Conformal fluid dynamics

We present a conformal theory of a dissipationless relativistic fluid in 2 space-time dimensions. The theory carries with it a representation of the algebra of 2-D area-preserving diffeomorphisms in the target space of the complex scalar potentials. A complete canonical description is given, and the central charge of the current algebra is calculated. The passage to the quantum theory is discussed in some detail; as a result of operator ordering problems, full quantization at the level of the fields is as yet an open problem. ∗ e-mail: [email protected] ∗∗ e-mail: [email protected] † on leave from NIKHEF, Amsterdam NL 1 Fluid dynamics as a lagrangean field theory Non-dissipative fluid mechanics in either a relativistic or non-relativistic framework can be formulated as a lagrangean field theory [1, 2]. In a relativistic context, the relevant physical degrees of freedom are described by the time-like four-current density j(x), related to the scalar density ρ(x) by gμν j j = −ρ. (1) The velocity field u(x) then is related to the current density by j = ρu, gμν u u = −1. (2) The essential physical aspects of fluid dynamics in Minkowski space are the conservation laws as represented by vanishing divergence of the fluid current and the energy-momentum tensor: ∂μj μ = 0, ∂μT μν = 0. (3) In the non-relativistic limit these reduce to the Bernouilli and Euler equations. In addition, there is an equation of state relating the pressure p and energy density ε = f(ρ). It is straightforward to generalize this description to a general relativistic context so as to include the gravitational field [3]. A lagrangean formulation of a fully relativistic theory, including the gravitational field, in n space-time dimensions is given by the action S = ∫ dx √−g ( − 1 8πG R + Lfluid ) , (4) where the first term is the Einstein-action for general relativity, and the lagangean density for the fluid is Lfluid = −j · (∂θ + iz̄∂z − iz∂z̄)− f(ρ). (5) Here (θ, z̄, z) are auxiliary fields acting as langrange multipliers imposing the correct physical conditions on the current density. Strictly speaking, the action (5) is motivated from fluid dynamics in n = 4, but it can consistently be continued to other values of n as well. The field equations derived from the action (4) by varying the current components j and the potentials (θ, z̄, z) lead to an equation for the current f (ρ) ρ jμ = ∂μθ + iz̄∂μz − iz∂μz̄, (6) subject to the conditions D · j = 0, j · ∂z = j · ∂z̄ = 0. (7) 1 These are supplemented by the Einstein equations Rμν − 1 2 gμν R = −8πGTμν , (8) with the energy-momentum tensor taking the perfect-fluid form Tμν = pgμν + (ε+ p)uμuν . (9) In this expression the energy density and pressure are given by ε = f(ρ), p = ρf (ρ)− f(ρ). (10) A typical equation of state is found by taking a power law for the energy density f(ρ) = αρ ⇒ p = η ε. (11) This is the type of equation of state familiar from applications in cosmology and astrophysics. Eqs. (7) for the potentials (z̄, z) state that these auxiliary fields are constant in a comoving frame: u · ∂z = 0 ⇔ dz dt = ∂z ∂t + v · ∇z = 0, (12) and similarly for the complex conjugate potential. As a result, there is an infinite set of covariantly conserved currents of the form [4] J [G] = 2G(z̄, z)jμ, D · J [G] = 0. (13) The factor 2 results from Noether’s theorem, applied to the invariance of the action under the infinitesimal transformations δGθ = 2G− zG,z − z̄G,z̄, δGz = −iG,z̄, δGz̄ = iG,z, (14) whilst δGj μ = δGgμν = 0. The commutator algebra of these infinitesimal transformations is closed, with the composition rule [δG, δG′] = δG′′ , G ′′ = i ( ∂G ∂z ∂G ∂z̄ − ∂G ∂z̄ ∂G ∂z ) . (15) It is readily checked that the transformations (14) represent the invariances of