Deforming the Maxwell-Sim Algebra

The Maxwell alegbra is a non-central extension of the Poincar\'e algebra, in which the momentum generators no longer commute, but satisfy $[P_\mu,P_\nu]=Z_{\mu\nu}$. The charges $Z_{\mu\nu}$ commute with the momenta, and transform tensorially under the action of the angular momentum generators. If one constructs an action for a massive particle, invariant under these symmetries, one finds that it satisfies the equations of motion of a charged particle interacting with a constant electromagnetic field via the Lorentz force. In this paper, we explore the analogous constructions where one starts instead with the ISim subalgebra of Poincar\'e, this being the symmetry algebra of Very Special Relativity. It admits an analogous non-central extension, and we find that a particle action invariant under this Maxwell-Sim algebra again describes a particle subject to the ordinary Lorentz force. One can also deform the ISim algebra to DISim$_b$, where $b$ is a non-trivial dimensionless parameter. We find that the motion described by an action invariant under the corresponding Maxwell-DISim algebra is that of a particle interacting via a Finslerian modification of the Lorentz force.


Introduction
A popular line of thought in theoretical physics is to start with a Lie algebra g or Lie group G, and then to construct from it the space or spacetime in which physical objects, for examples p-branes, move. Typically the spaces or spacetimes are cosets G/H. The dynamics of p-branes is then described as a map from the (p + 1)-dimensional world volume into G/H [1,2,3,4,5]. In the case of point particles, the dynamics is often thought of as geodesic motion, or some modification thereof by "forces," such as the Lorentz force on electrically charges particles in electromagnetism, with respect to a metric on G/H that is invariant under the left action of G on G/H. More generally, one is interested in invariant Lagrangians L(x, v) on the tangent space T (G/H), or Hamiltonians H(x, p) on the cotangent space T ⋆ (G/H). For a recent statement of this viewpoint in the context of quantum field theory see [6].
An alternative construction of a p-brane action in the space G/H is to consider the quotient (G/H)/K, where K is the stabilizer of the p-brane. The action of lowest order in derivatives is obtained by considering the pull-back to the world-volume of a (p + 1)form invariant under K [7] (see also [8], where one can find further references). The action contains extra Goldstone fields associated with the broken "rotations." In order to make contact with the geometrical Lagrangian L(x, v), we should eliminate the extra fields by their non-dynamical equations of motion, or more generally, by the inverse Higgs mechanism [9].
An early example of this programme followed the discovery of the three congruence geometries; hyperbolic or Lobachevsky space H 3 , Euclidean space E 3 , and spherical space S 3 . Helmholtz characterised these three possibilities physically in terms of axioms of the free mobility of rigid bodies [10]. Such bodies permit rotations about any point in space, and translations to any point in space. Thus he demanded that H = SO (3) and that G act transitively on G/H. He arrived, after some additional arguments, at the three possibilities G = SO (3,1) : An equivalent way of looking at this is to say that the configuration space Q of a rigid body with one point fixed admits a simply-transitive left action by SO (3), and may thus be identified with SO (3). Free motion of a rigid body is given by geodesic motion on SO(3) with respect to a left-invariant metric. If the body moves in ordinary Euclidean space, Q is enlarged to become the Euclidean group E (3). If the body moves in an inviscid fluid, conservation of momentum and angular momentum will still hold and the metric is then given by a general left-invariant metric on E (3). Correspondingly, geodesic motion on SO (3,1) or SO(4) with respect to a left-invariant metric gives the motion of a rigid body moving in a fluid in H 3 or S 3 respectively.
A slightly different strand of thought begins with the observation (originally due to Lambert [11]) that passing to S 3 or H 3 introduces a new parameter into physics: the radius of curvature. This new parameter is associated with the fact that the translations in SO3, 1) or SO(4) no longer commute. Expressed mathematically, the Lie algebras so (3,1) and so (4) are continuous deformations of e (3), and this suggests that in seeking new physical laws, a fruitful procedure is to look for continuous deformations of existing laws. In algebraic terms, this translates into looking for continuous deformations of the Lie algebra g that one begins with. Given that one has introduced a new physical parameter whose magnitude is arbitrary, it is natural to enquire whether it might be time-dependent. In the case of spatial curvature, just such a suggestion was made by Calinon long before General Relativity and the Robertson-Walker metric [12].
The group theory viewpoint came into its own with Einstein's theory of Special Relativity, for which G = E(3, 1) = ISO(3, 1), the Poincaré group. Indeed only retrospectively was the Galilei group recognised as its Wigner-İnönü [13] contraction. As with the Euclidean group, the Poincaré group admits two continuous deformations, to SO(4, 1) or SO (3,2), for which spacetime translations fail to commute. It was perhaps only the early death of Minkowski which delayed until after the advent of Einstein's General Relativity the implementation of Calinon's idea. de Sitter, seeking a covariant version of Einstein's Static Universe, introduced their cosets, de Sitter and anti-de Sitter respectively.
Einstein did not scruple to break boost invariance with his static universe [14], and this is a feature of all Robertson-Walker metrics except those of de Sitter [15]. A natural question, answered by Bacry and Levy-Leblond [16], is what other 10-dimensional kinematical algebras exist, that contain rotations, translations in time and space and boosts. All can be regarded as Wigner-İnönü contractions of the de Sitter and anti-de Sitter algebras.
Invariance under the local Lorentz group is extremely well attested by experiment, but nevertheless Cohen and Glashow [17] observed that if it is broken down to its fourdimensional maximal subgroup Sim(2) ⊂ SO(3, 1) it leaves invariant no spurion fields, merely leaving fixed a null direction n ν ≡ λn ν , λ = 0, where η µν n µ n ν = 0. It may play a role linking small neutrino masses, and is compatible with all present day tests of violations of Lorentz invariance. Thus they proposed in their Very Special Relativity theory that the fundamental local symmetry group is the semi-direct product of Sim (2) and the translations, known as ISim(2) ⊂ ISO (3,1). In recent work, in an attempt to obtain non-commuting translations, and hence spacetime curvature [18], we studied the continuous deformations of ISim(2) and found a two-parameter family, one of which was rejected because the deformation of the SO(2) rotation generator ceased to be compact. The remaining one-parameter deformed group DISim(2) b depends on a dimensionless parameter b, and coincides with one introduced by Bogoslovsky [19] in his proposal for an anisotropic Finslerian spacetime.
