We ask whether Cohen and Glashow's Very Special Relativity model for Lorentz
violation might be modified, perhaps by quantum corrections, possibly producing
a curved spacetime with a cosmological constant. We show that its symmetry
group ISIM(2) does admit a 2-parameter family of continuous deformations, but
none of these give rise to non-commutative translations analogous to those of
the de Sitter deformation of the Poincar\'e group: spacetime remains flat. Only
a 1-parameter family DISIM_b(2) of deformations of SIM(2) is physically
acceptable. Since this could arise through quantum corrections, its
implications for tests of Lorentz violations via the Cohen-Glashow proposal
should be taken into account. The Lorentz-violating point particle action
invariant under DISIM_b(2) is of Finsler type, for which the line element is
homogeneous of degree 1 in displacements, but anisotropic. We derive
DISIM_b(2)-invariant wave equations for particles of spins 0, 1/2 and 1. The
experimental bound, $|b|<10^{-26}$, raises the question ``Why is the
dimensionless constant $b$ so small in Very Special Relativity?''Comment: 4 pages, minor corrections, references adde

C. N. Pope

G. W. Gibbons

G. Y. Bogoslovsky

Joaquim Gomis

P. A. M. Dirac

English

arXiv

Gibbons, G. W.

Gomis, Joaquim

Pope, C. N.

Texas A&M University

General very special relativity is Finsler geometry
G. W. Gibbons,1 Joaquim Gomis,2 and C. N. Pope3
1DAMTP, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 OWA, United Kingdom
2Departament ECM, Facultat de Fı´sica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
3George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University,
College Station, Texas 77843-4242, USA
(Received 20 August 2007; published 15 October 2007)
We ask whether Cohen and Glashow’s very special relativity model for Lorentz violation might be
modified, perhaps by quantum corrections, possibly producing a curved space-time with a cosmological
constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family of continuous
deformations, but none of these give rise to noncommutative translations analogous to those of the
de Sitter deformation of the Poincare´ group: space-time remains flat. Only a 1-parameter family
DISIMb2 of deformations of SIM(2) is physically acceptable. Since this could arise through quantum
corrections, its implications for tests of Lorentz violations via the Cohen-Glashow proposal should be
taken into account. The Lorentz-violating point-particle action invariant under DISIMb2 is of Finsler
type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive
DISIMb2-invariant wave equations for particles of spins 0, 12 , and 1. The experimental bound, jbj<
1026, raises the question ‘‘Why is the dimensionless constant b so small in very special relativity?’’
DOI: 10.1103/PhysRevD.76.081701 PACS numbers: 03.30.+p, 02.20.Sv, 11.30.Cp, 11.30.Er
Local Lorentz and CPT invariance are fundamental
assumptions in almost all current physical theories. It is
important to test these assumptions experimentally, lest
evidence of new physics beyond the standard model be
overlooked. Current experimental limits on violations of
local Lorentz and CPT invariance are extremely stringent.
Thus what is required are novel alternative non-Lorentz
invariant theories, capable of circumventing these tight
limits. Recently, Cohen and Glashow [1] have made the
ingenious proposal that the local laws of physics need not
be invariant under the full Lorentz group, generated by
M, but rather, under a SIM(2) subgroup, whose Lie
algebra is generated by (Mi;Mij;M  M03), (with i
and j ranging over the values 1 and 2) [2]. This they
referred to as very special relativity. Taking the semidirect
product with the translations P; P; Pi gives an 8-
dimensional subgroup of the Poincare´ group called
ISIM(2) [3].
The great merits of Cohen and Glashow’s suggestion are
that CPT symmetry is preserved and that ISIM(2) leaves
invariant no vector or tensor fields, known as ‘‘spurion
fields.’’ For example, a spurionic vector field may be
thought of as the 4-velocity of the æther [4]. In fact
SIM(2) consists of those Lorentz transformations 
leaving invariant the null direction n  , i.e., such
that n  n for some  which depends on .
The generator M acts by sending n ! n. This
scaling symmetry implies that one cannot take n to define
the actual 4-velocity of the æthereal motion, but only its
direction, thus rendering the presence of such an æther
more difficult to detect. A theory of this kind appears to be
compatible with all current experimental limits on viola-
tions of Lorentz invariance and spatial isotropy [1,5].
Subsequently, Cohen and Freedman [6], and later
Lindstro¨m and Rocˇek [7], showed that ISIM is compatible
with supersymmetry. There are several ways one might try
to incorporate gravity. One is where we make local the
ISIM(2) algebra [8]. Another is to consider a global space-
time and to see if it is compatible with very special rela-
tivity ideas. In this case there appear to be difficulties [9].
For example, the maximal subgroup of SO4; 1, the
isometry group of de Sitter space-time, is SIM(3), which
is 7-dimensional [10]. SIM(3) contains SIM(2) as a sub-
group, but the stabilizer, or tangent-space group, is SO(3),
not SIM(2).
Alternatively, we recall that Poincare´ group admits a
unique deformation into the de Sitter (anti-de Sitter) group
[11], with P;P  13M, where the parameter  is
the cosmological constant. One may ask whether ISIM(2)
admits a similar deformation, such that the translations P
become noncommutative. If so, the coset of the deformed
group divided by SIM(2) (or its deformation) could be
thought of as a curved space-time.
Here we show that there are indeed continuous defor-
mations of ISIM(2), but in all of them the translations
remain commutative. Among them is a 1-parameter family
of deformations, which we denote by DISIMb2. For any
values of b this is an 8-dimensional subgroup of the 11-
dimensional Weyl group, i.e., the semidirect product of
dilatations with the Poincare´ group. Subgroups with differ-
ent values of b are not isomorphic. Interestingly, if one
constructs a point-particle action for the deformed groups
DISIMb2, using the methods of nonlinear realizations
[12], one arrives at Lagrangians of Finsler form, first
proposed by Bogoslovsky (see [13] and references
therein). Therefore the deformation of very special relativ-
PHYSICAL REVIEW D 76, 081701(R) (2007)
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ity leads in a natural way to Finsler geometry. In the
remainder of this paper we shall outline the derivation of
these results, and comment on their physical significance.
Continuous deformations of Lie algebras have been
extensively explored, by both mathematicians and physi-
cists, under the rubric of Lie-algebra cohomology [11].
Here we give an elementary account based on the
Cartan-Maurer equations, which provides a simple and
easily automated scheme for determining the deformations
of a given Lie algebra g with structure constants Cabc. We
suppose there exists a family of deformed Lie algebras gt
parametrized by a continuous variable t, with structure
constants
 C^ a
b
ct  Cabc  tAabc  t2Babc     : (1)
We are only interested in deformations which do not arise
merely from a (t-dependent) change of basis:
 C^ a
b
ct  SbeCdefS1daS1fc; Sab 2 GLn;R:
(2)
Expanding the Jacobi identity C^deatC^bdct  0 in
powers of t gives rise at linear order to
 Cd
e
aAb
d
c  AdeaCbdc  0: (3)
A first-order deformation A will be trivial if Sabt  ab 
tab     and
 Aa
b
c  beCaeb  Cebcea  Cabeec: (4)
Introducing a basis a of left-invariant 1-forms of the
original algebra, such that da   12Cbacb ^ c, we
define vector-valued 1-forms and 2-forms a 	 bb
and Aa 	 12Abacb ^ c and a matrix-valued 1-form
Cab 	 cCcab. Defining D 	 d C ^ , the first-order de-
formation equations (3) may then be written as
 DA  0; A  D; (5)
where the second equation expresses the requirement of
nontriviality of the deformation. Because D2  0 as a
consequence of the Jacobi identities of the undeformed
algebra, dC C ^ C  0, the differential D may be re-
garded as a coboundary operator acting on g-valued forms.
The nontrivial linearized deformations are therefore in 1-1
correspondence with the second cohomology group
H2g;g.
If a nontrivial linear deformation A is found, the next
step is to investigate the Jacobi identities at order t2. These
read
 Cd
e
aBb
d
c  BdeaCbdc  AdeaAbdc  0; (6)
and can be reexpressed in terms of the vector and matrix
valued forms as
 DB A 
 A  0;
A 
 Ae 	 12AdeaAbdca ^ b ^ c:
(7)
This equation can only be solved if DA 
 A  0, which
implies that A 
 A must be both D closed and exact. Thus
there is a potential obstruction to finding a deformation at
quadratic order: A 
 A should not have a projection in the
third cohomology group H3g; g. There are analogous
equations to (7) at higher orders in t. If H3g; g vanishes,
then the equations may be solved at all nonlinear orders. If
H3g; g is nonzero, then the higher-order analogues of
(A 
 A) should have no projections into it.
We start with the ISIM(2) Cartan-Maurer relations
 
