Fortschritte der Physik / Progress of Physics: Volume 32, Number 6 [Reprint 2022 ed.] 9783112656167

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 32 1984 Number 6

Board of Editors

F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J. Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J. topuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J. Zinn-Justin, Saclay

CONTENTS: E . J . FERRER, V . DE LA INCERA, and A . A . SHABAD

Covariant Decomposition and Polarizational Selection Rules for Interaction Among Photons in a Moving Medium

261-279

H . J . W . MULLER-KIRSTEN, and A. WIEDEMANN

Application of the Method of Collective Coordinates to the Quantisation of the Two-dimensional Higgs Model

281-314

AKADEMIE-VERLAG • BERLIN ISSN 0015-8208

Fortschr. Phys., Berlin 32 (1984) 6, 2 6 1 - 3 1 4

EVP 1 0 , - M

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in lenght. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the author's name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Be, Im, sin, cos, exp, ...): black underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as cG,kK,oO,p P; s 8, uU,vV,w W, xX,yY,z Z). I t will help the printer if the position of subscripts and superscripts is marked with pencil in the following way: at, b>, Mil, Please differentiate between following symbols: a, may bring some of them to the opposite side of the cut in the &0-planes. For instance, after the analytical continuation eq. (8) turns out to read —k; u) = k; u) where R and A denote retarded and advanced polarization tensors, respectively. Consider the time reflection. Under this operation the density matrix (6) transforms to

Ue T{u(), u)

= Q(U0, -U)

(9)

since P 0 and N are invariant while UP TU~ 1 = — P. (Here U is the unitary operator realizing the T-inversion and the superscript T designates the transposition operation). Equation (9) implies that under the time inversion the density matrix changes into that for the same system but moving in the opposite direction. In the rest frame, where 3

) We do not distinguish between n R and its one-particle-irreducible part n as long as properties under discrete transformations are concerned since the latter are identical for the both quantities.

1*

264

E . F E R R E R , V . D E LA INCERA, a n d A . E . STTABAD, C o v a r i a n t D e c o m p o s i t i o n

u0 = 1, u = 0, the density matrix is T-invariant. Under the time reflection the 4-current operator changes into Uj*(x0,x)

U'1 = ( - 1 )"kin);

-«)

•

(12)

In the thermodynamical region of variables where the subscript A may be dropped relation ( 1 2 ) agrees with the conclusion drawn by P E R E Z R O J A S and S H A B A D [ 5 ] who considered the thermodynamical perturbation expansion for quantum electrodynamics of electron-positron gas with the use of the density matrix (5) and established that the parity in u of any diagram is (—1)®, where s is the number of elementary electronphoton vertices (including vertices of coupling with an external field, if any) i.e. the parity in u coincides with that in the electron charge while (—1)® = (—1)". Expression (4) makes a real contribution into the effective Lagrangian A ^ x . . . , •A,,n(zn) n ^ - p j x i , ...,%„; u)

(13)

(where A^Xi) are real vector-potentials) in the region of momenta, in which the relation »W./JAi,

...,k„;u)

=

: u ) = n wi( k > —k;u).

(15)

An important property is the four-dimensional transversality of the polarizability tensors (Fradkin, 1965) •••> k i i ) >.... * ( B ) ;«) = 0,

i — 1,2,...,%

(16)

which provides invariability of the contribution (13) under the gauge transformation of potentials Ap(x) Ap(x) + 8l8xltA(x). Finally, the generalized Furry theorem (charge invariance) requires [4] that all the amplitudes (7) with odd n vanish in the limit of zero chemical potential, i.e. for the charge-symmetric case, in particular, in the vacuum. Let us turn to building the tensors of multiphoton amplitudes (5) in the Minkowskian space using the photon 4-momentum vectors the medium 4-velocity vector Up and the unit (metric) tensor gM,. To satisfy condition (16) we shall build the necessary tensors as products of transverse vectors and, if necessary, of transverse modification of the unit tensor.

3.

