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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.
arXiv:2003.07282v1 [math-ph] 16 Mar 2020
Aspects of aperiodicity and randomness in
theoretical physics
Leonardo Ort´ız
, Marcelo Amaral and Klee Irwin
Quantum Gravity Research, Los Angeles, CA, U.S.A
March 17, 2020
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.
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
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
G(γ) = Zγ
where σis an hypersurface, γis its boundary and Sis 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]
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-
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
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
(or heat) equation in Rd
∂t =1
2ϕ, (2)
with the initial condition ϕ(x, 0) = ϕ0(x) is given by
ϕ(y, t) = 1
The function ϕ0(x) is interpreted as the initial distribution of particles at
time t= 0, and |xy|denote the Euclidean distance between xand yin
The kernel Kt(x, y) of the operator et
2, is called the heat kernel and is
given by
Kt(x, y) = 1
and represents the probability density of finding the particle at yat time t
given its location at xat time 0. From the simigroup property
e(t+s)∆ =etes,(5)
for s,t0 we have
Kt(x, y) = Zdx1...dxN1Kt/N (xN, xN1)...Kt/N (x1, x0) (6)
for each N1, where we have set x0=xand xN=y.
There is an obvious one-to-one correspondence between configurations
(x1, ..., xN1) and parametrized piecewise linear paths ω: [0, t]Rdfrom x
to yconsisting of line segments [x0, x1], [x1, x2],...,[xN1, xN], such that the
segment [xi1, xi] is parametrized linearly by s[i1
Nt, i
Nt]. We denote the
collection of all such paths by ΩN,t(x, y). Hence we may consider
tω= (2πt
as a measure on the finite dimensional space ΩN,t (x, y).
Noting that
t/N =
t/N )2=Zt
0|˙ω(s)|2ds, (8)
where ˙ωis the piecewise constant velocity of the trajectory ω, hence we can
Kt(x, y) = Z(x,y)
where the suffix (x, y) indicates that paths are restricted to go from xto y.
We refer to this equation as a random walk representation of Kt(x, y) on
N,t(x, y).
More generally, given an action functional Son a piecewise linear parametrized
paths, we call the equation
t(x, y) = Z(x,y)
a random walk representation of the kernel HN
t(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 αis a irrational number. Now if we interpret this function as given
the position of a particle after a time τthen 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{−x2(τ)
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
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 Xnwhich can be any integer. Now suppose this
walker can move to the left or to the right with equal probability21/2 and
the length of the step being l. We would like to know the probability that
the walker is nRsteps to the right and nLsteps to the left. And also the
probability of being a distance mfrom the origin after nRsteps to the right.
This problem is discussed in [9] and now we will give the solution.
Since each step has length lthe location of the walker must be of the form
x=ml where mis 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 Nsteps 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 Nsuch atoms?
b) Diffusion of a molecule in a gas: A given molecule travels in three
dimensions a mean distance lbetween 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 Nsteps of length lthe particle is located at x=ml where
NmN. We want to calculate the probability PN(m) of finding the
particle at x=ml after Nsteps. The total number of steps is N=nL+nR
and the net displacement in units of lis given by m=nRnL. If it is known
that in some sequence of Nsteps the particle has taken nRsteps to the right,
then its net displacement from the origin is determined. Indeed
m=nRnL=nR(NnR) = 2nRN. (13)
2The probabilities can be different, for example in the case we have a slope.
This shows that if Nis odd then mis odd and if Nis even then mis even
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 nRsteps to the right and nLstep
to the left is given simply by multiplying the probability of each step and is
given by
There are several ways to take nRsteps to the right and nLsteps to the left
in Nsteps. By known combinatorial calculus this number is given by
Hence the probability WN(nR) of taking nRsteps to the right and nL=
NnRsteps to the left in Ntotal steps is given by
WN(nR) = N!
This probability function is known as the binomial distribution. The reason
is because the binomial expansion is given by
We already pointed out that if we know that the particle has made nRsteps
to the right in Ntotal steps then we know its net displacement m. Then the
probability of the particle being at mafter Nsteps is
PN(m) = WN(nR).(20)
We find explicitly that
Hence, in general we have that
PN(m) = N!
((N+m)/2)!((Nm)/2)!p(N+m)/2(1 p)(Nm)/2.(22)
In the special case when p=q= 1/2 then
PN(m) = N!
