In this work we explore how the heat kernel, which gives the solution to the diffusion equation and the Brownian motion, would change when we introduce quasiperiodicity in the scenario. We also study the random walk in the Fibonacci sequence. We discuss how these ideas would change the discrete approaches to quantum gravity and the construction of quantum geometry.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

### Tag Cloud

theoretical physics

Leonardo Ort́ız∗, Marcelo Amaral and Klee Irwin

Quantum Gravity Research, Los Angeles, CA, U.S.A

March 17, 2020

Abstract

In this work we explore how the heat kernel, which gives the solu-

tion to the diffusion equation and the Brownian motion, would change

when we introduce quasiperiodicity in the scenario. We also study the

random walk in the Fibonacci sequence. We discuss how these ideas

would change the discrete approaches to quantum gravity and the

construction of quantum geometry.

Keywords: Heat kernel, Fibonacci sequence, quantum gravity, aperiodic.

∗leonardoortizh@gmail.com

1

1 Introduction

In this paper we discuss the notion of quantum geometry [1] using tools from

quasicrystals and quasiperiodic functions [2] as an alternative to canonical

quantization presented in [3], mainly the modifications of quantities of inter-

est, such as the heat kernel and entropy, due to the introduction of quasiperi-

odicity in the setting. Although these modifications seem to be mild, they

have the potential to be important in derivations of thermodynamical quan-

tities which we plan to develop in the future.

In [1] a kind of quantum geometry is constructed in one and two di-

mensions. This is done using random walks and random two-dimensional

surfaces. This is done with the idea of obtaining a quantum description of

gravity at least in two dimensions. This is limited in several aspects with the

main limitation that a realistic theory of gravity should be four-dimensional.

The main idea of the present work is to make something similar to [1] but

not only with random walks and random two-dimensional surfaces but also

with quasiperiodic trajectories as described in [2]. In this manuscript we will

consider one and two dimensional quasiperiodic trajectories but one of our

goals is to make it with quasiperiodic trajectories inherited from quasicrys-

tals in several dimensions. Also once we have this under control we will try

to not only have nonperiodicity but also stochasticity. In the near future

we will try to do what is done with the Polyakov action in [4] but this time

with the Einstein-Hilbert action. Also instead of letting the lattice size go

to zero we will probe the geometry with a massive particle so we can feel the

granular structure of spacetime.

In this context the quasiperiodicity will be given by the trajectory of the

quasiparticle.

It is important to note that in this step we can introduce

nonperidicity and stochasticity.

In the scenario we are describing here we would like to obtain the analo-

gous to the Einstein field equations as something emergent in the same spirit

as thermodynamics is obtained from the microscopic statistical mechanics.

This manuscript is organized as follows: In section 2 we describe briefly

the idea of quantum geometry. In section 3 we introduce the necessary math-

ematics from quasicrystals for our purposes. In section 4 we describe our

approach to the build up of quantum geometry with a discussion of the heat

kernel. In section 5 we study the random walk in a Fibonacci chain in one

and its generalization to two dimensions. Finally in section 6 we discuss fur-

ther ideas and our final comments. In the appendix A we discuss the relation

2

of the Hamiltonian in usual quantum theory and in the Euclidean setting,

since this ideas are related to the main work.

2 A first look at quantum geometry

The idea of quantum geometry can be very intuitive. The concept of cur-

vature of a manifold is studied in semiriemannian geometry. In this context

the manifold -the spacetime- is smooth, however if nature is quantum at the

most fundamental level then the smooth spacetime should be quantized-the

notion of quantum geometry.

Let us describe the idea of spacetime a little more deeply. In general rela-

tivity (GR) the spacetime is made of events, these events can be in principle

anything: the explosion of a bomb, the hand shake of two friends, the click

one do on the mouse, etc. However if we think careful on this definition of

spacetime one realizes that something estrange happens if we want to de-

scribe events with quantum systems as for example with a transition of one

level of energy to another in the hydrogen atom1. Clearly this happens be-

cause the idea of spacetime described in standard GR books is classical. But

then we face a conceptual problem similar to the one of the measurement

problem in quantum mechanics, as the question where lies the boundary

between the machine which measures and the system under study. In the

proposal of the Quantum Gravity Research group -called emergence theory-

this problem does not arises because the spacetime would be constructed

from the “trajectory” of the phason quasiparticles. The challenge then is to

obtain in certain limit something analogous to the Einstein field equations.