It is straight forward to generalise the DIsim(2) b group to k + 2 spacetime dimensions.
We denote the resulting group DISim b (k). It is interesting to note [20] that the DISim b (k) is then isomorphic to the extended Schrödinger groupS ch(k) [21] which has resurfaced in recent studies of non-relativistic holography in k spatial dimensions (see e.g. [22] and references therein). Since this topic is not strictly connected with the Maxwell algebra which is the main concern of the present paper, we relegate the details to appendix B.
Since Maxwell's equations are invariant under DISim(2) b , the dispersion relation for photons is the standard one, and hence these theories are consistent with the recent highprecision test of Lorentz violation using the gamma-ray burst GRB090510 [23].
The advent of quantum mechanics led to the realisation that not only are deformations of algebras important, but so also are extensions, especially central extensions. The Urexample is the Heisenberg algebra However, a more relevant example for our purposes is the motion of a particle of charge e in a uniform time-independent magnetic field. The minimally-coupled Lagrangian is where A i = − 1 2 F ij x j , and T is the kinetic energy. If p i ≡ ∂T /∂ẋ i (sometimes called the mechanical momentum), then the canonical momentum is Assuming that π i and x j satisfy the standard Poisson algebra {x i , π j } = δ i j , {x i , x j } = {π i , π j } = 0, then p i and x j satisfy the centrally-extended algebra The action associated to (1.5) is invariant, up to a boundary term, under constant translations: The Noether charges associated with these spatial translations arẽ It follows from the equations of motion that dp i /dt = 0. One finds the following non-trivial Poisson brackets: (1.10) Note that thep i and p i momenta Poisson commute: The constancy of thep i is now seen to follow from the fact that the Hamiltonian obtained by taking the Legendre transform of the Lagrangian (1.5) is a function only of the mechanical momentum p i , and independent of the spatial coordinates x i .
One can think of e as a central element −Z in a finite-dimensional Poisson algebra, which commutes with all other generators. The moment maps p i andZ generate a group of transformations on phase space, whose Lie algebra is (1.12) (The relative sign between Lie algebra brackets and Poisson brackets is a consequence of our general conventions, which are detailed in appendix A.) Note that whereas the "central term" in the Poisson algebra (1.12) is cohomologically trivial (it can be removed by the local redefinition of generators that maps from x i and p i to x i and π i ), the central term in the Lie algebra (1.12) is cohomologically non-trivial. The reason for the difference is that the x i are included as generators in the Poisson algebra, but not in the Lie algebra.
Inclusion of a uniform time-independent electric as well as a magnetic field generalises (1.12) to the Lorentz-covariant Lie algebra This five-dimensional algebra has as group manifold G the five-dimensional space with coordinates x µ and θ, conjugate to P µ and Z respectively. It turns out, in Kaluza-Klein fashion, that geodesic motion on G projects down onto the coset G/H, where H ≡ R is generated by the central element Z, and electric charge is the conserved momentum in that direction. Particles with magnetic as well as electric charge (dyons) may also be catered for, by passing to six dimensions and replacing (1.13) by where F ⋆ µν = 1 2 ǫ µναβ F αβ is the Hodge dual of F µν . A different approach is to consider non-central extensions of the fundamental algebra g. This is by now standard in supersymmetric p-brane theories, following the pioneering work of van Proeyen and van Holten [24]. However the simplest example, which is purely bosonic and predates their work, is the 16-dimensional Maxwell algebra, which is a noncentral extension of the Poincaré algebra with six tensorial charges Z µν arising through a non-commutativity of the momentum generators, (1.15) One now introduces six angles conjugate to Z µν . These angles are dynamical variables with a non-trivial evolution.
Note that for any particular solution of the relevant equations of motion,Z µν = −eF µν , spontaneous symmetry breaking will occur and the symmetry will be reduced to the subgroup of the Poincaré group leaving the background F µν invariant. This is the kinematical group in this context.
The aim of this paper is to study in the framework of the Very Special Relativity [17], or its deformation [18], the motion of a bosonic charged massive particle in the presence of a constant electromagnetic field. This will done by constructing the non-central extensions and deformations of ISim(2). As we shall see, the Maxwell-Sim algebra is constructed from the translation generators P µ and the non-central extension Z µν = [P µ , P ν ], together with the Sim(n) generators (M +i , M +− , M ij ).
Later we study the deformations of the Maxwell-Sim algebra. In general dimensions we find two deformations, with parameters b and c. The deformation parametrized by c is analogous to the k deformation of the Maxwell algebra found in [25] (now restricted to the 14 generators of Maxwell-Sim(2)), which gave SO(3, 2) × SO(3, 1) or SO(4, 1) × SO(3, 1), depending on the sign of k. The b-deformation of Maxwell-Sim produces the Maxwell extension of the DIsim b algebra, which is related to Finslerian geometry.
In order to construct the particle models with the previous symmetries, we use two different approaches: one based in the Lagrangian formalism and the non-linear realization approach [26], and the other based on the Hamiltonian formalism constructed from the momentum maps.
In the case of Maxwell-DISim b the motion is given by a Finslerian Lorentz force, while for the undeformed Maxwell-Sim we obtain the ordinary Lorentz force. Therefore the study of anisotropies of a massive particle in an electromagnetic field could provide a test of a possible Finsler geometry.
The organization of the paper is as follows. In section 2 we review the Maxwell algebra, and in section 3 we recall the basic facts about the ISim algebra. In section 4 we construct the Maxwell-Sim algebra, and then in section 5 we the study its deformations. The particle Lagrangians are constructed in section 6, and in section 7 we perform the Hamiltonian analysis. The paper ends with conclusions. There is also an appendix about the Hamiltonian formalism and momentum maps.

