Lifshitz and Schrdinger Algebras and Dynamical Realizations 1 - - PowerPoint PPT Presentation

Lifshitz and Schrödinger Algebras and Dynamical Realizations 1

Joaquim Gomis

Departament d’Estructura i Constituents de la Matéria Universitat de Barcelona

University of Crete Heraklion, May 17, 2011

1Based on G. Gibbons, J. Gomis, C. Pope, arXiv:0910.3220[hep-th],

Phys.Rev. D82 (2010) 065002

• R. Casalbuoni, J. Gomis J. Gomis, and K. Kamimura work in progress.

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Outline

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

AdS/CFT correspondence

BMN sector of AdS5 × S52 Strings in a maximal Susy plane wave Susy plane algebra is obtained by contraction of AdS5 × S53 Non-relativistic limit of AdS5 × S54 Symmetry algebra String super Newton-Hooke (NH) atring algebra

2Berenstein, Maldacena, Nastase (02) 3Hatsuda, Kamimura, Sakaguchi (02) 4Gomis, Gomis, Kamimura 05 Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Theories with violation of Lorentz symmetry

Examples Lifshitz theories of gravity5 Anisotropic general relativity Very Special Relativity6 The relativity group is a subgroup of Lorentz group preserving a light-like direction Deformations of Very Special Relativity7 General Very special Relativity

5Horava (09) 6Cohen and Glashow (06) 7Gibbons, Gomis, Pope (07) Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz scalar field theories

In non-relativistic condensed matter theories with k spatial dimensions, one is interested in the behaviour of physical quantities under Lifshitz scaling t → λzt , x → λx where t is the time variable and x = (x1, x2, . . . , xk) is the spatial position vector. Consider the action8 S = 1 2

• dt dkx
• ˙

φ2 − φ(△)zφ

• where △ =

∇2. The scaling dimension of the Lifshitz scalar [φ] = k−z

2 .

Compared to the relativistic scalar in the same space-time the Lifhsitz scalar has an improved UV behavior. For z=3 φ is dimensionless in 3+1 dimensions, therefore any non-linear polinomial interaction are power counting renormalizable.

8Lifshitz (41) Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Non-relativistic adS/Condensed Matter Correspondence

Relativistic metrics with non-relativistic isometries like the Lifshitz and Schrödinger symmetries. These metrics could be dual we to some non-relativistic theories in CMP9

9Son (08), Balasubramanian McGreevy (08), Herzog, Rangamani, Ross (08) Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz algebra

If D generates scalings or dilatations we may combine this with space translations Pa, spatial rotations, Mab and time translations H, to obtain the Lifshitz Algebra, lif z(k) in k spatial dimensions,

• D, Mab
• = 0 ,
• D, Pa
• = Pa ,
• D, H
• = zH ,

If a = 1, 2, . . . , k lif z(k) has dimension 1

2k(k + 1) + 2, then the quotient

lif z(k)/so(k) has dimension k + 2 and it represents the Lifshitz spacetime .

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz spacetime

Lifshitz spacetime is a k + 2 dimensional spacetime equipped with a metric invariant under the left action of the (k + 2)-dimensional group generated by Pi, H and D. A Maurer-Cartan basis f is er = dr r , ea = dxa r , e0 = dt r z . The Lifshitz metric is then ds2

k+2 = L2

−dt2 r 2z + dxadxa r 2 + dr 2 r 2

• ,

with Killing vector fields corresponding to Mab = −(xa∂b − xb∂a) , Pi = −∂a , H = −∂t , D = −(zt∂t + xa∂a + r∂r) .

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz spacetime

The boundary metric at infinity is obtained by taking out a factor of r 2 and letting r → 0: ds2

k+2 = L2

r 2

• −

dt2 r 2(z−1) + dxadxa + dr 2 Thus ds2

boundary = dxadxa − r 2(1−z)dt2 ,

the speed is c(r) = r (1−z), and

◮ If z > 1, we obtain infinite speed (the boundary lightcone opens out to a

plane), Galilean theories

◮ If z = 1, we obtain finite speed (the boundary lightcone remains a cone),

Relativistic theories

◮ If z < 1, we obtain zero speed (the boundary lightcone closes up to a

half line ), Carroll theories

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

The boost-extended Lifshitz algebra

One may extend the Lifshitz algebra to include boosts. The scaling dependence of Ka is then determined by its commutation relations. Since Ka is a vector we have Ka.

