A note on DSR-like approach to space-time

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by  R. Aloisio

A note on DSR-like approach to space-time
R. Aloisio
INFN - Laboratori Nazionali del Gran Sasso, SS. 17bis, 67010 Assergi (L’Aquila) - Italy
A. Galante
INFN - Laboratori Nazionali del Gran Sasso, SS. 17bis, 67010 Assergi (L’Aquila) - Italy and
Dipartimento di Fisica, Universit`a di L’Aquila, Via Vetoio 67100 Coppito (L’Aquila) - Italy
A. Grillo
INFN - Laboratori Nazionali del Gran Sasso, SS. 17bis, 67010 Assergi (L’Aquila) - Italy
E. Luzio
Dipartimento di Fisica, Universit`a di L’Aquila, Via Vetoio 67100 Coppito (L’Aquila) - Italy
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0 F. M´endez
0 INFN - Laboratori Nazionali del Gran Sasso, SS. 17bis, 67010 Assergi (L’Aquila) - Italy
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In this note we discuss the possibility to define a space-time with a DSR based approach. We
n
showthatthestrategyofdefininganonlinearrealizationoftheLorentzsymmetrywithaconsistent
a
J vector composition law cannot be reconciled with the extra request of an invariant length (time)
scale. The latter request forces to abandon the group structure of the translations and leaves a
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space-timestructurewherepointswithrelativedistancessmallerorequaltotheinvariantscalecan
2
not be unambiguously defined.

1
v
I. INTRODUCTION It has also been claimed that κ-Minkowski [8] gives a
9
7 possible realization of the DSR principles in space-time
0 [9],howevertheconstructionisstillnotsatisfactorysince
It is widely believed that the space-time, where phys-
1 the former is only compatible with momentum composi-
ical process and measurements take place, might have a
0 tion law non symmetric under the exchange of particles
structure different from a continuous and differentiable
5
labels (see discussions in [10]). In this work we are deal-
0 manifold, when it is probed at the Planck length ℓp. For
ing with non linear realizations of the Lorentz algebra
/ example, the space-time could have a foamy structure
c which induce symmetric composition law and therefore
[1], or it could be non-commutative in a sense inspired
q it is not compatible with the κ-Minkowski approach.
- bystringtheoryresults[2]orinthesenseofκ-Minkowski
r approach [3]. The main results of our studies are: i) the strategy of
g
defining a non linear realization of the Lorentz symme-
: If this happens in the space-time, in the momentum
v try with a consistent vector composition law cannot be
space there must also be a scale, let say p , that signs
i p reconciled with the extra request of an invariant length
X this changeofstructure of the space-time, evenif the in-
r terplaybetweenlengthandmomentum(p∼c~λ−1)will (time) scale; ii) the request of an invariant length forces
a presumably change when we approach such high energy to abandon the group structure of the translations and
leaves a space-time structure where points with relative
scales.
distances smaller or equal to the invariant scale can not
Onecouldarguethat,ifthePlancklengthgivesalimit
be unambiguously defined.
at which one expects that quantum gravity effects be-
InthenextsectionwewillexploretheapproachtoDSR
come relevant, then it would be independent from ob-
in the momentum space and will implement these ideas
servers, and one should look for symmetries that reflect
in the space-time sector. In the final section conclusion
this property. Such argument gave rise to the so called
and discussions are presented.
DSR proposals, that is, a deformation of the Lorentz
symmetry (in the momentum space) with two invariant
scales: the speed of light c and p (or E ) [4, 5, 6].
p p
II. NON LINEAR REALIZATION OF LORENTZ
In this note, we will discuss this class of deformations
GROUP APPROACH TO SPACE-TIME
oftheLorentzsymmetryanditsrealizationinthespace-
time. Approachestotheprobleminspiredbythemomen-
tum space formulation have been presented [6, 7], but InthissectionwewillfirstreviewtheapproachtoDSR
ourapproachisquitedifferentfromthesebecausewede- asanonlinearrealizationoftheLorentztransformations
mand the existence of an invariant measurable physical in the energy-momentum space, and then try to apply
scalecompatiblewiththedeformationofthecomposition these ideas to the space-time. For a more generalreview
law of space-time coordinates induced by the non linear of DSR see for example [11] and references therein.