The Maxwell Algebra
The name Maxwell algebra appears to originate with Glashow, as reported in [27] in connection with the behaviour of matter in extremely strong magnetic fields such as are found in neutron stars. Nowadays one might think of magnetars. However it is Schrader [28] who seems to have been the first to study it systematically. Other earlier work applying group theoretic methods to uniform electromagnetic fields is in [29,30]. This often-cited work assumes a constant c-number background field F µν , and is largely concerned with what they called the kinematical group, i.e. with the 6-dimensional subgroup of the Poincaré group that leaves F µν invariant, generated by P µ and two commuting Lorentz generators This gives the algebra where we have defined F ⋆ µν = 1 2 ǫ µνρσ F ρσ . We shall refer to these 6-dimensional kinematical algebras as the Bacry-Combe-Richards or BCR-algebras. The BCR group is E(2) × E(1, 1), the product of the two-dimensional Euclidean group E(2) with the two-dimensional Poincaré group E(1, 1).
If we use light-cone coordinates where x µ = (x − , x + , x i ), define ǫ +−12 = +1, and take the Maxwell field to be zero except for F +− = 1, then G = M +− , G ⋆ = M 12 , and the algebras associated with the two factors are generated by For more details on the BCR group, the reader may consult [31].
According to [32], the BCR algebra has three central extensions, so that There also four other non-central charges Z µν (in the complement of Z elec µν = Z elec F µν and Z mag µν = Z mag F ⋆ µν ): In total there are six extensions which we can decompose in representations of G and G ⋆ .
The difference with respect to the Maxwell case to be treated later is the presence of two central charges. The central charges are present because the Lorentz group has been reduced to the abelian subgroup G 2 generated by G and G ⋆ . (Note that this 2-dimensional abelian group G 2 is not to be confused with the 14-dimensional non-abelian simple Lie group of the Cartan/Dynkin classification!) In [32,33] arguments are given to the effect that if the rotation in G, G ⋆ is to be a compact generator then a should vanish. That leaves Z elec and Z mag which, as the notation suggests, may be identified with electric and magnetic charge respectively. In what follows we shall refer to the 8-dimensional doubly-extended kinematic algebra generated by The EBCR algebra is a direct sum of two subalgebras, each of which has a nontrivial quadratic Casimir. In the case where the only non-vanishing component of F µν is given by F +− = 1, the generators of the two subalgebras are {G ⋆ , P 1 , P 2 , Z mag } and {G, P + , P − , Z elec }. The two Casimirs are By contrast, the Maxwell algebra is a 16-dimensional extension of the Poincaré algebra with the non-vanishing brackets where Z αβ = −Z βα . There are two generic Casimirs (which exist for the Maxwell algebra in any dimension) [28], and a third quadratic Casimir that exists only in the special case of four dimensions: 14) The Casimirs are, of course, elements of the universal enveloping algebra of the Maxwell Lie algebra.
The six additional generators Z µν are on the same footing as the Poincaré generators The relevant algebra will then reduce to the EBCR algebra that leaves invariant the background field F µν .
A set of left-invariant 1-forms are (we omit those for the Lorentz sub-algebra) with generators of right actions being given by (2.18) they satisfy P µ , P ν = Z µν .
The 1-forms (2.15) are invariant under which are generated by the vector fields (2.20) The ten-dimensional subalgebra, which is obtained by taking the quotient with respect to the Lorentz subalgebra, is spanned by P µ and Z µν , and closes on the generalised Heisenberg algebra. The associated coset is thus also a group manifold, sometimes called a superspace, and has as coordinates x µ and θ µν . This 10-dimensional superspace, which is fibred over Minkowski spacetime with flat six-dimensional fibres, carries a natural Lorentz-invariant metric:

Quantisation
The obvious approach to quantisation is to consider wave functions Ψ(x µ , θ µν ) depending upon both x µ and θ µν . A generalised Klein-Gordan or Dirac equation may readily be written in the usual way using the differential operators The equations can be solved using Fourier transforms, and the solutions used to construct one-particle Hilbert spaces. The Maxwell group acts on these wave functions by pull-back, and in this way one obtains a projective representation of the Maxwell group. For details of the procedure, including the calculation of the relevant co-cycles, the reader is referred to Schrader's paper [28].

Deformations and Contractions
In general dimensions, the Maxwell algebra admits a unique deformation parameter k. For [34,25]. Conversely, it may be regarded as a Wigner-İnönü contraction [25] such that where k has the dimensions of length and the sign choice is made depending on whether we consider the AdS or dS part.

The ISim Algebra
We may consider a generalisation of the discussion of the Maxwell algebra of [35], where the starting point is taken to be the ISim algebra rather than the Poincaré algebra. The ISim generators are The ISim algebra, with the conventions we are using, is given in [18].
We define left-invariant 1-forms λ as in (A.1), but now, for convenience, we denote them by λ a = (P µ , M +i , M +− , M ij ), and so In terms of these, the ISim(n) algebra is given by

The Maxwell-Sim Algebra
The Maxwell-Sim algebra can be constructed in complete analogy to the Maxwell algebra discussed previously. One way to describe this it that we start with the P µ generators alone, obtain the central extension in which [P µ , P ν ] = Z µν , and then append the Sim(n) generators (M +i , M +− , M ij ) to form the Maxwell-Sim(n) algebra. At the level of the left-invariant 1-forms, this means that we augment the ISim(n) relations (3.3) by

