Foldy-Wouthuysen Transormaton wth Drac Matrces n Chral Representaton V.P.Neznamov RFNC-VNIIEF, 679, Sarov, Nzhny Novgorod regon Abstract The paper oers an expresson o the general Foldy-Wouthuysen transormaton n the chral representaton o Drac matrces and n the presence o boson elds B ( xt, ) nteractng wth ermon eld ( x, t) ψ.
The papers [], [] dscuss the theory o nteractng quantum elds n the Foldy-Wouthuysen representaton [3]. These papers oer, n partcular, the relatvstc nonlocal Hamltonan H n the orm o a seres n terms o powers o charge e. Quantum electrodynamcs n the Foldy-Wouthuysen ( ) representaton has been ormulated usng Hamltonan H and some quantum electrodynamcs processes have been calculated wthn the lowest-order perturbaton theory. As a result, the concluson has been made that the representaton descrbes some quas-classc states n the quantum eld theores. Both partcles and antpartcles are avalable n these states. Partcles, as well as antpartcles, nteract wth each other. However, there s no nteracton o real partcles wth antpartcles such nteracton s possble only n ntermedate (vrtual) states. The representaton modcaton s requred to take nto account real partcle/antpartcle nteractons. In the papers [], [] such modcaton has been made usng the symmetry dentcal to the sotopc spn symmetry owng to nvarance o nal physcal results under change o sgns n the mass terms o Drac Hamltonan H D and Hamltonan H. In the moded Foldy- Wouthuysen representaton, real ermons and antermons can be n two states 3 characterzed by the values o the thrd component o the sotopc spn T = ± ; real ermons and antermons nteractng wth each other must have opposte sgns o 3 T. Quantum electrodynamcs n the moded representaton s nvarant under P -, C -, T -transormatons. Volatons o the ntroduced symmetry o the sotopc spn lead to the correspondng volaton o CP -nvarance. The Standard Model n the moded representaton was ormulated n the papers [], [4]. It has been shown that ormulaton o the theory n the moded representaton doesn t requre that Hggs bosons should oblgatorly nteract wth ermons to preserve the SU()- nvarance, whereas all the rest theoretcal and expermental mplcatons o the Standard model obtaned n the Drac representaton are preserved. In such a case, Hggs bosons are responsble only or the gauge nvarance o the boson sector o the ± theory and nteract only wth gauge bosons W,Z, gluons and photons.
3 In the papers mentoned above, the energy representaton o Drac matrces derved by Drac hmsel s used: σ I I α =, β γ, γ, γ γ α σ = = = =. () I I The queston arses: what changes o the Foldy-Wouthuysen transormaton orm wll result rom usng the chral representaton o Drac matrces? σ I I α =, β γ, γ, γ γ α σ = = = = () I I The chral representaton () s commonly used n the modern gauge eld theores and n the Standard Model, n partcular. To answer the queston above, rst consder the structure o equatons descrbng the components o the wave unctons ψ D ( x ) or the two representatons o Drac matrces consdered n the paper. In relatons (), () and below the system o unts wth = c = s used; x, p, A k k are 4-vectors; the nner product s taken as x y = x y x y, =,,,3, k =,,3;, k = p = ; σ are Paul matrces; α = ; ψ D ( x) s the ourcomponent wave uncton, ϕ( x), χ( x), ψ ( x), ψ ( x) are the two-component x α, = k, k =,,3 wave unctons. R The ollowng operator relatons are vald or the ree Drac equaton wth representaton (): ϕ( x) pψd( x) = ( αp+ βm) ψd( x); ψd( x) = ; χ( x) pϕ( x) = σ pχ( x) + mϕ( x) ; pχ( x) = σ pϕ( x) mχ( x) (3) χ = p + m σ pϕ; ϕ = p m σ pχ ( ) ( ) Wth representaton (), relaton (3) looks lke ψ R ( x) pψd( x) = ( αp+ βm) ψd( x); ψd( x) = ; ψ L( x) L
pψr( x) = σ pψr( x) + mψl( x) ; pψl( x) = σ pψl( x) + mψr( x) p σ p ψl( x) = ψr( x) = ( p + σ p) mψr( x) m p + σ p ψr( x) = ψl( x) = ( p σ p) mψl( x) m Relatons (4) use the operator equalty: p = E = p + m. 4 (4) Comparson between relatons (3) and (4) shows that wth the substtuton below, m σ p, β γ () these relatons transorm nto each other. The Foldy-Wouthuysen transormatons or the energy and chral representatons o Drac matrces also transorm nto each other, substtuton () s made. The Foldy-Wouthuysen transormatons or ree moton o ermons (wthout boson elds B ( x) ) look lke ( U ) m βγ σ p = + E m en ( U ) σ p γβm = + (7) E σ p chr Expresson (6) s wrtten or the energy representaton o Drac matrces; expresson (7) s the desred one descrbng the Foldy-Wouthuysen transormaton wth Drac matrces n the chral representaton; n expressons (6), (7) E = p + m. Matrces (6), (7) are untary and U α p+ βm U = βe; en en ( ) ( )( ) chr U α p+ βm U = γ E chr ( ) ( )( ) ; I boson elds B ( x) nteractng wth ermon eld ψ ( x ) are present, the desred Foldy-Wouthuysen transormaton wth Drac matrces n the chral representaton chr chr 3 ( U ) and the ermon Hamltonan ( H = γ qk + q K + q K +...; qs the couplng 3 (6) (8) (9)
constant) o the correspondng orm can be obtaned usng the algorthm descrbed n the Res.[], [] along wth substtuton (). For example, expressons or operators C and N ormng the bass or Hamltonan o nteracton n the Foldy-Wouthuysen representaton (see [], []) can be wrtten n the ollowng orm or Drac matrces n the chral representaton: chr chr even C = ( U ) qα B ( U ) = qr( B LB L) R qr( αb LαBL) R; γβ m σ p () L=, R= σ p E chr chr ( ) ( ) odd N = U qα B U = qr( LB B L) R qr( LαB αbl) R () even, odd n expressons (), () denote the even and odd parts o the correspondng operator (see [], []). Thus, the general Foldy-Wouthuysen transormaton wth Drac matrces n the chr o chr сhr сhr сhr chral representaton U = ( U ) ( + δ + δ + δ3 +...), as well as the ermon Hamltonan n the Foldy-Wouthuysen representaton chr chr chr 3 chr = γ + + + 3 +... H E qk q K q K en en can be obtaned rom the correspondng expressons or U, H wth Drac matrces n the energy representaton (see [], []) wth substtuton m σ p, β γ. Certanly, physcal results do not depend on the chosen representaton o Drac matrces. A researcher chooses a certan representaton or the sake o convenence. The Appendx below gves examples o calculatons or two quantum eletrodynamcs processes n Foldy-Wouthusyen representaton usng Drac matrces n the chral representaton.