Just to put things on perspective we are aiming to construct something

like

G(γ) =

∫

γ

Dσe−S(σ),

(1)

where σ is an hypersurface, γ is its boundary and S is the action of the

system. Most of the actions constructed so far are geometric, but since we

are constructing a theory more general than the ones we have at the moment

we will not attach from the beginning to geometric actions. Clearly the

two challenges in this aim are the construction of the measure Dσ and the

action S(σ). We are working with a kind of Euclidean action, which is not a

1Similar ideas are consider in [5]

3

limitation because we want to have spacetime emergent in our model . Also

it is worthwhile to mention that from G(γ) we expect to obtain a kind of

generalized partition function.

3 Some mathematical tools from quasicrys-

tals

A quasicrystal is an object that has order but not periodicity. The math-

ematics to study these object is very rich and very well developed, see for

instance [2], [6] and [7] just to mention a few references.

In the study of quasicrystals, quasiperiodic functions are relevant. The

idea of this work is to construct a spacetime foam-like model. First in one

dimension with quasiperiodic functions, as the ones shown in [2]. Later we

will introduce stochasticity too. So our quantum geometry will be the result

of nonperiodicity and stochasticity.

4 First steps in the construction of Quantum

Geometry

The main idea of this section is to replace the stochastic process such a

random walk used for example in [8] by a quasiperiodic random process

described by a quasiparticle. However as a first step in this direction we will

study the quasiperiodic process described by a quasiperiodic function as the

one given in [2].

4.1 The random walk representation of the heat kernel

and quasiperiodicity

In order to have an idea on how to implement quasiperiodicity in the models

of quantum geometry let us study some aspects of the random walk repre-

sentation of the heat kernel associated with the diffusion equation when we

introduce quasiperiodicity in the model. This discussion rather than new is

pedagogical, for more details see [1].

Let ∆ denote the Laplace operator in Rd. The solution to the difussion

4

(or heat) equation in Rd

∂ϕ

∂t

=

1

2

∆ϕ,

(2)

with the initial condition ϕ(x, 0) = ϕ0(x) is given by

ϕ(y, t) =

1

(2πt)d/2

∫

Rd

dxe−

|x−y|2

2t ϕ0(x).

(3)

The function ϕ0(x) is interpreted as the initial distribution of particles at

time t = 0, and |x− y| denote the Euclidean distance between x and y in

Rd.

The kernel Kt(x, y) of the operator e

t

2

∆, is called the heat kernel and is

given by

Kt(x, y) =

1

(2πt)d/2

e−

|x−y|2

2t

,

(4)

and represents the probability density of finding the particle at y at time t

given its location at x at time 0. From the simigroup property

e(t+s)∆ = et∆es∆,

(5)

for s, t≥ 0 we have

Kt(x, y) =

∫

dx1...dxN−1Kt/N (xN , xN−1)...Kt/N (x1, x0)

(6)

for each N ≥ 1, where we have set x0 = x and xN = y.

There is an obvious one-to-one correspondence between configurations

(x1, ..., xN−1) and parametrized piecewise linear paths ω : [0, t] → Rd from x

to y consisting of line segments [x0, x1], [x1, x2],...,[xN−1, xN ], such that the

segment [xi−1, xi] is parametrized linearly by s∈[ i−1N t, iN t]. We denote the

collection of all such paths by ΩN,t(x, y). Hence we may consider

DNt ω = (2π

t

N

)−

d

2

Ndx1...dxN−1

(7)

as a measure on the finite dimensional space ΩN,t(x, y).