d~~  ~~i ^ i  ~~ ^ ~~; d  ~~ ^ ;
di  ij12 ^ j   ^ ~~i;
d~~i  ij12 ^ ~~j   ^ ~~i; d~~ 0;
d12  0; (8)
where g1dg  ~~P  P  iPi  ~~iMi 
M  12M12. Defining N 	 M and J 	 M12,
the corresponding nontrivial Lie brackets are therefore
 N;P  P; N;Mi  Mi;
J; Pi  ijPj; J;Mi  ijMi;
Mi; P  Pj; Mi; Pj  ijP:
(9)
Expanding the vector-valued 2-form Aa on a basis of 2-
forms and solving the resultant linear equations in (5)
reveals that there is a 2-parameter family of nontrivial
solutions, i.e., H2isim2; isim2 is 2-dimensional.
Substituting this linearized solution into the full Jacobi
identities, we find that it gives a 2-parameter family of
exact Lie algebras of the leading-order form (8), with
additional terms as follows:
 d ! d  a12 ^   b ^ ; (10)
where   ;; i. Here a and b are arbitrary constant
parameters.
As in the undeformed case, the algebra here has the
structure of a semidirect sum of sim2 and the trans-
lations R4, i.e., sim2 32 R4. While the Mi act on the
translations as in the undeformed case, the adjoint action of
the generators N and J is given by N;P  PCN and
J; P  PCJ, where the matrices CN and CJ are
given, respectively, by
G. W. GIBBONS, JOAQUIM GOMIS, AND C. N. POPE PHYSICAL REVIEW D 76, 081701(R) (2007)
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b 1 0 0 0
0 b 1 0 0
0 0 b 0
0 0 0 b
0
BBB@
1
CCCA;
a 0 0 0
0 a 0 0
0 0 a 1
0 0 1 a
0
BBB@
1
CCCA:
Minkowski space-time may be thought of as the symmetric
space E3; 1=SO3; 1 with SO3; 1 playing the role of
the tangent-space group, and the tangent space being
spanned by the translations P. In our case we wish to
replace SO3; 1 by SIM(2). However, it follows by expo-
nentiating CJ that J does not generate a compact SO2
subgroup unless the deformation parameter a vanishes.
From now on we shall restrict attention to this a  0
case, for which we denote the deformed algebra by
disimb2.
The nontrivial Lie brackets for disimb2 are given by
 N;P  b 1P; N;Pi  bPi;
N;Mi  Mi; J; Pi  ijPj;
J;Mi  ijMi; Mi; P  Pj;
Mi; Pj  ijP:
(11)
The deformed group DISIMb2 is a subgroup of the Weyl
group with an action on Minkowski space-time given by
translations, and boosts in the i directions, together with
a combination of a boost in the  direction and a
dilatation. Specifically, the deformed generator N acts as
 xi ! bxi; x ! 1bx; x ! 1bx:
(12)
The group DISIMb2 does not leave invariant the stan-
dard Minkowski line element ds  dxdx1=2, but
rather, the Finslerian line element
 ds  2dxdx  dxidxi1b=2dxb
 dxdx1b=2ndxb: (13)
This is of the form first suggested by Bogoslovsky [13].
We shall now construct, using the theory of nonlinear
realizations, a DISIMb2-invariant Lagrangian for a point
particle. We parametrize the coset DISIMb2=SO2 as
 g  exPewiMiewN; (14)
which implies that
 g1dg  dwN  ewdwiMi  ew1bdxP
 ew1bdx  widxi  12wiwidxP
 ewbdxi  widxPi
 N  ~~iMi  P  ~~P  iPi;
(15)
where ; ~~i; ~~; ; i are the restrictions (or pull-
backs) of the invariant 1-forms on the group DISIMb2 to
the coset DISIMb2=SO2.
In order to construct a DISIMb2-invariant Lagrangian
with worldline reparametrization invariance, we allow the
Goldstone coordinates w;wi; x to depend on the world-
line coordinate  (see, for example, [14]). We shall restrict
our attention to Lagrangians that are linear in the left-
invariant 1-forms pulled back to the particle’s worldline.
Requiring invariance under SO2 then implies that we
must discard i and i, and thus we consider the
Lagrangian
 
