00073$0.00051 are both compatible with zero. For the length

one

observes a power law behavior with exponentsA"0.49971$0.00052 andd"1.9988$0.0021.

This is consistent withd"2 andA"1/2 which implies the scaling law Eq. (2.33). In summary, the

numerical results obtained for quantum mechanics in imaginary time are consistent with Hausdor!dimensiond

"2 for unmonitored paths of free motion and con"rm also the scaling law.

2.3.2. Local potentials: Coulomb potential and harmonic oscillator

For the case of unmonitored paths, the analytic calculation of Ref. [1] has been generalized byCampesino-Romeo et al. [26] to include a harmonic oscillator potential, giving also d

"2. For

the case of monitored paths, there is Feynman and Hibbs [56] calculation which holds forarbitrary local potentials. Thus, based on the scaling relation Eq. (2.32), Feynman and Hibbs resultimpliesd

"2 for any local potential. KroKger et al. [91] have done Monte Carlo simulations for

the Euclidean path integral in the presence of local potentials like the harmonic oscillator and theCoulomb potential. The numerical results are compatible with d

"2 (see Fig. 7). Thus there is

indication ford"2 for both monitored and unmonitored paths in the presence of arbitrary local

potentials.



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Fig. 6. Average length and length squared for quantum paths versus number of time steps. (a) Lengthand (b) lengthsquaredSfor free motion, (c) length in presence of harmonic oscillator potential.

One might ask: What is the relationship between monitored and unmonitored paths and whyshould d

coincide for both? First of all, the employed de"nition of length is di!erent. For

monitored paths, the authors of Refs. [1,26] have de"ned length l as the usual quantummechanical expectation value of the absolute value of an increment of position. This corresponds toa state in which the wave function has been evolved for an increment of timetfrom an originalwave function being characterized by localization uncertainty x (Eq. (2.19)). For unmonitoredpaths one is interested in the length of the total path. This involves a number of intermediate times.This leads to Feynman and Hibbs'transition element, which we has been employed in the length

de"nition given in Eq. (2.36). Thus the length de"nition and average are di!erent. However, there isa common link, which may be considered as physical origin of fractal paths: Heisenberg'suncertainty relation and behind this the fundamental commutator relation between position andmomentum. For the case of the monitored paths, the process of localization leads via theuncertainty principle to erratic paths. But also for the case of unmonitored paths, Feynman andHibbs'calculation shows that the scaling relation betweenxandt, Eq. (2.31), is directly relatedto the commutator relation between position and momentum, which again is directly related to theuncertainty principle. But why then should the outcome ofd

agree? One can speculate on this

coincidence. For unmonitored paths the scaling relation Eq. (2.31) is valid for a very large class of



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Fig. 7. Average length and length squared for quantum paths of velocity dependent action ("1/2) versus number oftime steps. (a) Length and (b) length squared S.

interactions, namely all local potentials. Also the numerical simulations in Ref. [91] have given,within statistical errors, d

"2 for all investigated local potentials. In other words there is

indication thatd"2 holds for unmonitored paths in the presence of arbitrary local potentials. On

the other hand, monitoring a path means interaction by measurement. If one assumes that suchinteraction is represented by a local potential it is plausible that d

coincides for monitored and

unmonitored paths.

2.3.3. Velocity-dependent potentials in condensed matter and nuclear physics

We have seen above that a local potential which in#uences the dynamics of the quantum system,

like the spectrum or scattering phases, does not in#uence the Hausdor!dimension. If we identifythe Hausdor!dimension with a critical exponent, and use the language of critical phenomena onecan state this as follows: All local potentials fall into the same universality class as the free system.This leads to the question: Are there interactions which do not fall into this class? Let us considervelocity-dependent interactions. For instance, the action for a massive charged particle interactingwith an electro-magnetic "eld

S"dtm

2x !q#

q

cA ) * (2.41)



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contains a velocity-dependent term. An example from nuclear physics is Brueckner's [24] theory ofnuclear matter using a momentum-dependent potential of the form


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Fig. 8. Critical exponents d

and C versus potential parameter for velocity dependent action. The solid linecorresponds to the estimated exponents.

For"2, the velocity-dependent term is equivalent to the kinetic term. In this case the estimatedHausdor!dimension of the velocity-dependent term agrees with the Hausdor!dimensiond

"2

of the kinetic term.Numerical simulations (in imaginary time) with the velocity dependent action, Eq. (2.42) have

given the following results [91], presented in Figs. 7 and 8. Fig. 7 showsandSas function ofthe number of time steps ("N

t). The full lines are"ts using the estimated power-law behavior,

Eq. (2.47). For a velocity dependent term corresponding to "1/2, one observes the criticalexponents A"0.5012$0.0023 and d

"2.005$0.009 as well as B"0.0031$0.0079 and

C"!0.0062$0.0159. This is compatible with A"1/2, d"2 and B"C"0. This is not in

agreement with Eq. (2.48) for"1/2, but it is for"2. The explanation is that the action has twovelocity-dependent terms, the kinetic term ("2) and the model.

TheX>model is de"ned on a 2-Dlattice as follows. Let us consider a regular lattice of spacing a.The lattice sites are denoted byx

. Lets

denote a classical spin, described a vector of length unity in

the 2-D plane being de"ned on each lattice site. Then the Hamiltonian is given by

H"!J

2

s)s

"!J

2

cos(!