Noting that

N∑

i=1

|xi − xi−1|2

t/N

=

N∑

i=1

t

N

(

|xi − xi−1|

t/N

)2 =

∫ t

0

|ω̇(s)|2 ds,

(8)

5

whereω̇ is the piecewise constant velocity of the trajectory ω, hence we can

write

Kt(x, y) =

∫

(x,y)

DNt ω exp(−

1

2

∫ t

0

|ω̇(s)|2 ds),

(9)

where the suffix (x, y) indicates that paths are restricted to go from x to y.

We refer to this equation as a random walk representation of Kt(x, y) on

ΩN,t(x, y).

More generally, given an action functional S on a piecewise linear parametrized

paths, we call the equation

HNt (x, y) =

∫

(x,y)

DNt ωe

−S(ω)

(10)

a random walk representation of the kernel HNt (x, y) on ΩN,t(x, y).

In is clear from the expressions for the heat kernel that the introduction

of quasiperiodicity in the partition of the intervals will bring new features

that is worth to be investigated.

4.1.1 Quasiperiodic Brownian movement

As a warm up let us write down the transition probability when a parti-

cle follows a quasiperiodic Brownian motion. The quasiperiodicity can be

introduced with a concrete function such as [2]

x(τ) = cos(2πτ) + cos(2πατ),

(11)

where α is a irrational number. Now if we interpret this function as given

the position of a particle after a time τ then according to the well know

evolution of this movement we have that the probability of being at (τ, x(τ))

if at τ = 0 it was at x = 0 is given by [8]

W (x(τ), τ ; 0, 0) =

1

√

4πDτ

exp{−x

2(τ)

4Dτ

}.

(12)

5 Random walk on a Fibonacci chain

In this section we will review the general random walk procedure and then

apply it to the Fibonacci chain as preparation for studying random walks on

6

more involved geometries. We will restrict ourselves to the random walk in

one dimension.

Let us suppose we have a random walker which can move on a line. Let

us denote its position as Xn which can be any integer. Now suppose this

walker can move to the left or to the right with equal probability2 1/2 and

the length of the step being l. We would like to know the probability that

the walker is nR steps to the right and nL steps to the left. And also the

probability of being a distance m from the origin after nR steps to the right.

This problem is discussed in [9] and now we will give the solution.

Since each step has length l the location of the walker must be of the form

x = ml where m is an integer. A question of interest is the following: after

N steps what is the probability of being located at the position x = ml?

One can readily generalize this one-dimensional problem to more dimen-

sions. One again asks for the probability that after N steps the walker is

located at certain distance from the origin, however this distance is no longer

of the form ml. Also on higher dimensions we add vectors of equal length

in random directions and then we ask the probability of the resultant vector

being in certain direction and certain magnitude. This is exemplified by the

following two examples:

a) Magnetism: An atom has spin 1/2 and magnetic moment µ; in accor-

dance with quantum mechanics, its spin can point up or down with respect

to certain direction. If both possibilities are equally likely, what is the net

total magnetic moment of N such atoms?

b) Diffusion of a molecule in a gas: A given molecule travels in three

dimensions a mean distance l between collisions with other molecules. How

far is likely to have gone after N collisions?

The random walk problem illustrates some very fundamental results of

probability theory.The techniques used in the study of this problem are pow-

erful and basic, and recur again and again in statistical physics.

After a total of N steps of length l the particle is located at x = ml where

−N ≤ m ≤ N . We want to calculate the probability PN(m) of finding the

particle at x = ml after N steps. The total number of steps is N = nL + nR

and the net displacement in units of l is given by m = nR−nL. If it is known

that in some sequence of N steps the particle has taken nR steps to the right,

then its net displacement from the origin is determined. Indeed

m = nR − nL = nR − (N − nR) = 2nR −N.

(13)

2The probabilities can be different, for example in the case we have a slope.

7

This shows that if N is odd then m is odd and if N is even then m is even

too.

A fundamental assumption is that successive steps are statistically inde-

pendent. Thus we can assert simply that, irrespective of past history, each

step is characterized by the respective probabilities

p = probability that the step is to the right

(14)

q = 1− p = probability that the step is to the left.