Deformations of Maxwell-Sim(n)
We follow the method for finding the general non-trivial deformation of an algebra that is described in [18] (see [36] for further details). This entails first finding the second cohomology class H 2 (g, g), which determines the non-trivial deformations at the linear level.
If H 3 (g, g) is trivial, then there must exist, possibly after making (trivial) redefinitions, an extension of the linearised deformations that is valid to all orders. (This is checked by verifying that the deformed algebra satisfies the Jacobi identities.) If, on the other hand, is non-trivial, then the extension beyond the linearised level may not be possible.
For the generic case of the Maxwell-Sim(n) algebra, we find that there are two distinct 1-parameter non-trivial deformations. We denote these by the b-deformation and the c-deformation, where b and c are the respective constants parameterising the two deformations. 2

The b-deformation
In the b-deformation, the Maxwell-Sim(n) algebra defined by (3.3) and (4.1) is modified by the following additions to dP µ and dZ µν : where the "· · · " terms represent the usual right-hand sides of the undeformed Maxwell-Sim(n) algebra. The Sim(n) relations in (3.3) are unmodified. 2 We have performed some of the calculations with differential forms with the aid of the EDC Mathematica package [37].

The c-deformation
In the c-deformation, the Maxwell-Sim(n) algebra defined by (3.3) and (4.1) is modified by the following additions to dP µ and dZ µν : In the special case of Maxwell-Sim (2), we find that there is an additional non-trivial deformation characterised by a parameter a, which can be turned on simultaneously with the b-deformation. Thus in place of the b-deformation given by (5.1), for Maxwell-Sim (2) we may have A calculation of the cohomology group H 3 (g, g) for Maxwell-Sim (2) shows that it is non-trivial, and of dimension 3.
Note that the deformation parameterised by c in (5.2) is analogous to the k deformation of the Maxwell algebra found in [25], Integration of (6. 2) gives f µν = f 0 µν , and such a solution spontaneously breaks the Lorentz symmetry into a subalgebra of the Maxwell algebra (namely the EBCR algebra discussed earlier). Substituting this solution into equation (6.4) gives the motion of a particle in a constant electromagnetic field.
Alternatively, since we know how to construct Lorentz scalars, we can construct a Lagrangian without the introduction of the new dynamical variables f µν as (6.5) The quantitiesθ µν + 1 2 (x µẋν − x νẋµ ) are constants of motion. If we choose them equal to 1 2 f 0 µν , we recover the same equation of motion of a particle moving in a constant electromagnetic field that we obtained above.
Another way to construct the Lagrangian is to consider the coset (Maxwell)/(Rotations).
This coset is useful for the construction of massive particle Lagrangians when the tensor calculus is not known. (For example, in the case of ISim, we may consider the coset (ISim)/(Rotations), rather than (ISim)/(Sim), because we do not know a priori what is the length element; in order words, we do not have an obvious tensorial calculus. A more striking example is the case of the deformed ISim algebra DISim b , discussed in [18]. We obtain left-invariant 1-forms, by first defining where and w i , i = 1, 2, 3, are the Goldstone bosons associated with the broken boost generators.
The left-invariant 1-forms λ may then be read off from Defining we have The Lorentz transformations generated by U may be used to define Λ ν µ (w i ): The left-invariant 1-forms λ µ P , λ ij R and λ µν Z are then given by The 1-forms λ i M , are given by where the * indicates that the 1-forms are pulled back onto the world-line: [dx µ ] * ≡ẋ µ (τ ) dτ , etc. The coefficient α is constant, whilstf µν (τ ) is a dynamical field that depends upon τ .
We see that (6.14) may be written as where Λ µ ν is a general Lorentz boost transformation and depends on the non-dynamical coordinates w i . We have also introduced the tensor field f µν , which is related tof µν by We now define the particle momentum p µ in the canonical way: Because Λ 0 µ is a timelike Lorentz vector, we have Introducing e as a Lagrange multiplier to enforce the mass-shell condition (6.18), we arrive at the Lagrangian Varying with respect to p µ giveṡ Substituting for p µ in (6.19), and then varying with respect to e to obtain e = − √ −ẋ 2 m , (6.21) we finally arrive at the Lagrangian (6.1). In section 7 we shall see how the non-linear realisation method and coadjoint orbit technique gives the same results.

