Canonical Formalism of Non-Relativistic Theories Coupled to Newton-Cartan Gravity

In this short note we perform canonical analysis of Schrodinger field and non-relativistic electrodynamics coupled to Newton-Cartan gravity. We identify physical degrees of freedom and analyze constraints structure of these theories.


Introduction and Summary
There is a renewed interest in Newton-Cartan (NC) geometry which has been observed in the last years. The first significant paper was [1] which introduced NC to field theory that analyzes strongly correlated electrons. It was further shown in [2,3] that NC geometry with torsion naturally emerges as the background boundary geometry in holography for z = 2 Lifshitz geometries, for relevant works, see [4,5,6,7] and for review and extensive list of references, see [8]. In fact, NC geometry is non-relativistic background geometry to which non-relativistic field theories can be covariantly coupled, see for example [5,9,10,11,12,13]. In particular, it was shown in significant paper [11] how non-relativistic electrodynamics can couple to the most general NC geometry with torsion. Further, non-relativistic scalar fields coupled to NC geometry and background electromagnetic field were also analyzed there.
Since these results are very interesting non-relativistic theories in NC background certainly deserve to be studied in more details further. In this short note we focus on canonical analysis of non-relativistic scalar field and non-relativistic electrodynamics coupled to NC geometry. It turns out that this is rather non-trivial problem with interesting property that the constraints explicit depend on time. In more details, we start with the action for Schrödinger field in the background NC geometry and background non-relativistic electromagnetic field. Such an action was derived in [11] with the help of null reduction of complex scalar field in higher dimensional space-time 2 . Then in order to find Hamiltonian form of this action we have to impose important restriction on the NC space-time in the sense that it has to have a notion of foliation by spatial surfaces that are orthogonal to one form τ µ where τ µ is known as clock form. This form defines a preferred notion of spatial direction at each point and also arrow of time in the sense that vector field t µ is said to be future directed if it obeys the condition τ µ t µ > 0. τ µ defines a pointwise notion of spatial direction with the help of the vectors w µ that obey the condition τ µ w µ = 0. However this notion can be integrated to a local codimension one subspace when τ µ obeys Frobenious condition τ ∧ dτ = 0 where τ = τ µ dx µ . Then we define causal space-times as space-times where this condition holds everywhere, for more detailed analysis and discussion, see for example [9,12]. For such space-time we will be able to find Hamiltonian for the Schrödinger field in NC background. However we also find that when we write the complex scalar field in polar form as ψ = √ ρe iS that the momentum conjugate to ρ is zero which is first primary constraint of the theory. Further, in case of the momentum conjugate to S we find that it is determined by second primary constraint that explicitly depends on time. This is very interesting situation that deserves careful treatment. For that reason we perform analysis of constraint systems with explicit time dependent primary constraints in appendix 3 . Taking into account explicit time dependence of the constraints we will be able to derive canonical equations of motion that reproduce Lagrangian equations of motion which is a nice consistency check.
As the next step we extend this analysis to the case of non-relativistic electrodynamics in Newton-Cartan background. Since canonical analysis is based on an existence of Lagrangian we start with the non-relativistic electrodynamics action in NC background that is derived using null dimensional reduction [11]. We again restrict to the case of causal space-time and in the first step we determine set of primary constraints which Poisson commute among themselves. This is different situation than in case of the scalar field where the primary constraints were the second class constraints. Then the requirement of the preservation of the primary constraints gives set of secondary constraints which together with the primary constraints form set of the second class constraints. As a result we find that gauge field and corresponding conjugate momenta can be eliminated from the theory at least in principle. We also determine Lagrange multipliers corresponding to the primary constraints using the equations of motion for gauge field and we show that the resulting equations of motion coincide with the equations of motion derived by variation of action.
Let us outline main results derived in this paper. We obtain Hamiltonian form of non-relativistic theories on NC background and we determine physical degrees of freedom. This is very important result since we show that in case of non-relativistic electrodynamics the only physical degree of freedom is the scalar field and conjugate momenta. We also discuss the problem of the constraint structure in case of theories with explicit time dependent constraints.
The structure of this paper is as follows. In the next section (2) we review basic facts about NC geometry and introduce an action for Schrödinger field in the NC background and background non-relativistic electromagnetic field through null dimensional reduction. Then we perform Hamiltonian analysis of this theory and determine structure of constraints. In section (3) we analyze non-relativistic electrodynamics in NC background. We firstly perform canonical analysis of nonrelativistic electrodynamics in flat background and then we extend this analysis to the case of non-relativistic electrodynamics in NC background. Finally in appendix (A) we study constrained systems with explicit time dependence and discuss their properties.
2 Hamiltonian Analysis of Schrödinger field in NC background

