High Energy Commutators in Particle, String and Membrane Theories

0.21 MB·English

by  W. Chagas-Filho

High Energy Commutators in Particle,
String and Membrane Theories
5 W. Chagas-Filho
0
Departamento de Fisica, Universidade Federal de Sergipe, Brazil
0
2
February 1, 2008

n
a
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Abstract
7

We study relativistic particle, string and membrane theories as defin-
1
ing field theories containing gravity in (0+1), (1+1) and (2+1) space-
v
timedimensions, respectively. Weshowhowanoff shellinvarianceofthe
5
4 masslessparticleactionallowstheconstructionofanextensionofthecon-
0 formalalgebraandinducesatransitiontoanon-commutativespace-time
1 geometry. Thisnon-commutativegeometryisfoundtobepreservedinthe
0 space-timesupersymmetricmasslessparticletheory. Itisthenshownhow
5 thebasicbosoniccommutatorswefoundforthemasslessparticlemayalso
0 be encountered in the tensionless limit of string and membrane theories.
h/ Finally we speculate on how the non-locality introduced by these com-
t mutators could beused to construct a covariant Newtonian gravitational
-
p field theory.
e
h
1 Introduction
:
v
i
X Since some time ago there has been some interest in explaining a very small
r measured value for the cosmological constant. It is believed that this tinny
a
value is a reflection of the presence of large amounts of dark matter in the
universe [1,2]. Because the measured value is positive, dark matter should be
contributingwithpositivegravitationalpotentialenergyasaconsequenceofits
interactionwithusualmatter. Thisimpliesthatdarkmattershouldrepelusual
matter. We may add to this situation the observation[3], using high resolution
measurementsofthe cosmicmicrowavebackgroundradiation,thatthe universe
is effectively flat. On theoreticalgrounds there is an interesting result obtained
bySiegel[4],basedinscalarparticletheory,thattherearestringycorrectionsto
gravitationandthatthesecorrectionsmakethegravitationalforcelessattractive
at very short distances. This work is an attempt to sketch a mathematical
pictureinwhichalltheabovementionedsubjectswouldoccupymathematically
natural and consistent positions. The central mathematical concept in this
attempt will be non-commutativity of space-time coordinates
1
Non-commutativity is the central mathematical concept expressing uncer-
tainty in quantum mechanics, where it applies to any pair of conjugate vari-
ables. But there are other situations where non-commutativity may manifest
itself. For instance, in the presence of a magnetic field even momenta fail to
mutually commute. The modern trend is to believe that one might postulate
non-commutativity for a number of reasons. One of these is the belief that
in quantum theories including gravity, space-time must change its nature at
the Planck scale. This is because quantum gravity has an uncertainty princi-
ple which prevents one from measuring positions to better accuracies than the
Planck length. The momentum and energy required to make such a measure-
ment will itself modify the space-time geometry at this scale [5]. This effect
could then be modeled by a non-vanishing commutation relation between the
space-time coordinates.
For most theories, postulating an uncertainty relation between space-time
coordinates will lead to a non-local theory and this may conflict with Lorentz
invariance. However, there is a remarkable exception to this situation: string
theory. String theory is not local in any sense we now understand and indeed
it has more than one parameter characterizing this non-locality. One of these
is the string length l , the average size of a string. As we will see in this work,
s
another useful parameter is the string tension T , the string analogue of the
particle’s mass. The string tension defines an energy scale in string theory and
the vanishing tension limit is expected to describe the string dynamics at the
Planck scale [6].