The Maxwell-Sim Lagrangian
We start with the coset (Maxwell-Sim)/SO(2), and then construct the left-invariant 1-forms from the coset representative Following the same steps as in the Maxwell case, we have The left-invariant 1-forms λ µ P and λ µν Z are then given by where The Lorentz transformation Λ µ ν (w i , w) is given by where we order the spacetime coordinates in the sequence x µ = (x − , x + , x i ), i = 1, 2.
The 1-forms λ i M and λ N are given by (6.28) A particle Lagrangian that is invariant under SO(2) (generated by J = M 12 ) is given by As in the Maxwell case the coefficients α and β are constants, whilstf µν (τ ) is a dynamical field that depends upon τ . We see that (6.29) may be written as where The particle momentum p µ is given by Noting that Λ + µ Λ +µ = Λ − µ Λ −µ = 0 and Λ + µ Λ −µ = 1, we see that where we have defined the mass parameter as m = 2α β . (6.34) Introducing e as a Lagrange multiplier to enforce the mass-shell condition (6.33), we arrive at the Lagrangian Varying with respect to p µ giveṡ Substituting for p µ in (6.35), and then varying with respect to e we get the Lagrangian (6.1). Thus the undeformed Maxwell-Sim algebra gives the same particle Lagrangian as the Maxwell algebra based on the full Poincaré group.

The Maxwell-DISim b Lagrangian
The left-invariant 1-forms λ µ P of the DISim b algebra are given by where the matrixΛ is The 1-forms λ i M and λ N are given by We wish to construct a particle Lagrangian that is invariant under SO(2) (generated by J = M 12 ). Thus we begin by writing where as before the * indicates that the 1-forms are pulled back onto the world-line. The coefficients α and β are constants. We see that (6.40) may be written as whereΛ µ ν depends on the non-dynamical coordinates w and w i , and is given by (6.38).
We now define the particle momentum p µ in the canonical way: we have the constraint With α = −m(1 − b) and β = − 1 2 m(1 + b) we obtain equation (18) of [18]: Introducing e as a Lagrange multiplier to enforce the mass-shell condition (6.45), we arrive at the Lagrangian . (6.46) Varying with respect to p µ giveṡ If we solve for p µ and substitute into (6.46), we obtain Varying this with respect to e we get For the Maxwell-DISim b case, following the same steps as for the Maxwell-Sim case, we with the non-trivial algebra and metric The metric (7.3) is invariant under the left action of the Heisenberg group, and an additional outer action of the abelian subgroup G 2 ⊂ SO(3, 1) generated by G and G ⋆ . We may identify Z with Z elec introduced earlier. If we consider the coset (EBCR)/(G 2 , Z mag ), then the quadratic combination P 2 +Z 2 is invariant under the stability group. The corresponding metric is (7.3) and therefore it is invariant under the whole EBCR group.
A convenient matrix representation of the Heisenberg group is given by The phase or cotangent space T ⋆ (G) ≡ G × g of the Heisenberg algebra has coordinates (x µ , θ, p µ , p θ ). (P µ ,M µν ,Z) are the corresponding moment maps generating right actions, and are given byP The non-vanishing Poisson brackets of the generators of the right actions of the Heisenberg group are P µ ,P ν = −F µνZ .
The equation for θ, conjugate toZ µν isθ = 1 mZ . (7.10) The moment maps that generate left translations are given bȳ where β is an arbitrary constant.
We may also include a magnetic charge by adding an extra central extension The Hamiltonian µP µ (7.14) will now lead to the constancy of bothZ andZ ⋆ and the equation of motioṅ If we identify Z ⋆ with Z mag , then then the six-dimensional Heisenberg algebra with two central charges may be identified with the coset (EBCR)/G 2 . Note that the presence of magnetic and electric charges is due to the presence of central charges in the 8-dimensional EBCR algebra. These central charges are absent in the Maxwell algebra.

