Fluid dynamics of relativistic quantum dust

Abstract

The microscopic transport equations for free fields are solved using the Schwinger function. Thus, for general initial conditions, the evolution of the energy-momentum tensor is obtained, incorporating the quantum effects exactly. The result for relativistic fermions differs from classical hydrodynamics, which is illustrated for Landau and Bjorken type initial conditions in this model of exploding primordial matter. Free fermions behave like classical dust concerning hydrodynamic observables. However, quantum effects which are present in the initial state are preserved. PACS numbers:Often the complicated time dependent dynamics of quantum many-body systems or fields is approximated by a perfect fluid model. Since the seminal work by Fermi and Landau this approach has been applied successfully, in order to study global features, such as multiplicity distributions and apparently thermal transverse momentum spectra of produced particles, in high-energy collisions of strongly interacting matter [1,2,3,4]. Similarly, the hydrodynamic approximation is often invoked in astrophysical applications and cosmological studies of the early universe [5] Recently it has been shown that a free scalar field indeed behaves like a perfect fluid in the semiclassical (WKB) regime [6]. More generally, the mechanisms of quantum decoherence and thermalization in such systems which can be described hydrodynamically, i.e. the emergence of classical deterministic evolution from an underlying quantum field theory, are of fundamental interest [7,8,9,10].The limitations of the fluid picture, however, have rarely been explored in the microscopic or high energy density domain. Difficulties reside in the derivation of consistent transport equations and in the amount of computation required to find realistic solutions; see Refs. [11], for example, for a review and recent progress concerning selfinteracting scalar particles and the quark-gluon plasma, respectively. More understanding of related hydrodynamic behavior, if any, seems highly desirable.Presently, we study the relation between relativistic hydrodynamics and the full quantum evolution of a free matter field. In the absence of interactions, decoherence or thermalization may be present in the initial state, corresponding to an impure density matrix, but is followed by unitary evolution. We consider this as a "quantum dust" model of the expansion of matter originating from a high energy density preparation phase, which the Landau and Bjorken models describe classically [2,3].Our approach is independent of the nature of the field, as long as it obeys a standard wave equation. To be definite, we choose to work with Dirac fermions and comment about neutrinos later. We introduce the spinor Wigner function, i.e., a (4x4)-matrix depending on space-time and four-momentum coordinates:

Nonequilibrium Quantum Field Theory

“…If we understand hydrodynamics as stated in the first paragraph of this chapter, then a system defined in terms of a quantum field may not have a hydrodynamic limit. This has been shown in [Elz02] for the case of a free Fermi field. However, since the hydrodynamic description seems justifiable when applied to the physics of quark-gluon plasmas (see the discussion in Chapter 14) and early universe cosmology [Hu82,Hu83,CalGra02], we shall accept as a working hypothesis that for "interesting" systems whose fundamental description involves quantum fields there is a local thermal equilibrium limit where the system may be described as a fluid.…”

Klein–Gordon equation in hydrodynamical form

“…A 0 (r, t) = d 3 r 2 n + (r 2 , t)e + + n − (r 2 , t)e − v 0 (r, r 2 ), (27) n ν = a n aν φ 2 aν , and n = n + + n − .…”

“…The foundation of hydrodynamics is usually presented within the framework of kinetic theory, in which particles and antiparticles are described as distinct constituents and their interactions are weak [24][25][26][27]. In such a description, particles are considered to be approximately on-the-mass-shell, and their inter-particle collisions lead to thermalization.…”

“…As the foundation of hydrodynamics is usually presented within the framework of the kinetic theory [24][25][26][27], it is instructive to investigate how the present quantum probability density approach and the kinetic theory approach are connected. We shall study how the equation of motion for the Wigner function, derived from the time-dependent Schrödinger equations representing the Klein-Gordon equation, can be related to the equation of motion for the phase space distribution function in the kinetic theory, in the classical weak-field and collisionless limit.…”

Relativistic Quantum Transport Theory

“…In this section, we will study the relation between relativistic hydrodynamics and the full quantum evolution of a free matter field [29]. In particular, we try to answer how a free fermion field and its energy-momentum tensor will evolve, given arbitrary initial conditions and especially those of the Landau and Bjorken models.…”