(15)

Now, the probability of a given sequence of nR steps to the right and nL step

to the left is given simply by multiplying the probability of each step and is

given by

pp...p

︸ ︷︷ ︸

nRfactors

qq...q

︸ ︷︷ ︸

nLfactors

= pnRqnL.

(16)

There are several ways to take nR steps to the right and nL steps to the left

in N steps. By known combinatorial calculus this number is given by

N !

nR!nL!

.

(17)

Hence the probability WN(nR) of taking nR steps to the right and nL =

N − nR steps to the left in N total steps is given by

WN(nR) =

N !

nR!nL!

pnRqnL.

(18)

This probability function is known as the binomial distribution. The reason

is because the binomial expansion is given by

(p+ q)N =

N∑

n=0

N !

n!(N − n!)p

nqN−n.

(19)

We already pointed out that if we know that the particle has made nR steps

to the right in N total steps then we know its net displacement m. Then the

probability of the particle being at m after N steps is

PN(m) = WN(nR).

(20)

We find explicitly that

nR =

1

2

(N +m)

nL =

1

2

(N −m).

(21)

8

Hence, in general we have that

PN(m) =

N !

((N +m)/2)!((N −m)/2)!p

(N+m)/2(1− p)(N−m)/2.

(22)

In the special case when p = q = 1/2 then

PN(m) =

N !

((N +m)/2)!((N −m)/2)!(1/2)

N .

(23)

5.1 Generalized random walk and the Fibonacci chain

case

Now we will study the generalized random walk. The random walk can be

studied in several dimensions, and we will do this up to a certain point and

later we will focus on one dimension and finally on the random walk on the

Fibonacci chain. In this subsection we mainly follow [10].

Let Pn(r) denote the probability density function for the position Rn of

a random walker, after n steps have been made. In other words, the prob-

ability that the vector Rn lies in an infinitesimal neighbourhood of volume

δV centered on r is Pn(r)δV . The steps are to be taken independent ran-

dom variables and we write pn(r) for the probability density function for the

displacement of the nth step. Then the evolution of the walk is governed by

the equation

Pn+1(r) =

∫

pn+1(r− r′)Pn(r′)ddr′,

(24)

where the integral is over all of d-dimensional space. This equation is an

immediate consequence of the independence of the steps.

It is important to note that, by hypothesis, the probability density func-

tion for a transition from r′ to r is a function of r − r′ only, and not on r

and r′ separately. In other words, the process is translationally invariant; it

is the relative position, not absolute location, which matters. The analysis

become much harder when pn+1(r− r′) must be replaced by pn+1(r, r′).

The assumed translational invariance ensures that the formal solution

of the problem is easily constructed using Fourier transform. The Fourier

transformp̃(q) of a function p(x) is defined as

p̃(q) =

∫ ∞

−∞

eiqxp(x)dx.

(25)

9

Under appropriate restrictions on the function p(x), there exist an inversion

formula:

p(x) =

1

2π

∫ ∞

−∞

e−iqxp̃(q)dq.

(26)

These equations are easily generalized to d dimensions. The Fourier trans-

form becomes

p̃(q) =

∫ ∞

−∞

eiq·rp(r)ddr,

(27)

where ddr denotes de d-dimensional volume element and the integral is taken

over all of d-dimensional space. Similarly the inversion formula becomes

p(r) =

1

(2π)d

∫ ∞

−∞

e−iq·rp̃(q)ddq.

(28)

The convolution theorem for the Fourier transform states that under modest

restrictions on g and h

k(x) =

∫ ∞

−∞

g(x− x′)h(x′)dx′

corresponds to

k̃(q) =g̃(q)h̃(q).

(29)

The generalization of the convolution theorem to d dimensions is straightfor-

ward:

k(r) =

∫ ∞

−∞

g(r− r′)h(r′)ddr′

corresponds to

k̃(q) =g̃(q)h̃(q). (30)

Taking the Fourier transform of our equation for the probabilities we have

that

P̃n+1(q) =p̃n+1(q)P̃n(q).