Summary of Newton-Cartan Geometry
We start this analysis with the brief review of Newton-Cartan geometry in d + 1 dimensions. Newton-Cartan background in d + 1 dimensions is given by a set of one forms (τ µ , e a µ ) where a = 1, . . . , d and where µ, ν = 0, 1, . . . , d. We also have one form M µ . We define inverse vielbeins v µ and e a µ through the relations v µ e a µ = 0 , v µ τ µ = −1 , e µ a τ µ = 0 , e µ a e b µ = δ b a . (1) The determinant of the (d + 1) × (d + 1) matrix (τ µ , e a µ ) is denoted by e. With the help of vierbeins we can construct degenerative "spatial metric" By definitions, one forms τ µ , e a µ and M µ transform under diffeomorphism as usual but they also transform under various local transformations: Galilean boosts with λ a as local parameter, local SO(d) rotations which is parameterized by λ ab = −λ ba and U(1) σ gauge transformation that is parameterized by σ where we have The inverse vielbein e µ a is invariant under local Galilean transformations. Note that we have an important relation which will be useful below. It is also useful to define objects that are invariant under local Galilean transformationsv µ ,ê µ a ,ĥ µν and Φ defined aŝ It is important to stress thatĥ µν =ê a µê b ν δ ab . Instead, using the definition ofê a µ given above, we obtain following relation Finally note that hatted objects obey following the relationŝ After this review of NC geometry we proceed to the Hamiltonian analysis of Schrödinger field.

Schrödinger Field in NC geometry through Null Dimensional Reduction
We would like to find Hamiltonian formulation of scalar field on Newton-Cartan background with fixed electromagnetic background. The most convenient way how to find such an action is to perform null dimensional reduction, see for example [9,11]. Let us consider an action for complex scalar field in d + 2 dimension in the form where Let us now consider a background metric which possesses a null isometry that is generated by coordinates ∂ u Then Since the metric γ AB is non-singular we can easily find inverse metric with components wherev µ and Φ are defined in (6). We further presume that the gauge field has the form A A = (A u , A µ ) = (ϕ,Ā µ − ϕM µ ). Now with the help of this metric we perform dimensional reduction of the action. To do this we have to presume that all fields do not depend on u. We impose following ansatz for the scalar field Ψ and insert it to the action. Then also using (13) we obtain where The action (15) is the action for Schrödinger field of the mass m and charge q in Newton-Cartan background and in the background electromagnetic field where the electromagnetic field has components A µ . Note that ψ couples to ϕ through the combination m − qϕ and hence ϕ effectively shifts the mass of the scalar field and hence it is natural to call it as mass potential [11]. Our goal is to find Hamiltonian from the action (15).

Hamiltonian Analysis
We would like to work with real variables rather than with complex ones. For that reason se introduce following parameterization of the scalar field ψ as ψ = √ ρe iS so that the action has the form where It is clear that the previous action is well defined for general Newton-Cartan background for arbitrary τ µ apart from the fact that τ µ has to obey Newton-Cartan compatibility condition. On the other hand in order to have well defined Hamiltonian formulation we have to have a notion of foliation by spatial surfaces that are orthogonal to τ µ . This is guaranteed when we impose hypersurface orthogonality condition τ [µ ∂ ν τ ρ] = 0 on the whole space-time M. This condition is known as Frobenius condition and for more detailed discussion of causality in Newton-Cartan background, see [9,12]. Space M that obeys this condition is called as causal. Since τ µ is nowhere non-zero we can write it as τ 0 = e −Φ L where Φ L is known as Luttinger potential. In what follows we restrict to such space-time. Since τ i = 0 we obtain following consequences on the form of the metric h µν thanks to the condition Explicitly, for ν = 0 this equation implies τ 0 h 00 = 0 and hence we have to have h 00 = 0 while for ν = i we have τ µ h µi = τ 0 h 0i = 0 which again implies that h 0i = 0. Then the action I sch simplifies considerably wherev µ = v µ − h µν M ν has generally non-zero all its components. Before we proceed to the Hamiltonian formulation of the theory we derive equa-tions of motion for ρ and S from (20) Now we are ready to proceed to the Hamiltonian formalism. From (20) we obtain following conjugate momenta From these two equations we see that there are two primary constraints while the bare Hamiltonian is equal to We see that generally G S ≈ 0 and H B explicit depend on time. This is not usual situation and we discuss theory of constraints systems with explicit time dependence in more details in Appendix (A).
As the next step we calculate Poisson bracket between G S and G ρ and we obtain which show that they are two second class constraints. Note that the inverse matrix has the form △ ρS = − 1 2e(m−qϕ)v 0 δ(x − y). As a result we can eliminate p ρ = 0 and p S = 0 from the set of canonical variables when we introduce Dirac bracket between ρ and S defined as As was explicitly shown in Appendix (A), in the presence of the time dependent constraints the equations of motion for canonical variables have the form 4 that can be rewritten into more symmetric form that coincides with the first equation of motion given in (21). Let us now proceed to the canonical equation of motion for S that can be again rewritten into the form that coincides with the second equation of motion given in (21).
In summary, we found the Hamiltonian formulation of Schrödinger field in NC background. We found that the dynamical fields are ρ and S that have non-zero Dirac bracket (26). Then we derived their canonical equations of motion and found that they coincide with the equations of motion derived from Lagrangian.