Historically,the firstuse of non-commutativegeometry in string theory was
in Witten’s formulation of open string field theory [7]. Witten’s formulation
of non-commutative string field theory is based in the introduction of a star
product which is defined as the overlap of half of two strings. The central idea
in this formulation is to decompose the string coordinates xµ[σ] as the direct
sum xµ[σ]=lµ[σ] rµ[σ] , where lµ[σ] and rµ[σ] are respectively left and right
⊕
coordinatesofthestring,definedrelativetothemidpointatσ =π/2.Ifwenow
consider the fields of two strings, say A(lµ[σ],rµ[σ]) and B(lµ[σ],rµ[σ]), then
1 1 2 2
Witten’s star product gives
A(lµ[σ],rµ[σ])⋆B(lµ[σ],rµ[σ])=C(lµ[σ],rµ[σ]) (1.1)
1 1 2 2 1 2
where the field C(lµ[σ],rµ[σ]) is given by the functional integral
1 2
C(lµ[σ],rµ[σ])= [dz]A(lµ[σ],zµ[σ])B(zµ[σ],rµ[σ]) (1.2)
1 2 Z 1 2
and zµ[σ] = rµ[σ] = lµ[σ] corresponds to the overlap of half of the first string
1 2
with half of the second string.
Witten’s formulation of open bosonic string field theory has experienced a
rebirth through new physical insights and technical advances. In particular, it
was shown by Bars [8] that Witten’s star product, originally defined as a path
integral that saws two strings into a third one, is equivalent to the Moyal star
product [9], which is the central mathematical concept in the familiar formu-
lation of non-commutative geometry. But this equivalence is rather subtle. As
2
shownin[8],Witten’sstarproductisequivalenttotheusualMoyalstarproduct
intherelativisticphasespaceofonlytheevenmodesofoscillationofthestring.
ThisisbecausethemappingoftheWitten⋆totheMoyal⋆necessarilyrequires
that the Fourier space for the odd positions be named as the even momenta
since (x ,p ) arecanonicalunder the Moyalstar-product. These developments,
e e
and many others that have recently appearedin the literature, led to the belief
that non-commutative geometry naturally arises in string field theory.
However,becauseitwasdefinedforapplicationsintraditionalquantumme-
chanics,the Moyalstar product leaves unchanged the Heisenberg commutation
relationsbetweenthecanonicaloperators. Intheusualformulationsoffieldthe-
ory on non-commutative spaces, the position coordinates satisfy commutation
relations of the form [10,11]
[qµ,qν]=iθµν (1.3)
whereθµν isaconstantanti-symmetrictensor. Theeffectofsuchamodification
isreflectedinthe momentumspaceverticesofthe theoryby factorsofthe form
[12]
exp iθµνp q (1.4)
µ ν
{ }
Thefirstquantizedstringinthe lightconegauge,asaperturbativenon-commu
tative theory based in a commutation relation such as (1.3), was studied in
[12]. It was found by the authors in [12] that, apart from trivial factors, the
structure of the non-commutative theory is identical to the structure of the
commutative one. It thus seems that nothing substantially new is introduced
instring theory by the use of a Moyalstar-productstructure, orby postulating
another star-product structure based on commutators such as (1.3).
In this workwe givea smallcontributionto this subject by showinghowwe
may use the special-relativistic ortogonality between velocity and acceleration
toinducetheappearanceofanewinvarianceofthemasslessrelativisticparticle
action. This invariance may then be used to generate transitions to new space-
time coordinates which, in the simplest case, satisfy commutator relations of
the form
[x ,x ] iM∗ (1.5)
µ ν ∼ µν
where M∗ is an off shell anti-symmetric operator that generates space-time
µν
rotations. This commutation relation is part of a modified Heisenberg com-
mutator structure which reduces to the usual canonical commutators when the
mass shell condition is imposed. Most part of this work is the explicit verifi-
cation that the basic commutator structure we found for the massless particle
mayalsobeencounteredinstringandmembranetheory. Aspointedoutin[11],
anyfinite-dimensionaldeformedPoissonstructurecanbecanonicallyquantized.
It may be then interesting to investigate what quantum effects we may find in
particle, string and membrane theories if we construct the corresponding non-
commutative theories based on commutators such as (1.5) instead of (1.3).