) , (6.45)



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where denotes the angle between spin s

and an arbitrary, but globally "xed direction. It is

evident that the Hamiltonian is invariant under global rotations. The model in 3-Dis given by thesame Hamilton function, except for the lattice sites being now de"ned on a cubic lattice. Vorticesare important in 2-D, where the Kosterlitz}Thouless phase transition is explained as a dissociationof vortex}antivortex pairs.

InD"3 dimensions, the vortex excitations are also called vortex loops. Contrary to the situation

in D"2, in D"3 there is a conservation law giving closed vortex loops. The current is de"nedto pass through a plaquette formed by four neighboring lattice sites, say i,j, k, l. Let

,2,

denotethe corrsponding spin angles, which take values in the interval (!, ]. The the current isde"ned by

J"1

2 [

!

] , (6.46)

where the sum runs over the four oriented links ij,jk,kl andli and the values in thebracket [ ] lie also in the interval (!, ]. ThusJ is integer valued. IfJO0, the#ux is associated

with a link of the dual lattice perpendicular to the plaquette. Such excited links form closed loops.The situation with the 3-DplanarX>model is similar to the Abelian ;(1) lattice gauge model in

4-D, where closed monopole current loops exist. Those loops have been shown to display fractalgeometry, depending on the phase of the model. The fractal dimension of the monopole currentloop has been shown to play the role of an order parameter. A similar situation prevails here. Themodel has a phase transition at

"0.4542, where "1/(k

). Pochinsky et al. [134] have studied

the geometry of such vortex loops from simulations on a lattice (7}14) and"nd the behavior for(

di!erent from that for'

. In Fig. 41, the fractal dimension is plotted versus the inverse

temperature. The fractal dimension of a vortex loop is de"ned here as the ratio of the number oflinks to the number of sites belonging to the vortex. One observes a behavior compatible withD

"1 for'

andD

'1 for(

.

7. Concluding discussion and outlook

We have given an account of a number of areas where fractal geometry plays a role in quantumphysics. In non-relativistic quantum mechanics, for the process of percolation, fractal geometry issuitable to describe the geometry of porous media. In other words, one considers quantummechanics with fractal boundary conditions. On the other hand, the dynamics of a free system

without boundary conditions can be characterized by fractal paths. Those paths for most interac-tions (local potentials) correspond to trajectories of Hausdor!dimensiond

"2. It is interesting to

consider situations, wheredO2. An example is a velocity dependent potential, like


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Fig. 41. Fractal dimension of vortex loops in 3-DX> model versus. Figure taken from Ref. [134].

space}time, is it possible then to establish a correspondence between the distribution of massivebodies and the Hausdor!dimension of the probing quantum particle? In other words, can one "nd

a map of the Hausdor!dimension?This leads us to the next question: Can one measure experimentally the Hausdor!dimension of

propagation of a quantum particle? In principle, one expects the answer to be yes. In practice, itseems to be quite complicated to devise such an experiment. One severe inherent problem is the factthat the determination of d

requires to use a resolution of length which goes to zero. Then

Heisenberg's uncertainty relation invokes a large uncertainty in momentum, which through themeasuring process interferes with the particle and `createsa an erratic path. We have discusseda proposal, how to avoid this problem, via the topological Aharonov}Bohm e!ect. With presentday progress in quantum optics and electron interferometry, one might think of doing somepreliminary experiment.

In many-body systems, like spin systems or quantum "eld theory, fractal geometry has proven tobe a useful tool, in particular for the analysis of critical phenomena. Using Sierpinski carpets, it hasbeen possible to simulate systems in non-integer space dimension and study critical behavior asa function of space dimension. Doing numerical simulations (Monte Carlo) on those latticesprovides non-perturbative information in non-integer dimensions. Fractal geometry has been usedto analyze the geometry of critical clusters at second order phase transitions. An interestingquestion is that of a relation between critical exponents and the Hausdor!dimension. However,there is no unique way to de"ne such a Hausdor! dimension. In quantum gravity, e.g., theHausdor!dimension has been de"ned via comparison of scaling exponents of two observables,



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which have naive dimension of length and volume, respectively. The Hausdor!dimension de"nedin such a way is closely related to the anomalous dimension of a vertex function. Another de"nitionof the Hausdor!dimension, based on the hopping expansion of a propagator has been suggested in"eld theory.

Another interesting subject is the role of fractal geometry at the decon"nement phase transitionat "nite temperature in lattice gauge theory. An observable has been found (the monople current

loop) which is non-fractal in the decon"ned phase, but becomes fractal in the con"ned phase. Thusit plays the role of an order parameter.

We have emphasized the correspondence between a fractal, being a geometrical object havingthe property of self-similarity, and the renormalization group in physics, which also has built in theproperty of self similarity (generating the running coupling constant).

Finally, let us recall that fractal dimension in classical physics tells us something about anobservable (e.g. the length of a coast line) when the resolution of length goes to zero. A similar

situation prevails when we consider hadron structure functions as function ofQ, where 1/Qisthe wave length of resolution. Also the formal instrument of operator product expansion makes

statements about the singularities of a theory at small distances. Thus it is interesting to speculateabout the role that fractal geometry might possibly play in the context of operator productexpansion and hadron structure functions.

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