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The algebraic structure of the generalized uncertainty principle
Phystcs I.xtters B 319 ( 19t}3 ) 83-86 Norlh-llolland
PHYSICS LETTERS B
The algebraic structure of the generalized uncertainty principle Michele Maggiore I.VF.V and Diparttmemo dl 11 ~u'a doll'[ "m~erstta, pm::a "li,rncelh 2. 1.56 I00 Ptm. Italy
ReccDed I 3 September 1993 Editor: R Gatto
Wc shov, that a deformation of the Hetscnber8 algebra which depends on a d~mcnstoaful parameter ~c~sthe al$cbraic structure v~hlch undcfltcs the gencrahzcd unccrtamt) prmople m quantum gravit). The deformed algebra and therefore the form of thc generahzcd uncertamu, pnnoplc are fixed umquel.~ b) rather simple assumptton,~. The stnng theor) result i$ reproduced expand. mg our result at fi~t order in ..~/.ttet We also bnefl) comment on possible implications for Lorentz mvariance at the Planck scale.
In recent years it has been suggested that measurements in q u a n t u m gravity are governed by a generalized uncertainty princ=ple h A~"i> -~p + const. × G A p
(I)
(G is Newton's constant ). At energies much below the Planck mass .l,[vt the extra term in eq. ( I ) is irrelevant and the Heisenberg relation is recovered. As we approach the Planck mass, this term becomes important and it is responsible for the existence of a m i m m a l observable length on the order o f the Pianck length. The result ( I ) was first suggested in the context of string theor). [ 1-4 ], in the kinematical region where 2 G E is smaller than the string length (for a review, see ref. [ 5 ] ). However. heuristic arguments [ 6 ] suggest that this formula might have a more general validity in q u a n t u m graver), and it is not necessaril) related to strings. It ~s therefore natural to ask whether there is an algebraic structure which reproduces eq. ( l ) (or, more m general, which reproduces the existence o f a m i n i m a l obserwable length), in the same way in which the Heisenberg uncertainty principle follows from the algebra Ix. p] = ih. In this Letter we answer (in the affirmative) this question. Our strateg~ ts as follows. Since it is relativel.~ clear that no Lie algebra can reproduce eq. ( 1 ), we turn out attention to deformed algebras. A de-
formed algebra is an associative algebra where a comm u t a t o r is defined which is n o n - l i n e a r in the elements o f the algebra; and there is a deformation p a r a m e t e r such that, in an a p p r o p r i a t e limit, a Lie algebra ~s recovered. We therefore look for the most general deformed algebra =* which can be constructed from coordinates x, and m o m e n t a p, (~ = I, 2 . 3 ). We restrict the range o f possibilities making the following assumptions. ( I ) The three-dimensional rotation group is not deformed; the angular m o m e n t u m d satisfies the undeformed S U ( 2 ) c o m m u t a t i o n relations, and coordinate and m o m e n t a satisfy the undeformed commutation relations [J,, x~] =icu~,'j,, [J,, p~] = i(,,kp~. (2) The m o m e n t a c o m m u t e s between themselves: [p,, pj] = 0 . so that also the translation group is not deformed. ( 3 ) The Ix. x] and Ix, p] c o m m u tators depend on a deformation p a r a m e t e r x with dimensions o f m a s s . In the hmit x.-,oc. (that is, x m u c h larger than any energy ), the canonical c o m m u t a t i o n relations are recovered. Note that we are ready to accept a non-zero comm u t a t o r between the x's. While this is at first surprising, we will see that i f x ~ Planck mass, the non-commutativity shov,.s up only at the level o f the Planck =t We roll not require the extstenor o f a coproduct and o f an antipode, v, luch would promote the deformed algebra to a quan. tum algebra, see below
0370.2693/9315 Oh.U0 ~, 1903 Elsc',ler Socnce Pubhshcr~ B.V All r=$hts rcsencd.