L  	ew1b _x  wi _xi  12wiwi _x  
ew1b _x
  _w; (16)
where 	, 
, and  are arbitrary constants. Since the last
term is a total derivative, we can discard it. Eliminating the
nondynamical Goldstone coordinates w and wi, one ob-
tains, in physical units, the Lagrangian
 L  m _x _x1b=2n _xb: (17)
Calculating the canonical momenta from (17), we obtain
the DISIMb2-invariant dispersion relation or
Hamiltonian constraint
 pp  m21 b2

 n
p
m1 b

2b=1b
: (18)
Note that for b  0 we recover the ordinary free relativistic
particle, which does not see the lightlike direction n. The
cases b  1 are special, and are best investigated directly
from Eq. (16). The conclusion is that for b  1 we obtain
the massless equation pp  0, while for b  1
we have _xi  0 and _x  0, and the dynamics is trivial in
this case.
Upon quantization, p ! i@, we obtain a general-
ized Klein-Gordon equation of the form
 m21 b2

in@
m1 b

2b=1b
  0: (19)
This is in general a nonlocal equation, since it involves
fractional derivatives. Although the special case b  1
appears to give a local modification of the usual Klein-
Gordon equation involving a term linear in n@ this is
really equivalent to the standard massless Klein-Gordon
equation (as discussed earlier for this special value of b).
GENERAL VERY SPECIAL RELATIVITY IS FINSLER . . . PHYSICAL REVIEW D 76, 081701(R) (2007)
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Specifically, the first-order term can be removed by making
the phase transformation ! eimnx=2.
The free Maxwell equations are also invariant under the
action of the Weyl group and so they too are clearly
invariant under DISIMb2. The invariance of Adx,
together with (12), implies that A; A; Ai !
b1A; b1A; bAi. Since d4x! 4bd4x any in-
variant action must have L! 4bL. Thus we can add a
mass term, giving
 L   1
4
FF
  1
2
m2
nA2
AA

b
AA
: (20)
If b  1 we can further include a non-Lorentz invariant
Chern-Simons term [15], Lcs  12 ‘1nAF,
where ‘ is an arbitrary length scale.
Since it is a subgroup of the Weyl group, DISIMb2
leaves invariant the massless Dirac Lagrangian.
Bogoslovsky and Goenner [16] have pointed out that add-
ing to the massless Dirac Lagrangian a term of the form
 m

in  
 
  