Animportantpointtoourconsiderationshereisthatspace-timenon-commu-
tativity introduces an associated non-locality [10]. In non-commutative gauge
3
theories based on the commutator (1.3) this non-locality allows the interpre-
tation of the gauge particles as dipoles carrying opposite charges of the corre-
sponding gauge theory [10, 12]. In the context of string theory, it was shown
[13] that an open string with both ends on a (D2-D0)-brane system in which
a magnetic field is defined looks like an electric dipole if a background electric
field is added. The dipole-background interaction gives the flux modifications
needed for the Born-Infeld action on the non-commutative torus and results in
a Hamiltonian that depends on two integers,n and m, whose ratio m gives the
n
density of D0-branes distributed on the D2-brane [13].
In this work we study relativistic particle, string and membrane theories as
gaugesystemsinwhichthegaugeinvarianceisgeneralcovariance. Weinterpret
these relativistic objects as defining field theories containing gravity in (0+1),
(1+1) and (2+1) space-time dimensions, respectively. The paper is organized
as follows: For the task of clarity, in section two we briefly review the concept
of a space-time conformalvector field and the associated conformalalgebra. In
section three we discuss particle theory and show how we may induce an off
shell invariance of the massless particle action and how this invariance allows
the construction of an off shell extension of the conformal algebra and how it
permits a transition to non-commutative space-time coordinates. In section
four we show that the bosonic commutator structure we found for the massless
particle is preserved in the space-time supersymmetric extension of the theory.
Section five deals with bosonic strings. We present an alternative string action
thatin principle does notrequirea criticaldimensionandthat allowsa smooth
transitiontothetensionlesslimitofstringtheory. Weshowhowthisalternative
stringactionmay be gauge-fixedto yielda non-commutativetheory atthe uni-
tary tension value and to give the extension of the particle commutators in the
vanishing tension limit. Section six extends these results to superstring theory
and section seven briefly describes the same situation in relativistic membrane
theory. We present our conclusions in section eight, where we also speculate
on the possibility of using the non-locality introduced by our commutation re-
lations to provide the conceptual basis for a covariant Newtonian gravitational
field theory.
2 Conformal Vector Fields
Consider the Euclidean flat space-time vector field
Rˆ(ǫ)=ǫµ∂ (2.1)
µ
such that
∂ ǫ +∂ ǫ =δ ∂.ǫ (2.2)
µ ν ν µ µν
The vector field Rˆ(ǫ) gives rise to the coordinate transformation
δxµ =Rˆ(ǫ)xµ =ǫµ (2.3)
4
Thevectorfield(2.1)isknownastheKillingvectorfieldandǫµ isknownasthe
Killing vector. One can show that the most general solution for equation (2.2)
in a four-dimensional space-time is
ǫµ =δxµ =aµ+ωµνx +αxµ+(2xµxν δµνx2)b (2.4)
ν ν
−
The vector field Rˆ(ǫ) for the solution (2.4) can then be written as
1
Rˆ(ǫ)=aµP ωµνM +αD+bµK (2.5)
µ µν µ
− 2
where
P =∂ (2.6)
µ µ
M =x ∂ x ∂ (2.7)
µν µ ν ν µ
−
D =xµ∂ (2.8)
µ
K =(2x xν δνx2)∂ (2.9)
µ µ − µ ν
P generatestranslations,M generatesrotations,D generatesdilatationsand
µ µν
K generates conformal transformations in space-time. The generators of the
µ
vector field Rˆ(ǫ) obey the commutator algebra
[P ,P ]=0 (2.10a)
µ ν
[P ,M ]=δ P δ P (2.10b)
µ νλ µν λ µλ ν
−
[M ,M ]=δ M +δ M δ M δ M (2.10c)
µν λρ νλ µρ µρ νλ νρ µλ µλ νρ
− −
[D,D]=0 (2.10d)
[D,P ]= P (2.10e)
µ µ
−
[D,M ]=0 (2.10f)
µν
[D,K ]=K (2.10g)
µ µ
[P ,K ]=2(δ D M ) (2.10h)
µ ν µν µν
−
[M ,K ]=δ K δ K (2.10i)
µν λ νλ µ λµ ν
−
5
[K ,K ]=0 (2.10j)
µ ν
The commutator algebra (2.10) is the conformal space-time algebra in four di-
mensions. Noticethatthecommutators(2.10a-2.10c)correspondtothePoincar´e
algebra. The Poincar´e algebra is a sub-algebra of the conformal algebra. Let
us now see how we can extend the conformal algebra, and how this extended
algebra is related to a non-commutative space-time geometry.