S~,DI O'~70.26o3( 93 )E 132~).S
83
Volume 319. number 1.2.3
PIIYSI('S LETIER% B
length. The idea is m the spirit o f t h e approach to the structure ofspacetime at small distances suggested by non-commutative geometr)., pioneered b) Connes [7]. which underlies much of the apphcatlons of quantum groups to gravlt). One should also note that our formalism is not Lorentz co~ariant. We ~tll come back to this important point below. With these assumptions, the most general form of the E-deformed algebra is
[.~,. x, ] =
F1:a(E)
i~,,~J~
(2 )
E;
[x,. p,] = it7,6,, f( E).
(3)
ttere a ( E ) and . f ( E ) are real. dimensionless functions of E/E, and E ' = p : + m=: the angular momentum ,/is defined as dimensionless, so on the righthand side the dimensions are carried by h and p,"only. The fact that this is the most general form compattble v~ith our assumptions is clear from the follo~ mg consideratlons: the factors of i are determined b) the condition of hermiticit) of x,, p, and J,. The pov~ers of fi and ~,"are dictated by dimensional anal)sis. The tensor ¢,,~ in eq. ( 2 ) appears because ge assume that the three-dimensional rotation group is undeformed and then it ts the only tensor antis.~ mmetric in t.j: it must be contracted with J~ rather than p~ or .x~ because of parit). One might also add to the right-hand side of eq. (2) a term proportional to x , p . , - x / p , . ( Note that I.~ = ~,,~ (.~, pj - xj p, ) cannot be identified gith the angular momentum in the spinless case. since it does not satlsf) the SU( 2 ) commutation relations unlessf(E) = I ). However such a term can be eliminated with a non-local redefinition of coordinates. ,~ = g ( E ) x , . with a suitable function g ( E ) . In the second equation, again 6,, must appear because it is the onl.~ avadable tensor under rotation. In order to recover the undeformed limit, we further require that . f ( 0 ) = I and that a ( E ) is less singular than E- : as E .0. We neglect the possibllit? that the functions a. f d e p e n d also on other scalars like x: or x'9. Of course, the form of the functions a ( E ) . . t ( E ) is severely restricted by the Jacobi identities. Let us consider first the Jacobi identit) [.L. [x,. x,l]+ cyclic = 0. Using Ix,. 1-'] = ifi.f( E ) p , / E , and [x,. a ( E ) ] = ihf(E) ( p , / E ) d a / d E , we get da
~-..p..t= 0. ~4
(4)
23 Dc~cmber 1993
Since the Jacobl identity must be satisfied independently of the particular representation of the algebra. that is |ndependently of whether the condition p . l = 0 holds or not. we conclude that a ( E ) =const. With a redefimtion of E we can set this constant to +_ 1. The Jacobi identity [x,, Ix,. p~ ] ] + cyclic = 0 gives
.f(E) d/E
dE
l
-:l:
--;.
E"
(5)
where the upper ( lower ) sign correspond to the choice a=+l(-I). Since f ( 0 ) = l , eq. (5) gives .f(E) = [I ~ (E2/E2)]L 2. All other Jacobi identities are automaticall) satisfied. Of course, to satisfy the Jacobi identities is in general highly non-trivial, especiall~ in a deformed algebra, it is remarkable that in our case (t) there exists a solution: thus. we can Edeform the Heisenberg algebra. (it) The solution is unique (within our assumptions and apart from the _ sign ). In the following we only consider the lower sign =: and write the m-deformed Heisenberg algebra as h2" [x,. x,] = -
~
'
E"
tE,.,,J,.
[x,,p,]=i~3,,\l( +
(6) (7)
Actuall). we already denved eqs. ( 6 ). ( 7 ) in ref. [ 8 ]. following a rather different route: we considered the ~-deformed Poincare algebra suggested in ref. [9 ] and we discussed ho~, to generalize the Newton-Wtgner localization operator to the E-deformed case. We then obtained the algebra (6), ( 7 ). with x, identified with the E-deformed Ne~xon-Wigner operator and p,. J, identified with the generators of spatial translations and rotations of the E-deformed P o m c a ~ algebra. ( To compare with the results of ref. [ 8 ] one must observe that the present definition of E differs b.~ a factor of 2 from that used in refs. [q.8 ] and that in eq. ( 7 ) E 2 ts a shorthand for p2+DI2, while in the E-Poincar~ algebra the dispersion relation is. with the present definition o f E, p : + m : = ~c2 sinh " ( E / K ). ) The derivation that we have presented in this Let[he ¢,olutmn wtth the upper sign ts how¢,,¢r quttc intriguing. since.f(/'.) ts real b.~ dcfin,tton, the solution is ',al,d onl) for /" ~ ~ and dev:nbes a s)stcm which obc.,,s standard quantum mcchamcs for I..,=: x and sattsfle* Ix. p] = 0 at F.= K.