2

b=2
  (21)
gives a nonlinear DISIMb2-invariant generalization of
the massive Dirac equation. This follows from the scalings
 ! 3b=2 , @ ! b@ under the action of the
generator M. As with the generalized Klein-Gordon
equation (19), the case b  1 is special: The additional
term may then be removed by a phase transformation of the
form  !  eimnx . Note however that, as discussed be-
low, experimental bounds constrain jbj to be very much
less than 1.
CPT will be preserved if an operator exists in the com-
plexification of DISIMb2 that reverses x. As discussed
for ISIM(2) in [1], a candidate CPT operator is eiJei	N.
This has the following action on the momenta:
 P;P;P1  iP2 ! eb	e	P; e	P; eiP1  iP2:
(22)
Requiring that P ! P implies that
 	  in  n;   2n3  n  n;
b  1 n  n
n  n ;
(23)
for integers n, n, and n3. Although b  1 p=q is
rational, with p odd, one may always choose n and n
so that b is arbitrarily close to any given real number.
We have shown that ISIM(2) admits no deformations
with de Sitter-like noncommutative translations. However,
it is interesting to note that ISIM(2), unlike the Poincare´
group but like the Galilei group, admits a central extension:
the cohomology group H2isim2;R is nontrivial. We
find that it is generated by  ^ 12, and so may adjoin to
isim2 a central element Z, whose only nontrivial Lie
bracket is
 N; J  Z: (24)
Thus unlike the translations, the boosts and rotations can
be rendered noncommutative. This also works for the full
2-parameter family of deformations of ISIM(2). Including
the extra generator Z, appending eZ, and proceeding with
the construction of an invariant particle Lagrangian leads
to unmodified equations of motion, since the only effect is
to add a total derivative _ to the Lagrangian.
It was argued in [13] that æther-drift experiments imply
jbj< 1010. However, it follows from (17) that every
particle has a mass tensor mij  1 bmij  bninj.
Hughes-Drever–type limits [17] on the anisotropy of iner-
tia then potentially imply that jbj< 1026. However, this
depends on the precise form of the interactions [18,19].
Since a nonvanishing b could arise through quantum cor-
rections, very special relativity faces the question, analo-
gous to the puzzle posed by the cosmological constant in
traditional relativity: ‘‘Why is b so small?’’
We are grateful to Dan Freedman and Jaume Gomis for
helpful discussions, and to the Galileo Galilei Institute in
Florence, the workshop on String and M-Theory
Approaches to Particle Physics and Cosmology, and the
INFN, for support and hospitality during the course of this
work. We thank George Bogoslovsky for a helpful discus-
sion about upper bounds on the parameter b. This work has
also been supported by the European EC-RTN project
MRTN-CT-2004-005104, MCYT FPA 2004-04582-C02-
01, CIRIT GC 2005SGR-00564. The research of C. P.
was supported in part by DOE Grant No. DE-FG03-
95ER40917.
[1] A. G. Cohen and S. L. Glashow, Phys. Rev. Lett. 97,
021601 (2006).
[2] We take the Minkowski metric to be ds2  dxdx 
2dxdx  dxidxi.
[3] Although not named, ISIM(2) appears to have been first
introduced in J. B. Kogut and D. E. Soper, Phys. Rev. D 1,
2901 (1970).
[4] P. A. M. Dirac, Nature (London) 168, 906 (1951); 169, 146
(1952); Sci. Mon. 78, 142 (1954).
[5] A. Dunn and T. Mehen, arXiv:hep-ph/0610202.
[6] A. G. Cohen and D. Z. Freedman, J. High Energy Phys. 07
(2007) 039.
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RAPID COMMUNICATIONS
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[7] U. Lindstro¨m and M. Rocˇek, arXiv:hep-th/0606093.
[8] G. W. Gibbons, J. Gomis, and C. N. Pope (work in
progress).
[9] D. Z. Freedman (private communication).
[10] J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, J.
Math. Phys. (N.Y.) 18, 2259 (1977).
[11] M. Levy-Nahas, J. Math. Phys. (N.Y.) 8, 1211 (1967).
[12] S. R. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177,
2239 (1969).
[13] G. Bogoslovsky, arXiv:0706.2621.
[14] J. Gomis, K. Kamimura, and P. West, Classical Quantum
Gravity 23, 7369 (2006).
[15] A. J. Hariton and R. Lehnert, Phys. Lett. A 367, 11 (2007).
[16] G. Y. Bogoslovsky and H. F. Goenner, Phys. Lett. A 323,
40 (2004).
[17] S. K. Lamoreaux et al., Phys. Rev. Lett. 57, 3125 (1986);
T. E. Chupp et al., Phys. Rev. Lett. 63, 1541 (1989).
[18] G. Y. Bogoslovsky, Nuovo Cimento Soc. Ital. Fis. B 77,
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186 (2007).
GENERAL VERY SPECIAL RELATIVITY IS FINSLER . . . PHYSICAL REVIEW D 76, 081701(R) (2007)
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Physical Review D

