New dynamical realizations of the Lifshitz group

The method of nonlinear realizations is applied to construct new dynamical realizations of the Lifshitz group in mechanics, hydrodynamics, and field theory.


Introduction
The Lifshitz group [1] (for a review see [2]) involves dilatation transformation under which the temporal and spatial coordinates scale differently: t ′ = λ z t, x ′ i = λ 1 2 x i . The parameter z is known as the dynamical critical exponent. In addition to the scaling transformations, the group also involves temporal and spatial translations and spatial rotations. If desirable, Galilei boosts can also be added. For z = 1, the group can be also extended by special conformal transformation, which links it to the Schrödinger group.
The Lifshitz group is of interest for several reasons. Originally, it played an important role in describing phase transitions in condensed matter physics [1]. More recently, it was used within the framework of the nonrelativistic AdS/CFT correspondence [3] (for a review see [2]). Anisotropic scaling of temporal and spatial coordinates underlies the Horava-Lifshitz gravity [4]. Its applications to cosmology are reviewed in [5].
When constructing dynamical realizations of the Lifshitz group in mechanics, a problem arises that the standard free action S = 1 2 dtẋ iẋi does not hold invariant under the anisotropic scaling t ′ = λ z t, x ′ i = λ 1 2 x i . One way to avoid the difficulty is to raise the kinetic term to an appropriate power (ẋ iẋi ) z 2z−1 (see e.g. [6]). Another option is to introduce a conformal compensator ρ(t) which transforms with respect to the dilations as follows ρ ′ (t ′ ) = λ z−1 ρ(t) [7]. Note that within the conventional group-theoretic framework [8] there is a natural room for the conformal compensator, where it shows up as a partner accompanying the dilatation generator in the conformal algebra. In recent works [7,9], the method of nonlinear realizations has been successfully applied to build some dynamical realizations of the Lifshitz group.
The goal of this work is to construct new dynamical realizations of the Lifshitz group in mechanics, hydrodynamics, and field theory by applying the conventional group-theoretic framework [8].
The work is organized as follows. In the next section, we consider a particular onedimensional mechanics, for which the spatial translations and rotations are inessential. In this case, the scaling symmetry can be realized as a subalgebra of a one-dimensional conformal algebra so(2, 1) [10]. It is demonstrated that the equations of motion have one-parameter ambiguity and for a special choice they reproduce the recent results in [7].
In Sect. 3, we consider a dynamical realization of the Lifshitz group in multidimensional mechanics. Such model describes a particle moving in d-dimensional space driven by a conformal mode, which can be treated as a dynamical system with varying mass. It is shown that in a closed system the conformal mode depends only on the modulus of the conserved momentum, while in a particular isotropic case z = 1 2 it satisfies the equation of motion of the so(2, 1) conformal mechanics in the harmonic trap.
Other many-body mechanics models with the Lifshitz symmetry are discussed in Sect. 4. Working within the Hamiltonian framework, conserved charges are constructed which obey the Lifshitz algebra. Partial differential equations are formulated which determine an interaction potential compatible with the Lifshitz symmetry. Their general solution is found. In the particular case of z = 1 they reduce to the Calogero-like models studied in [11,12].
In Sect. 5, dynamical realizations of the Lifshitz algebra, which is extended by the Galilean boost generator as well as constant accelerations up to some finite order N, are discussed. Using the group-theoretical approach [8], invariant derivatives and field combi-nations are determined, which can be used for model building. It is shown that in general such models involve higher derivatives.
In Section 6, perfect fluid equations with the Lifshitz symmetry are studied. Our consideration is similar to the construction of many-body mechanics. Introducing a conformal mode, one can construct a Hamiltonian and conserved charges which form the Lifshitz algebra under the Poisson bracket. In particular, our approach circumvents the problem of constructing the dilatation generator recently revealed in [13].
In Section 7, it is demonstrated that the group-theoretical approach can also be used in order to build Lifshitz-type scalar field theories [14][15][16][17][18].
In the concluding Section 8, we summarize our results and discuss possible further developments.