transformation. DSR principles are realized in the energy-momentum
2
space by means of a non-linear action of the Lorentz foranyx . Withrelation(4)wecandefinetheoperation
c
group [5, 12]. More precisely, if the coordinates of the oftranslationofallvectorsofourspacebyafixedvector
physical space P are p = {p ,p}, we can define a non- α:
µ 0
linearfunctionF :P →P,whereP isthespacewithco-
ordinates πµ = {ǫ,π}, on which the Lorentz group acts Tˆα(x) ≡ x+ˆα. (6)
linearly. We will refer to P as a classical momentum
space. With the above definition the translations behave as in
In terms of the previous variables, a boost of a single the standard case under the action of boosts:
particle with momentum 1 p to another reference frame,
where the momentum of the particle is p′, is given by B Tˆα(x) = TˆB(α)(B(x)). (7)
(cid:16) (cid:17)
p′ =F−1◦Λ◦F [p]≡B[p]. (1) It is easy to check that these transformations form a
group. We will call e the neutral element of the group
Finally, an addition law (+ˆ) for momenta, which is i.e. the neutral element for the composition law. From
covariant under the action of B, is the relations (4),(6) we read that the neutral element is
suchthatG[e]=0i.eitsimagecorrespondstotheorigin
pa+ˆpb =F−1[F[pa]+F[pb]], (2) of the classical space.
and satisfies B[p +ˆp ]=B[p ]+ˆB[p ] .
a b a b
In this formulation, the requirement of having an in- 1. Invariant scales
variant scale fixes the action of F on some points of the
real space P. Indeed, since the Lorentz transformation
With the previous definitions we want to addthe con-
leaves the points π = 0 and π = ∞ invariant, one sees
ditionthat,underthedeformedboosts,aninvariantmea-
that if we demand invariance of Planck momentum (p )
p surable physical scale (both a time or length scale) has
(or energy (E )) then F[p ] (or F[E ]) only can be 0 or
p p p to exist. In doing that we have in mind that, eventually,
∞[5]. Ageneraldiscussiononthe possibledeformations
thisscaleisrelatedwiththePlancklength(ortime). Let
is given in [13].
uscallδˆ thevectorwhichdefinestheinvariantscale. By
In the following we explore the possibility to extend p
this we mean that δˆ is any vector of the form
the above discussion generalizing it to define a non lin- p
ear realization of Lorentz symmetry in space-time. In
T t
analogy with the momentum space, we will consider a p or , (8)
real space-time X with coordinates xµ = {x0,··· ,x3} (cid:18) x (cid:19) (cid:18)xp (cid:19)
and will assume: a) the existence of an auxiliary space-
time X with coordinates ξµ = {ξ0,··· ,ξ3} (called clas- wherexp isanyspatialvectorwithmodulus equalto the
invariant scale and T is the invariant time scale. The
sical space-time), where the Lorentz group acts linearly p
invariance condition we impose is that a time (length)
and b) the existence of an invertible map G[x] such that
equaltotheinvariantscaleisnotaffectedbyboosts. This
G[x]:X →X.
corresponds to the physical intuition that any Planck-
Boosts in the space-time will be defined in the same
length segment (whatever his time position) or Planck-
way as DSR boosts in the momentum space, that is
timeinterval(whateverhisspaceposition)shouldremain
B =G−1◦Λ◦G, (3) unaffected by a change in reference frame: i.e.
T T t t′
whAersewΛasisdtohneeLinortehnetzenbeorogsyt-,mwohmicehntaucmtsslpinaecaer,lyweonwaXn.t B(cid:18) xp (cid:19)=(cid:18) xp′ (cid:19) and B(cid:18)xp (cid:19)=(cid:18)xp (cid:19). (9)
to define a space-time vector composition law covariant
In the classical space we have the invariant (ξ0)2−(ξ)2
under the action of deformed boost:
whose image in the physical space is obviously invariant
x +ˆx =G−1[G[x ]+G[x ]]. (4) under the action of deformed boost. This, together with
a b a b
the requirement of invariance under rotations, allows to
This definition implies that a vector can always be writ- demonstrate that the above relations have to satisfy t=
ten as the sum of two (or more) vectors and this decom- t′ and x=x′.
position is covariant under boosts: The above result is valid if we assume the existence of
i) only a temporal invariant scale, ii) only a spatial in-
δˆ =G−1[G[x ]−G[x ]]=δˆ +ˆ δˆ , (5) variantscale,iii)bothatemporalandaspatialinvariant
(ab) a b (ac) (cb)
scale.