General very special relativity is Finsler geometry

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:0707.2174v2  [hep-th]  20 Aug 2007DAMTP-2007-68 UB-ECM-PF-07-17 MIFP-07-18General Very Special Relativity is Finsler GeometryG.W. Gibbons1, Joaquim Gomis2 and C.N. Pope3∗1DAMTP, Centre for Mathematical Sciences, Cambridge University Wilberforce Road, Cambridge CB3 OWA, UK2Departament ECM, Facultat de F´ısica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain3George P. & Cynthia W. Mitchell Institute for Fundamental Physics,Texas A&M University, College Station, TX 77843-4242, USA(Dated: July 13, 2007)We ask whether Cohen and Glashow’s Very Special Relativity model for Lorentz violation mightbe modified, perhaps by quantum corrections, possibly producing a curved spacetime with a cos-mological constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family ofcontinuous deformations, but none of these give rise to non-commutative translations analogous tothose of the de Sitter deformation of the Poincare´ group: spacetime remains flat. Only a 1-parameterfamily DISIMb(2) of deformations of SIM(2) is physically acceptable. Since this could arise throughquantum corrections, its implications for tests of Lorentz violations via the Cohen-Glashow pro-posal should be taken into account. The Lorentz-violating point particle action invariant underDISIMb(2) is of Finsler type, for which the line element is homogeneous of degree 1 in displace-ments, but anisotropic. We derive DISIMb(2)-invariant wave equations for particles of spins 0,12and 1. The experimental bound, |b| < 10−26, raises the question “Why is the dimensionless constantb so small in Very Special Relativity?”PACS numbers: 11.25.-w, 98.80.Jk, 04.50.+hLocal Lorentz and CPT invariance are fundamentalassumptions in almost all current physical theories. Itis important to test these assumptions experimentally,lest evidence of new physics beyond the standard modelbe overlooked. Current experimental limits on viola-tions of local Lorentz and CPT invariance are extremelystringent. Thus what is required are novel alternativenon-Lorentz invariant theories, capable of circumventingthese tight limits. Recently, Cohen and Glashow [1] havemade the ingenious proposal that the local laws of physicsneed not be invariant under the full Lorentz group, gen-erated by Mµν , but rather, under a SIM(2) subgroup,whose Lie algebra is generated by (M+i,Mij ,M+− =M03, ) (with i and j ranging over the values 1 and 2) [2].This they referred to as Very Special Relativity. Takingthe semi-direct product with the translations (P+, P−, Pi)gives an 8-dimensional subgroup of the Poincare´ groupcalled ISIM(2) [3].The great merits of Cohen and Glashow’s suggestionare that CPT symmetry is preserved and that ISIM(2)leaves invariant no vector or tensor fields, known as “spu-rion fields.” For example, a spurionic vector field may bethought of as the 4-velocity of the æther [4]. In factSIM(2) consists of those Lorentz transformations Λµνleaving invariant the null direction nµ = δµ+, i.e. suchthat Λµνnν = λnµ for some λ which depends on Λµν .The generator M+− acts by sending nµ → λnµ. Thisscaling symmetry implies that one cannot take nµ to de-fine the actual 4-velocity of the æthereal motion, but onlyits direction, thus rendering the presence of such an æthermore difficult to detect. A theory of this kind appears tobe compatible with all current experimental limits on vi-olations of Lorentz invariance and spatial isotropy [1, 5].Subsequently, Cohen and Freedman [6], and later Lind-stro¨m and Rocˇek [7], showed that ISIM is compatiblewith supersymmetry. There are several ways one mighttry to incorporate gravity. One is where we make localthe ISIM(2) algebra [8]. Another is to consider a globalspace-time and to see if it is compatible with very specialrelativity ideas. In this case there appear to be difficul-ties [9]. For example, the maximal subgroup of SO(4, 1),the isometry group of de Sitter spacetime, is SIM(3),which is 7-dimensional [10]. SIM(3) contains SIM(2) asa subgroup, but the stabilizer, or tangent space group, isSO(3), not SIM(2).Alternatively, we recall that that Poincare´ group ad-mits a unique deformation into the de Sitter (anti-deSitter) group [11], with [Pµ, Pν ] =13ΛMµν , where theparameter Λ is the cosmological constant. One may askwhether ISIM(2) admits a similar deformation, such thatthe translations Pµ become non-commutative. If so, thecoset of the deformed group divided by SIM(2) (or itsdeformation) could be thought of as a curved spacetime.Here we show that there are indeed continuous defor-mations of ISIM(2), but in all of them the translationsremain commutative. Among them is a 1-parameter fam-ily of deformations, which we denote by DISIMb(2). Forany values of b this is an 8-dimensional subgroup of the11-dimensional Weyl group, i.e. the semi-direct productof dilatations with the Poincare´ group. Subgroups withdifferent values of b are not isomorphic. Interestingly, ifone constructs a point-particle action for the deformedgroups