The Maxwell algebra
The phase space, or cotangent space, T ⋆ (G) ≡ G × g of the Maxwell algebra has coordinates (x µ , θ µν , p µ , f µν ). The left-invariant Maurer-Cartan forms are given by (2.15) and (P µ ,M µν ,Z µν ) are the corresponding moment maps generating right actions. They are given byP The non-vanishing Poisson brackets are M αβ ,P γ = η βγPα − η αγPβ , (7.17) There are two generic Casimir functions, For the Hamiltonian, we take Thus the Euler equations implẏ 24) and the x µ equation of motion isẋ The equation for θ µν , conjugate toZ µν iṡ Thus we obtain the motion of a particle in a constant electromagnetic field, for which the momentum vectorP µ (τ ) undergoes a constant Lorentz transformation By contrast with the Kaluza-Klein approach, which gives the same equations for the x µ variables with an externally imposed constant Maxwell field F µν , in the Maxwell algebra approach we find that the Maxwell field must be constant as a consequence of the equations of motion. The equations for the six angles θ µν are also richer. They may be interpreted geometrically as follows. The curve in spacetime x µ = x µ (τ ) has a projection onto each µ-ν 2-plane. The curve sweeps out area at a rate In other words θ µν (τ ) is the total area A µν (τ ) swept out during the motion.
The canonical Lagrangian that reproduces the previous equation of motion is The equations of motion are the same as before except for those of the variables θ µν , which now satisfyθ The canonical Lagrangian gives, after eliminating the non-dynamical field F µν , the Lagrangian (6.5).

Other Hamiltonians
Those which admit a constantZ µν = F µν and are Lorentz-invariant are of the form 2mH =P µP µ + 1 2 αZ µνZ µν − βZ µνM µν . (7.34) The second term does not contribute, since it commutes with everything, and so we drop it. Hamilton's equations then giveṖ Note that in the special case β = 1, we findṖ µ = 0. This is not surprising, because in that case the Hamiltonian is bi-invariant, i.e. it is a Casimir, and hence generates no motion at all.

Maxwell-Sim and Maxwell-DISim b
The Maurer-Cartan forms of the coset (Maxwell-Sim)/(Sim) are the same as in the Maxwell case (2.15), and the moment maps are also given bȳ The geodesic Hamiltonian is given by and therefore reproduces the same dynamics as in the Maxwell case.
For the case of the coset (Maxwell-DISim b )/(Sim), the Maurer-Cartan forms and the momenta are the same as for the Maxwell case.

Hamiltonian Treatment of the Bogoslovsky-Maxwell algebra
In previous work [18] we obtained a Finslerian Lagrangian invariant under DISim(2) b , where b is the deformation parameter constructed from the Finslerian line element where the Finsler function F (v µ ) is homogeneous of degree 1 in the four-velocity v µ = dx µ /dτ . In general, if we were to use a multiple of the Finlser function F (v µ ) as a Lagrangian L(v µ ), then its Legendre transform would vanish, since a Lagrangian which is homogeneous of degree k in velocities gives, on taking a Legendre transform, a Hamiltonian = v µ ∂L ∂v µ − L (7.40) which is homogeneous of degree k k−1 in momenta p µ . If k = 2 we have and both are of degree two. Therefore it is customary in Finsler geometry to set For the case of Bogoslovsky's Finslerian geometry we would then have where n µ = η µν n ν is a constant future-directed null vector. The minus signs appear in (7.44) because v µ is assumed to be future-directed and timelike. With our signature convention, the inner product n · v = n µ v µ is then negative. We find that and Imposing the mass-shell condition H = − 1 2 m, i.e. F (v) 2 = 1, leads to equation (18) of [18]. In this case, the parameter τ coincides with the Finslerian measure of proper time along the world-line of the particle. 3 Equation (7.46) is also equivalent to the expression (6.44) . We may also give the expression for v µ = ∂H/∂p µ , finding We have also studied the motion of a massive particle interacting with a constant electromagnetic field with these symmetries. In the case of Maxwell-DISim b , the motion is given by a Finslerian Lorentz force, whilst by contrast for the undeformed Maxwell-Sim algebra we obtain the ordinary Lorentz force.

Acknowledgements
We

A Conventions
In this appendix we record some of our conventions and notation when working with Lie groups, Lie algebras and Poisson algebras.
The left and right invariant vector fields L µ a and R µ a dual to λ a µ and ρ a µ respectively, and respectively generate right and left translations on G.
Quantum mechanically, one often inserts i's so that ifR a = 1 TheR a andL a vector fields are then operators acting on complex-valued functions of the group coordinates x µ .
Thinking of G as a configuration space, we can pass to the phase space or cotangent space T G ⋆ ≡ G × g, with coordinates (x µ , p ν ). The actions of G on G then lift to T G ⋆ as canonical transformations, leaving the natural symplectic form dp µ ∧ dx µ invariant. Given the symplectic form, we can introduce the Poisson bracket as usual. In local Darboux coordinates (x µ , p ν ), it is given by Infinitesimally, the lifts of left and right actions are canonical transformations generated by "generating functions" or "moment maps." Because, in general, we have both left and right actions to take into account, we define two sets of moment maps into g ⋆ , the dual of the Lie algebra, with Poisson brackets which are readily seen to be