Hamiltonian Formalism for Electromagnetic Field in Newton-Cartan Gravity
In this section we focus on canonical analysis of non-relativistic electromagnetic field in NC background. We start with the simpler case of the action for non-relativistic electrodynamics in flat background.

Non-Relativistic Electrodynamics through Null Dimensional Reduction
Following [11] we derive an action for non-relativistic electrodynamics by performing a null reduction of the Maxwell action in one higher dimension. More precisely, let us consider d + 2 dimensional Maxwell action where η AB dx A dx B = 2dtdu + dx i dx i . Following [11] we set A u = ϕ , A t = −φ , A i = a i and presume that all fields do not depend on u. Since the inverse metric has the form η tu = η ut = 1 , η ij = δ ij we get (32) As a result we obtain an action for non-relativistic electrodynamics in flat background in the form where f ij = ∂ i a j −∂ j a i . From the action (33) we derive following conjugate momenta so that we have following primary constraints together with the bare Hamiltonian in the form and consequently the extended Hamiltonian is equal to where λ i , λφ are Lagrange multipliers corresponding to the constraints G i ≈ 0 and pφ ≈ 0. As the next step we have to ensure the preservation of all primary constraints. In case of the constraint G i ≈ 0 we obtain where G i II ≈ 0 are secondary constraints. In case of the constraint pφ ≈ 0 we obtain dpφ dt which is the generator of gauge transformations. In fact, if we define we obtain standard transformation rules Finally we have to ensure the preservation of the constraint G i II ≈ 0. To do this we have to calculate the Poisson bracket between constraints G i and G i II . After some calculations we obtain Let us introduce an inverse matrix D ij (x, y) that obeys the relation Since △ ij is given in (42) we find that D ij is a solution of the equation As the next step we determine canonical equations of motion for ϕ and p ϕ Finally the equation of motion for a i has the form so that the equation of motion for p ϕ can be written as whereẼ k = −∂ t a k − ∂ kφ . If we perform partial time derivation of the first equation in (45) and use the second one we obtain with agreement with the equation (2.11) in [11]. Further, if we apply the partial derivative ∂ i on the first equation in (45) we obtain with agree with the second equation in (2.3) in [11]. In the same way we find that the divergence of G i = 0 implies that agrees with the first equation in (2.3) [11]. Finally we should determine the Lagrange multiplier λ i using the requirement of the preservation of the constraint G i II but this is not necessary since we know that λ i = ∂ t a i . On the other hand since G i ≈ 0 and G i II ≈ 0 are two second class constraints they can be explicitly solved for π i and a i . In other words there is only one dynamical variable which is ϕ and its conjugate momentum p ϕ .

