Canonical analysis of non-relativistic particle and superparticle

The European Physical Journal C, Feb 2018

We perform canonical analysis of non-relativistic particle in Newton–Cartan Background. Then we extend this analysis to the case of non-relativistic superparticle in the same background. We determine constraints structure of this theory and find generator of \(\kappa \)-symmetry.

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Canonical analysis of non-relativistic particle and superparticle

Eur. Phys. J. C Canonical analysis of non-relativistic particle and superparticle 0 Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University , Kotlárˇská 2, 611 37 Brno , Czech Republic We perform canonical analysis of non-relativistic particle in Newton-Cartan Background. Then we extend this analysis to the case of non-relativistic superparticle in the same background. We determine constraints structure of this theory and find generator of κ -symmetry. 1 Introduction and summary 1 See also [34-38]. - Holography is very useful for the analysis of properties of strongly coupled quantum field theories. Recently these ideas were extended to non-relativistic theories since today it is well known that non-relativistic holography is very useful tool for the study of strongly correlated systems in condensed matter, for recent review see [1]. Non-relativistic symmetries also have fundamental meaning in the recent proposal of renormalizable quantum theory of gravity known today as Hoˇrava–Lifshitz gravity [2], for recent review and extensive list of references, see [3]. There is also an interesting connection between Hoˇrava–Lifshitz gravity and Newton– Cartan gravity [4, 5]. Newton–Cartan gravity is covariant and geometric reformulation of Newton gravity that is now very intensively studied, see for example [6–13].1 Concept of non-relativistic physics also emerged in string theory when strings and branes were analyzed at special backgrounds. At these special points of string moduli spaces non-relativistic symmetries emerge in natural way [14, 15] . These actions were obtained by non-relativistic “stringy” limit where time direction and one spatial direction along the string are large. The stringy limit of superstring in Ad S5 × S5 was also formulated in [19] and it was argued here that it provides another soluble sector of AdS/CFT correspondence, for related work, see [24, 25]. Non-relativistic limit was further extended to the case of higher dimensional objects in string theory, as for example p-branes [20–23].2 It is important to stress that these constructions of nonrelativistic objects are based on manifest separation of directions along which the non-relativistic limit is taken and directions that are transverse to them. The first step for the more covariant formulation which corresponds to the particle in Newton–Cartan background was performed in [30] and further elaborated in [10]. The structure of this action is very interesting and certainly deserves further study. In particular, it would be very useful to find Hamiltonian for this particle. Our goal in the first part of this article is to find the Hamiltonian formulation of the particle in Newton–Cartan and in Newton–Cartan–Hooke background. It turns out that this is