Volume 319. number 1.2.3
PHYSI('S LE'I'FERS B
ter is instead independent of whether we consider the usual or the deformed Poincar~ algebra. We do not attempt to define a coproduct on the deformed algebra given b.,, eqs. (6), ( 7 ), since a system composed of two parttcles localized at .x't~ and .r~:L respectively, cannot be considered as a single localized s,,stem and therefore we cannot reqmre, in general, that it can be described by a coordinate X = X ( . v ~'~, .x"~:' ) which, together with the total momentum, satisfies the same algebra as the constituents. Thus, the algebraic structure that we have presented is not a Hopfalgebra. nor a quantum algebra. From eq. (7) we immedtately derive the generalized uncertainty principle
ALAp,>~ ~d~,,: I +
;
(8)
Expanding the square root m powers of (F/K)-" and using (p2) =p:+ (.~p)'.. ~hcre (Ap)2= ~ (p<'p) ) : ) , at first order we obtain A~" APJ >~ .~ t~'t
2h"
]"
(9)
Thus. in the regime E,tzx. Ap<,x. we recover the string theoD result. The constant in eq. ( I ) is reproduced ff we identif.~ h/h" with the string length ;., times a numerical constant of order one. Note that if h/ x~,~, (and. as usual, the string length is larger than the Planck length ), then the condition E.*: x ~mplies the condition 2 G E ~ 2 , . If we consider instead the regime ( p > - ' ~ ( A p ) -~ : ~ x : we get h x
A.r I> const. X - .
( I0 )
while in the regime (p~2 :~ (.,.,~): ,(9>., :~ x2 wc get hl(p>l Ax">~const. X - - r Ap
h ::=' - . x
(II)
It is ~mportant to observe that the string theor) resuit. eq. ( i ). is reproduced in the appropriate kinematical limit after expanding our closed expression at first order in ,~o/x. However, only the full expression. eq. (8). has an underlying algebraic origin. We now comment on the issue o f Lorentz invariance. When we discuss the possible existence of a minimal observable length, we actually refer to a spa-
23 l)¢ccmlx.r IO93
tial length. Such a concept, of course, is not Lorentz mvariant. We can always perform a boost and squeeze any "minimal" length as much as v~.e like. Then, ifa minimal spatial length actually exists, we must consider the r e d interesting possibility that Lorentz invariance is not respected by quantum gravity at a scale on the order of h/x. In the formalism of the x-deformed Poincar6 algebra of ref. [ 9 ] this is explicit, since the Lorent,, algebra is deformed by the parameter x. In our approach, it is implicit in the use of the three-vectors x,, p, in place of the corresponding fourvectors. In th~s context it is interesting to mentmn that a breaking of Lorentz invariance at the string scale or at the Planck scale has been suggested vet). recently by Susskind in ref. [ 10], where it is argued that Lorentz contraction of particles must saturate as E ~ .lIVL, and a constant value should be approached (cir. our equations (10). ( ! l ) ). It might be interesting to obse~'e that the considerations of ref. [ 10] stem from Gedankcn experiments w~th black holes. and a (different) Gedanken experiment with black holes [6] was also the starting point of our investigation. Two final comments are in order. The first is that we have a scheme m which Newton's constant G (or equivalently x) enters the theoD at the kinematical level (see also ref. [ I I ] for early attempts in this direction ). The minimal length also emerges at a kinematical, rather than dynamical level. This feature is rather satisfying The second, related comment, is that this formalism can acquire substance onl) after a suitable dynamics has been added. This is particularly clear ffwe repeat our analysis in ( I + I ) dimensions. In this case only the equation Ix. p] =ihf(E) survives. Of course, we can ahvays perform a nonlocal redefinition of the coordinate, defining ,~= x / /~ E). so that [~, p] = ih. However, one would pay this at the level of the action, which would become nonlocal. The fact that a non-local particle Lagrangian can generate a mimmal observable length has been nicely shown in ref. [ 12 ]. 1 thank M. Mintchev, M. Shifman Vainshtein for useful discussions.
and
A.
