Fortschritte der Physik / Progress of Physics: Volume 34, Number 10 [Reprint 2022 ed.] 9783112613825, 9783112613818

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ISSN 0015 - 8208

Fortechr. Phye., Berlin 84 (1986) 10, 649-686

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 length. 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. 6. The titel of the paper should be followed by the authors 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, H e , . . . , n, p , . . . ) , elementary mathematical functions like Re, Im, sin, cos, e x p , . . . ) : black underlined Greek letters: red underlined Boldface Greek letters: red interlined 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 cC,kK,oO,pP,sS,uUfvV,v>W,xX,yY,z Z). I t will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, 6», Mil, Mij, Wnf Please differentiate between following symbols: a, ))>

y

(-^'t ~

z

*>)'>

=

2"

Zi

>)

then we get the standard form of the commutation relations of sp (2N) [cÂij, cAkl] = i(djkd.u [c/¿a,

=

[cAn, J„]

= i(ôikâu

— duu] = iô ABI^RT^SU + ^ST^RU + ÔruNgt + - Zkl] = l ip z

+ àuXik — SuXjk

ki\ =

z

—i{ô,kPji + ànXik

— ôuXjk

[Xii> Zkl\ =

—i{ôikZn

+ âjlZik — àilZjk —

àjkZu)

[Xii> ki] =

—i(àikZji

+ ôjiZ i k — (5 tl Z ik —

ôikZu)

[X-ij, ki\ =

—i(àik ji — ànZik

z

z

Z

[Xij> Zki] = i{ôikzn

— USV(2K)=,U(K)ZDO{K-R)

Table 3.B. Lie algebra dimension

Hermitean oscillator-like realization of generators

u.(2 K)

Xaßij

4 K2

X-aßij = SaiSß) ~t~ SßjStxi

usp ( 2 K )

= X

+ 1)

K{2K

M

ij

u (K) 2

K o

— i(SZi£pi

—

Sßfiai)

•

M

= -»("a)«/»

\ j = (ai)«ß

x

=

*ßij

Xij

=

M1j;

Xij

=

M\j

Xij

=

Mjj;

Xn

=

M\j

Xti

= Mij;

X^

=

Xaßij

(«3hß

Xaßij

X i j ^ M f j

(K;R) M\j

« , 0 = 1,2;

M = l, . .., K

This table is analogous to the Table 2.C. for the bosonic case. We would like to notice, t h a t by assumption we considered only the bilinears of the creation and annihilation operators. Hence, our realization of o (2N; R) is a reducible representation in the Fock space of the oscillator states. I t can be decomposed into two irreducible representations acting on the states with even or odd number of fermions. Both chain of inclusions one can immediately apply in atomic spectroscopy in order to investigate the structure of the atomic shells [22, 23]. The same group-theoretical tools are used for the analysis of nuclear specta [24, 25].

4.

Hermitean Realizations of Classical Superalgebras in Terms of Boson-Fermion Creation and Annihilation Operators

Our considerations of this Chapter we begin recalling the most important facts (i)

For given M pairs of the bosonic operators a f , a} [ai,a,j+]

(ii)

=

dij

(4.1.1)

i , j = i , . . . , M

their bilinears form the symplectic algebra sp (2M; C). For given N pairs of the fermionic operators £ m + , {fm, £n+}

=

dmn

171,71=1,

(4.1.2)

...,N T

their bilinears form the orthogonal algebra o (2A ; C). Introducing boson-fermion bilinears of the form (4.1.3)

Fortschr. Phys. 34 (1986) 10

665

one can extend both algebras, under graded commutation relations to the orthosymplectic superalgebra (Z 2 -graded algebra) osp (2N ; 2M\ C). The even sector (bosonic sector) of this superalgebra is a direct sum sp (2M ; C) © o (2M ; C) and the odd sector (fermionic sector) is given by the C-linear combinations of the boson-fermion bilinears (4.1.3). The superalgebra osp (2N ; 2M ; C) is the most general one obtained using bilinears of the creation and annihilation operators of all kinds. (a) The superalgebra osp (2N ; 2 M ; R ) The largest hermitean sub-superalgebra of osp (2N; 2M; C) is of course, the real orthosymplectic algebra osp (2 N ; 2 M, R) with the bosonic sector sp (2 M ;R) © o (2 N ; R). We choose its generators in analogy to (2.5) and (3.5), namely — the bosonic sector sp (2M ; R) : i(ai+aj

X{j = X^

=

a^cij

— a,+a,-)

Z

+

«¿a,-

+

i }

=

i(a ¡a

Zij =

«j«,-

—

a ^ a f ) +

«¡ a?-+

+

(4.2.1)

1,

i, j —

...,

M

— the bosonic sector o (2N; R) mn

mn

Ymn

'

^mn

£m =

^mn

n )

(4.2.2)

£m£n

^ ^

• • •:

^

— the fermionic sector Qim

=

i{a>i£m+

Qirn

~

a

— «i+£m)

i%m+

$im

a +

i £m

= =

&im

»(«¿fm a

»¡+£m+)

~ ~l~,ai+£m+

i$m

(4.2.3)

•

The choice of the generators of fermionic sector (supercharges) to be hermitean operators is not necessary but it is convenient. The generators of sp (2M ; R) and o (2N ; R) satisfy the commutation relations given by (2.6) and (3.6) respectively. The remaining graded commutation relations look as follows — the anticommutation relations : {Qim:

Qjn}

{Qim:

Sjn)

Qjn}

{Qim:

Qjn]

{Sim:

^ jn)

^mn^-ij ^mn-^-ij

Qim:

Sjn) —

^mn-^-ij

ij

S/ni

ij

H

^mn^-ij

{&im: Sjn}

{Qim:

^ij^mn -

^if^mn ÔjiTmn

(4.3.1)

^ij^mn

^ij^mn "f"

(4.3.2)

^ij^mn

{Qim: {Qim:

' ^jn)

^mn^ij

~

^ij^mn

^mn^ij

(4.3.3)

mn

— the covariance relations :

2

V^ij:

Qkn\

=

i(&kjQin

V^ij:

Qkn]

=

—¿(^ikQjn

Y^ij:

Qkn]

=

—i{^ikQjn

+

mn

L iumn xT1 r Jf

(A

^y

12 2^

— the sector u (1):

C=

WTrl+^Tr?

(4-12'3)

— the fermionic sector we choose as follows (see (4.10)): Q\im'i

i

Qlim

$2 im'i

(4.12.4)

$2im-

Using the relations (4.11) we get {Qlimi Qljn)

{$2imi

jn)

{Qlimt $2jn} {Qlimt

=

=

{Qlim> Qljn)

~

{Q\imt Qljn!

=

{^2,'TEI S2jn)

~

{$2imy ^2jn)

=

^mn^l2ij

=

$2jn)

r^ mn

^mnX-aij

-f-

dmnX22ij

(4.13.1)

dijYmn ^ij^mn

(4.13.2)

^ij^mn {Qlimi

&mn^l2ij

=

^mn-^-Wij

$2jn}

{Qlimt

^mn^\2ij

=

&2jn}

=

^mn^l2ii

(4.13.3) •

In order to simplify the formulas for graded commutation relations, it is useful to introduce the following notation : — the sector su (m, M) : Xab

=

(X'Uij,

Z'liij,

X