DISIMb(2), using the methods of non-linear real-isations [12], one arrives at Lagrangians of Finsler form,first proposed by Bogoslovsky (see [13] and referencestherein). Therefore the deformation of very special rela-tivity leads in a natural way to Finsler geometry. In theremainder of this letter we shall outline the derivation of2these results, and comment on their physical significance.Continuous deformations of Lie algebras have been ex-tensively explored, by both mathematicians and physi-cists, under the rubric of Lie-algebra cohomology [11].Here we give an elementary account based on the Cartan-Maurer equations, which provides a simple and easilyautomated scheme for determining the deformations of agiven Lie algebra g with structure constants Cabc. Wesuppose there exists a family of deformed Lie algebras gtparameterised by a continuous variable t, with structureconstantsCˆabc(t) = Cabc + t Aabc + t2Babc + · · · . (1)We are only interested in deformations which do not arisemerely from a (t-dependent) change of basis:Cˆabc(t) = SbeCdef (S−1)da (S−1)f c , Sab ∈ GL(n,R) .(2)Expanding the Jacobi identity Cˆde[a(t) Cˆbdc](t) = 0 inpowers of t gives rise at linear order toCde[aAbdc] +Ade[a Cbdc] = 0 . (3)A first-order deformation A will be trivial if Sab(t) =δab + tΦab + · · · andAabc = Φbe Caeb − Cebc Φea − CabeΦec . (4)Introducing a basis λa of left-invariant 1-forms of theoriginal algebra, such that dλa = − 12Cbac λb∧λc, wedefine vector-valued 1-forms and 2-forms Φa ≡ Φb λband Aa ≡ 12Abac λb∧λc and a matrix-valued 1-formCab ≡ λcCcab. Defining D ≡ d + C∧ , the first-orderdeformation equations (3) may then be written asDA = 0 , A 6= −DΦ , (5)where the second equation expresses the requirement ofnon-triviality of the deformation. Because D2 = 0 as aconsequence of the Jacobi identities of the undeformedalgebra, dC + C∧C = 0, the differential D may be re-garded as a co-boundary operator acting on g-valuedforms. The non-trivial linearised deformations are there-fore in 1-1 correspondence with the second cohomologygroup H2(g; g).If a non-trivial linear deformation A is found, the nextstep is to investigate the Jacobi identities at order t2.These readCde[aBbdc] + Bde[a Cbdc] +Ade[aAbdc] = 0 , (6)and can be re-expressed in terms of the vector and matrixvalued forms asDB +A •A = 0 , (A •A)e ≡ 12Ade[aAbdc]λa∧λb∧λc .(7)This equation can only be solved if D(A •A) = 0, whichimplies that A•Amust be bothD-closed and exact. Thusthere is a potential obstruction to finding a deformationat quadratic order: A • A should not have a projectionin the third cohomology group H3(g; g). There are anal-ogous equations to (7) at higher orders in t. If H3(g; g)vanishes, then the equations may be solved at all non-linear orders. If H3(g; g) is non-zero, then the higher-order analogues of (A • A) should have no projectionsinto it.We start with the ISIM(2) Cartan-Maurer relationsdλ+ = λ+i∧λi + λ+−∧λ+ , dλ− = −λ+−∧λ− ,dλi = ǫijλ12∧λj + λ−∧λ+i ,dλ+i = ǫijλ12∧λ+j + λ+−∧λ+i ,dλ+− = 0 , dλ12 = 0 , (8)where g−1dg = λ+P+ + λ−P−+ λiPi + λ+iM+i +λ+−M+− + λ12M12. Defining N ≡ M+− and J ≡ M12,the corresponding non-trivial Lie brackets are therefore[N,P±] = ∓P± , [N,M+i] = −M+i ,[J, Pi] = ǫijPj , [J,M+i] = ǫijM+i ,[M+i, P−] = Pj , [M+i, Pj ] = −δijP+ . (9)Expanding the vector-valued 2-form Aa on a basis of 2-forms and solving the resultant linear equations in (5) re-veals that there is a 2-parameter family of non-trivial so-lutions, i.e., H2(isim(2); isim(2)) is 2-dimensional. Sub-stituting this linearised solution into the full Jacobi iden-tities, we find that it gives a 2-parameter family of exactLie algebras of the leading-order form (8), with additionalterms as follows:dλµ −→ dλµ + aλ12∧λµ + bλ+−∧λµ , (10)where µ = (+, −, i). Here a and b are arbitrary constantparameters.As in the undeformed case, the algebra here has thestructure of a semi-direct sum of sim(2) and the trans-lations R4, i.e. sim(2)⋉R4. While the M+i act on thetranslations as in the undeformed case, the adjoint actionof the generators N and J is given by [N,Pµ] = Pν CNνµand [J, Pµ] = Pν CJνµ, where the matrices CN and CJare given respectively by−b+ 1 0 0 00 b− 1 0 00 0 b 00 0 0 b ,−a 0 0 00 −a 0 00 0 −a −10 0 1 −a .Minkowski spacetime may be thought of as the symmet-ric space E(3, 1)/SO(3, 1) with SO(3, 1) playing the roleof the tangent-space group, and the tangent space beingspanned by the translations Pµ. In our case we wish toreplace SO(3, 1) by SIM(2). However, it follows by expo-nentiating CJ that J does not generate a compact SO(2)subgroup unless the deformation parameter a vanishes.3From now on we shall restrict attention to this a = 0 case,for which we denote the deformed algebra by disimb(2).The non-trivial Lie brackets for disimb(2) are given by[N,P±] = −(b± 1)P± , [N,Pi] = −bPi ,[N,M+i] = −M+i , [J, Pi] = ǫijPj ,[J,M+i] = ǫijM+i , [M+i, P−] = Pj ,[M+i, Pj ] = −δijP+ , (11)The deformed group DISIMb(2) is a subgroup of the Weylgroup with an action on Minkowski spacetime given bytranslations, and boosts in the +i directions, togetherwith