non-trivial task even in the bosonic case due to the complicated structure of the action. On the other hand when we determine Hamiltonian constraint we find that the canonical structure of this theory is trivial due to the fact that there is only one scalar first class constraint. A more interesting situation occurs when we consider supersymmetric generalization of the non-relativistic particle. As the first case we study Galilean superparticle whose action was proposed in [30]. We find its Hamiltonian form and identify primary constraints. We show that fermionic constraints are the second class constraints that can be solved for the momenta conjugate to fermionic variables when we also obtain non-trivial Dirac brackets between fermionic variables. Next we consider more interesting case corresponding to the κ -symmetric non-relativistic particle action [30]. We again determine all primary constraints. Then the requirement of the preservation of the fermionic primary constraints determines corresponding Lagrange multipliers. In fact we find a linear combination of the fermionic constraints that is the first class constraint and that can be interpreted as the generator of the κ -symmetry. Since it is the first class constraint it can be 2 For recent works, see [16–18,27–29]. fixed by imposing one of the fermionic variables to be equal to zero and we return to the previous case. Finally we perform Hamiltonian analysis of superparticle in Newton–Cartan background. This action was found in [10] up to terms quadratic in fermions. We again determine Hamiltonian constraint which however has much more complicated form due to the presence of fermions . We also determine two sets of primary fermionic constraints. As the next step we study the requirement of the preservation of these constraints during the time evolution of the system. It turns out that this is rather non-trivial and complicated task in the full generality and hence we restrict ourselves to the simpler case of the background with the flat spatial sections. In this case we show that the Hamiltonian constraint is the first class constraint with vanishing Poisson brackets with fermionic constraints. We also identify linear combination of the fermionic constraints which is the first class constraint and that can be interpreted as the generator of κ-symmetry. The extension of this paper is obvious. It would be very interesting to analyze constraint structure of superparticle in Newton–Cartan background in the full generality. Explicitly, we should analyze the time evolutions of all constraints and determine conditions on the background fields with analogy with the case of relativistic superparticle as was studied in [33]. We hope to return to this problem in future. The organization of this paper is as follows. In the next Sect. 2 we perform Hamiltonian analysis of particle in Newton–Cartan background. Then in Sect. 3 we generalize this analysis to the case of particle in Newton–Cartan–Hooke background. In Sect. 4 we perform Hamiltonian analysis of Galilean superparticle. Finally in Sect. 5 we analyze nonrelativistic superparticle in Cartan–Newton background. 