• Kc, Mab
• = −
• δcaKb − δcbKa
• For the Galilei group,
• Ka, Pb
• =

0 ,

• Ka, H
• =

Pa , which implies that we must take

• D, Ka
• = (1 − z)Ka .

For the Carroll group

• Ka, Pb
• =

δabH ,

• Ka, H
• =

0 , which implies that we must take

• D, Ka
• = (z − 1)Ka .

In the case of the Poincaré group there is no choice, and one must take z = 1.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

The Schrödinger and Extended Schrödinger algebras

In k spatial dimensions, the centrally extended ( 1

2k(k + 1) + k + 3)

dimensional Schrödinger algebra which we denote ˜ schz(k), is obtained by adjoining Galilean boosts Ki, and a central term N to the of translations, rotations and time translations, such that

• Mab, Kc
• =
• δacKb − δbcKa
• ,
• Pa, Kb
• =

−δabN ,

• H, Ka
• =

−Pa . One then adjoins a dilatation D,

• D, Ka
• = (1 − z)Ka ,
• D, N
• = (2 − z)N .

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Schrödinger group

If k = 3 this is 12-dimensional, whereas what has been called the Schrödinger group, i.e. the conformal symmetry group of the free Schrödinger (corresponding to z = 2) is 13 dimensional. This is because the special conformal or temporal inversion operator has been left out. This transformation sometimes called expansion is given

• x′

=

• x

1 − kt t′ = t 1 − kt where the k is the parameter of the expansion. The infinitesimal generator of this special conformal transformation is given by C = t2 ∂ ∂t + t xi ∂ ∂xi

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Schrödinger group

The new commutation relations are [C, Pa] = Ba, [C, Ba] = 0, [C, H] = − D, [C, D] = − C. (H, C, D) form the conformal algebra in one dimensions SO(1, 2). Notice the difference with the Galilean Conformal algebra obtained by contraction from the relativistic conformal algebra which has 15 generators

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

ISIM and DISIMb

Cohen and Glashow have made the proposal that the local laws of physics need not be invariant under the full Lorentz group, generated by Mµν, but rather, under a SIM(2) subgroup, M+a, Mab, M+− = M03, (with i and b ranging over the values 1 and 2) . This they referred to as Very Special Relativity. Taking the semi-direct product with the translations (P+, P−, Pa) gives an 8-dimensional subgroup of the Poincaré group called ISIM(2) [M+−, P±] = ∓P± , [M+−, M+a] = −M+a , [J, Pa] = ǫabPb , [J, M+a] = ǫabM+b , [M+a, P−] = Pa , [M+a, Pb] = −δabP

+.

where J ≡ Mab.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

DISIMb(2)

In order to see if very special relativity with curved space, one can find the deformations of ISM(2) algebra. One obtain DISIMb(2)10 [M+−, P±] = −(b ± 1)P± , [M+−, Pa] = −bPa , which does not describe a curved since the translations commute. Howvever

• ne can show that11

˜ schz(k) ≡ disimb(k) , b = 1 1 − z . To see this, one must identify the generators as follows; H ↔ P− , N ↔ −P+ Pa ↔ Pa , Ka ↔ M+a . and D ↔ (z − 1)M+− .