a combination of a boost in the +− direction and adilatation. Specifically, the deformed generator N acts asxi → λ−b xi , x− → λ1−b x− , x+ → λ−1−b x+ . (12)The group DISIMb(2) does not leave invariant the stan-dard Minkowski line element ds = (ηµνdxµdxν)1/2, butrather, the Finslerian line elementds = (2dx+dx− + dxidxi)(1−b)/2 (dx−)b ,= (ηµνdxµdxν)(1−b)/2 (nρdxρ)b . (13)This is of the form first suggested by Bogoslovsky [13].We shall now construct, using the theory of non-linear realisations, a DISIMb(2)-invariant Lagrangianfor a point particle. We parameterise the cosetDISIMb(2)/SO(2) asg = exµPµewiM+iewN , (14)which implies thatg−1dg = dwN + ewdwiM+i + e−w(1−b)dx−P−+ew(1+b)(dx+ + widxi − 12wiwidx−)P++ewb(dxi − widx−)Pi , (15)= λ+−N + λ+iM+i + λ−P−+ λ+P+ + λiPi ,where (λ+−, λ+i, λ+, λ−, λi) are the restrictions (or pull-backs) of the invariant 1-forms on the group DISIMb(2)to the coset DISIMb(2)/SO(2).In order to construct a DISIMb(2)-invariant La-grangian with worldline reparameterisation invariance,we allow the Goldstone coordinates (w,wi, xµ) to dependon the worldline coordinate τ (see, for example, [14]).We shall restrict our attention to Lagrangians that arelinear in the left-invariant 1-forms pulled back to the par-ticle’s worldline. Requiring invariance under SO(2) thenimplies that we must discard λ+i and λi, and thus weconsider the LagrangianL = αew(1+b)(x˙++wix˙i− 12wiwix˙−)+βe−w(1−b)x˙−+γw˙ ,(16)where α, β and γ are arbitrary constants. Since the lastterm is a total derivative, we can discard it. Eliminatingthe non-dynamical Goldstone coordinates w and wi, oneobtains, in physical units, the LagrangianL = −m(−ηµν x˙µx˙ν)(1−b)/2 (−nρx˙ρ)b . (17)Calculating th canonical momenta from (17), weobtain the DISIMb(2)-invariant dispersion relation orHamiltonian constraintηµνpµpν = −m2(1− b2)(−nνpνm(1− b))2b/(1+b). (18)Note that that for b = 0 we recover the ordinary free rela-tivistic particle, which does not see the light-like directionnµ. The cases b = ±1 are special, and are best investi-gated directly from equation (16). The conclusion is thatfor b = 1 we obtain the massless equation ηµνpµpν = 0,whilst for b = −1 we have x˙i = 0 and x˙− = 0, and thedynamics is trivial in this case.Upon quantisation, pµ → −i∂µ, we obtain a gener-alised Klein-Gordon equation of the form−φ+m2(1− b2)( inµ∂µm(1− b))2b/(1+b)φ = 0 . (19)This is in general a non-local equation, since it involvesfractional derivatives. Although the special case b = 1appears to give a local modification of the usual Klein-Gordon equation involving a term linear in nµ∂µ this isreally equivalent to the standard massless Klein-Gordonequation (as discussed earlier for this special value of b).Specifically, the first-order term can be removed by mak-ing the phase transformation φ→ φ e−imnµxµ/2.The free Maxwell equations are also invariant un-der the action of the Weyl group and so they too areclearly invariant under DISIMb(2). The invariance ofAµdxµ, together with (12), implies that (A+, A−, Ai) →(λb+1A+, λb−1A−, λbAi). Since d4x → λ−4bd4x any in-variant action must have L → λ4bL. Thus we can add amass term, givingL = − 14FµνFµν − 12m2( (nµAµ)2AνAν)bAρAρ . (20)If b = 1 we can further include a non-Lorentz invari-ant Chern-Simons term [16],Lcs =12ℓ−1 ǫµνρσnµAνFρσ,where ℓ is an arbitrary length scale.Since it is a subgroup of the Weyl group, DISIMb(2)leaves invariant the massless Dirac Lagrangian. Bo-goslovsky and Goenner [15] have pointed out that addingto the massless Dirac Lagrangian a term of the formm[( inµψ¯γµψψ¯ψ)2]b/2ψ¯ψ (21)gives a non-linear DISIMb(2)-invariant generalisation ofthe massive Dirac equation. This follows from the scal-ings ψ → λ3b/2ψ, γµ∂µ → λbγµ∂µ under the action of thegenerator M+−. As with the generalised Klein-Gordon4equation (19), the case b = 1 is special: The additionalterm may then be removed by a phase transformationof the form ψ → ψ eimnµxµ. Note however that, as dis-cussed below, experimental bounds constrain |b| to bevery much less than 1.CPT will be preserved if an operator exists in the com-plexification of DISIMb(2) that reverses xµ. As discussedfor ISIM(2) in [1], a candidate CPT operator is eiφJeiαN .This has the following action on the momenta:(P+, P−, P1 + iP2)→ e−bα (e−αP+, eαP−, eiφ(P1 + iP2)) .(22)Requiring that Pµ → −Pµ implies thatα = iπ(n−−n+) , φ = π(2n3−n+−n−) , b =1+ n+ + n−n+ − n−,(23)for integers n+, n− and n3. Although b = 1 + p/q isrational, with p odd, one may always choose n+ and n−so that b is arbitrarily close to any given real number.We have shown that ISIM(2) admits no deformationswith de Sitter-like non-commutative translations. How-ever, it is interesting to note that ISIM(2), unlike thePoincare´ group but like the Galilei group, admits a cen-tral extension: the cohomology group H2(isim(2),R) isnon-trivial. We find that it is generated by λ+−∧λ12, andso may adjoin to isim(2) a central element Z, whose onlynon-trivial Lie bracket is[N, J ] = Z . (24)Thus unlike the translations, the boosts and rotationscan be rendered non-commutative. This also works