2 Hamiltonian of Newton–Cartan particle By Newton–Cartan particle we mean a particle that moves in Newton–Cartan background with an action invariant under general coordinate transformations. In order to describe Newton–Cartan background we need a temporal vielbein τμ and spatial vielbein eμa , μ = 0, . . . , d −1, a = 1, . . . , d −1. We also need a central charge gauge field mμ. Then the action for particle in Newton–Cartan background has the form [10] where λ is a parameter that labels the world-line of the particle and x˙μ = ddxλμ . From ( 1 ) we derive following conjugate momenta pμ = m eμa x˙ν eνbδab x˙ρ τρ m x˙ρ eρa x˙ν eνbδab − mmμ − 2 (x˙ρ τρ )2 Using ( 2 ) it is easy to see that the bare Hamiltonian is equal to zero HB = pμx˙μ − L = 0 as we could expect since the action ( 1 ) is manifestly invariant under world-line reparameterization λ = λ (λ), x μ(λ ) = x μ(λ). The fact that the theory is invariant under world-line diffeomorphism suggests that there should exist corresponding Hamiltonian constraint. In order to find it we use following relations ( 1 ) hμν = eμa eνbδab. As the next step we would like to present an alternative way how to derive an action for Newton–Cartan particle. We start with an action for relativistic massive particle in general background that couples to external gauge field Mμ ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 10 ) τμ. ( 2 ) S = −M dλ −ηAB EμA EνB x˙μx˙ν − Mμx˙μ , ( 11 ) τ μτμ = 1, τ μe a μ = 0 in ( 2 ) and we obtain m x˙μeμa x˙ν eνbδab τ μ( pμ + mmμ) = − 2 (x˙ρ τρ )2 , eμa ( pμ + mmμ) = m δabeνb x˙ν . τρ x˙ρ If we combine these results together we obtain following primary constraint Hλ = eμa ( pμ + mmμ)eνb( pν + mmν )δab + 2mτ μ( pμ + mmμ) ≈ 0 which is desired result. Then extended Hamiltonian has the form HT = λλHλ, p˙μ = where where λλ is a Lagrange multiplier corresponding to the first class constraint Hλ. The equations of motion for x μ, pμ have the form x˙μ = x μ, HT = 2λλ hμν ( pν + mmν ) + mτ μ , pμ, HT = −λλ ∂μhρσ ( pρ + mmρ )( pσ + mmσ ) + 2m∂μmρ hρσ ( pσ + mmσ ) + 2m∂μτ ρ ( pρ + mmρ ) + 2m2τ ρ ∂μmρ , where EμA is vielbein where A = 0, . . . , d − 1 and where exists inverse vielbein E μA defined as EμA E μB = δB , EμA E νA = δμ. A ν Now we find corresponding canonical form of the action ( 11 ). To do this we derive conjugate momenta from ( 11 ) Hλ = 2m( pμ+mmμ)τ μ+( pμ+mmμ)hμν ( pν +mmν ) ≈ 0. We see that the Hamiltonian constraint ( 20 ) coincides with the constraint ( 7 ). Using ( 20 ) we find the canonical form of the Newton–Cartan particle action S = dλ( pμx˙μ − vλHλ). It is instructive to find corresponding Lagrangian. Using ( 20 ) we obtain x μ ˙ = Now we are ready to find Hamiltonian constraint for Newton– Cartan particle using following form of the vielbein EμA that was introduced in [10] Eμ0 = ωτμ + 21ω mμ, Eμa = eμa , where a = 1, . . . , d − 1 and where ω is a free parameter and we take ω → ∞ in the end. An inverse vielbein to ( 15 ) has the form E μa = eμa + O(ω−2) , E μ0 = ω1 τ μ + O(ω−3), where eμa , τ μ are defined as [10] eμa eμb = δb , τ μτμ = 1 , τ μe a a μ = τμeμa = 0, eρa eμa = δμ − τμτ ρ . ρ Note that ( 16 ) contain term linear in ω1 . In principle it contains infinite number of terms of corrections ω1n that will give vanishing contributions in the limit ω → ∞. Finally the gauge field has the form [10] 1 Mμ = ωτμ − ω mμ. Inserting ( 16 ) together with ( 18 ) into ( 14 ) and rescaling M as M = ωm we obtain Hλ = − ω12 pμτ μ pν τ ν + pμeμa δabeνb pν + 2mpμτ μ 2 + 2mpμhμν mν − ω2 mpμτ μτ ν mν − m2ω2 + 2m2mμτ μ − mω22 (mμτ μ)2 + m2mμeμa δabeνbmb + m2ω2 ≈ 0 that in the limit ω → ∞ simplifies considerably ( 12 ) ( 13 ) ( 14 ) ( 15 ) ( 16 ) ( 17 ) ( 18 ) ( 19 ) ( 20 ) ( 21 ) ( 22 ) ( 23 ) ( 24 ) ( 25 ) ( 26 ) L = pμx˙μ−HT = vλ( pμ+mmμ)hμν ( pν +mmν )−mmμx˙μ. Of course, this is not final form of the Lagrangian since it has to be function of x μ and x˙μ instead of pμ. In order to find this form we multiply ( 22 ) with τμ and use ( 17 ) so that we can express vλ as vλ 1 = 2m τμx˙μ. Further, if we multiply ( 22 ) with hμν we obtain 21 hμν x˙ν = vλ( pμ + mmμ) − vλτμτ ρ ( pρ + mmρ ) Inserting this result into ( 23 ) and using again ( 17 ) together with ( 24 ) we finally obtain m x˙μhμν x˙ν L = 2 x˙μeμa x˙ν eνbδab x˙ρ τρ x μeμa x ν eνbδab R2 where the Ad S radius R2 is related to the cosmological constant , < 0 as R2 = − 1 . From ( 27 ) we find conjugate momenta Following analysis performed in previous section we determine various projectors of the combination pμ + mmμ m x˙ ρ eρa x˙ ν eνbδab τ μ( pμ + mmμ) = − 2 (x˙ ρ τρ )2 m − 2 R2 x ρ eρa x ν eνbδab, eμa ( pμ + mmμ) = m x˙ ν eνbδba . x˙ ρ τρ If we combine these two relations we obtain following primary Hamiltonian constraint for the Newton–Cartan–Hooke particle Hλ = 2mτ μ( pμ + mmμ) + ( pμ + mmμ)hμν ( pν + mmν ) m2 + R2 hμν x μ x ν ≈ 0. ( 30 ) From ( 32 ) we determine corresponding conjugate momenta ∂ L ∂ t˙ i = − m2 x˙t 2x˙i , ˙ pα− = ∂ L x pi = ∂ x˙ i = m ˙t˙i , i = 1, 2, pt = ∂∂Lθ˙−Lα = m2 (θ¯−γ 0)α , α = 1, 2, where ∂ L means left-derivative. As usually we find that the bare Hamiltonian is zero HB = x˙ i pi + t˙pt + θ˙−α pα− − L = 0 while the theory possesses three primary constraints Gα− ≡ pα− − m2 (θ¯−γ 0)α = pα− + m2 θ−β δβα ≈ 0, Hλ ≡ pi pi + 2m pt ≈ 0. It is important to stress that pα− and θ−β are Grassmann odd variable with the graded Poisson brackets θ T 0 ¯− = θ− γ . 4 Supersymmetric generalization In this section we proceed to the Hamiltonian analysis of non-relativistic superparticles whose actions were derived in [30] and in [10]. We begin with the simplest case which is Galilean Superparticle. We restrict ourselves to the case of three dimensions as in [30]. 4.1 The Galilean superparticle The action for Galilean Superparticle has the form3 3 Note that we use Majorana representation where all gamma matrices are real γ μ = (i σ2, σ1, σ3), or explicitly γ 0 = −01 01 , γ 1 = 0 1 1 0 , γ 2 = 1 0 0 −1 γ bγ a = 2ηabI, ηab = diag( −1, 1, 1 ), I = that θ− is real Majorana spinor so that that obey the standard relation γ a γ b + 1 0 0 1 . Then we presume θ α , pβ− − = p−, θ α β − = δαβ . Gα−, Gβ− = mδαβ . Then it easy to find following non-zero Poisson bracket In other words Gα− are second class constraints. As a result we can eliminate all conjugate momenta pα− with the help of Gα− = 0. Then of course we have to replace Poisson brackets between θ α , θ β with corresponding Dirac brackets − − θ α , θ β − − D = θ α , θ β − − 1 δγ δ × m − α θ−, Gγ− Gδ−, θ β − = − m1 δαβ . 