# Canonical analysis of non-relativistic particle and superparticle

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## Abstract

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.

## 1 Introduction and summary

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 Hořava–Lifshitz gravity [2], for recent review and extensive list of references, see [3]. There is also an interesting connection between Hořava–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, 7, 8, 9, 10, 11, 12, 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 \(AdS_5\times S^5\) 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, 21, 22, 23].^{2}

It is important to stress that these constructions of non-relativistic 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 \(\kappa \)-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 \(\kappa \)-symmetry. Since it is the first class constraint it can be 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 \(\kappa \)-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 non-relativistic superparticle in Cartan–Newton background.

## 2 Hamiltonian of Newton–Cartan particle

*M*as \(M=\omega m\) we obtain

## 3 The Newton–Cartan–Hooke particle

*AdS*radius \(R^2\) is related to the cosmological constant \(\Lambda , \Lambda <0\) as \(R^2=-\frac{1}{\Lambda }\). From (27) we find conjugate momenta

## 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

^{3}

### 4.2 \(\kappa \)-symmetric Galilean superparticle

## 5 Non-relativistic superparticle in Newton–Cartan background

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. We expect that the requirement of the existence of three first class constraints, where one is Hamiltonian constraint while remaining two constraints correspond to generators of \(\kappa \)-symmetry, will impose some restriction on the background fields. We hope to return to this analysis in near future.

## Footnotes

- 1.
- 2.
- 3.Note that we use Majorana representation where all gamma matrices are real \(\gamma ^\mu =(i\sigma _2,\sigma _1,\sigma _3)\), or explicitly \(\gamma ^0=\left( \begin{array}{cc} 0 &{} 1 \\ -1 &{} 0 \\ \end{array}\right) , \gamma ^1=\left( \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \\ \end{array}\right) , \gamma ^2= \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} -1 \\ \end{array}\right) \) that obey the standard relation \(\gamma ^a\gamma ^b+\gamma ^b\gamma ^a=2\eta ^{ab}\mathbf {I} , \eta ^{ab}=\mathrm {diag}(-1,1,1), \mathbf {I}=\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \\ \end{array}\right) \). Then we presume that \(\theta _-\) is real Majorana spinor so that

## Notes

### Acknowledgements

This work was supported by the Grant Agency of the Czech Republic under the Grant P201/12/G028.

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