3 Relativistic Particles
A relativistic particle describes is space-time a one-parametertrajectory xµ(τ).
The dynamics of the particle must be independent of the parameter choice. A
possible formof the actionis the one proportionalto the arclength traveledby
the particle and given by
S = m ds= m dτ x˙2 (3.1)
− Z − Z −
p
In this work we take the parameter τ to be the particle’s proper time, m is the
particle’s mass and ds2 = δ dxµdxν. A dot denotes derivatives with respect
µν
−
to τ and we use units in which h¯ =c=1.
Action(3.1)isobviouslyinadequatetostudythemasslesslimitofthetheory
andsowemustfindanalternativeaction. Suchanactioncanbeeasilycomputed
by treating the relativistic particle as a constrained system. In the transition
to the Hamiltonian formalism action (3.1) gives the canonical momentum
m
p = x˙ (3.2)
µ µ
√ x˙2
−
and this momentum gives rise to the primary constraint
1
φ= (p2+m2)=0 (3.3)
2
WefollowDirac’s[14]conventionthataconstraintissetequaltozeroonlyafter
allcalculationshavebeenperformed. ThecanonicalHamiltoniancorresponding
toaction(3.1),H =p.x˙ L,identicallyvanishes. Thisisacharacteristicfeature
−
of reparametrization-invariantsystems. Dirac’s Hamiltonian for the relativistic
particle is then
1
H =H +λφ= λ(p2+m2) (3.4)
D
2
where λ(τ) is a Lagrange multiplier. The Lagrangianthat correspondsto (3.4)
is
L = p.x˙ H
D
−
1
= p.x˙ λ(p2+m2) (3.5)
− 2
6
Solvingthe equationofmotionfor p that followsfrom(3.5)andinsertingthe
µ
result back in it, we obtain the particle action
1 1
S = dτ( λ−1x˙2 λm2) (3.6)
Z 2 − 2
The great advantage of action (3.6) is that it has a smooth transition to the
m=0 limit.
Action (3.6) is invariant under the Poincar´etransformations
δxµ =aµ+ωµxν (3.7a)
ν
δλ=0 (3.7b)
Invariance of action (3.6) under transformation (3.7a) implies that we can con-
struct a space-time vector field corresponding to the first two generatorsin the
right of equation (2.5). These generators realize the Poincar´e algebra (2.10a-
2.10c).
Now we make a transition to the massless limit. This limit is described by
the action
1
S = dτλ−1x˙2 (3.8)
2Z
The massless limit is usually considered to describe the high energy limit of
relativistic particle theory. The canonical momentum conjugate to xµ is
1
p = x˙ (3.9)
µ µ
λ
The canonical momentum conjugate to λ identically vanishes and this is a pri-
maryconstraint,p =0. ConstructingthecanonicalHamiltonian,andrequiring
λ
the stability of this constraint, we are led to the mass shell condition
1
φ= p2 =0 (3.10)
2
Let us now study which space-time symmetries are present in this limit. Being
the m=0 limit of (3.6), action(3.8) is also invariantunder the Poincar´etrans-
formation(3.7). Themasslessaction(3.8)howeverhasalargersetofspace-time
invariances. It is also invariant under the scale transformation
δxµ =αxµ (3.11a)
δλ=2αλ (3.11b)
where α is a constant, and under the conformal transformation
δxµ =(2xµxν δµνx2)b (3.12a)
ν
−
7
δλ=4λx.b (3.12b)
where b is a constant vector. Invariance of action (3.8) under transformations
µ
(3.7a), (3.11a) and (3.12a) then implies that the full conformal field (2.5) can
be defined in the massless sector of the theory.