4.2 κ -symmetric Galilean superparticle It is well known that the relativistic superparticle is invariant under additional fermionic symmetry called κ -symmetry [31, 32]. In case of strings and branes an existence of κ symmetry is necessary for the correct counting of degrees of freedom. In the non-relativistic case it is known from the work [20] that the κ -symmetry is just a Stückelberg symmetry that acts as a shift on one of the fermionic coordinates, see also [30]. Then this symmetry can be easily fixed by setting this fermionic coordinate to be equal to zero. It is instructive to see how it works in the canonical description so that we now perform Hamiltonian analysis of κ -symmetric version of the flat Galilean superparticle action that depends on an additional fermionic coordinate θ+. The action has the form [30] ( 33 ) ( 34 ) ( 35 ) ( 36 ) ( 37 ) ( 38 ) (39) where the line elements π 0, π i are defined as Now we proceed to the canonical formulation of this theory. From (39) we obtain pi = ∂∂x˙Li = m ππi0 , pt = ∂∂Lt˙ = − m2 (ππi 0π)i2 , pα− = ∂∂Lθ˙−Lα = − m4 (θ¯+γ i )α ππi0 + (θ¯−γ 0)α, pα+ = ∂∂Lθ˙+Lα = m8 (θ¯+γ 0)α (ππi0π)i2 − m4 ππ 0i (θ¯−γ i )α. Last two equations imply two sets of primary constraints Gα− = pα− + 41 (θ¯+γ i )α pi − m2 (θ¯−γ 0)α ≈ 0, Gα+ = pα+ + 41 (θ¯+γ 0)α pt + 41 (θ¯−γ i )α pi ≈ 0 with following non-zero Poisson brackets 1 Gα−, Gβ− = mδαβ , Gα−, Gβ+ = 2 (γ 0γ i )αβ pi , Gα+, Gβ+ 1 = − 2 δαβ pt . (41) (42) (43) (44) (45) (46) Finally the theory possesses the Hamiltonian constraint in the form Hλ = 2mpt + pi pi ≈ 0 that is clearly first class constraint Hλ, Gα+ From the last equation we obtain λα+ = p1t δαβ (γ 0γ i )βγ λγ− pi . Inserting this result to the first Eq. (47) we find that it is obeyed since it is proportional to ( pi pi + 2mpt ) = Hλ ≈ 0 so that λβ is not specified and it is a free parameter. Then − it is natural to introduce following linear combination of the fermionic constraints 1 α = Gγ+δγβ (γ 0γ i )βα p + Gα−. t Then it is easy to see that α, Gβ− α, Gβ+ 1 = 2 pt ( pi pi + 2mpt )δαβ ≈ 0, = 0 which implies that α, β ≈ 0 and hence α ≈ 0 is the first class constraint while Gα− ≈ 0 is the second class constraint. Let us then calculate the Poisson brackets between all canonical variables and α α, θ+β = (γ 0γ i )βα ppi , t α, θ−β β = δα α, xi = − 41pt (θ−γ i γ j )α p j − 41 (θ¯+γ i )α, { α, t } = 41 (θ¯+γ j )α ppj . t With analogy with the Lagrangian description we would like to define κ-transformation that acts on the θ+ as a shift. For that reason we define κ-transformation in the following way (52) p j (κγ j γ 0)α { α, X } ≡ κα { α, X } . δκ X = 2m Then using (52) we easily find δκ θ+α = κα, δκ θ−α = 2pmj (κγ j γ 0)α, δκ xi = 41 κ¯ γ i θ− − 8pmj κγ j γ i θ+, δκ t = − 41 κθ+ that are correct κ-symmetry transformations. Fixing the κsymmetry can be done by imposing the gauge fixing condition and hence we see that they are second class constraints that can be explicitly solved for θ+α and pα+. Further, the constraints Gα− ≈ 0 are two second class constraints that can be solved for pα− exactly in the same way as in previous section. 