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Motion in Quantum Gravity 551

The interpretation of the formula (30) is clear. Note, however, an important point:

The formula (30) deﬁnes a second-order operator in the sense that it involves

j 1=2

and

j C1=2

that are not nearest matrices but second nearest matrices.

The derivative operator is obviously necessary for deﬁning a dynamics in the

noncommutative fuzzy space. The ambiguity in the deﬁnition of P

implies imme-

diately an ambiguity in the dynamics. For instance, the fact that C

.j; k/ relates

matrices

with

j ˙1=2

only is a consequence of the choice of P

that is in fact a

function whose non-vanishingFourier modes are the matrix elements of a dimension

2 matrix: indeed, P

.u/ D `

1

1=2

u/. Another choice would lead to a different

dynamics and then there is an ambiguity. Such ambiguities exist as well in full Loop

Quantum Gravity [23].

3.2.3 Free Field: Solutions and Properties

Now, we have all the ingredients to study dynamics on the quantum space. Due to

the fuzzyness of space, equations of motion will be discrete and therefore, there is in

general no equivalence between the Lagrangian and Hamiltonian dynamics. Here,

we choose to work in the Euler–Lagrange point of view, that is, the dynamics are

governed by an action of the type:

Œ˚; J  D

.2/





˚?@



˚ C V.˚;J/



.x/; (31)

where V is the potential that depends on the ﬁeld ˚ and eventually on some exterior

ﬁelds J . The action has been written in the C

/ formulation to mimic easily

the classical situation.

Obviously, ﬁnding the equations of motions reduces to extremizing the previous

action, but with the constraint that ˚ belongs to C

/: in particular, ˚ (as well

as the exterior ﬁeld) admits two independent components ˚

, which are classical

functions on E

whose spectra are bounded. The action (31) couples these two com-

ponents generically. Even when one of the two ﬁelds vanishes, for instance ˚



D 0,

it happens in general that the extrema of the functional SŒ˚ differ from the ones

that we obtain for a classical ﬁeld ˚ whose action would be formally the same

functional, but deﬁned with the pointwise product instead of the ? product. This

makes the classical solutions in the deformed and nondeformed cases different in

general. Let us state this point more precisely. When the ﬁeld is free, in the sense

that V is quadratic (with a mass term), deformed solutions are the same as classical

ones. However, solutions are very different when the dynamics are nonlinear, and

the differences are physically important.

First, let us consider the case of a free ﬁeld: we assume that V.˚/ D 

˚?˚,

where  is a positive parameter. Equations of motion are:



C 

D 0 for all spin j: (32)

552 K. Noui

Due to the quite complicated expression of the derivative operator, it appears more

convenient to solve this set of equations in the C.S U.2// representation. Indeed,

these equations are equivalent to the fact that  D F

1

Œ˚ has support in the

conjugacy classes  2 Œ0; 2Œ such that sin

.=2/ D `



. Thus, a solution exists

only if   `

1

, in which case we write  D `

1

sin.m=2/ with 0<m<.Then

the solutions of the previous system are given by

˚ D



with:

d.u/ I

.u/



˛.u/D

.u/ C ˇ.u/D

.u/



; (33)

where ˛ and ˇ are SU.2/ complex valued functions; the notation I

holds for the

characteristic functions on the conjugacy class  D m (for the C sign) and  D

2  m (for the  sign). These functions are normalized to one according to the

relation

d.u/I

.u/ D 1.Iftheﬁelds˚

.x/ are supposed to be real, the matrices

are hermitian, and then ˛ and ˇ are complex conjugate functions. As a result,

we obtain the general solution for the noncommutative free ﬁeld written in the fuzzy

space representation.

Using the mapping m, one can reformulate this solution in terms of functions on

. The components of ˚ are given by:

.x/ D

16

sin

cos

pı.p /



.p/e

ipx

C ˇ

.p/e

ipx



; (34)

where B

is the Planck ball, ˛

.p/ D ˛.u.p// where u.p/ is the inverse of p.u/

when u is restricted to the sets U

; a similar deﬁnition holds for ˇ

. We recover

the usual solution for classical free scalar ﬁelds with the fact that the mass has

an upper limit given by `

1

. Therefore, the Planck mass appears to be a natural

UV cutoff.