Conformal mode
We start with a simple but instructive example of one-dimensional mechanics with anisotropic scaling symmetry. Omitting all generators carrying vector indices in the Lifshitz algebra, one is left with the generator of temporal translation H and dilatation D obeying Although in one dimension z can be removed by redefining D, keeping in mind forthcoming multi-dimensional generalizations, we prefer to keep it explicit. Introducing a temporal variable t, the algebra (2.1) can be realized by the operators Let us apply the group-theoretic approach [8] to (2.1) with the aim to construct onedimensional mechanics. Let us consider the group-theoretic element where 3) The forms are invariant under the transformation of the group-theoretic element g ′ = e iβH e iλD · g with real parameters β and λ, which imply Taking into account (2.3), one can build the invariant derivative and field combinations The latter can be used to construct the equation of motion

5)
γ and a being arbitrary constants.
Making the field redefinition, ρ = e zu 2 , for which the dilatation transformation reads ρ ′ (t ′ ) = e zλ 2 ρ(t), one brings (2.5) to the form In order for the corresponding action functional to be invariant under the dilatation transformation, one has to set a = − z 2 so that the second term in (2.6) drops out. The resulting action reads Up to a redefinition of the coupling constant, this coincides with the one-dimensional conformal mechanics [10]. By applying the Noether theorem to (2.7), one gets conserved charges In order for the energy H to be positive-definite, it is necessary to assume zγ 2 > 0. Having at our disposal two integrals of motion, one can construct the general solution to (2.7) by purely algebraic means As compared to the previous studies in [7], where the uniform field redefinition ρ = e u 2 was implemented, in our approach the new field ρ depends on z explicitly which facilitates the explicit construction of a conserved charge corresponding to the scaling symmetry. After such a redefinition, one-dimensional mechanics with the Lifshitz symmetry coincides with so(2, 1) conformal mechanics for arbitrary z, not just for z = 1 as in [7].

Dynamical realizations of the Lifshitz group in mechanics
Let us now turn to the full Lifshitz algebra, which in addition to H and D also includes the generators of spatial translations C i and spatial rotations M ij , i, j = 1, ..., d. The corresponding structure relations read z being the critical dynamical exponent. In a non-relativistic space-time parameterized by (t, x i ), i = 1, ..., d, the algebra can be realized by the operators Let us construct multi-dimensional mechanics systems with the Lifshitz symmetry (3.1) by applying the group-theoretic approach. A similar consideration of the ℓ-conformal Galilei group has been given in [19][20][21] One starts with the coset space element which give rise to the Maurer-Cartan one-forms where The forms are invariant under the transformation of the group-theoretic element g ′ = e iβH e ia i C i e iλD · g with real parameters β, λ and a i . The latter implies The transformation laws of x i (t) indicate that they should be interpreted as describing particle's orbit. From (3.2) one obtains the invariants which can be used to build equations of motion.
For the conformal mode ρ = e zu 2 it seems natural to choose the equation (2.7) constructed in the previous section. For field variables x i (t) we take the most general linear expression quadratic in derivatives where a is a free parameter which will be fixed later. Taking into account ρ = e zu 2 and (3.4) the equation (3.5) can be rewritten as Similar model was considered in [7], which describes an oscillator with a time-dependent frequency and damping in the co-moving framex i (t) = ρ − 1 z x i (t). Note thatx i (t) remains inert under the dilatation 1 . In the conventional coordinate frame x i (t), the equation (3.6) can be obtained from the action functional where we redefined the free parameter α = (2a+2z−1) z . Finally, the parameter α can be fixed by demanding the dilatation invariance of (3.7): α = 2(z−1) z . Note that R(t) = ρ α can be interpreted as a cosmic scale factor entering the spatial metric g ij = R(t)δ ij (see e.g. the discussion in [7]). Alternatively, m(t) = ρ α can be regarded as a varying mass.
To summarize, the equations of motion of the multi-dimensional mechanics invariant under the Lifshitz group read The second equation describes the dynamics of a particle in d spatial dimensions x i (t), which is driven by the conformal mode ρ(t) satisfying the first equation. The general solution to the first equation was found in the preceding section (2.8), which yields where C i is a constant of integration.
The system of equations (3.8) is not Lagrangian. In order to obtain a Lagrangian mechanics, we consider the modified action functional which gives the equations of motion Noether's integrals corresponding to the Lifshitz symmetry read Note that the first equation in (3.10) does not explicitly depend on x i , it depends on the square of the conserved momentum of the system C i = ρ 2(z−1) zẋ i and can be written in the formρ For z = 1 the second term on the left-hand side drops out and the equation describes the so(2, 1) conformal mechanics discussed in 2. At the same time, for z = 1 2 the equation (3.13) describes conformal mechanics in a harmonic trap which is so(2, 1)-invariant as well. Its general solution describes oscillations around the and reduces to (2.8) As compared to [7], we have constructed the complete Lagrangian dynamical realization of the Lifshitz group (3.9) by modifying the equation of motion for the conformal mode ρ(t), in which ρ is not a background field in which a particle parametrized by x i (t) moves, but rather it makes part of a closed system.
In the next section we consider a generalization of the action (3.9) to the case of manybody dynamical systems.

