Field strength for graded Yang–Mills theory
The graded field strength is defined for osp(2/1;C) non-degenerate gauge algebra. We show that a pair of Grassman odd scalar fields find their place as a constituent part of the graded gauge potential on the equal footing with an ordinary (Grassman even) one-form taking values in the proper Lie suba...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | Field strength for graded Yang–Mills theory / K. Ilyenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 74-75. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-794272015-04-02T03:02:00Z Field strength for graded Yang–Mills theory Ilyenko, K. Quantum field theory The graded field strength is defined for osp(2/1;C) non-degenerate gauge algebra. We show that a pair of Grassman odd scalar fields find their place as a constituent part of the graded gauge potential on the equal footing with an ordinary (Grassman even) one-form taking values in the proper Lie subalgebra, su(2), of the graded Lie algebra. Some possibilities of constructing a meaningful variational principle are discussed. 2001 Article Field strength for graded Yang–Mills theory / K. Ilyenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 74-75. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 11.10.Ef, 11.15.-q http://dspace.nbuv.gov.ua/handle/123456789/79427 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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The graded field strength is defined for osp(2/1;C) non-degenerate gauge algebra. We show that a pair of Grassman odd scalar fields find their place as a constituent part of the graded gauge potential on the equal footing with an ordinary (Grassman even) one-form taking values in the proper Lie subalgebra, su(2), of the graded Lie algebra. Some possibilities of constructing a meaningful variational principle are discussed. |
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Field strength for graded Yang–Mills theory |
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Field strength for graded Yang–Mills theory |
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Field strength for graded Yang–Mills theory |
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Field strength for graded Yang–Mills theory |
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field strength for graded yang–mills theory |
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Field strength for graded Yang–Mills theory / K. Ilyenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 74-75. — Бібліогр.: 8 назв. — англ. |
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Вопросы атомной науки и техники |
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FIELD STRENGTH FOR GRADED YANG–MILLS THEORY
K. Ilyenko
Institute for Radiophysics and Electronics, NASU
12 Ak. Proskura St., Kharkiv - 61085, Ukraine
The graded field strength is defined for osp(2/1;C) non-degenerate gauge algebra. We show that a pair of Grass-
man odd scalar fields find their place as a constituent part of the graded gauge potential on the equal footing with an
ordinary (Grassman even) one-form taking values in the proper Lie subalgebra, su(2), of the graded Lie algebra.
Some possibilities of constructing a meaningful variational principle are discussed.
PACS: 11.10.Ef, 11.15.-q
1. INTRODUCTION
As well-known the number of gauge bosons of a the-
ory is given by the number of generators of the corre-
sponding gauge Lie algebra. The requirement of being a
Lie algebra stems from the reality of the action function-
al which follows from the properties of the Fermat prin-
ciple in optics and the Feynman path integral. In particu-
lar, this leads to unitary evolution of a system with such
an action functional. Nevertheless, no one is restricted
from looking for not necessarily of Lie-type algebras as
the gauge algebras provided they are physically mean-
ingful.
2. GRADING OF GAUGE ALGEBRA
Such an algebra is advocated in the current contribu-
tion. This is a graded extension of su(2) gauge algebra
by a pair of odd generators, Aτ , which anticommute
with one another and commute with the three even gen-
erators, aT , of su(2). We use the square brackets to de-
note both commutation and anticommutation operations
of the generators with understanding of their proper us-
age. The defining relations have the following form, [1-3]:
cabcba TiTT ε=],[ , B
B
AaAaT τγτ )(],[ 2
1= ,
aABa
i
BA T)(],[ 2 γττ −= (1)
Lowercase Roman indices run from 1 to 3; uppercase
Roman indices run over 1, 2; BC
C
AaABa є)()( γγ = ; abcε (
123ε = 1) and ABє ( 12є = 1) are the Levi-Civita totally
antisymmetric symbols in three and two dimensions; the
Clifford matrices B
Aa )(γ and ABє are given by
( ) ( ) ( )[ ];)(
0
0
,
01
10
,
10
01
i
iB
Aa −−
=γ
( ).
01
10
−
=ABє (2)
In the adjoint representation the matrices aT and Aτ can
be written as follows:
,
210000
021000
0000
0000
00000
1
−
−
= i
i
T
,
021000
210000
0000
00000
0000
2
−= i
i
T
.
021000
210000
00000
0000
0000
3
−
−
=
i
i
T (3)
The non-degenerate super Killing form, ),( βα TTB , is
defined by
==
AB
ab
єi
TTstrTTB
0
0
)(),(
δ
βαβα , (4)
where the supertrace operation is adopted from [4] and
the Greek indices run over the whole set of the graded
Lie algebra generators. It turns out that all of the genera-
tors are grade star Hermitian: the even ones being just
Hermitian in an ordinary sense while the odd generators
obey more complicated relations (cf. [5,6]). We assign a
degree, αTdeg , 0 to the even and 1 to the odd genera-
tors.
3. THE GAUGE POTENTIAL
Given an su(N) Lie algebra one defines a gauge po-
tential, which takes values in the algebra, by introducing
nN ×− )1( 2 one-forms µ
µ dxxAa )( , n being the dimen-
sions of space-time, and subtracting them with the alge-
bra generators µ
µ dxxAT a
a )( . Note that from the stand-
point of graded Lie algebra we have a composite object
of the degree (0,1), the first position shows that genera-
tor aT is an even element of the algebra and the second
position corresponds to the fact that µ
µ dxxAa )( is a one-
form of the algebra of exterior forms on Mn. This has a
suggestive generalization to the case when one has the
degree 1 odd part of a gauge algebra: (s)he needs to con-
74 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 74-75.
struct the homogeneous compliment of the expression
above, namely, the element of degree (1,0). It has the
form )(xA
AΦτ and must be added to the element of de-
gree (0,1) to form the complete graded gauge potential
A (x) = )(xA
AΦτ + (5)
homogeneous = odd⊗even + even⊗odd
Here )(xAΦ are zero-forms on Mn. Thus one obtains a
proper element A (x) of the degree one of the direct
product of two graded algebras (compare with [7],
p. 629).