X5
Volume 319, numl:)cr 1.2.3
PH YSIC:S LE IH'ERS B
References [ I ] G. Venczlano. Europh)s. I..ctt 2 ( 19861 199. Pro(: To,as Supcrslnng Workshop ( 1989 }; D. Gross. Pro<' ICHEP (Munich 19851. [2] D. Amatt. M. ('tafalon, and G. Vc'nezlano. Phys. L~I. B 19",' (19871 81. B 216 (19891 41. Intcrn. ] Mod. Phys. A 3 ( 19881 1615 Nucl Ph.,,s. B 347 (1990) 530. [ 3 ] D J . Gross and P.F Mcndc. Ph)s. Lelt B 197(19871 129; Nucl. Phys. B 303 ( 19881 407 [4] K. Komsht. G Paffutl and P. Provero. Ph)s Let|. B 234 (1990) 276 R. (imda. K. Komshs and P I)ro,,ero. Mod Ph}.s. Left. A 6 (1991) 1487
86
2 )) [.)cccmbcr 1993
[ 5 ] M. Ctafalon,. prcpnnl DFF 17219/92. [6] M. Maggiore, Phys Left. B 304 ( 19931 65. [ 7 } A Connes. Non.commutatsv¢ differentlaJ geometry. IHES Publ No. 62 (1986)
[8] M MagglO~. p~pnnt IFLIP-TH 19/93, hel>-th/9305163 (May 19931. [9] J. L.uk~er~t..-%.Nowtck~ and H. Ruegs. Phys. Lctt. B 293 ( 19921 344. [10] L. Susskmd. prepnnt SU-ITP-93-21. hep-lh/9308139 (Augusl 1993 ) [I I ] P.K. Townsend. Phys. Rev. D 15 (1977) 2795. [12]M. Kato, Phys. Lctt B245 (1990143.
PHYSICS LETTERS B
The algebraic structure of the generalized uncertainty principle Michele Maggiore I.VF.V and Diparttmemo dl 11 ~u'a doll'[ "m~erstta, pm::a "li,rncelh 2. 1.56 I00 Ptm. Italy
ReccDed I 3 September 1993 Editor: R Gatto
Wc shov, that a deformation of the Hetscnber8 algebra which depends on a d~mcnstoaful parameter ~c~sthe al$cbraic structure v~hlch undcfltcs the gencrahzcd unccrtamt) prmople m quantum gravit). The deformed algebra and therefore the form of thc generahzcd uncertamu, pnnoplc are fixed umquel.~ b) rather simple assumptton,~. The stnng theor) result i$ reproduced expand. mg our result at fi~t order in ..~/.ttet We also bnefl) comment on possible implications for Lorentz mvariance at the Planck scale.
