A recent publication [Phys. Rev. Lett. 92, 020404 (2004)] raises the possibility of momentum transfer from zero-point quantum fluctuations to matter, controlled by applied electric and magnetic fields. We present a Lorentz-invariant description using field-theoretical regularization techniques. We find no momentum transfer for homogeneous media, but predict a very small transfer for a Casimir-type geometry. DOI: 10.1103/PhysRevLett.96.130402 PACS numbers: 03.50.De, 42.50.Nn, 42.50.Vk The Feigel hypothesis [1] (FH hereafter) suggests a nonzero total momentum density of zero-point fluctuations [2] in magneto-electric (ME) matter. ME effects can occur in media associated with a few specific symmetry classes [3], but can be induced in any medium, including the real quantum vacuum [4], by an external electric field E 0 and magnetic field B 0 . The FH, based on the manifestation of ME in optical birefringence, is controversial [5][6][7], yet important since it raises the possibility to transfer momentum from zero-point fluctuations to matter, controlled in direction and magnitude by the externally applied fields.In ME media, photons with wave vectors k and ÿk behave differently, though independent of their polarization. Optical ME effects have been observed in birefringence [8][9][10] and in absorption [11][12][13]. The final result of Ref.[1] is an expression for the momentum density v (mass density , velocity v) of a homogeneous medium in crossed, uniform, stationary fields E 0 and B 0 :Here and " are the usual optical constants, is the ME coupling parameter, and ! c is a cutoff frequency. The first problem is the Lorentz variance of Eq. (1). All quantities in the second factor of Eq. (1) In this Letter we address all four problems. To cope with the diverging vacuum without breaking Lorentz invariance we shall apply field regularization techniques [14]. To describe external fields that can be slowly switched on we shall consider total momentum balance, which is also free from the Abraham-Minkowski controversy [15]. The simplest inhomogeneous situation that can be regularized while respecting Lorentz invariance is the Casimir geometry [16], for which we shall make a precise prediction that is quite different from the FH. An important restriction of our work is the use of the macroscopic Maxwell equations, assuming that vacuum fluctuations are governed by the ''macroscopic'' properties of matter. Effects of dispersion and absorption caused by microscopic resonances cannot be described.The optics of ME media is described by the constitutive relations of a bi-anisotropic form [3]The tensors " and are assumed real valued and symmetric, the ME tensor only real valued, all frequency independent. All can be spatially varying and time dependent, but no dispersion or optical absorption is considered. In bianisotropic media, momentum conservation is expressed by,It features the symmetric stress tensor T 0 , with tensor elements 4T 0;ij E i E j B i B j ÿ 1 2 E E B B ij . Contrary to pseudomomentum, the ''momentum'' R drE B...