4. GRADED FIELD STRENGTH
The graded field strength, F (x), is defined by means
of exterior derivative of the graded gauge potential,
A (x), and adding its wedge product with itself,
A (x) ∧ A (x):
F (x) = dA (x) + (ig/2)A (x) ∧ A (x). (6)
Here we must clarify the meaning of both operations.
As to the first term on the right-hand side of (6) we
obtain
dA (x) = )(xd A
A Φτ + νµ
µ ν dxdxxFT a
a ∧)(!2
1 . (7)
Here aF = )(!2
1 cbabcaa AAgAA νµµννµ ε−∂−∂ νµ dxdx ∧ is
just the Yang-Mills field strength.
The second term on the right-hand side of (6) needs
more explanations. Firstly, as in the case with Yang-
Mills theory, one defines the wedge product for algebra
valued forms A (x). Secondly, we have to deal with the
odd part of gauge algebra and zero-forms involved in
the graded gauge potential. Thus we define, [7, p. 632],
A ∧ A = (Tα⊗Aα) ∧ (Tβ⊗Aβ)
≡ βα
βα
β
α
AATTTA ∧⊗− • ],[)1( degdeg . (8)
Here Aα comprises the whole set of fields {Φ A, aA }
and the deg Aα means the degree in the exterior algebra
of differential forms on Mn. One can easily see that this
straight generalization is bilinear, goes into usual Yang-
Mills result for ordinary Lie algebras and makes possi-
ble the graded Jacobi identity for the direct product of
two graded algebras: the exterior and matrix ones.
Finally, we unite (7) and (8) in one expression:
F = ])()!21([ BA
AB
aa
a
A
A FT ΦΦγ++ϕτ , (9)
where aF is defined after formula (7), and
)()()()()( !2
1 xxAxdx BaA
Ba
AA ΦγΦϕ += . (10)
We intend to preserve the graded property of gauge po-
tential in the expression for the field strength. The first
summand in (9) is of the degree (1,1), the expression
a
a FT is of the degree (0,2), while the expression
BA
AB
a
aT ΦΦγ )( falls out of this pattern. Allowing for
the symmetric property of the Clifford matrices
BA
a
AB
a )()( γγ = , we are forced to assume the functions
)(xAΦ as being Grassman-valued odd variables, which
obey the property ABBA ΦΦΦΦ −= . Then, equation (9)
takes the form
F (x)= )()( xFTx a
a
A
A +ϕτ . (11)
Thus, we have built a homogeneous expression of de-
gree two representing the field strength.
5. DISCUSSION AND OUTLOOK
The very possibility to define the graded four-poten-
tial and field strength opens up a number of further
questions.
Firstly, a problem of graded gauge invariance arises.
We hope that existing in the literature (see [8]) introduc-
tion of Grassman-odd parameters of transformations for
odd generators of the algebra could provide a sensible
solution to this problem.
Secondly, as mentioned at the beginning of the cur-
rent contribution, one is interested in a definition of a
real-valued Lagrangian density. This would lead to the
physically meaningful Euler-Lagrange equations. In par-
ticular, we are going to consider an invariant with re-
spect to the gauge algebra automorphisms expression:
S= ∫ str (F ∧*F ) (12)
Thus we intend to undertake an extensive research on
the opportunities, which follow from the given develop-
ment.
ACKNOWLEDGMENTS
The author would like to thank Yu.P. Stepanovsky and
V. Pidstrigach for many helpful discussions.
REFERENCES
1. V. Kac. Representations of classical Lie superalge-
bras. Differentiat Geometrical Methods in Mathe-
matical Physics II, edited by K. Bleuler, H.R. Petry
and A. Reete. Lecture Notes in Mathematics. 1977,
676 p.
2. J.W. Hughes. Representations of osp(2,1) and the
metaplectic representation // J. Math. Phys. 1981,
v. 22, p. 245-250.
3. R. Brooks, A. Lue. The monopole equations in topo-
logical Yang-Mills // J. Math. Phys. 1996, v. 37,
p. 1100-1105.
4. J.F. Cornwell. Group Theory in Physics, part 3, L.,
1989, 628 p.
5. M. Scheunert, W. Nahm, Y. Rittenberg. Graded Lie
algebras:generalization of Hermitian representations
// J. Math. Phys. 1977, v. 18, p. 146-154.
6. M. Scheunert, W. Nahm, Y. Rittenberg. Irreducible
representations of the osp(2,1) and spl(2,1) graded
Lie algebras // J. Math. Phys. 1977, v. 18, p. 155-
162.
7. S. Lang. Algebra. 3-nd edition, R., 1985, 829 p.
8. V.I. Ogievetskii, L. Mizinchesku. Symmetries be-
tween boson and fermion superfields // Uspekhi Fiz.
Nauk. 1975, v. 117, p. 637-683 (in Russian).
K. Ilyenko
Institute for Radiophysics and Electronics, NASU
12 Ak. Proskura St., Kharkiv - 61085, Ukraine
PACS: 11.10.Ef, 11.15.-q
1. INTRODUCTION
2. GRADING OF GAUGE ALGEBRA
3. THE GAUGE POTENTIAL
4. GRADED FIELD STRENGTH
5. DISCUSSION AND OUTLOOK
ACKNOWLEDGMENTS
REFERENCES
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