In recent years it has been suggested that measurements in q u a n t u m gravity are governed by a generalized uncertainty princ=ple h A~"i> -~p + const. × G A p
(I)
(G is Newton's constant ). At energies much below the Planck mass .l,[vt the extra term in eq. ( I ) is irrelevant and the Heisenberg relation is recovered. As we approach the Planck mass, this term becomes important and it is responsible for the existence of a m i m m a l observable length on the order o f the Pianck length. The result ( I ) was first suggested in the context of string theor). [ 1-4 ], in the kinematical region where 2 G E is smaller than the string length (for a review, see ref. [ 5 ] ). However. heuristic arguments [ 6 ] suggest that this formula might have a more general validity in q u a n t u m graver), and it is not necessaril) related to strings. It ~s therefore natural to ask whether there is an algebraic structure which reproduces eq. ( l ) (or, more m general, which reproduces the existence o f a m i n i m a l obserwable length), in the same way in which the Heisenberg uncertainty principle follows from the algebra Ix. p] = ih. In this Letter we answer (in the affirmative) this question. Our strateg~ ts as follows. Since it is relativel.~ clear that no Lie algebra can reproduce eq. ( 1 ), we turn out attention to deformed algebras. A de-
formed algebra is an associative algebra where a comm u t a t o r is defined which is n o n - l i n e a r in the elements o f the algebra; and there is a deformation p a r a m e t e r such that, in an a p p r o p r i a t e limit, a Lie algebra ~s recovered. We therefore look for the most general deformed algebra =* which can be constructed from coordinates x, and m o m e n t a p, (~ = I, 2 . 3 ). We restrict the range o f possibilities making the following assumptions. ( I ) The three-dimensional rotation group is not deformed; the angular m o m e n t u m d satisfies the undeformed S U ( 2 ) c o m m u t a t i o n relations, and coordinate and m o m e n t a satisfy the undeformed commutation relations [J,, x~] =icu~,'j,, [J,, p~] = i(,,kp~. (2) The m o m e n t a c o m m u t e s between themselves: [p,, pj] = 0 . so that also the translation group is not deformed. ( 3 ) The Ix. x] and Ix, p] c o m m u tators depend on a deformation p a r a m e t e r x with dimensions o f m a s s . In the hmit x.-,oc. (that is, x m u c h larger than any energy ), the canonical c o m m u t a t i o n relations are recovered. Note that we are ready to accept a non-zero comm u t a t o r between the x's. While this is at first surprising, we will see that i f x ~ Planck mass, the non-commutativity shov,.s up only at the level o f the Planck =t We roll not require the extstenor o f a coproduct and o f an antipode, v, luch would promote the deformed algebra to a quan. tum algebra, see below
0370.2693/9315 Oh.U0 ~, 1903 Elsc',ler Socnce Pubhshcr~ B.V All r=$hts rcsencd.
S~,DI O'~70.26o3( 93 )E 132~).S
83
Volume 319. number 1.2.3
PIIYSI('S LETIER% B
length. The idea is m the spirit o f t h e approach to the structure ofspacetime at small distances suggested by non-commutative geometr)., pioneered b) Connes [7]. which underlies much of the apphcatlons of quantum groups to gravlt). One should also note that our formalism is not Lorentz co~ariant. We ~tll come back to this important point below. With these assumptions, the most general form of the E-deformed algebra is
[.~,. x, ] =
F1:a(E)
i~,,~J~
(2 )
E;
[x,. p,] = it7,6,, f( E).
(3)
ttere a ( E ) and . f ( E ) are real. dimensionless functions of E/E, and E ' = p : + m=: the angular momentum ,/is defined as dimensionless, so on the righthand side the dimensions are carried by h and p,"only. The fact that this is the most general form compattble v~ith our assumptions is clear from the follo~ mg consideratlons: the factors of i are determined b) the condition of hermiticit) of x,, p, and J,. The pov~ers of fi and ~,"are dictated by dimensional anal)sis. The tensor ¢,,~ in eq. ( 2 ) appears because ge assume that the three-dimensional rotation group is undeformed and then it ts the only tensor antis.~ mmetric in t.j: it must be contracted with J~ rather than p~ or .x~ because of parit). One might also add to the right-hand side of eq. (2) a term proportional to x , p . , - x / p , . ( Note that I.~ = ~,,~ (.~, pj - xj p, ) cannot be identified gith the angular momentum in the spinless case. since it does not satlsf) the SU( 2 ) commutation relations unlessf(E) = I ). However such a term can be eliminated with a non-local redefinition of coordinates. ,~ = g ( E ) x , . with a suitable function g ( E ) . In the second equation, again 6,, must appear because it is the onl.~ avadable tensor under rotation. In order to recover the undeformed limit, we further require that . f ( 0 ) = I and that a ( E ) is less singular than E- : as E .0. We neglect the possibllit? that the functions a. f d e p e n d also on other scalars like x: or x'9. Of course, the form of the functions a ( E ) . . t ( E ) is severely restricted by the Jacobi identities. Let us consider first the Jacobi identit) [.L. [x,. x,l]+ cyclic = 0. Using Ix,. 1-'] = ifi.f( E ) p , / E , and [x,. a ( E ) ] = ihf(E) ( p , / E ) d a / d E , we get da
~-..p..t= 0. ~4
(4)
23 Dc~cmber 1993
Since the Jacobl identity must be satisfied independently of the particular representation of the algebra. that is |ndependently of whether the condition p . l = 0 holds or not. we conclude that a ( E ) =const. With a redefimtion of E we can set this constant to +_ 1. The Jacobi identity [x,, Ix,. p~ ] ] + cyclic = 0 gives
.f(E) d/E
dE
l
-:l:
--;.