For example, if we assume an invariant time scale T ,
p
the vector (T ,x) (with arbitrary x) under the action
p
1 Fromherewewillomitalltheindexes. Whennecessarywewill of a boost transformation has to be modified in such
usetheconvention aµ=(a,a) a way to keep Tp fixed as well as to leave the Casimir
3
C(T ,x)=(ξ0)2−(ξ)2 =(G0(T ,x))2−(G(T ,x))2 in-
p p p
variant. Since, in physically interesting cases, C(Tp,x) G[T]
depends on the spatialcoordinatesonly via the modulus
|x|, we get C(T ,|x|) = C(T ,|x′|) i.e. |x′| = |x|. We T
p p
getthe interesting resultthat, assuming onlya temporal
invariant, all the vectors with time component equal to
the invariant quantity (and any x) keep the modulus of -T_p
the spatialcoordinate unchangedwhen boosted. Clearly T_p
this does not imply that any length scale is an invari-
ant one since, in general, B(t,x) = (t′,x′) with x 6= x′
if t 6= T . The same analysis shows that, if we assume
p
the existence of a lengthinvariant,the Casimir invariant
implies B(t,x )=(t,x ) for any t value.
p p
The final result is that our invariant vector(s) has to
satisfy the following relation
FIG. 1: An example of thefunction G0
Bδˆ =δˆ , (10)
p p
or, equivalently,
scale is reached. It is clear that both the images of T
p
ΛG[δˆp]=G[δˆp]. (11) and −Tp are zero: we loose the uniqueness (G is not in-
vertible at this points) and an invertible extension of G
The invariant points of standard Lorentz transforma- can not be defined on points in the interval [−Tp,Tp].
tions are0 and∞(i.e.vectorswhereallthe components What is happening is that, to enforce a physical in-
are zero or infinity) and we can only use one of this two variant, we had to choose a G function such that all the
points to ensure the condition (10). In DSR in the mo- vectors of the classical space (the vertical axis in fig. 1)
mentum space we have a similar situation: in that case map,in the physicalspace (the horizontalaxis infig. 1),
the fixedpointatinfinity isusedto guaranteeinvariance into vectors with spatial or temporal length larger than
of (high) energy or momentum scales. Since we demand the physical invariant scale. When we try to consider
ourmodel tobe equivalentto usualspace-time whenthe vectors in the physical space with spatial or temporal
distances(times)aremuchbiggerthantheinvariantscale length equal or smaller than the invariant scale we have
we expect G to approach the identity in this conditions no counterpart in the classical space. For example, if we
and this implies G[∞]=∞. want to consider a vector x that is half of the invariant
Then the only possibility left is that the image of the vector we should write
invariant vectors under G must be 0. This condition is
equivalent to the condition we used to define the neu- x+ˆx=δˆp (13)
tral element of translations e. This result indicates that
andthisequationcorrespondstoG[x]=−G[x]or,equiv-
at this point we are getting in troubles: infinitely many
different physicalvectorsare mapped to the same vector alently, G[x]=0, i.e. x=δˆp.
in the classical space i.e. the function G can not be any With a speculative attitude we can imagine that the
more invertible when we approach the invariant scale. model suggests that a process of space (or time) mea-
Forexampleifweconsidertwodistinctpointsx ,x and surement can never give a result with a precision better
a b
assumethat their distance (inspace and/ortime) equals than the invariant length since all points that differ by
the invariant quantity we have to write an invariant length are practically indistinguishable.
Inthe particularexampleoffig. 1,ifT isa time inter-
x −ˆx =δˆ (12) valmeasuredinagivenspacepoint,wehavetoconclude
b a p
that we can not speak of kTk≤T or, equivalently, this
p
and, after applying the G function to previous relation, canbeunderstoodasanindeterminationinthemeasure-
we get G[x ]=G[x ] in contrastwith our assumption of mentoftimewhenthePlanckscaleisreached. Therefore
a b
G invertible. We conclude that, to satisfy the invariance we can imagine this signals an intrinsic obstruction to
condition,weloosetheuniquenessoftheneutralelement measure a time (distance) when it becomes of the order
of translations and therefore its group structure. of the invariant scale.