(31)

With P0(r) the probability density function for the initial position of the

walker, andP̃0(q) its Fourier transform, we have that

P̃n(q) =P̃0(q)

n∏

j=1

p̃j(q).

(32)

Taking the inverse Fourier transform of both sides of this equation, we find

the solution for the probability density function for the position after n steps:

Pn(r) =

1

(2π)d

∫

e−iq·rP̃0(q)

n∏

j=1

p̃j(q)d

dq.

(33)

10

When all steps have the same probability density function p(r) and the walk

is taken to commence at the origin of coordinates, so that

P0(r) = δ(r)

P̃0(q) = 1,

(34)

then we have

Pn(r) =

1

(2π)d

∫

e−iq·rp̃(q)nddq.

(35)

There are very few cases in which this integral can be evaluated in terms

of elementary functions. However, much useful information can still be ex-

tracted.

Now we will see one of the cases where this integral can be reduced a

elementary functions. For a random walk in one dimension with different

length steps we have that

p(x; ln) =

1

2

(δ(x− ln) + δ(x+ ln)).

(36)

Using that

δ(x− ln) =

1

2π

∫ ∞

−∞

dkeik(x−ln),

(37)

then we have that

p̃(q) =

∫ ∞

−∞

eiqxp(x)dx =

1

2

(eiqln + e−iqln) = cos(qln).

(38)

Hence

Pn(x; ln) =

1

2π

∫

e−iqx cosn(qln)dq.

(39)

In the case of the Fibonacci sequence we have

ln+1 = ln + ln−1 with

l0 = 0, l1 = 1.

(40)

So in this case we can solve the problem completely.

There is a subtlety with this expression for the probability, it diverges.

The problem is that we are dealing with distributions and classical analysis

does not work here. So we have to use the distribution theory. From p. 63

of [10] we know that the correct expression for the probability is

Pr {Xn = lln} =

ln

2π

∫ π/ln

−π/ln

e−illnξ cosn(lnξ)dξ,

(41)

11

where l ∈ Z. It is interesting that if we change variables as lnξ = k then

Pr {Xn = lln} =

1

2π

∫ π

−π

e−ilk cosn kdk,

(42)

and there is no dependence of ln in the integral.

5.2 The random walk in a two dimensional Fibonacci

lattice

Now let us consider a infinite two dimensional Fibonacci lattice. Then in

this case the probability density is given by

p(x, y; lnx, lny) =

1

4

(δ(x− lnx) + δ(x+ lnx) + δ(y − lny) + δ(y + lny)). (43)

Then following the one dimensional case we have that in the present case the

probability function is given by

Pn(x, y; lnx, lny) =

1

8π

(

∫

e−iqx cosn(qlnx)dq +

∫

e−ipx cosn(plny)dp).

(44)

Here q and p are variables in the Fourier space and lnx and lny are Fibonacci

numbers.

Making the corresponding manipulations we did in the 1-dimensional Fi-

bonacci sequence, now we obtain in this case

Pr {Xn = llnx, Yn = mlny} =

1

8π

(

∫ π

−π

e−ilk cosn kdk +

∫ π

−π

e−imk cosn kdk),

(45)

where l, m ∈ Z.

6 A kind of partition function

One of the main object in our approach is the a kind of partition function

which in certain limit should be reducible to the Einstein-Hilbert action and

in other limit to the partition function of quantum statistical mechanics. In

order to construct this partition function we will follow the ideas explained

in [1], [11] and [12].

12

Let us give a simple example of the kind of things we are working with.

One possible action for a piecewise constant path is [1]

A =β̃

n∑

i=1

|xi − xi−1| ,

(46)

where we will suppose thatβ̃ is a generalized inverse of the temperature.

Then the partition function3 associated with this action is

Z = e−β̃

∑n

i=1|xi−xi−1|.