E"
(5)
where the upper ( lower ) sign correspond to the choice a=+l(-I). Since f ( 0 ) = l , eq. (5) gives .f(E) = [I ~ (E2/E2)]L 2. All other Jacobi identities are automaticall) satisfied. Of course, to satisfy the Jacobi identities is in general highly non-trivial, especiall~ in a deformed algebra, it is remarkable that in our case (t) there exists a solution: thus. we can Edeform the Heisenberg algebra. (it) The solution is unique (within our assumptions and apart from the _ sign ). In the following we only consider the lower sign =: and write the m-deformed Heisenberg algebra as h2" [x,. x,] = -
~
'
E"
tE,.,,J,.
[x,,p,]=i~3,,\l( +
(6) (7)
Actuall). we already denved eqs. ( 6 ). ( 7 ) in ref. [ 8 ]. following a rather different route: we considered the ~-deformed Poincare algebra suggested in ref. [9 ] and we discussed ho~, to generalize the Newton-Wtgner localization operator to the E-deformed case. We then obtained the algebra (6), ( 7 ). with x, identified with the E-deformed Ne~xon-Wigner operator and p,. J, identified with the generators of spatial translations and rotations of the E-deformed P o m c a ~ algebra. ( To compare with the results of ref. [ 8 ] one must observe that the present definition of E differs b.~ a factor of 2 from that used in refs. [q.8 ] and that in eq. ( 7 ) E 2 ts a shorthand for p2+DI2, while in the E-Poincar~ algebra the dispersion relation is. with the present definition o f E, p : + m : = ~c2 sinh " ( E / K ). ) The derivation that we have presented in this Let[he ¢,olutmn wtth the upper sign ts how¢,,¢r quttc intriguing. since.f(/'.) ts real b.~ dcfin,tton, the solution is ',al,d onl) for /" ~ ~ and dev:nbes a s)stcm which obc.,,s standard quantum mcchamcs for I..,=: x and sattsfle* Ix. p] = 0 at F.= K.
Volume 319. number 1.2.3
PHYSI('S LE'I'FERS B
ter is instead independent of whether we consider the usual or the deformed Poincar~ algebra. We do not attempt to define a coproduct on the deformed algebra given b.,, eqs. (6), ( 7 ), since a system composed of two parttcles localized at .x't~ and .r~:L respectively, cannot be considered as a single localized s,,stem and therefore we cannot reqmre, in general, that it can be described by a coordinate X = X ( . v ~'~, .x"~:' ) which, together with the total momentum, satisfies the same algebra as the constituents. Thus, the algebraic structure that we have presented is not a Hopfalgebra. nor a quantum algebra. From eq. (7) we immedtately derive the generalized uncertainty principle
ALAp,>~ ~d~,,: I +
;
(8)
Expanding the square root m powers of (F/K)-" and using (p2) =p:+ (.~p)'.. ~hcre (Ap)2= ~ (p<'p) ) : ) , at first order we obtain A~" APJ >~ .~ t~'t
2h"
]"
(9)
Thus. in the regime E,tzx. Ap<,x. we recover the string theoD result. The constant in eq. ( I ) is reproduced ff we identif.~ h/h" with the string length ;., times a numerical constant of order one. Note that if h/ x~,~, (and. as usual, the string length is larger than the Planck length ), then the condition E.*: x ~mplies the condition 2 G E ~ 2 , . If we consider instead the regime ( p > - ' ~ ( A p ) -~ : ~ x : we get h x
A.r I> const. X - .