B Lifshitz and Schrödinger algebras
In this appendix we shall describe the connection between the deformed inhomogeneous Sim algebra disim b (k) and the Lifshitz, Schrödinger and extended Schrödinger algebras, lif z , sch z (k) and sch(k) respectively. We start with

B.1 Lifshitz scaling
In non-relativistic theories with k spatial dimensions, one is interested in the behaviour of physical quantities under what has come to be called Lifshitz scaling, i.e. under where t is the time variable and x = (x 1 , x 2 , . . . , x k ) is the spatial position vector.
If D generates scalings or dilatations we may combine this with space translations P i , spatial rotations, M ij and time translations H, to obtain the Lifshitz Algebra, lif z (k) in k spatial dimensions, where the obvious brackets for M ij have been omitted. The Lie algebra spanned by D, P i , and H is therefore invariant under the adjoint action of the rotation subalgebra so(k) generated by M ij . If i = 1, 2, . . . , k, then lif z (k) has dimension 1 2 k(k+1)+2 and the quotient lif z (k)/so(k) has dimension k + 2.

B.2 Lifshitz spacetime
This is a k + 2 dimensional spacetime equipped with a metric invariant under the left action of the (k + 2)-dimensional group generated by P i , H and D. A Maurer-Cartan basis for this solvable group is The Lifshitz metric is then with Killing vector fields corresponding to • As r → ∞ we approach a singular horizon (IR limit) .
• As r → 0 we approach infinity (UV limit) The boundary metric at infinity is obtained by taking out a factor of r 2 and letting r → 0: Thus the speed is c(r) = r (1−z) , and • If z > 1, we obtain infinite speed (the boundary lightcone opens out to a plane) • If z = 1, we obtain finite speed (the boundary lightcone remains a cone) • If z < 1, we obtain zero speed (the boundary lightcone closes up to a half line ) Strictly speaking, in the z > 1 case, we need to consider the inverse metric when taking the limit r → 0.

B.3 The boost-extended Lifshitz algebra
One may extend the Lifshitz algebra to include boosts K i . The scaling dependence of K i is then determined by its commutation relations. Since K i is a vector we have For the Galilei group, which implies that we must take which implies that we must take D, K i = (z − 1)K i . (B.14) In the case of the Poincaré group there is no choice, and one must take z = 1.

B.4 DISim b (k)
Recall that DISim b (k) is a deformation of the ISim(k) subgroup of the Poincaré group in (k + 2) spacetime dimensions, depending on a parameter b, which may be regarded as a subgroup of the inhomogeneous Weyl group or Causal group (i.e the semi-direct product of Poincaré with dilatations), in which the actions of a boost and dilations are identified up to a factor [18]. It is thus of dimension 1 2 k(k + 1) + k + 3. The so(k) rotations have the standard brackets and act on P i and M +i as vectors. The boost generator M +− acts on (k + 2)-dimensional Minkowski spacetime as If b = 0, then M +− acts as an ordinary boost..
the conformal symmetry group of the free Schrödinger equation (corresponding to z = 2) is 13-dimensional 4 . This is because the special conformal or temporal inversion operator has been left out.
One may consistently drop the central extension N from the Bargmann algebra to get the Galilei algebra, and then the extended Schrödinger algebra sch(k) reduces to the ( 1 2 k(k + 1) + k + 2) dimensional unextended Schrödinger algebra sch(k). If one then drops the boost generator K i one gets the Lifshitz algebra lif z (k).
It is well known that non-relativistic symmetries and non-relativistic conformal symmetries (Schrödinger algebras) in k spatial dimensions may be thought of as subgroups of relativistic or conformal symmetries in k + 2 dimensional Minkowski spacetime which commute with light-like translations. Thus it is no surprise that To see this, one must identify the generators as follows;

B.6 Schrödinger spacetime
This is (k + 3)-dimensional, and has metric with Killing vectors In the cases z > 1, we may regard the boundary as the (k +2)-dimensional Duval-Kunzle spacetime whose null reduction produces the (k +1)-dimensional Newton-Cartan spacetime.
Strictly speaking we need to consider the inverse metric when taking the limit.