5.1 Generalized random walk and the Fibonacci chain
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 Rnof
a random walker, after nsteps have been made. In other words, the prob-
ability that the vector Rnlies in an infinitesimal neighbourhood of volume
δV centered on ris 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) = Zpn+1 (rr)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 rto ris a function of rronly, and not on r
and rseparately. 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(rr) 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
transform ep(q) of a function p(x) is defined as
ep(q) = Z
eiqxp(x)dx. (25)
Under appropriate restrictions on the function p(x), there exist an inversion
p(x) = 1
eiqx ep(q)dq. (26)
These equations are easily generalized to ddimensions. The Fourier trans-
form becomes
ep(q) = Z
where ddrdenotes de d-dimensional volume element and the integral is taken
over all of d-dimensional space. Similarly the inversion formula becomes
p(r) = 1
The convolution theorem for the Fourier transform states that under modest
restrictions on gand h
k(x) = Z
g(xx)h(x)dxcorresponds to e
k(q) = eg(q)e
The generalization of the convolution theorem to ddimensions is straightfor-
k(r) = Z
g(rr)h(r)ddrcorresponds to e
k(q) = eg(q)e
Taking the Fourier transform of our equation for the probabilities we have
that e
Pn+1(q) = epn+1(q)e
With P0(r) the probability density function for the initial position of the
walker, and e
P0(q) its Fourier transform, we have that
Pn(q) = e
j=1 epj(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 nsteps:
Pn(r) = 1
j=1 epj(q)ddq.(33)
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)e
P0(q) = 1,(34)
then we have
Pn(r) = 1
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-
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(xln) + δ(x+ln)).(36)
Using that
δ(xln) = 1
then we have that
ep(q) = Z
eiqxp(x)dx =1
2(eiqln+eiqln) = cos(qln).(38)
Pn(x;ln) = 1
2πZeiqx cosn(qln)dq. (39)
In the case of the Fibonacci sequence we have
ln+1 =ln+ln1with 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
P r {Xn=lln}=ln
eillnξcosn(lnξ)dξ, (41)
where lZ. It is interesting that if we change variables as lnξ=kthen
P r {Xn=lln}=1
eilk cosnkdk, (42)
and there is no dependence of lnin the integral.
5.2 The random walk in a two dimensional Fibonacci
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(xlnx) + δ(x+lnx ) + δ(ylny) + 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(Zeiqx cosn(qlnx )dq +Zeipx cosn(plny )dp).(44)
Here qand pare variables in the Fourier space and lnx and lny are Fibonacci
Making the corresponding manipulations we did in the 1-dimensional Fi-
bonacci sequence, now we obtain in this case
P r {Xn=llnx, Yn=mlny }=1
eilk cosnkdk +Zπ
eimk cosnkdk),
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].
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]
i=1 |xixi1|,(46)
where we will suppose that ˜
βis a generalized inverse of the temperature.
Then the partition function3associated with this action is
The energy associated with this partition function is
βln Z=
i=1 |xixi1|(48)
and the entropy is
S=E+ ln Z= (1 ˜
i=1 |xixi1|.(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
Swith 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
βshould 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
If we consider the Fibonacci chain in 1-dimension, we can define a partition
function as
P r {Xn=lln}.(50)
3Here we are thinking the action as an effective action which coincides at zero loops
with the classical action.
Analogously in the 2-dimensional case we have then
P r {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)hXni+ ln Z, (52)
where F(ln) is a function which we should determine using plausible argu-
ments and hXniis the expectation value of Xn.
Analogously, in the two dimensional case we have
S=G(lnx, lny )hXn, Yni+ ln Z, (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
A On the Euclidean action and the Boltzman
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:
eit ˆ
Here β=κ
T, with Tthe temperature and κthe Boltzman’s constant. If
we make t= , where τis real and periodic with period of βthen 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
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 Sof 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=ZDxe S
Whereas the Boltzman factor is
kT .(56)
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
kT .(57)
The similarity with the path integral is obvious. Now the normal distribution
has the following form
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 Poincar´e 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=EV, (59)
where Eis the energy and Vthe potential (energy). Hence we see that the
Lagrangian is a kind of generalized energy. The action is
S=ZLdt. (60)
4See for example [17] where the relationship between the path integral and the effective
action is displayed.
Hence when we make timaginary 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
Then the partition function is
ZBT Z (T) = exp (πl)2T
where lis the AdS radius. The expectation value of the energy is
∂β ln Z=M. (63)
Whereas the entropy is given by
SBT Z =βEBT Z + ln ZB T Z = 4πr+=A
Which is the result one expects on the grounds of Beckenstein ideas on en-
tropy of black holes.
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
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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Ambos volúmenes están dedicados a las "Integrales de Caminos" con aplicaciones en F'isica. Si bien esta clase de integrales no está del todo estudiada, en F'isica tienen bastante utilidad. El primer volumen está dedicado a Procesos Estocásticos y Mecánica Cuántica, el segundo volumen muestra las aplicaciones en la Teor'ia de Campos Cuánticos, F'isica Estad'istica y otros temas más actuales.\ Recomendado para estudiantes de Pregrado y de Posgrado