( I0 )
while in the regime (p~2 :~ (.,.,~): ,(9>., :~ x2 wc get hl(p>l Ax">~const. X - - r Ap
h ::=' - . x
(II)
It is ~mportant to observe that the string theor) resuit. eq. ( i ). is reproduced in the appropriate kinematical limit after expanding our closed expression at first order in ,~o/x. However, only the full expression. eq. (8). has an underlying algebraic origin. We now comment on the issue o f Lorentz invariance. When we discuss the possible existence of a minimal observable length, we actually refer to a spa-
23 l)¢ccmlx.r IO93
tial length. Such a concept, of course, is not Lorentz mvariant. We can always perform a boost and squeeze any "minimal" length as much as v~.e like. Then, ifa minimal spatial length actually exists, we must consider the r e d interesting possibility that Lorentz invariance is not respected by quantum gravity at a scale on the order of h/x. In the formalism of the x-deformed Poincar6 algebra of ref. [ 9 ] this is explicit, since the Lorent,, algebra is deformed by the parameter x. In our approach, it is implicit in the use of the three-vectors x,, p, in place of the corresponding fourvectors. In th~s context it is interesting to mentmn that a breaking of Lorentz invariance at the string scale or at the Planck scale has been suggested vet). recently by Susskind in ref. [ 10], where it is argued that Lorentz contraction of particles must saturate as E ~ .lIVL, and a constant value should be approached (cir. our equations (10). ( ! l ) ). It might be interesting to obse~'e that the considerations of ref. [ 10] stem from Gedankcn experiments w~th black holes. and a (different) Gedanken experiment with black holes [6] was also the starting point of our investigation. Two final comments are in order. The first is that we have a scheme m which Newton's constant G (or equivalently x) enters the theoD at the kinematical level (see also ref. [ I I ] for early attempts in this direction ). The minimal length also emerges at a kinematical, rather than dynamical level. This feature is rather satisfying The second, related comment, is that this formalism can acquire substance onl) after a suitable dynamics has been added. This is particularly clear ffwe repeat our analysis in ( I + I ) dimensions. In this case only the equation Ix. p] =ihf(E) survives. Of course, we can ahvays perform a nonlocal redefinition of the coordinate, defining ,~= x / /~ E). so that [~, p] = ih. However, one would pay this at the level of the action, which would become nonlocal. The fact that a non-local particle Lagrangian can generate a mimmal observable length has been nicely shown in ref. [ 12 ]. 1 thank M. Mintchev, M. Shifman Vainshtein for useful discussions.
and
A.
X5
Volume 319, numl:)cr 1.2.3
PH YSIC:S LE IH'ERS B
References [ I ] G. Venczlano. Europh)s. I..ctt 2 ( 19861 199. Pro(: To,as Supcrslnng Workshop ( 1989 }; D. Gross. Pro<' ICHEP (Munich 19851. [2] D. Amatt. M. ('tafalon, and G. Vc'nezlano. Phys. L~I. B 19",' (19871 81. B 216 (19891 41. Intcrn. ] Mod. Phys. A 3 ( 19881 1615 Nucl Ph.,,s. B 347 (1990) 530. [ 3 ] D J . Gross and P.F Mcndc. Ph)s. Lelt B 197(19871 129; Nucl. Phys. B 303 ( 19881 407 [4] K. Komsht. G Paffutl and P. Provero. Ph)s Let|. B 234 (1990) 276 R. (imda. K. Komshs and P I)ro,,ero. Mod Ph}.s. Left. A 6 (1991) 1487
86
2 )) [.)cccmbcr 1993
[ 5 ] M. Ctafalon,. prcpnnl DFF 17219/92. [6] M. Maggiore, Phys Left. B 304 ( 19931 65. [ 7 } A Connes. Non.commutatsv¢ differentlaJ geometry. IHES Publ No. 62 (1986)
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