3.3 Particles Evolving in the Fuzzy Space

Important discrepancies between classical and fuzzy dynamics appear when one

considers nonlinear interactions. In the case where we study the dynamics of a sole

ﬁeld , we may introduce self-interactions. However, even in the standard clas-

sical commutative space E

, classical solutions of self-interacting ﬁeld cannot be

written in a closed form in general; and then one cannot expect to ﬁnd explicit so-

lutions for the self-interacting ﬁeld evolving in the fuzzy background. Faced with

such technical difﬁculties (which we postpone for future investigations), we will

consider simpler models. We will perform symmetry reductions in order that the

ﬁeld  depends only on one coordinate out of the three. We will interpret this model

as describing one particle evolving in (Euclidean) fuzzy space-time.

Motion in Quantum Gravity 553

3.4 Reduction to One Dimension

Let us deﬁne the algebra C

of symmetry reduced ﬁelds and its different repre-

sentations: the group algebra, the matrix, and the continuous formulations.

First of all, C

can be identiﬁed to the convolution algebra C.U.1//



of U.1/

distributions. In particular, a function ' 2 C

is a function of  2 Œ0; 2.The

matrix representation reduces to the Fourier representation of C.U.1//



W C.U.1//



! Diag

.C/; ' 7!

˚ with

 '

Dh'; e

ia

i; (35)

where h; i is the duality bracket between U.1/ distributions and U.1/ functions and

Diag

.C/ is the algebra of inﬁnite-dimensional diagonal complex matrices. This

identity reduces to the following more concrete relation when ' is supposed to be a

function:

2

2

d './e

ia

: (36)

The algebraic structure of Diag

.C/ is induced from the convolution product ı and

is simply given by the commutative discrete pointwise product:

8 ';'

2 C.U.1//



.' ı '

D '

: (37)

Let us now construct the mapping between the convolution algebra C.U.1// and

the algebra C

/, which has to be understood for the moment as the one-

dimensional analogue of C

/. We proceed in the same way as in the full theory:

1. ﬁrst, we cut U.1/  Œ0; 2 in two parts, U

 



;



Œ and U







;

3

where the symbol  means equal modulo 2;

2. then, we construct two bijections between U

and B

  `

1

Œ by as-

signing to each  2 U

amomentumP./ D `

1

sin ;

3. the third step consists in associating to any function ' 2 C.U.1// apairof

functions '

2 C.U

/, and a pair of functions

2 C.R/ induced by the

previous bijections as follows

.

/.p/ 

.p/ D

d

2

ı.p  `

1

sin /'

./I (38)

4. ﬁnally, we make use of the standard one-dimensional Fourier transform F

construct the mapping m

D m

˚ m



W C.U.1// ! C

/ where the

components m

D F

ı a

are given by:

/.t/  ˚

.t/ D

2

d

2

./ exp.iP./t/: (39)

554 K. Noui

The space C

/ is the image of C.U.1// by m and therefore is deﬁned by

C.U

/ ˚

C.U



/ where

C.U

/ are the images by F

of C.U

/.Asinthe

full theory, this construction can be extended to the algebra C.U.1//



of distri-

butions.

The link between the discrete and the continuous representations of C

is given by:

.t/ D

.t/; (40)

where the functions K

.t/ are deﬁned by the integrals

.t/ 

d

2

iaCiP . /t

D .˙1/



d



cos



a 

sin 



: (41)

As in the general case, the relation (40) is invertible. The integral deﬁning K

is a simpliﬁed version of the general formula (22) and one can view these func-

tions as the components of the element K DK

˚ K



2 Diag

.C/ ˝ C

Furthermore, K

˚ K



is the image by m

of the (discrete) plane waves

exp.ia/. As a ﬁnal remark, let us underline that K

and K



are closely related

by the property K



.t/D.1/

.t/. This implies that the functions ˚

are

also closely related: if we assume for instance that '

2nC1

D0 for any n 2 Z then



.t/D˚

.t/; if we assume on the contrary that '

D0 for any n 2 Z then



.t/D˚

.t/. Such a property will have physical consequences as we will

see in the sequel.

Let us give some physical interpretation of the formula (40). One can view it as a

way to extend '

, considered as a function on Z, into the whole real line R.Inthat

sense, this formula is a link between the discrete quantum description of a ﬁeld and

a continuous classical description. One sees that any microscopic time a contributes

(positively or negatively) to the deﬁnition of a macroscopic time t with an amplitude

precisely given by K

.t/. At the classical limit `

! 0, K

.t/ are maximal for

values of the time t D˙`

a. In other words, the more the microscopic time a`

is close to the macroscopic time t, the more the amplitude K

.t/ is important.

Concerning the reduced ?-product, it is completely determined by the algebra of

the functions K

viewed as elements of C

/ and a straightforward calculation

leads to the following product between K

type functions:

 m

.exp.ia/ ı exp.ib// D ı

: (42)

This result clearly illustrates the non-locality of the ?-product.