6 Reerences. V.P.Neznamov. Physcs o Partcles and Nucle,Vol.37, N, pp.86-3, (6).. V.P.Neznamov. Voprosy Atomno Nauk I Tekhnk. Ser.: Teoretcheskaya Prkladnaya Fzka. 4. Issues -. P.4; hep-th/4. 3. L.L.Foldy and S.A.Wouthuysen, Phys.Rev 78, 9 (9). 4. V.P.Neznamov. The Standard Model n the moded Foldy-Wouthuysen representaton, hep-th/447, ().
7 Appendx Calculaton o some quantum electrodynamcs processes n representaton wth Drac matrces n the chral representaton [], []. The matrx elements and calculatons below are gven usng the notatons rom Ze. Electron scatterng n Coulomb eld s A(x) =. 4π х Feynman dagram o the process s shown n Fg.. Fgure. Electron scatterng n Coulomb eld S = d x( Ψ (x,p,s )) K A Ψ (x,p,s ) = 4 ( + ) ( + ) δ (E E ) o U s p C A p U s = < > = ( π ) Ze δ(e E ) E σ p γ βm γ βm + E + σ p = U s U s,q = p p. q ( π ) E E + σ p E + σ p E K The entry K A made or the sake o convenence actually means that wth A ( x) =. In other words, A ( ) occupes the postons determned by A K expresson (6) rom the paper []. The same s vald or the entry C A. The transton rom K A to C A has been made accordng to expresson (39) rom the paper []. x
8 Then, havng the matrx element S and usng the common methods one can obtan the derental Mott scatterng cross-secton that takes the orm o the Rutherord cross-secton n non-relatvstc case. dσ 4Z α E E σ p m + σ p m = Sp 4 + + dω q E ( E σ p )( E σ p) E ( E σ p)( E σ p ) + + + + σ p ; E p e E = E = E,p= p = p,p p = p cos θ, β =, α = ; E 4π Further transormatons gve us the derental Mott scatterng cross-secton: dσ Z α θ = β sn d 4θ Ω. 4p β sn. The sel-energy o electron n the second-order perturbaton theory. Feynman dagrams o the processes are shown n Fg.. Fgure. The electron sel-energy
4 ( ) dk ( p) = K 4 ( p; p k; ν ) K ( p k; p; ) = ν = k p k βe p k ( π) β E( p) k E( p k) 9 ( ) ( + β ) K ( p; p k; ) ν = K ( p k; p; ν = ) + ( + β ) ( ; ) + C p p k C ( p k; p) + β E( p) k E( p k) ( β ) + N ( p; p k) ( ; ) ( ) ( ) N p E p k E p k k p β + ( + ) ( + ) In vew o βψ ψ,) or a ree electron (p =m ) the rst two tems n the = ntegrand are mutually annhlatng and expressons C ( p;p )C ( p ;p), N ( p;p )N ( p ;p) have the orm C ( p;p )C ( p ;p) = E σ p m E σ p m + + σ p = E + ( E σ p)( E σ p) E + + + ( E+ σ p)( σ p) E σ p m m E + σ p m m E α α α α ( E σ p) ( E σ p) E + + ( E+ σ p) ( σ p) σ p = ( EE+ pp+ m ), E EE p = p k, E = m + ( p k ) ; N ( p;p )N ( p ;p) = E σ p γβm γβm E σ p γβm γβm + + σ p = E ( E σ p) ( E σ p) E ( E σ p) ( E σ p) + + + + E σ p γβm γβm E+ σ p γβm γβm α α α α E E + σ p E+ σ p E E+ σ p E + σ p σ p = ( EE+ pp+ m ). E EE
Hence, 4 () e d k pk+ m ( р) = 4 E( p) ( π ) k [( p k) m ] and, wth regard to spnor normalzaton, ths expresson s smlar to the expresson descrbng the mass operator n Drac representaton.