10Gibbons, Gomis, Pope (07) 11Gibbons, Gomis, Pope (09) Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Dynamical realizations. Lifshitz particle

The starting point is the Lifshitz algebra with generators H, P, Mab, D associated to spacetime translations, rotations and dilatations. The algebra is given by [Mab, Mcd] = −i (ηbcMad − ηacMbd − ηbdMac + ηadMbc) , [Pa, Mbc] = −i (ηabPc − ηacPb) , [D, H] = −iz H, [D, Pa] = −i Pa We consider the coset

Lifshitz Rotations that we locally parametrize as

g = e−iHt eiPaxa eiDσ

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz particle

The Maurer Cartan is given by Ω = −ig−1dg = HLH + PaLa

P + DLD

where LH = −e−zσ dt, La

P

= e−σ dxa, LD = dσ. The global symmetries are H ; δt = ǫ0, Pa ; δxa = ǫa, D ; δσ = ǫ, δt = zǫt, δxa = ǫxa

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz particle

An action invariant under Lifshitz and diffeomorphism with the lowest order in derivative is given by L = a LH + b

• L2

P

= −a e−zσ (˙ t) + b e−σ ( ˙ xa)2. where the coefficients a and b are dimension full parameters, [a] = Mz, [b] = M The momenta are pt + a e−zσ = 0, pa = b e−σ ˙ xa

• ( ˙

xa)2 , → p2

a − b2 e−2σ = 0,

pσ = 0. We have two second class constraints pσ = 0, e−zσ = −pt a ,

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz particle

If a = bz the first class constraint can be written as φ = 1 2

• p2

t − (

p2)z = 0. There is a certain degree of arbitrariness in presenting the power of pt. Canonical form of the action L = −pt(σ, x) ˙ t + p(σ, x) ˙

• x

that we can write as12 LC = −pt ˙ t + p ˙

• x − e

2

• p2

t − (

p2)z

12S.Kalyana Rama, (2010), D. Capasso, A.P

. Polychronakos (2009) T. Suyama (2009)

• L. Sindoni (2009), A. Mosaffa (2010), M. Eune, W. Kim (2010)]

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Quantization Lifshitz particle

Following the Dirac’s procedure of quantization we should impose the first class constraint on the physical states. We have (∂2

t − △z)φ(

x, t) = 0 which gives the equation of motion of the free Lifshitz scalar.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz particle

Since the σ variable is non dynamical we can use their equations of motion to eliminate it. The lagrangian depending only on x’s and t is13 L ∼ ( ˙

• x ˙
• x)

z 2(z−1) ˙

t

1 1−z

which is a Finslearian lagragian. The Finslerian line element is given by ds = (δijdxidxj)

z 2(z−1) dt 1 1−z 13Romero, Cuesta, Garcia, Vergara 2009 Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz particle

Note there are not WZ terms. If we consider higher order derivatives we could have add the extrinsic curvature term of the spatial dimensional curve x(τ) 1 ( ˙ x2)

• (¨

x ˙ x)2 − (¨ x2)( ˙ x2). The lagrangian in this case will be L = −a e−zσ (˙ t) + b e−σ ( ˙ xa)2 + 1 ( ˙ x2)

• (¨

x ˙ x)2 − (¨ x2)( ˙ x2).

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz particle

Note that if we add to the Lifshitz algebra, the boost generators K and the "central" element N, we obtain the schrodingerz algebra which is isomorphic to DISim(b). We know that in this case the line element is also Finslearian.14 L = −m(−ηµν ˙ xµ ˙ xν)(1−b)/2 (−nρ ˙ xρ)b . The Schrodinger algebra is generated by (H, Pa, Ba, Ja, Z, D, C). Since (Ba, Ja, Z, C) have a closed subalgebra we can consider a coset Schrodinger

(Ba,Ja,Z,C).

For z=2 the particle invariant under the Schrodinger algebra is L = ˙ x2 ˙ t

14Bogoslovsky(06), Gomis, Gibbons, Pope (07) Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

z=1 Lifshitz particle

For z=1 we should consider the initial lagrangian and (˙ t) +

• ( ˙

xa)2 = 0, a = b, as L = = e−σ(− (˙ t) +

• ( ˙

xa)2) = e−σ ( (˙ t) +

• ( ˙

xa)2) (− (˙ t)2 + ( ˙ xa)2) = e−σ ( (˙ t) +

• ( ˙

xa)2) ( ˙ xµ ˙ xµ) ≡ ˙ xµ ˙ xµ 2˜ e . It is the massless relativistic particle lagrangian if we regard the coefficient as ein-bein. Note that in this case dilatation invariance implies conformal invariance

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Uniqueness of Lifshitz particle action

