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Laboratoire de Physique M ol ecul aire et des C oll isions, 1, boul evard A rago, 57078 M etz, France Starti ng from a H am i l toni an descri pti on ofthe photon w i thi n the set ofB argm ann-W i gner equati ons w e deri ve new sem i cl assi calequati ons ofm oti on for the photon propagati ng i n stati c gravi tati onal el d. T hese equati onsw hi ch are obtai ned i n the representati on di agonal i zi ng the H am i l toni an atthe order h,presentthe rstordercorrecti onsto the geom etri calopti cs. T he photon H am i l toni an show s a new ki nd ofhel i ci ty-torsi on coupl i ng. H ow ever,even for a torsi onl ess space-ti m e,photons do not fol l ow the usualnul lgeodesi c as a consequence of an anom al ous vel oci ty term . T hi s term i s responsi bl e for the gravi tati onalbi refri ngence phenom enon: photons w i th di sti nct hel i ci ty fol l ow di erent geodesi cs i n a stati c gravi tati onal el d.PA C S num bers:In thel astfew yearsm any studi esfocused on thetransport ofquantum parti cl es w i th spi n. Indeed,m ani pul ati ng spi n pol ari zati on ofel ectrons i s a chal l engi ng goali n sem i conductor spi ntroni cs. A chi evi ng thi s goalrequi res the understandi ng of the spi n transport m echani sm i n system s w i th spi n orbi t (SO ) i nteracti on. It was found thati n such a system a B erry phase i n m om entum space pl ays an i m portant rol e by a ecti ng both parti cl e phase and i ts transport properti es [ 1] . It i s wel lknow n si nce the sem i nalwork ofB erry [ 2] ,thatw hen a quantum m echani calsystem has an adi abati c evol uti on,a wave functi on acqui res a geom etri c phase. It i s onl y recentl y that the possi bl e i n uence of the B erry phase on transport properti es (i n parti cul ar on the sem i cl assi caldynam i cs) ofseveralphysi calsystem s has been i nvesti gated. Sem iconductors,havi ng SO coupl i ngs greatl y enhanced w i th respectto thevacuum case,requi rea theory ofspi n transport. H owever, even i n the vacuum , new fundam ental resul ts concerni ng the sem i cl assi calequati ons ofm oti on ofel ectronswererecentl y deri ved.Fori nstancei n [ 3]and [ 4] ,consi deri ng the D i rac equati on i n an externalpotenti al ,i t was show n that the posi ti on operator acqui res a spi n-orbi t contri buti on w hi ch turns out to be a B erry connecti on renderi ng the coordi nate al gebrai c structure non-com m utati ve.T hi sdrasti cal l y m odi esthesem i cl assi calequati ons ofm oti on and i m pl i es a topol ogi calspi n transportsi m i l arto the i ntri nsi c spi n H al le ecti n sem iconductors [ 1] . A si m i l ar non-com m utati ve al gebra has been al so found i n the context ofel ectrons i n m agneti c B l och bands [ 5] ,l eadi ng to an anom al ous vel oci ty term .D espi te i ts very di erent nature,the photon di spl ays m any si m i l ar behavi ors w i th el ectroni c phenom ena such as energy bands i n photoni c crystal s and l ocal i zati on. T hese si m i l ari ti es stem from the wave-l i ke nature of quantum parti cl es. B ecause a photon i s al s...

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at orderh for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.