(47)

The energy associated with this partition function is

E = − ∂

∂β̃

lnZ =

n∑

i=1

|xi − xi−1|

(48)

and the entropy is

S = E + lnZ = (1−β̃)

n∑

i=1

|xi − xi−1| .

(49)

Here the xi’s are an homogeneous partition of the path. In this sense it

is a periodic partition. It is clear that if now we assume that the xi’s are

quasiperiodic then the entropy will change. It is not difficult to imagine how

hard it would be to solve if instead of having a one-dimensional path we

have a surface or a volume. It could be interesting to compare the entropy

S with the entropy of a elastic string. If we want the discrete action to

go to the continuous action as the size of the partition goes to zero then

β̃ should depend on the size of the partition function [1]. Then clearly in

this case if we choose a quasicrystalline partition then the entropy and other

thermodynamical quantities will be impacted.

6.1 Partition function and entropy of the Fibonacci

lattice

If we consider the Fibonacci chain in 1-dimension, we can define a partition

function as

Z =

∑

n

Pr {Xn = lln} .

(50)

3Here we are thinking the action as an effective action which coincides at zero loops

with the classical action.

13

Analogously in the 2-dimensional case we have then

Z =

∑

n

Pr {Xn = llnx, Yn = mlny} .

(51)

If these definitions are correct, then it is a matter of brute force to calculate

the analogous of thermodynamical quantities.

For example let us do this for the 1-dimensional Fibonacci chain. In this

case we would have that the entropy is given by

S = F (ln) 〈Xn〉+ lnZ,

(52)

where F (ln) is a function which we should determine using plausible argu-

ments and 〈Xn〉 is the expectation value of Xn.

Analogously, in the two dimensional case we have

S = G(lnx, lny) 〈Xn, Yn〉+ lnZ,

(53)

where G(lnx, lny) is a function we have to propose. For example, if we agree

that with a new step there is an increasing of information then these functions

should be decreasing functions of the lengths.

7 Further ideas and final comments

It is clear that the introduction of aperiodicity in the framework of quantum

gravity would give substantially different results compare with the standard

approaches. Hence it would be interesting in the future to do something

similar with other quantum gravity approaches.

From the considerations in this work it is clear that our approach is closer

to the standard path integral approach than to the Hilbert space framework.

In this sense it would be interesting if with our approach we can recover the

well known results from Euclidean quantum gravity as explained in [13], [14].

Acknowledgments: This work is fully sponsored by Quantum Gravity

Research.

14

A On the Euclidean action and the Boltzman

factor

A.1 Introduction

One of our goals is to construct an object that in one limit gives the General

Relativity action (classical and quantum) and in the other side gives the

quantum mechanical statistics partition function.

In the book [15] Huang says that it is a deep mystery of physics that the

Hamiltonian operator appears in the evolution operator in quantum mechan-

ics and in the partition function in quantum statistical mechanics:

e−itĤ

e−βĤ .

(54)

Here β = κ

T

, with T the temperature and κ the Boltzman’s constant. If

we make t = −iτ , where τ is real and periodic with period of β then both

expressions become the same.

The purpose of this appendix is to comment on this deep mystery and to

try to elucidate, at least partially, why this occurs.

We think this discussion is important since important results such as

the entropy of black holes in euclidean quantum gravity [13] uses this deep

mystery.

The organization of this appendix is as follows: In the section 2 we discuss

how the Boltzman factor is related to the action, in the section 3 we explain

how the entropy of the BTZ black hole is obtained in Euclidean Quantum

Gravity, and finally in the section 4 we give our final comments.

A.2 On the action and the Boltzman factor

It is interesting to note that the action S of a system appears in the path

integral [8], [12], the partition function [1] and the Hamilton-Jacobi equation

[16]. Also it is interesting the similarity between the Boltzman factor and

the normal distribution. Let us elaborate on these two ideas.

In the Euclidean setting we have the path integral

A =

∫

Dxe

−S

~ .

(55)

Whereas the Boltzman factor is

Bi = e

−Ei

kT .