Another possible invariant action, for z > 1, is L =

• βL2

P − αL2 H

=

• βe−2σ ˙
• x

2 − α e−2zσ ˙

t2. where α, β are positive dimensionfull parameters [α] = M2z, [β] = M2. pt = −α e−4zσ˙ t L , pa = β e−2σ ˙ xa L , → φ = 1 β ( pa e−σ )2 − 1 α( pt e−zσ )2 − 1 = 0 pσ = 0.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Uniqueness of Lifshitz particle

There appear two primary constraints and one secondary from ˙ pσ = 0 1 β ( pa e−σ )2 − z 1 α( pt e−zσ )2 = 0. If we choose α = βzzz( 1 z − 1)

1 z−1

we have the first class constraint p2

t − (

p2)z = 0 It gives the same dispersion relation as the previous lagrangian

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz Lagrangian with further breaking of anisotropy

Let further break the anisotropy among the spatial directions t → ezǫt, x1 → ez1ǫx1, xi → eǫxi ( j = 2, ...d − 1). the coset is given by g = e−itHeix1P1eixi Pi eiσD The MC forms are LH = −e−zσ dt, L1

P

= e−z1σ dx1, Li

P

= e−σ dxi, LD = dσ. An invariant lagrangian is L = a LH + bL1

P + c

• L2

P = −a e−zσ (˙

t) + b e−z1σ ( ˙ x1) + c e−σ ( ˙ xi)2.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Lifshitz particle

there are two first class constraints, a = cz, a = b φ = 1 2

• p2

t − (p2 a)z

, pt = (−p1)

z z1 .

For the quantization one should impose the two first class constraints on the physical states.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Bosonic Lifshitz string

To construct a Lifshitz string we first consider the analogous of the string Galilei algebra. We distinguish among the longitudinal directions, x0, x1, and transverse directions of the string. A Lifshitz string algebra is [D, Pa] = Pa, [D, H] = izH, [D, P1] = izP1 The string theory is invariant under xµ → ezǫxµ, xa → eǫxa, (µ = 0, 1; a = 2, ...d − 1). We consider the coset

Lifshitzstring TransverseRotations that we local parametrize as

g = eixµPµeixaPaeiσD

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Bosonic Lifshitz string

The Maurer Cartan is given by Ω = eµPµ + eiPi = −e−zσdxµPµ + eσdxaPa + dσD An action invariant under Lifshitz and diffeomorphism with the lowest order in derivative is given by S = S + S⊥ =

• dτdσ
• a e−2zρ

− det g + b e−2ρ det g⊥

• ,

gij = ηµν∂ixµ∂jxν = −(ǫµν∂ixµ∂jxνǫij)2, g⊥ij = ηab∂ixa∂jxb.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Bosonic Lifshitz string

The Hamiltonian becomes H =

• dσ
• λ0

H∗ + λT + λ1 ⊥T⊥

• where

H∗ = 1 2(p2

+ a2

• p2

⊥

b2 x′

⊥ 2

z x′

• 2) = 0,

T = px′

= 0,

T⊥ = p⊥x′

⊥ = 0,

One can prove that

• p2

⊥

b2 x′

⊥ 2

• is a constan of motiont. For z=1 does not

reproduce the tensionless string.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Finslearian string

Now we would like to construct the Lifshitz string in terms in the x’s and t space, e−σ =

• −b
• detgij

aǫµν∂ixµ∂jxνǫij

• 1

2(z−1)

using this expression we get L∗ ∼ (

• detgij)

z z−1 (ǫµν∂ixµ∂jxνǫij) 1 z−1

which for z=2 gives L∗ ∼ detgij ǫµν∂ixµ∂jxνǫij = detgij

• −detgµν

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Non-relativistic String

It does not coincide with the non-relativistic string15 S = −1 2T

• d2σ
• − det g gij∂iX a∂jX bδab

where gij is the inverse of the two dimensional metric gij = ∂ixµ∂jxνηµν, the worldsheet coordinates are τ and σ, σi = (τ, σ), i = 1, 2. The NR string can be obtained as semiclassical expansion along an straight Wilson line16