In figure 1 we give an explicit realization of the tem- Two comments are in order here. We have demon-
poral part of the G function for a vector with a fixed stratedtheimpossibilityofimplementingaminimalscale
spatial part and a varying temporal component in order (in space, time or both) in the space-time by means of a
to represent it as an ordinary one-dimensional function. DSR-like approach. However, it is clear that –as occurs
As required, we recover the standard Lorentz transfor- in DSR in the momentum space– it is always possible to
mation for large times i.e. the G function becomes the map the invariant scale (space or time) to infinity in the
identity for t >> T while goes to 0 once the invariant classical space time and to map the zero of the physical
p
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space time to the zero of the classical space time. The We understand why we encounter differences respect
resultingspace-timewilldifferfromtheLorentzinvariant totheDSRapproachinmomentumspace. Forthelatter
one at large scales but it will not suffer the problems we the invariant momentum is realized mapping the phys-
discussed above: it will have a maximal scale (and pos- ical momentum space up to the maximum momentum
sibly a minimum momentum) and will mirror the usual (energy) to the entire classical space (i.e. we obtain the
DSR in the momentum space. invariantscale using the standard Lorentz fixed point at
Finally,letussaythatthestatementthattheapproach infinity). In present case instead, we are forced to map
with a minimal scale is not possible, but the one with a the invariant scale to the first Lorentz fixed point: the
maximalscaleisallowed,canbe understoodbyadimen- originofvectorspace(recallthatbothincoordinateand
sional argument. If we assume: i) a continuous differ- momentum space the Lorentz transformations are linear
entiable manifold structure for the space time, ii) the and the only two fixed point are zero and infinity). This
the existence of a length scale ℓ, it is always possible to procedure unavoidably leaves all vectors with spatial or
express any quantity depending on the coordinates as a temporal length smaller or equal to the invariant length
seriescontainingonlynegativepowersofℓ. Ifweputthe without counterpart in the classical space.
extra condition that iii) it should exist a smooth limit
Themaindifferenceofourresultwithotherapproaches
towardthe undeformedspacetime, itis clearthe smallℓ
in the literature [7], resides in the definition of the com-
limitcannotbeaccepted. Thelimitoflargeℓ,instead,is
position law (4), which has been introduced in order to
welldefinedandthe interpretationofthe scaleasamax-
extend the notion of covariance. We think that the as-
imal scale becomes clear. These arguments were already
sumption of the standard composition law for vectors in
discussed in [14].
the physical space is not correct. To assume (4) is an
unavoidable step if we want to consistently construct a
non linear realization of Lorentz invariance.
III. CONCLUSIONS
At the end of our construction we arrive to inconsis-
tencies. A firstpossibility is, of course,to reject the idea
Inthisnotewehaveexploredapossiblescenarioforthe
that a DSR-like transformation can be defined in a co-
a space-time with an invariant scale in a DSR based ap-
ordinate space-time with an invariant scale. Indeed the
proach. We started constructing a non linear realization
connectionofDSRmodels(definedinmomentumspace)
of Lorentz transformations defining a non linear, invert-
with coordinate space is unclear [15] and may be this
ible map between the physical space and an auxiliary
connection will be realized only via a completely differ-
space where the Lorentz group acts linearly. In doing
ent approach.
thatweintroduceadeformedcompositionlawforvectors
Inanycase,sincethenon-linearrealizationsofLorentz
in the physical space to guarantee its invariance under
symmetry have been very successful in constructing new
boosts. Up to this point we can still define a translation
versions of particle’s momentum space, we think it is
operator compatible with the deformed action of boosts,
worthwhile to explore the same technique further in the
and this translations define a group.
attempt to understand the possible structure of space-
Then we try to impose the physical condition that
time.
some (small) measurable physical length (in space or
time) should remain invariant under boosts. To define Apossiblewayoutistoacceptthattranslationsdonot
the space length we use the standard expression for the formanymoreagroupandinterpretthe peculiarbehav-
modulus of a vector (againin full analogy with the DSR iorofthenonlinearmappingGasaclueforafundamen-
approach in momentum space). We showed that the in- tal indetermination at the scale of the invariant length.
variance requirement is incompatible with i) a well de- This interpretation suggests a sort of (space-time) un-
fined (and invertible) map for all the physical space vec- certainty, something that resembles what is expected to
torsandii)with the groupstructure forthe translations happen in a non commutative space-time. Alternatively
since the neutral element (and consequently the inverse we can speculate that defects (as in a worm-hole QG
of any given translation) can not be unique. vacuum [16]) might be present in space-time.
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New developments in fundamental interaction theories ,