Itisconvenientatthispointtorelaxthemassshellcondition(3.10)because
the massless particle will be off mass shell in the presence of interactions [15].
We may now use the fact that we are dealing with a special-relativistic system.
Special relativity has the characteristic kinematical feature that the relativistic
velocity is always ortogonal to the relativistic acceleration ( see, for instance,
[16]). Whilethemathematicsinvolvedatthispointisverysimple,thephysical
implications of this ortogonality condition are rather unexplored. In this work
wemakeanattempttopartiallyclarifythesephysicalimplicationsinthecontext
of massless relativistic particle theory. We start by adopting the point of view
that specialrelativity automaticallyimplies the ortogonalityconditionbetween
the velocity and the acceleration. Reasoning in this way, we find that we may
use this ortogonalityto induce the presence of a new invariance of the massless
particleaction. Then,asaconsequenceofthefactthataction(3.8)isaspecial-
relativistic action, it is invariant under the transformation
xµ x˜µ =exp β(x˙2) xµ (3.13a)
→ { }
λ exp 2β(x˙2) λ (3.13b)
→ { }
where β is an arbitrary function of x˙2. We emphasize that although the or-
togonality condition must be used to get the invariance of action (3.8) under
transformation(3.13),this conditionisnotanexternalingredientinthe theory.
Infact,the ortogonalitybetweenthe relativisticvelocityandthe accelerationis
an unavoidable condition here, it is an imposition of special relativity. We are
just using this aspect of special relativity to get an invariant action.
Now, invariance of the massless action under transformations (3.13) means
that infinitesimally we can define the scale transformation δxµ = αβ(x˙2)xµ ,
where α is the same constant that appears in equations (2.4) and (2.5). These
transformations then lead to the existence of a new type of dilatations. These
new dilatations manifest themselves in the fact that the vector field D of equa-
tion (2.8) can be changed according to
D =xµ∂ D∗ =xµ∂ +β(x˙2)xµ∂ =D+βD (3.14)
µ µ µ
→
Infact,becauseallvectorfieldsinequation(2.5)involvepartialderivativeswith
respecttoxµ andβ isafunctionofx˙µ only,wecanalsointroducethegenerators
P∗ =P +βP (3.15)
µ µ µ
M∗ =M +βM (3.16)
µν µν µν
8
K∗ =K +βK (3.17)
µ µ µ
and define the new space-time vector field
1
V∗ =aµP∗ ωµνM∗ +αD∗+bµK∗ (3.18)
0 µ − 2 µν µ
The generators of this vector field obey the algebra
[P∗,P∗]=0 (3.19a)
µ ν
[P∗,M∗ ]=(δ P∗ δ P∗)+β(δ P∗ δ P∗) (3.19b)
µ νλ µν λ − µλ ν µν λ − µλ ν
[M∗ ,M∗ ]=(δ M∗ +δ M∗ δ M∗ δ M∗ )
µν λρ νλ µρ µρ νλ− νρ µλ− µλ νρ
+β(δ M∗ +δ M∗ δ M∗ δ M∗ ) (3.19c)
νλ µρ µρ νλ− νρ µλ− µλ νρ
[D∗,D∗]=0 (3.19d)
[D∗,P∗]= P∗ βP∗ (3.19e)
µ − µ − µ
[D∗,M∗ ]=0 (3.19f)
µν
[D∗,K∗]=K∗+βK∗ (3.19g)
µ µ µ
[P∗,K∗]=2(δ D∗ M∗ )+2β(δ D∗ M∗ ) (3.19h)
µ ν µν − µν µν − µν
[M∗ ,K∗]=(δ K∗ δ K∗)+β(δ K∗ δ K∗) (3.19i)
µν λ λν µ− λµ ν λν µ− λµ ν
[K∗,K∗]=0 (3.19j)
µ ν
Noticethatthevanishingbracketsoftheconformalalgebra(2.10)arepreserved
as vanishing in the above algebra, but the non-vanishing brackets of the con-
formal algebra now have linear and quadratic contributions from the arbitrary
functionβ(x˙2). Algebra(3.19)is anoffshellextensionofthe conformalalgebra
(2.10).