15Gomis Ooguri (2000), Guijosa et al (2000)

Gomis, Gomis, Kamimura (2005)

16Sakaguchi and Yoshida (2007) Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Other bosonic string actions

We could also consider the NG like string action S =

• dτdσ
• − det gmn =
• dτdσ
• ( ˙

yy ′)2 − ˙ y 2y ′2, where ˙ y 2 = −e−2zσ˙ t2 + e−2σ˙

• x

2

y ′2 = −e−2zσt′2 + e−2σ x′ x′ ˙ yy ′ = −e−2zσ˙ tt′ + e−2σ˙

• x

x′ For z=1 S =

• dτdσ e−σ

( ˙ xx′)2 − ˙ x2x′2, whic reproduces the Poincare tensionless string

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Super Lifhsitz algebras

The superLifshitz algebra in 10 dimensions is given [Mab, Mcd] = −i (ηbcMad − ηacMbd − ηbdMac + ηadMbc) , [Pa, Mbc] = −i (ηabPc − ηacPb) , [Q±, Mab] = i 2Q±Γab, {Q−, Q−} = −2

• CΓ0P−
• H,

{Q+, Q−} = 2

• CΓaP−
• Pa,

and [D, H] = −inH H, [D, Pa] = −inP Pa, [D, Q±] = −in± Q±. The Jacobi identities require the weights of D as nH = 2 n−, nP = n− + n+. For example nH = z, nP = 1, n− = z 2 n+ = 1 − z 2.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Super Lifhsitz algebras

We use projection operators P± = 1 2 (1 ± Γ∗) , (Γ∗ = Γ0Γ11, η = (− + + + ...), η11 = +1), to define Q± = QP±.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Super Lifshitz particle in 10 dimensions

g = e−iHt eiPaxa eiQ−αθα

− eiQ+αθα + eiDσ

we get LH = e−nHσ (dt + i θ−Γ0dθ−), La

P

= e−nPσ (dxa + 2i θ+Γadθ−), Lab = 0, Lα

+

= e−n+σ dθ+, Lα

−

= e−n−σ dθ−, LD = dσ. and satisfy the MC equation

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Super Lifshitz particle in 10 dimensions

Global symmetries are H ; δt = ǫ0, Pa ; δxa = ǫa, Q+ ; δθα

+ = ǫα +, δxa = 2iθ−Γaǫ+

Q− ; δθα

− = ǫα −, δt = iθ−Γ0ǫ−, δxa = 2iθ+Γaǫ−

D ; δσ = ǫ, δθ± = n±ǫ θ±, δt = nHǫ t, δxa = nPǫ xa. A possible invariant action for a particle is (using second parametrization) L = a LH + b

• L2

P

= a e−nHσ (˙ t + i θ−Γ0 ˙ θ−) + b e−nPσ

• ( ˙

xa + 2i θ+Γa ˙ θ−)2. = a e−nHσ ut + b e−nPσ (ua)2. No WZ term and no kappa symmetry for general z.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Super Lifshitz particle in 10 dimensions

pt − a e−nHσ = 0, pa = b e−nPσ ua

• (ua)2 ,

→ p2

a − b2 e−2nPσ = 0,

ζ− − i ¯ θ−Γ0pt − 2i ¯ θ+Γapa = 0, ζ+ = 0, pσ = 0. Four of them are second class pσ = 0, e−nHσ = pt a , ζ− = i ¯ θ−Γ0pt + 2i ¯ θ+Γapa, ζ+ = 0,

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Super Lifshitz particle in 10 dimensions

and one is the first class φ = 1 2

• p2

t

nP −

• p2

a

nH = 0. Therefore there is no kappa symmetriy in this model.

Joaquim Gomis

Motivation for Non-relativistic algebras Lifshitz , Schrodinger, DIsimb algebras Dynamical realizations Conclusions

Conclusions

We have constructed two actionsion for a Lifshitz bosonic particle that leads to the same Lifshitz dispersion relation We have also constructed actions for a Lifshitz bosonic strings We have constructed a Super Lifshitz algebra and a super particle realization

Joaquim Gomis