Spectra and Polarizations of Linear Waves

We begin with the tensor of the second rank, i.e. the forward scattering amplitude or the polarization operator which is responsible for the ordinary linear polarizability and is the kernel of the equation of the photon propagator. The tensor of rank two depends on one momentum kp. The only transverse vector, which can be built of the characteristic vectors of the problem, is dp = [(uk) kp - k2Up]l[k2(k2 -

(uk)2)]112,

(ak) = 0 .

(17)

(It is normalized to the unity a 2 = 1). In the two-dimensional space orthogonal to a,p and to kp, one may choose arbitrarily one basic vector 6M: (bk) = (6a) = 0, 62 = 1, then the second basic vector is defined as the "vector product" in the four-dimensional space dp = Spvxybvaiky(k2y112. Here eM,xy is the fully antisymmetric unit pseudo-tensor of rank four, (db) = (da) = (dk) = 0; d2 = 1. The vectors kp, ap, bp, dp form an orthonormalized basis in the Minkowskian space. Apart from the tensor . Designate this set by V 2 ) ,V 2 ) >? (/(i-3))2 p,q,S= 1,2 (36) = (/(«•*>)2 (d«) • (&(1,2))2 according to the rule jfc(i.2)|2 (¿0)2. Finally, the remaining and newly appeared fc0 b™ is obtained from (29) by the substitution k' 1 ' -> k, k jfci1», b'J', i = 1, 3, 4 , . . . , n is obtained from (29) by the replacement if 1 ) &. The third transverse vector h^ (instead of d c a n be formed as follows: h ^ is obtained from b^ by the substitution ¿(2) _> ¿(3); the rest of the vectors except for i = 3, are obtained from by the substitution A*1' ; finally, the vector hff is obtained from hi2} by the interchange ¿(2) ->¿(3); ¿(3) The vectors formed in this way are transverse a^k'j} = b^k^ = AJ/'AJ," = 0 although not orthogonal at each i). The polarizability tensor of w-th rank (n ^ 4) has the decomposition » « . . . „ ( i « , • •* ( E w) " \i=1 /

(48)

3) where IT p. 1= 1, 72, ?3, J c' 1(| ' =( «:, c'2)ft{= fe'*', lit'c' ft = H 7 and the invariant functions w niPl Pn) depend on all possible scalars (k^k^), (uk ), i,'j = 1 , . . . , » . As distinct from Sees. 3, 4, we did not make decomposition (48) orthogonal, lest the form of the basic vectors become too cumbersome. However, in some important cases the aforesaid becomes invalid and we may face again the situation of Sec. 4. This takes place when we consider the interaction of photons taken from one or more (non-more than three) classical (i.e. multiphoton) electromagnetic waves. All the photons taken from one wave have the same 4-momentum. Accordingly we have no more than two independent 4-momenta, say, k ^ and kpl2\ all the rest being formed as their linear combination with rational coefficients. Now we must again use the pseudovector, and it must enter even (including zero) number of times in the tensor of any rank. The use of the pseudovector is also necessary 'for building the four-photon vertex (light-light scattering) in the vacuum, since in this case the vector u^ is absent. The set of three orthogonal, transverse vectors in the first sector (fa) can be obtained from

273

Fortschr. Phys. 82 (1984) 6

(28), (29), (30) by the replacement Up -> k ^ = — (V X ) + V + V 3 ) ) w e should restore preliminarily u2 in (29) before (fcW&f2)) within the second brackets and before (if1*)2 within the third brackets). In the second and third sectors the desired sets are obtained in a similar way from (32), (33), b™ and are obtained from (28), (29), (30) by the replacement uM &(3>. In this case all the vectors d ^ are proportional to = e ^ i y k / ^ k x ^ k y ^ K In the vacuum on the mass shell ( k ) = 0 the vectors a ^ become proportional to k ^ , the field strengthes corresponding to such polarizations vanish (the absence of longitudinal photons) and therefore the vectors a ^ should not be used in forming the four-photon vertex whereas the vectors dM (and, consequently, bp) necessarily enter even-fold. In this way the eight structures are formed: d d d d , d d b b , d b d b , d b b d , b d d b , b d b d , b b d d , bbbb, whose coefficients are not independent, but restricted by the conditions (7), (12). 1

6.