the one-form J = dθ + iz̄dz − izdz̄. (16) This also implies the existence of an invariant two-form A = 1 2 d ∧ J = idz̄ ∧ dz, (17) representing the area in the space of complex potentials. Area-preserving diffeomorphisms have been studied in a different context as a symmetry of the base-space manifold of the relativistic membrane in ref. [5, 6]. In contrast, here the diffeomorphisms are realized directly on the dynamical variables. It is an interesting question what happens to these transformations in a quantum theory, where these variables become operators. 2 2 Conformal fluid dynamics For specific equations of state, the classical relativistic fluid models (5) are conformally invariant. Indeed, it is easily established that the trace of the energymomentum tensor (9) vanishes for fluids in n space-time dimensions for energy densities ε = f(ρ) = αρ n n−1 ⇒ T μ μ = 0. (18) For such fluids the equation of state parameter is η = 1 n− 1 . (19) In 4 space-time this implies η = 1 3 ⇒ p = 1 3 ε (n = 4). (20) This is the indeed the equation of state for a gas of free massless particles, like photons or massless neutrinos. In two space-time dimensions a conformal fluid model has a somewhat unusual equation of state with η = 1: p = ε (n = 2). (21) The action in this case reads S = ∫ dx √−g [−j (∂μθ + iz̄∂μz − iz∂μz̄) + αgμνjj ] ≃ − 1 4α ∫ dx √ −gg (∂μθ + iz̄∂μz − iz∂μz̄) (∂νθ + iz̄∂νz − iz∂ν z̄) , (22) where the last line is obtained by algebraic elimination of the independent current vector field j by jμ = 1 2α (∂μθ + iz̄∂μz − iz∂μz̄) . (23) If the normalization of the current is chosen such that α = 1/2, one obtains standard kinetic terms for the scalar potential θ. A further important aspect of this 2-dimensional model is that one can extend the model with a Wess-Zumino term in which the real scalar θ couples to the invariant target-space area 2-form: ∆S = 2iλ ∫ dx εθ ∂μz̄∂νz. (24) For 2αλ = ±1 this implies that the current becomes self dual, or anti-self dual, respectively. It is easily verified that the action (22) and the Wess-Zumino term are invariant under local Weyl rescaling gμν(x) → g μν(x) = egμν(x), (25) 3 keeping the scalars (θ, z̄, z) unchanged; this confirms the conformal invariance of the model. Remarkably, the infinite set of transformations (14) also leaves the Wess-Zumino term invariant, up to a boundary term. In part, this results from the invariance of the area two-form A; the non-trivial part is, that the transformation δGθ by itself produces a total derivative. Finally we observe, that the energy-momentum tensor of this theory is not changed by the addition of the Wess-Zumino term and with α = 1/2 takes the Sugawara form Tμν = jμjν − 1 2 gμνg jκjλ, (26) which is traceless in 2-dimensional space-time. 3 Canonical light-cone formulation In this section we present the canonical formulation of 2-dimensional conformal fluid dynamics in the light-cone formulation, using light-cone co-ordinates and derivatives x = x ± x √ 2 , ∂± = ∂0 ± ∂1 √ 2 , (27) such that in the conformal gauge the space-like line element is ds = −2e dxdx. (28) Because of the conformal invariance, the action is independent of the 2-dimensional conformal gravitational field component φ(x). Therefore, using the canonical value α = 1/2, the full action for the conformal theory takes the form Slc = ∫ dxdx [(∂+θ + iz̄∂+z − iz∂+z̄)(∂−θ + iz̄∂−z − iz∂−z̄) + 2iλθ (∂+z̄∂−z − ∂−z̄∂+z)] . (29) The field equations then include the self dual/anti-self dual current conservation laws ∂−J+ = 0, λ = +1; ∂+J− = 0, λ = −1, (30) where the current components J± represent the fixed expressions J± = ∂±θ + iz̄∂±z − iz∂±z̄. (31) These equations lead to the result ∂+J− = 2i (∂+z̄∂−z − ∂−z̄∂+z) , λ = +1; ∂−J+ = −2i (∂+z̄∂−z − ∂−z̄∂+z) , λ = −1. (32)