(56)

15

We know the action has units of energy times time. So if we multiply in the

Boltzman factor the energy and the kT term by some time we have an term

with units of action. Now the partition function is

Z =

∑

i

Bi =

∑

i

e

−Ei

kT .

(57)

The similarity with the path integral is obvious. Now the normal distribution

has the following form

N = ne−

x2

D .

(58)

Clearly if we multiply the square term by one over time square, and also the

D, then we have energy units. In one step further we can have a kind of action

in the normal distribution. Now, the Boltzman distribution is ubiquitous in

statistical mechanics and so is the normal distribution in several natural

processes. From this point of view the normal distribution is analogous

to the expression of the effective action4. So one may wonder if there is

a deep connection between these three expressions. One might wonder if

we can make up a mechanical toy model where in one side one has the

normal distribution and on the other end tha path integral and in the middle

the partition function obtained from the Boltzman factor. If we impose a

periodicity in the Euclidean amplitude A then with the correct units we have

the well know temperature of Black Holes. This periodicity when seen from

a discrete system can be related to the Poincaré recurrence theorem.

This toy model seems to be relevant for the unification physics since in

one hand one has a discrete system (similar to a quantum geometry) and in

the other hand a continuous system (module some metric issues) similar to

a topological quantum field theory.

It is also interesting that the action appears in the Hamilton-Jacobi equa-

tion whose quantum limit is the Schrodinger equation and it can branch to

classical mechanics, gravitational physics and electromagnetic theory.

Just to finish this section we note that the Lagrangian is given as

L = E − V,

(59)

where E is the energy and V the potential (energy). Hence we see that the

Lagrangian is a kind of generalized energy. The action is

S =

∫

Ldt.

(60)

4See for example [17] where the relationship between the path integral and the effective

action is displayed.

16

Hence when we make t imaginary and periodic, with the correct period in,

for example, black holes then everything about time drops and we have the

partition function of statistical mechanics. It is as if there were hidden a

symmetry related with time. Here we have taken the simplest Lagrangian

however it is not difficult to see that for example the scalar field the situation

is very similar.

The above discussion makes clear why the Euclidean path integral coin-

cides with the partition function when the time is periodic, in some sense the

partition function is hidden in the path integral.

A.3 On the entropy of black holes

As one example of some of the ideas presented in the previous section now

we will explain how the entropy of some black hole can be obtained using

the effective action.

It is well know, see for example [18], that at zero order the effective action

Γ[Φ] coincides with the classical action A[Φ] evaluated on the mean field Φ.

The evaluation of the black hole entropy of the Kerr black hole can be

consulted [13], and now we will show how the entropy for the BTZ black hole

is obtained.

We follow mainly [19]. The Euclidean action of the BTZ black hole is

IE = βM −

A

4G

.

(61)

Then the partition function is

ZBTZ(T ) = exp

(πl)2T

2G

,

(62)

where l is the AdS radius. The expectation value of the energy is

EBTZ = −

∂

∂β

lnZ = M.

(63)

Whereas the entropy is given by

SBTZ = βEBTZ + lnZBTZ = 4πr+ =

A

4G

.

(64)

Which is the result one expects on the grounds of Beckenstein ideas on en-

tropy of black holes.

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It is interesting to note that there are at least three other values of the

BTZ black hole entropy obtained in [20], [21] and [22]. In the first in loop

quantum qravity, the second in standard statistical field theory and in the

third in the brick wall model. In the first two models it does not coincide

with the value given in [19] whereas in the brick wall model it coincides with

[19].

A.4 Final comments

It is interesting to note the following: The result of [19] is classical, although

using a quantum framework, the result of [20] is quantum but it does not

give the expected result, the result of [21] is semiclassical and gives a close

result to the one expected, and finally the result of [22] is quantum and gives

the expected result but the entropy is of the scalar field living on the BTZ

black hole. Hence there is no a consensus about this entropy. Just to finish

up we note that the temperature of a black hole does not make sense without

a field living on it, so, after all the brick wall model could be the one closer

to the origin of the BTZ black hole entropy.

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