Null Reduction of Maxwellian Electromagnetism in NC Background
We determine action for electromagnetic field in Newton-Cartan background again with the help of null dimensional reduction, following [11]. We start with the action for electromagnetic field in d + 2 dimensions that has the form Our goal is to dimensionally reduce this action along null isometry so that we will presume that A M do not depend on u. We further write A M = (A u , A µ ) and define A µ ≡ ϕ. Since the gauge field transforms under U(1) transformations as it is clear that ϕ is invariant under gauge transformation since Λ does not depend on u. On the other hand the gauge field A µ transform as In order to perform null dimensional reductions we use the components of metric inverse given in (13) and we obtain the action in the form It is convenient to use slightly different form of the action which depends on v µ instead ofv µ . Following [11] we introduce vector fieldĀ µ defined as Performing this substitution in the action (54) we find whereF The action (57) will be the starting point for the Hamiltonian formulation of the theory. As we argued above we restrict to causal space-time with non-zero τ 0 only.
Then v µ has generally non-zero all components with v 0 = −τ 0 . In case of causal space-time the action has the form Note that from this action we also obtain equations of motion in the form Let us now proceed to the canonical analysis. From (59) we obtain following conjugate momenta so that we have following explicitly time dependent primary constraints together with familiar constraint G 0 ≡ π 0 ≈ 0. Further, with the help of (61), we obtain the bare Hamiltonian in the form Now we have to analyze the requirement of the preservation of primary constraints G i ≈ 0 , G 0 ≈ 0. Note that the extended Hamiltonian has the form In case of G 0 ≈ 0 we obtain that the requirement of its preservation during the time development of the system implies standard Gauss law constraint while in case of G i we get where we used the fact that {G i (x), G j (y)} = 0. We see that the requirement of the preservation of the constraints G i ≈ 0 implies the second set of the constraints G i II ≈ 0. It is again easy to see that G i ≈ 0 , G j II ≈ 0 are two sets of second class constraints with rather complicated Poisson bracket between them. Then it is difficult to determine Lagrange multipliers λ i from the requirement of the preservation of the constraints G i II ≈ 0 during the time evolution of the system. On the other hand, as we will show below, these Lagrange multipliers can be determined with the help of the equations of motion forĀ i . Further, it is easy to see that G 0 and G II ≈ 0 are first class constraints where G II ≈ 0 is generator of gauge transformations.
As we argued above G i ≈ 0 , G i II ≈ 0 are two sets of second class constraints where G i = 0 can be solved for π i while G i II = 0 can be solved forĀ i at least in principle. On the other hand when we try to write equations of motion for ϕ and p ϕ it is convenient to express Lagrange multiplier λ i as a function of non-dynamical variableĀ i using its equation of motion that implies that λ i =F 0i . Then we can write canonical equations of motion for ϕ and p ϕ as If we combine these two equations together we obtain that coincides with the first equation in (60). Further, it is easy to see that the second equation in (60) coincides with the secondary constraint G i II = 0. Finally, the last equation in (60) is equivalent to the combination of the primary constraint G II and G i since In summary, we have shown that canonical equations of motion and constraints reproduce Lagrangian equations of motion. We have also determined physical degrees of freedom of non-relativistic electrodynamics and we have shown that there are only two phase space physical degrees left corresponding to ϕ and p ϕ . Acknowledgements: I would like to thank M. Chaichian for his comments and remarks. This work was supported by the Grant Agency of the Czech Republic under the grant P201/12/G028.

A Appendix: Systems with explicit time dependent constraints
Let us consider phase space system with variables p m , q m , m = 1, . . . , N, bare Hamiltonian H B and set of primary constraints φ j = φ j (p, q, t), j = 1, . . . , J, that explicitly depend on time t. Then the phase space action with primary constraints included has the form where λ j are independent variables known as Lagrange multipliers. Variation of the action with respect to p m , q m and λ j we obtain following set of equations of motioṅ Introducing standard Poisson bracket they can be written aṡ In order to determine Lagrange multipliers λ j we demand that the constraints φ j = 0 are preserved during the time evolution of the system. Note that it is clear from the form of the equations of motion written above that we have to firstly calculate Poisson bracket between canonical variables and φ j and then we can impose the condition φ j = 0. This is the reason why we write φ j ≈ 0 instead of φ j = 0. Now the time evolution of the constraint φ i is equal tȯ If we impose the condition that the constraint φ i is preserved during the time evolution of the system we find that the conditionsφ i = 0 provide J equations for J unknown λ i . Let us now presume non-degenerative case when {φ i , φ j } = △ ij is non-singular matrix so that it has an inverse △ jk , △ ij △ jk = δ k i . Then (74) can be solved as As a result we find that the time evolution of the phase space variables q m and p m is governed by equationṡ that can be written in an equivalent forṁ where we introduced Dirac bracket between two phase space functions defined as

A.1 Secondary time dependent constraints
Let us now consider situation when the primary constraints φ j (p, q, t) have weakly vanishing Poisson bracket among themselves. Then the requirement of their preservation during the time development of the system has the form where now φ II i (p, q, t) are secondary constraints. Finally we have to ensure the preservation of the secondary constraints which implies We will presume that the matrix φ II i , φ j ≡ △ ij is non-singular and hence the equation above can be solved for λ i as and equivalent one for p m . Let us outline results of the analysis performed in Appendix. We have shown that in case of the explicit time dependent constraints, either primary or secondary, there are additional terms in the equations of motion for canonical variables which are proportional to explicit time derivative of these constraints. These terms are crucial for the equivalence between Lagrangian and Hamiltonian equations of motion.