Before going to the dynamics, let us give the expression of the derivative oper-

ator @

. As for the general case, @

is a ﬁnite difference operator whose action on

Diag

.C/ is given, as expected, by the following formula:

aC1

 '

a1

/: (43)

Motion in Quantum Gravity 555

This expression is highly simpliﬁed compared to the more general one introduced in

the previous section. However, we still have the property that @

is in fact a second-

order operator for it relates a C 1 and a  1. An important consequence would be

that the dynamics (of the free ﬁeld) will decouple the odd components '

and the

even components '

2nC1

of the discrete ﬁeld. Then, we will have two independent

dynamics that could be interpreted as two independent particles evolving in the

fuzzy space. In particular, one could associate the continuous ﬁelds ˚.t/

odd

and

˚.t/

even

, respectively, to the families .'

/ and .'

2nC1

/. It is clear that ˚.t/

odd

and

˚.t/

even

are completely independent of one another and, using the basic properties

of K

, we ﬁnd that the ˙ components of each ﬁeld are related by:

odd



.t/ D ˚

odd

.t/ and ˚

even



.t/ D˚

even

.t/: (44)

Thus, ˚

and ˚



fundamentally describe two “mirror” particles.

3.4.1 Dynamics of a Particle: Linear versus Nonlinear

We have now assembled all the ingredients for studying the behavior of the one-

dimensional ﬁeld '. When written in the continuous representation, its dynamics

are governed by an action of type (31), but only one-dimensional, with no external

ﬁeld J , and the potential is supposed to be monomial, that is, of the form V.˚/ D

"=.˛ C 1/˚

?.˛C1/

, with ˛ C 1 a non-null integer. The equations of motions are

given by:

˚ C "˚

?˛

D 0; (45)

where  D @

. In the fuzzy space formulation, these equations read:

aC2

 2'

C '

a2

D"'

: (46)

As was previously emphasized, we note that these equations do not couple odd and

even integers a. For simplicity, we will consider only even spins, that is, we assume

that '

2nC1

D 0 for all integer n.

The linear case has already been studied in the previous section. For present

purposes, we consider the dynamics (46) with ˛  2 and we look for perturbative

solutions in the parameter ". The corresponding classical solution ˚

reads at the

ﬁrst order

.t/ D vt  "

˛C2

.˛ C 1/.˛ C 2/

C O."

/; (47)

where we assume for simplicity that ˚

.0/ D 0 and ˚

.0/ D v.

556 K. Noui

The perturbative expansion of the fuzzy solution is obtained using the same

techniques. We look for solutions of the type '

D a C "

where a D 2k by

assumption,  is a real number, and  must satisfy the following relation:



 

2k2

D



k1

nD1

.2n/

D

.2/



.k  1/

˛C1

˛ C 1

.k  1/

˛.k  1/

˛1



˛.˛  1/.˛  2/

720

.k  1/

˛3

˛.˛  1/.˛  2/.˛  3/.˛  4/

30240

.k  1/

˛5

C



The solution is in general complicated. To be explicit, we will consider the

case ˛ D2. The formula simpliﬁes considerably and, after some straightforward

calculations, one can show that



D



.k  1/.k C 1/ D



 k

/: (48)

In order to compute the C

/ representation of this solution, one uses the fol-

lowing relations for any integer n

.n/

.t/ 

kD1

.t/ D

2.2i/

d

exp.iP . /t /j

11



: (49)

Applying this formula for n D 1; 2,and4

.1/

.t/ D˙`

1

t; S

.2/

.t/ D 2`

2

.4/

.t/ D 2.4`

4

C `

2

/ (50)

one shows, after some simple calculations, that ˚

and ˚



are simply related by

.t/ D ˚



.t/ and ˚

is given by:

.t/ D 2`

1

t  "



.2`

4

C `

2

/ C O."

/: (51)

To compare it with the classical solution ˚

computed above (47), we impose the

same initial conditions, which leads to  D v`

=2 and then the solution reads:

.t/ D vt  "

 "

C O."

/: (52)

Motion in Quantum Gravity 557

Let us interpret the solution. First, let us underline that ˚

and ˚



are related

by ˚

.t/ D ˚



.t/: thus, it seems that ˚



corresponds to a particle evolving

backward compared to ˚

. In that sense, the couple ˚

behaves like a parti-

cle and a “mirror” particle: the presence of the mirror particle is due to quantum

gravity effects. Second, we remark that the solution for ˚

differs from its classical

counterpart at least order by order in the parameter ". At the no-gravity limit

! 0, ˚

tends to the classical solution (47). Therefore, we can interpret these

discrepancies as an illustration of quantum gravity effects on the dynamics of a ﬁeld.

3.4.2 Background Independent Motion

We ﬁnish this example with the question concerning the physical content of this

solution. For the reasons given in the previous section, we concentrate only on the

component ˚

. Can one interpret ˚

.t/ as the position q.t/ of a particle evolving in

the fuzzy space? If the answer is positive, it is quite confusing because the position

should be discrete valued whereas ˚

takes value in the whole real line a priori.

In fact, we would like to interpret ˚

.t/ D Q.t/ 2 R as the extension in the

whole real line of a discrete position q.t/ 2 Z. More precisely, we suppose that the

space where the particle evolves is one-dimensionaland discrete, and then its motion

should be characterized by a Z-valued function q.t/. If we restore the discreteness of

the time variable, then the motion of the particle should, in fact, be characterized by

a set of ordered integers fq.2k`

/; k 2 Zg. To make this description more concrete,

we make use of the identity satisﬁed by S

.1/

(50), which implies that

Q.t/ D

kD1

.2`

k/ K

.Q.t//: (53)

This identity makes clear that Q.t/ can be interpreted as a kind of continuation in

the whole real line of a set of discrete positions and K

.Q.t// gives the (positive

or negative) weight of the discrete point 2`

k in the evaluation of the continuous

point Q.t/. Therefore, one can associate an amplitude P.kj/ to the particle when

it is at the discrete position Q D 2`

k and at the discrete time t D 2`

 (in Planck

units) in the fuzzy space. This amplitude is given by:

P.kj/ D

.Q.2`

//

j D1

.Q.2`

//

D K

.Q.2`

//; (54)

because the normalization factor equals one. These amplitudes cannot really be

interpreted as statistical weights because they can be positive or negative. Never-

theless, they contain all the information of the dynamics of the particle in the sense

that one can reconstruct the dynamics from these data. Therefore, we obtain a back-

ground independent description of the dynamics of the particle that can be a priori

anywhere at any time: its position 2k`

at a given time 2`

is characterized by the

558 K. Noui

amplitude previously deﬁned. Furthermore, the amplitude is maximum around the

classical trajectory, that is, when Q.2`

/ D 2`

k, and gives back the classical

trajectory at the classical limit deﬁned by k;  !1, `

! 0 with the products

and `

, respectively, ﬁxed to the values t (classical time) and Q (classical

position).

4 Discussion

In this article, we have tackled the question of motion in Quantum Gravity. As we

have already emphasized, it is certainly too early to discuss this question in detail.

However it is at least possible to raise some preliminary problems that one needs

to resolve if one aims at understanding what motion means at the Planck scale.

Among the most fundamental problems are the questions of the deep structure of

space-time and those of the description of matter ﬁelds in Quantum Gravity. Loop

Quantum Gravity proposes a very clear answer to these questions (even if the spe-

ciﬁc viewpoint adopted here has warranted extensive discussion).

For this reason, we think that Loop Quantum Gravity presents a useful frame-

work for discussing the question of motion at very short distances. We started with

a very brief review of Loop Quantum Gravity, insisting on the kinematical aspects:

the description of the kinematical states in terms of spin-networks and the compu-

tation of the spectrum of the so-called area and volume operators. We explained in

what sense space appears discrete in Loop Quantum Gravity. We ﬁnished by men-

tioning the fundamental problem of the dynamics that, so far, no one knows how to

solve, namely, the remaining Hamiltonian constraint. Nonetheless, different promis-

ing strategies have been developed to solve this issue. For the moment, this relative

failure prevents us from discussing the question of the motion that is intimately

linked to the question of the dynamics.

As a consequence, in a second part, we presented a toy model where the dynam-

ics are very well understood: three-dimensional Euclidean quantum gravity with no

cosmological constant. This model is exactly solvable and shares several character-

istics with Loop Quantum Gravity, including the discreteness of space. Furthermore,

the coupling to a matter ﬁeld is very well understood and leads to a description of

scalar ﬁelds in terms of complex matrices evolving in a noncommutative fuzzy ge-

ometry. Therefore, the question of motion at the Planck scale reduces in that case

to the resolution of ﬁnite difference equations involving matrix coefﬁcients, which

are the quantum analogue of the equations of motions. We propose a solution of

these equations in simple examples: the free ﬁeld and the one-dimensional ﬁeld.

These examples are nice illustrations of what could represent motion more gener-

ally in Quantum Gravity.

It is nonetheless clear that we are far from a precise description of motion in full

Loop Quantum Gravity. We hope that the examples we have developed in this paper

will shed light on this problem.

Motion in Quantum Gravity 559

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