forthe full 2-parameter family of deformations of ISIM(2).Including the extra generator Z, appending eθZ, and pro-ceeding with the comstruction of an invariant particle La-grangian leads to unmodified equations of motion, sincethe only effect is to add a total derivative θ˙ to the La-grangian.It was argued in [13] that æther-drift experiments im-ply |b| < 10−10. However, it follows from (17) that everyparticle has a mass tensor mij = (1 − b)m(δij + bninj).Hughes-Drever type limits [17] on the anisotropy of iner-tia then potentially imply that |b| < 10−26. However, thisdepends on the precise form of the interactions [18, 19].Since a non-vanishing b could arise through quantum cor-rections, Very Special Relativity faces the question, anal-ogous to the puzzle posed by the cosmological constantin traditional relativity: “Why is b so small?”We are grateful to Dan Freedman and Jaume Gomisfor helpful discussions, and to the Galileo Galilei Insti-tute in Florence, the workshop on String and M-TheoryApproaches to Particle Physics and Cosmology, and theINFN, for support and hospitality during the course ofthis work. We thank George Bogoslovsky for a help-ful discussion about upper bounds on the parameter b.This work has also been supported by the European EC-RTN project MRTN-CT-2004-005104,MCYT FPA 2004-04582-C02-01, CIRIT GC 2005SGR-00564.∗ Research supported in part by DOE grant DE-FG03-95ER40917.[1] A.G. Cohen and S.L. Glashow, Very special relativity,Phys. Rev. Lett. 97, 021601 (2006), hep-ph/0601236.[2] We take the Minkowski metric to be ds2 = ηµνdxµdxν =2dx+dx− + dxidxi.[3] Although not named, ISIM(2) appears to have been firstintroduced in J.B. Kogut and D.E. Soper, Quantum elec-trodynamics in the infinite momentum frame, Phys. Rev.D1, 2901 (1970).[4] P.A.M. Dirac, Is there an æther?, Nature 168, 906(1951); Nature 169, 146; Quantum mechanics and theæther, The Scientific Monthly 78, 142 (1954).[5] A. Dunn and T. Mehen, Implications of SU(2)LxU(1)symmetry for SIM(2) invariant neutrino masses,hep-ph/0610202.[6] A.G. Cohen and D.Z. Freedman, SIM(2) and SUSY,hep-th/0605172.[7] U. Lindstro¨m and M. Rocˇek, SIM(2) and superspace,hep-th/0606093.[8] Work in progress[9] D.Z. Freedman, private communication.[10] J. Patera, R.T. Sharp, P. Winternitz and H. Zassen-haus, Continuous subgroups of the fundamental groupsof physics. III. The de Sitter groups, J. Math. Phys. 18,2259 (1977).[11] M. Levy-Nahas, Deformation and contraction of Lie al-gebras, J. Math. Phys. 8, 1211 (1967).[12] S.R. Coleman, J. Wess and B. Zumino, Structure of phe-nomenological Lagrangians. 1, Phys. Rev. 177 (1969)2239.[13] G. Bogoslovsky, Some physical displays of the spaceanisotropy relevant to the feasibility of its being detectedat a laboratory, arXiv:0706.2621 [gr-qc].[14] J. Gomis, K. Kamimura and P. West, The construc-tion of brane and superbrane actions using non-linearrealisations, Class. Quant. Grav. 23, 7369 (2006),hep-th/0607057.[15] G.Y. Bogoslovsky and H.F. Goenner, Concerning thegeneralized Lorentz symmetry and the generalizationof the Dirac equation, Phys. Lett. A323, 40 (2004),hep-th/0402172.[16] A.J. Hariton and R. Lehnert, Spacetime symmetriesof the Lorentz-violating Maxwell-Chern-Simons model,hep-th/0612167.[17] S.K. Lamoreaux et al., Phys. Rev. Lett. 57, 3125 (1986);T.E. Chupp et al., Phys. Rev. Lett. 63, 1541 (1989).[18] G.Y. Bogoslovsky, On the local anistropy of space-time,inertia and force field, Nuovo Cimento B77, 181 (1983).[19] J. Fan, W.D. Goldberger and W. Skiba, Spin dependentmasses and Sim(2) symmetry, Phys. Lett. B649, 186(2007), hep-ph/0611049.

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We ask whether Cohen and Glashow’s very special relativity model for Lorentz violation might be modified, perhaps by quantum corrections, possibly producing a curved space-time with a cosmological constant. We show that its symmetry group ISIM(2) does admit a 2-parameter family of continuous deformations, but none of these give rise to noncommutative translations analogous to those of the de Sitter deformation of the Poincaré group: space-time remains flat. Only a 1-parameter family 
DISIM
b
(
2
)
 of deformations of SIM(2) is physically acceptable. Since this could arise through quantum corrections, its implications for tests of Lorentz violations via the Cohen-Glashow proposal should be taken into account. The Lorentz-violating point-particle action invariant under 
DISIM
b
(
2
)
 is of Finsler type, for which the line element is homogeneous of degree 1 in displacements, but anisotropic. We derive 
DISIM
b
(
2
)
-invariant wave equations for particles of spins 0, 
1
2
, and 1. The experimental bound, 
|
b
|
<
10
−
26
, raises the question “Why is the dimensionless constant 
b
 so small in very special relativity?

Gomis Torné, Joaquim

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General Very Special Relativity is Finsler Geometry

General Very Special Relativity is Finsler Geometry

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Diposit Digital de la Universitat de Barcelona