Many-body mechanics with Lifshitz symmetry
In order to construct models of many-body mechanics with the Lifshitz symmetry, it proves convenient to pass to the Hamiltonian formalism. The phase space of n identical particles is defined by the canonical pairs (ρ, π) and (x A i , p A i ), A = 1, ..., n with the standard Poisson bracket The dynamics is governed by the Hamiltonian where V (ρ, x A i ) is a potential. Conserved charges corresponding to dilatation, spatial translations and spatial rotations have the form Together with the conserved energy H defined by the Hamiltonian (4.1) they satisfy the Lifshitz algebra (3.1) under the Poisson brackets provided the potential V (ρ, x A i ) obeys the first order partial differential equations The first two conditions arise from the invariance under spatial translations and rotations, while the latter guarantees the dilatation symmetry. The first equation in (4.2) implies that the potential V must depend on the combinations In total, for an n-particle system there are n − 1 independent combinations which can be parameterized as xÂ = x 1Â ,Â = 2, 3, ..., n. The third equation in (4.2) restricts the form of the potential as follows Finally, in order to take into account the second equation in (4.2), one has to form SO(d) Here δ ij and ε i 1 i 2 ...i d are the Euclidean metric and the Levi-Cevita symbol in R d which are SO(d)-invariant. Note that ωÂ 1 ...Â d is actually a pseudoscalar and in d dimensions there is a restriction on a minimal number of particles n min = d + 1. Thus, a solution to (4.2) can be written in the form In order to obtain reasonable models from an infinite number of solutions, one can impose additional restrictions on V such as homogeneity, parity (P -symmetry), permutation symmetry, supersymmetry, integrability, and etc. Let us give two examples of potentials which are invariant under permutations where the first example represents the only possibility of a potential built independent on ρ.
Both examples for z = 1 reproduce the Calogero-like models in [12].

Dynamical realizations of the extended Lifshitz group
The Lifshitz algebra can be extended by adding Galilei boost and a finite number of constant acceleration generators C i (n) , n = 0, 1, ..., N. The corresponding structure relations read where C i (n) for n = 0, 1 correspond to spatial translations and Galilei boosts, while for n > 1 they link to constant accelerations. The algebra can be realized by the operators Repeating the steps in the previous section, one constructs the coset space element where the summation over n is understood, and derives the Maurer-Cartan one-forms The forms are invariant under the transformation g ′ = e iβH e ia (n) i C (n) i e iλD · g with real parameters β, λ and a (n) i , which implies where m = 0, 1, ..., N. one readily gets

From (5.3), one gets the invariant derivative and field combinations
i . By analogy with the case (3.5), let us consider the equation of motion where a is a free parameter. Taking into account ρ = e zu 2 , (5.4) and constraints (5.5) the equation can be rewritten in the form where the free parameter was redefined as follows α = 2(a+z[N +1]− 1 2 ) z . To fix α we need a suitable action functional. In the particular case α = 0 the equation (5.6) describes a free particle with a constant acceleration of order N, which has a larger ℓ-conformal Galilei symmetry for an arbitrary ℓ = N +1 2 [22]. In the general case, the equation has the Lifshitz symmetry extended by the Galilei boosts and constant accelerations. Demanding ρ(t) to obey (2.8), the general solution to (5.6) can be written in the form where C i is the integral of motion One more dynamical realization of the extended Lifshitz algebra (5.1) is described by the invariant action functional The latter gives rise to the equations of motion which can be rewritten in terms of the invariant objects (5.4) and the constraints (5.5) Note that (5.6) and (5.7) represent higher derivative generalizations of (3.7) of the order N + 2 and 2(N + 1), respectively.