5 Non-relativistic superparticle in Newton–Cartan background In this section we perform Hamiltonian analysis of the nonrelativistic superparticle in Cartan–Newton background. The action for this particle was derived in [10] and it has the form πˆ a = x˙μ eμa − 21 θ¯+γ a ψμ− − 21 θ¯−γ a ψμ+ where Dˆ is covariant derivative with respect to spatial rotation 1 x˙μωμabγabθ−. Dˆ θ− = θ˙− − 4 Finally it is understood that we are interested in terms up to second order in fermions in πˆ a πˆ b. It is important to stress that spin connections ωμa , ωμab depend on e, τ, m, ψ . The explicit form of this dependence can be found in [10]. Now we can proceed to the Hamiltonian formalism. First of all we rewrite the line elements πˆ a and πˆ 0 as 1 1 + 4 θ¯+γ a Dˆ θ− + 4 − θ¯ γ a Dˆ θ+ 1 + 8 θ¯+γ a γb0θ+x˙μωμb, πˆ a = x˙μMμa + 41 θ¯+γ a θ˙− + 41 θ¯−γ a θ˙+, πˆ 0 = x˙μVμ + 41 θ¯+γ 0θ˙+, where Mμa = eμa − 21 + a 1 θ¯ γ a ψμ+ θ¯ γ ψμ− − 2 − 1 θ¯ γ a ωμcd γcd θ− − 16 + 1 1 − 16 − θ¯ γ a γb0θ+ωμb, θ¯ γ a ωμcd γcd θ+ + 8 + Vμ = τμ − 21 + 0 1 θ¯ γ 0ωμabγabθ+. θ¯ γ ψμ+ − 16 + Then the condition MμbMμa = δba implies MμbMμa = eμbeμa + eμbVμa + Wμbeμa + O(θ 4) = δba so that we obtain the condition Wμbeμa = −eμbVμa . In order to determine Wμa we multiply the last equation with eνa and we obtain Wμb(δμν − τμτ ν ) = −eμbVμa eνa that has solution Wμa = −eνa Vνbeμb (57) (58) (59) μ since e bτμ = 0. Using this notation we determine corresponding conjugate momenta pα− = ∂∂Lθ˙−Lα = − m4 (θ¯+γ b)αδba ππˆˆ b0 − m2 θ−β δβα, pα+ = ∂∂Lθ˙+Lα = − m4 (θ¯−γ a )α δabπˆ b π 0 ˆ m πˆ a πˆ bδab (θ¯+γ 0)α. + 8 (πˆ 0)2 To proceed further we have to introduce vector Tμ that obeys the condition To proceed further we now presume an existence of the inverse matrix Mμb to Mμa that obeys the relation MμbMμa = δb . a In fact, since we are interested in terms up to second order in fermions we derive this matrix as follows. Let us denote original matrix and its inverse as (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (60) TμMμa = 0. (61) τ μV a μ μ = −Wμe a With analogy with the pure bosonic case we presume that it has the form Tμ = τ μ + Wμ, where Wμ is quadratic in fermions. Then the condition (69) implies that can be again solved as Wμ = −τ ν Vνa eμa . Finally note that generally TμVμ = 1. (72) (73) (74) (75) (76) (77) (78) (79) With the help of the knowledge of Tμ, Mμa we can proceed to the derivation of corresponding Hamiltonian constraint. Using Tμ and Mμa we find m πˆ a πˆ bδab 2 (πˆ 0)2 1 = − TμVμ Mμa μ = m δabπˆ b π 0 ˆ where we defined μ as Tμ μ, m πˆ a πˆ bδab Mμa Vμ, − 2 (πˆ 0)2 m m μ = pμ + mmμ + 4 ωμa θ¯+γa θ− − 8 θ¯−γ 0ωμabγabθ−. Inserting the first equation in (74) into the second one we obtain Tμ Mμa − Tν Vν Mρa Vρ μ = m δabπˆ b π 0 ˆ so that we obtain following Hamiltonian constraint Hλ = Mμa Tσ Vσ − TμMρa Vρ μδab Mνa Tσ Vσ − Tν Mσa Vσ ν + 2Tρ Vρ Tμ μ ≈ 0. Further, using (74) in the second and third equation in (68) we also find fermionic primary constraints Gα− = pα− + m2 θ−β δβα + 41 (θ¯+γ a )α × Tμ Mμa − Tν Vν Mρa Vρ μ ≈ 0, Gα+ = pα+ + 41 (θ¯−γ a )α Tμ Mμa − Tν Vν Mρa Vρ μ + 41 (θ¯+γ 0)αTμ μ ≈ 0. As the next step we should analyze the requirement of the preservation of all constraints during the time evolution of the system. It is a difficult task in the full generality due to the complicated form of these constraints. For that reason we restrict ourselves to the special case when some of the background fields vanish ψμ− = ψμ+ = ωμcd = ωμa = 0 Gα+, Hλ = 0 and also Gα−, Gβ− Gα−, Gβ+ G˙α− = Gα−, HT G˙α− = Gα−, HT + mλβ−δβα = 0, β 1 = λ+ 2 = mδαβ , Gα+, Gβ+ 1 = 2 (γ 0γ a )αβ eμa ( pμ + mmμ). = − 21 δαβ τ μ( pμ + mmμ), Finally we have to determine the time evolution of these constraints when we take into account that the total Hamiltonian is the sum of all primary constraints HT = λλHλ + λα−Gα− + λα+Gα+. Hλ is preserved automatically while the time evolution of G−, G+α is governed by following equations α when we have Mμa = eμa , Vμ = τμ . Then all constraints simplify considerably Hλ = ( pμ + mmμ)hμν ( pν + mmν ) + 2mτ μ( pμ + mmμ) ≈ 0, Gα− = pα− + m2 θ−β δβα + 41 (θ¯+γ a )αeμa ( pμ + mmμ) ≈ 0, Gα+ = pα+ + 41 (θ¯−γ a )αeμa ( pμ + mmμ) + 41 (θ¯+γ 0)ατ μ( pμ + mmμ) ≈ 0. Now we have to calculate Poisson brackets between all constraints. We start with the following one 1 Gα−, Hλ = 4 (θ¯+γ a )α((∂ρ eμa + ∂ρ eσa )hρσ − eρa ∂ρ hμσ ) ×( pμ + mmμ)( pσ + mmσ ) m + 2 (θ¯+γ a )α(∂ρ eμa τ ρ −eρa ∂ρ τ μ)( pμ+mmμ). It is important to stress that the fact that we restrict to the case ωμab = ωμa = 0 implies, since they depend on spatial derivatives of eμa , τ μ, that we should impose the condition that eμa , mμ, τ μ do not depend on x μ as well. Then the equation given above implies that Gα−, Hλ = 0. In the same way we find that (80) (81) (82) (83) (84) (85) (γ 0γ a )βαeμa ( pμ + mmμ) = 21 λβ−(γ 0γ a )αβ eμa ( pμ + mmμ) 1 β − 2 λ+δαβ τ μ( pμ + mmμ) = 0. (86) From the last equation we express λβ as + λβ+ = δβγ (γ 0γ a )γ δ λδ−eμa ( pμ + mmμ) 1 τ μ( pμ + mmμ) and inserting back to the first equation we obtain G˙α− = (( pμ + mmμ)hμν ( pν + mmν ) + 2mτ μ( pμ + mmμ))λβ−δβα = Hλλ−δβα ≈ 0 (88) β and we see that this equation is obeyed for all λβ on the con− straint surface Hλ ≈ 0. This fact again implies an existence of the first class constraint in the form α = Gα−τ μ( pμ +mmμ) +Gβ+δβγ (γ 0γ a )γ α eμa ( pμ +mmμ) that has following Poisson brackets with canonical variables α , x μ α , θ α , θ β + β − = − 41 (θ¯+γ a )α eμa τ ν ( pν + mmν ) + 41 τ μ(θ¯+γ b)α eνb( pν + mmν ) − 41 (θ−γ a γ b)α eμa eνb( pν + mmν ), = (γ 0γ a )αωδωβ eμa ( pμ + mmμ), = δβ τ μ( pμ + mmμ). α Now we define κ variation as 1 δκ X = − 2mτ μ( pμ + mmμ) (κ¯ γ a )α eμa ( pμ+mmμ) { α , X } . Then with the help of (90) we obtain following transformation rules δκ θ+α = κ α , δκ θ−α = −(κ¯ γ a )α eμa ( pμ + mmμ), δκ x μ = 81m (κ γ a γ bθ+)eνa ( pν + mmν )eμb − 41 τ μκ θ 1 a − 4 (κ¯ γ θ−)eμa + that are covariant form of κ -transformations (54). To see this explicitly note that in the flat non-relativistic spacetime we can choose τ 0 = 1, τ i = 0, eμa = δia and the κ transformations given in (92) have explicit form δκ θ+α = κ α , α δκ θ− = − κ¯ γ i α ( pi + mmi ), j i κ γ γ κ¯ γ i θ − , ( p j + mm j ) 1 δκ t = − 4 κ θ+ 1 = 8m 1 − 4 δκ x i (93) that coincide with the transformations (54) which is a nice consistency check. In this section we performed Hamiltonian analysis of the non-relativistic superparticle in Newton–Cartan background. We derived general form of the Hamiltonian and fermionic constraints. Then we studied their properties for special configurations of background fields. It would be extremely interesting to analyze this theory for general background. 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Josef Klusoň. Canonical analysis of non-relativistic particle and superparticle, The European Physical Journal C, 2018, 117, DOI: 10.1140/epjc/s10052-018-5609-3