Now consider the commutator structure induced by transformation(3.13a).
We assume the usual commutation relations between the canonical variables,
[x ,x ] = [p ,p ] = 0 , [x ,p ] = iδ . Taking β(x˙2) = β(λ2p2) in transfor-
µ ν µ ν µ ν µν
mation (3.13a) and transforming the p in the same manner as the x , we find
µ µ
that the new transformed canonical variables (x˜ ,p˜ ) obey the commutators
µ µ
[p˜ ,p˜ ]=0 (3.20)
µ ν
9
[x˜ ,p˜ ]=iδ (1+β)2+(1+β)[x ,β]p (3.21)
µ ν µν µ ν
[x˜ ,x˜ ]=(1+β) x [β,x ] x [β,x ] (3.22)
µ ν µ ν ν µ
{ − }
written in terms of the old canonical variables. These commutators obey the
non trivial Jacobi identities (x˜ ,x˜ ,x˜ ) = 0 and (x˜ ,x˜ ,p˜ ) = 0. They also
µ ν λ µ ν λ
reduce to the usual canonical commutators when β(λ2p2)=0.
Itmaybequestionedhere,onthebasisofthedefinition(3.9)oftheclassical
canonical momentum, if the quantum momentum should not transform in the
inverse way of the xµ under (3.13), as is the case for the classical momemtum.
Thisisaninterestingpointinthemathematicaldevelopmentshere. Infact,the
identificationofthe correctquantumcanonicalmomentumis a typicalproblem
in field theories defined over a non-commutative space-time [24]. An example
is non-commutative quantum string field theory, where the momentum space
Fouriertransformoftheoddvibrationalpositionmodesbehaveasthequantum
canonicalmomentaconjugateto theevenvibrationalpositionmodes [8]. Inthe
context of this work, transforming the quantum momenta as the classical ones
implies substituting the commutator (3.21) by the commutator
[x˜ ,p˜ ]=(1+β) iδ (1 β) [x ,β]p
µ ν µν µ ν
{ − − }
ThiscommutatoralsoreducestotheusualHeisenbergcommutatorwhenβ(λ2p2)=
0. However, the Jacobi identity (x˜ ,x˜ ,p˜ ) = 0, based on commutator (3.22)
µ ν λ
and on the above commutator does not seem to close. If this is the case, the
classical and quantum canonical momenta of the massless particle on a non-
commutativespace-timearerelatedbyascaletransformationofthetype(3.13).
That is, p =exp 2β p .
q c
{ }
The simplest example of non-commutative space-time geometry induced by
transformation(3.13)is the casewhenβ(λ2p2)=λ2p2. The newpositions then
satisfy
[x˜ ,x˜ ]= 2iλ2M∗ (3.23)
µ ν − µν
where M∗ is the extended off shell operator of Lorentz rotations given by
µν
equation (3.16). The commutator (3.23) satisfies
Tr[x˜ ,x˜ ]=0 (3.24)
µ ν
Z
asis the casefora generalnon-commutativealgebra[10]. Fromequation(3.23)
we may say that there exists an inertial frame in which the uncertainty in-
troduced by two simultaneous position measurements is an off shell rotation
in space-time. This appears to confirm the observation in [25] that the non-
commutative space-time is to be interpreted as having an internal angular mo-
mentum throughout.
Wemaynowconsiderthequestionofifwecanfindamathematicalstructure
that wouldbehave like a dipole in the context ofmassless particle theory. Such
10