Perfect fluid equations with the Lifshitz symmetry
It is known that perfect fluid equations supplemented with an appropriate equation of state enjoy the Schrödinger symmetry [24,25]. Generalized perfect fluid equations with the ℓconformal Galilei symmetry and their Hamiltonian formulation were recently considered in [13,26,27]. In this section, we discuss perfect fluid equations with the Lifshitz symmetry.
In a recent work [13], the invariant equations were proposed where we omitted the conventional spatial rotations. The fields ̺(t, x) and υ i (t, x) transform nontrivially only under the dilatations (for more details see [13]) However, the construction of conserved charge corresponding to dilatation faces a problem. Below, we construct a one-parameter generalization of (6.1), for which all conserved charges can be constructed explicitly.
In addition to the Lifshitz symmetry, the equations (6.1) hold invariant under the Galilei boosts x ′ i = x i + b i t. Discarding this symmetry, one has larger freedom in formulating the Lifshitz invariant equations of motion.
By analogy with our consideration above, in order to construct a perfect fluid equations for an arbitrary z it suffices to introduce the conformal compensator ρ(t). It seems natural to keep the continuity equation intact, while the Euler equation and the equation of state are modified as follows where α is an arbitrary constant. The compensator ρ(t) transforms under the dilatations as follows ρ ′ (t ′ ) = e zλ 2 ρ(t). Note that equations (6.6) are Lifshitz invariant for arbitrary α and (6.1) is recovered for α = 0. Because ρ(t) cannot be removed by a field redefinition, it would be interesting to study its thermodynamic interpretation. We leave this issue for future study.
It is straightforward to construct the Hamiltonian which gives rise to the fluid equations (6.6) and simultaneously determines the dynamics of the conformal mode ρ(t) The corresponding Poisson brackets read Above we redefined the velocity vector fieldυ i = ρ α υ i , in terms of which the Poisson brackets (6.8) coincide with those in [24]. Because under the dilatation the Hamiltonian must scale as H ′ = e −zλ H, the value of the parameter α is fixed to be α = 2(z−1) z . Note that (6.7) has the same structure as (4.1) for the case of an infinite number of particles. The Hamiltonian (6.7) determines the dynamics of the conformal modė and reproduces the fluid equations of motion (6.6) provided the potential V obeys the equation Taking into account p = ν̺ 1+ 2z d , one finds V = d 2z p. Conserved charges corresponding to the dilatation, spatial translation and rotation read They form the Lifshitz algebra under the Poisson bracket Given the conformal mode ρ(t), one can construct a more general equation of state which will preserve the Lifshitz symmetry where µ is an arbitrary constant. This equation of state via (6.11) determines the potential V = d z(2+µ) p and for µ = 0 it reproduces the third equation in (6.6).

Dynamical realizations of the Lifshitz group in field theory
In this section, we focus on dynamical realizations of the Lifshitz group in field theory. Let us consider again the non-extended Lifshitz algebra and introduce the coset space element The Maurer-Cartan one-forms (3.2) Then one builds the invariant temporal and spatial derivatives as well as field combinations Aftewards, one considers the most general equation, which is quadratic in the derivatives and fixes the parameters Then the equation (7.1) takes the form and the corresponding action functional reads where we redefined the coupling constantγ 2 = (2z−d) 2 2(2z+d) γ 2 . Using Noether's theorem one can construct the integrals of motion When verifying conservation of over time, the identity proves useful, which holds provided the equations of motion are satisfied. A scalar field theory described by the action functional (7.2) corresponds to a particular case studied in [16]. For z = 1 2 and α = −1, the action (7.2) describes the relativistic conformal scalar field.

Conclusion
To summarize, in this work we constructed new dynamical realizations of the Lifshitz group in mechanics, hydrodynamics, and field theory. The main ingredient of our consideration was the conformal compensator ρ(t). Using the group-theoretic approach, in which ρ(t) naturally arises as a partner of the dilatation, the following new results were obtained. It was shown that one-dimensional mechanics with the Lifshitz symmetry by a proper field redefinition can be linked to so(2, 1) conformal mechanics for arbitrary z, not just for z = 1 as in [7]. In multidimensional mechanics, a complete Lagrangian formulation was attained, in which ρ(t) and x i (t) are treated on equal footing. Many-body examples were also constructed and studied. Generalized Calogero-like models involving arbitrary z were built. Two higher derivative examples, in which the Lifshitz algebra is extended by the Galilean boost generator as well as constant accelerations, were formulated. Perfect fluid equations with the Lifshitz symmetry and its Hamiltonian formulation were built. It was demonstrated that the grouptheoretical approach can also be used in order to construct Lifshitz-type scalar field theories.
As a possible further development it would be interesting to study in more detail the geometric aspects of the extended Lifshitz algebra(5.1) as well as to construct its dynamical realizations in hydrodynamics and field theory. Causality of the field theory (7.2) deserves a separate investigation.