2

Polarizational Selection Rules for Photons

The requirement that the pseudovector dp should appear in the photon vertex only even number of times results in the polarizational selection rules. For the processes in which three photons are involved, i.e. splitting of a photon into two and junction of two photons into one, the probability amplitudes are determined by the coefficients of the decomposition (34), when polarizations of the incoming and outgoing photons are given by the corresponding 4-vectors a,p, bp, dp. Imagine, that in asymptotically remote regions the medium is linear and that the non-linearity responsible for the interaction is switched on as photons draw together. This corresponds to the traditional formulation of the scattering problem in quantum field theory. We shall characterize the initial and final asymptotical states of each photon by the 4-polarization vectors (28), (29), (30), (32), (33), which are eigenvectors of the polarization operator (19). By doing this we fix the arbitrariness in the choice of the polarization state of the asymptotic photon (caused by the degeneracy discussed in Section 3) in a way appropriate for considering the interaction between the photons: the direction in the subspace formed by the vectors bp, dp of Section 3 is specialized by the 4-momentum of the photon to interact with the one whose state is being fixed. The plane, where the three wave vectors of the photons = — (fc +fc) lie will be refered to as the reaction plane. At first we consider the case when the medium moves along the reaction plane. In particular, this situation includes also the case of resting medium. According to (30) and the equality dp = dpi1) = —dpi2) = —dp we have d0 = 0 (since the 3-vectors it,fc*1*,feare coplanar, by the convention) and, as has been considered in detail in Sec. 3, each of the three photons with the polarization? c^1-2-3) has its electric vector (23) orthogonal to the planes (tt, fc(1))> (u, fc(2>), (u,fc = —&0(i)u + fe(i)w0 in the t'-th plane (u, fc(i)) each and polarized as eb(i) =fc60 _L + II •«-»• JL + J_ etc. are prohibited. In the general case the selection rules also exist but their geometry is more complicated. As follows from (48), the selection rules under consideration are not extended to w-photon processes with n > 3 in a medium and with n > 4 in the vacuum (except for degenerate situations to be discussed below in Sec. 7), since in this case there exists a third independent vector, and it is no longer necessary to use the pseudovector dp. For the same reason the selection rules established are fulfilled only to the extent to which the medium may actually be considered homogeneous, since other wise the equality &(3> + &C1) + = 0 does not hold and the momentum of the third photon becomes independent. It is clear from the abovesaid that the selection rules obtained are the manifestation of the T and P invariance. They can be violated for a gas with an admixture of the weak interaction among its particles. 7.

Polarizational Selection Rules for Intense Waves6)

The conclusions drawn in the previous section for the elementary three-quantum processes can be applied to the three and two-wave processes described classically. Consider the process in which two monochromatic, parallely or perpendicularly polarized waves with the 4-momenta ¿t1', and given amplitudes ^ ' ^ and A ^ , respectively, interact to form the third wave with the amplitude A^3). Imagine that each n i n eq. (1) enters with the small coefficient e". Then, bearing in mind that the nonlinear terms of eq. (1) should be switched off at infinity one may be looking for solution of equation (1) as the power series expansion OO

(50)

4.(3) = I e»an(3) + A„( 1) + A„(2) with AM( 1) and AM(2) satisfying the linear equation : [(¿G2 9W* 6

8

M

A x x

" ( i) + /

x

*)

d*x2 = 0.

(51)

) In this Section we are dealing for simplicity with the case when the medium moves in the reaction plane or is at rest.

Fortschr. Phys. 32 (1984) 6

275

Substituting (50) into (1) and using (51) one sees that an is represented by diagramms like the one shown in Fig. 1, where the triangle r^...^ represents (generally, onephoton-reducible) vertices of the » t h order that may be formed by using the Kernel i • /