Fortschritte der Physik / Progress of Physics: Band 28, Heft 2 1980 [Reprint 2021 ed.] 9783112522820, 9783112522813

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 2 • 1980 • B A N D 28

A K A D E M I E

- V E R L A EVP lo,— M 31728

G

•

B E R L I N

BEZÜGSMÖGl.ICHKEIl

EN

Bestellungen sind zu richten — in der D D R an das Zeitungsvertriebsamt, an eine Buchhandlung -:(x) and AM(x): l(x)=l(Ali(x),V(x)].

(2.1)

This should be invariant under the local gauge transformation A^x)

A^x)

+

(2.2a) (2.2b) (2.3)

Xp —

Xif V

^3) —

i

xP = Q** XV = ( — X0, Xlf X2, X3),

(2.4a) (2.4b) (2.5a) (2.5b)

where gf v and g^,, are the metric tensors (2.6) 4*

70

H . MATSUMOTO, G . S E M E N O F F , M . T A C H I K I a n d H . U M E Z A W A

We use the summation convention where repeated Greek indices are summed over 0, 1, 2, 3 and Roman indices over 1, 2, 3. We also employ the systems of units where h = c= 1. The Lagrangian has the form [23] *(«) = -"J" V '

+

(0, + ieA,) y>(x), F„v(x))

(2.7)

where I n the parabolic band model which we study in the following sections, J%(x) is given by jf4 = ift(8° + ieA°) y — -L-[(p

- iel) xp\P + ie2) y> -

¿¡ytyty]

—eneA0 — V(y>, yt, F^).

(2.8)

I n the F-term, ip and yfi do not carry a n y derivatives. I n (2.8), the constant (—ene) is the positive ion charge density. The electromagnetic interaction in the first term is called the minimal electromagnetic interaction, while the electromagnetic interaction in the F-term is called the magnetic interaction. A complication in quantum electrodynamics arises from the fact t h a t 1 in (2.7) does not supply us with any canonical conjugate of A0. To take advantage of the canonical formalism we follow the method used in refs. [3, 9,10]. We thus introduce a supplementary field B(x) which acts as the canonical conjugate of A0. To do this we need the presence of the B(x) 80A0-teTm in the Lagrangian. Such an additional term leads to the unreasonable gauge condition 80A0 = 0. We therefore introduce a more flexible form of the additional Lagrangian which contains some unknown derivative operators; these derivative operators will be determined self-consistently. We thus modify the Lagrangian as follows: Xn(x) = Ie(x) + B(x) Dh{8) A"(x) - ±-£(x)

a(-ii?)

B(x).

(2.9)

To make B{x) canonical conjugate of Ag, D0(8) should be simply the time derivative. We write Df(8) as =

(2.10)

We do not permit «(—iV) to contain time derivatives, because we do not want to upset the simple canonical relation between A0 and B. Note that any derivative operator is defined through its Fourier form. For example, v\~iV)

etf* = v2(p) e*?*.

(2.11)

The Lagrangian (2.9) leads to D„{8) Ao{x) =