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Journal ArticleDOI

Ghost-creating gauges in Yang-Mills theory

01 Jan 1985-Nuclear Physics (North-Holland)-Vol. 261, Iss: 1, pp 55-65

AbstractWe study the ghost-antighost symmetry of the extended BRS equations, discuss the geometrical interpretation of the formalism and define a new class of gauges in which the ghost number is only conserved modulo two.

Summary (1 min read)

5. CURCI FERRARI GAUGES

  • In their formalism, the authors may understand the breaking of unitarity algebraically.
  • In the usual construction, the physical states are defined as cohomology classes of the BRS operator:.
  • It follows that the BRS operator can no longer be used to remove the longitudinal degrees of freedom and the formalism collapses.

CONCLUSION

  • The new gauges considered in this note provide a systematic generalization of several earlier studies [2, 7, 10, 21] .
  • Then, abandoning ghost conservation, the authors have constructed a Feynman gauge where ghost and antighosts pairs are emitted by longitudinal photons, the Sp(2) symmetry is explicit and the 4-ghost interaction is absent.
  • The ghost creating gauges, and the relation to the geometry of the Lie groups, are certainly amusing and curious.

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Lawrence Berkeley National Laboratory
Recent Work
Title
GHOST CREATING GAUGES IN YANG-MILLS THEORY
Permalink
https://escholarship.org/uc/item/2b1194jt
Author
Thierry-Mieg, J.
Publication Date
1985
eScholarship.org Powered by the California Digital Library
University of California

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t
LBL-19049
Preprint
('.
~
Lawrence
Berkeley
Laboratory
UNIVERSITY
OF
CALIFORNIA
Physics Division
Submitted
for
publication
LA\VR;::NCE
e:::;:?l(FLF.v !
.AEv~r.;o!lTnRY
MAY
1 6
1985
LiBRARY AND
DOCUMENTS SE.CTION
GHOST
CREATING
GAUGES
IN
YANG-MILLS
THEORY
J.
Thierry-Mieg
January
1985
TWO-WEEK
LOA
Prepared
for
the U.S. Department
of
Energy under Contract DE-AC03-76SF00098

DISCLAIMER
This document was prepared
as
an account
of
work sponsored
by
the United States
Government. While this document is believed to contain correct information, neither the
United States Government nor any agency thereof, nor the Regents
of
the University of
California, nor any
of
their employees, makes any warranty, express or implied, or
assumes any legal responsibility for the accuracy, completeness, or usefulness
of
any
information, apparatus, product, or process disclosed, or represents that its use would not
infringe privately owned rights. Reference herein to any specific commercial product,
process, or service
by
its trade name, trademark, manufacturer, or otherwise, does not
necessarily constitute or imply its endorsement, recommendation, or favoring by the
United States Government or any agency thereof, or the Regents
of
the University
of
California. The views and opinions
of
authors expressed herein do not necessarily state or
reflect those
of
the United States Government or any agency thereof or the Regents
of
the
University
of
California.

January
1985
Ie
LBL-19049
GHOST
CREATING
GAUGES
IN
YANG-MILLS
THEORy
1
By
Jean
Thierry-Mieg
2
CNRS-Observatoire
de Meudon,
France
and
Lawrence
Berkeley
Laboratory
University
of
California
Berkeley,
Calfiornia
94720, U.S.A.
Abstract
We
study
the
ghost
antighost
symmetry
of
the
extended
BRS
equations,
discuss
the
geometrical
interpretation
of
the
formalism
and
define
a
new
class
of
gauges
in
which
the
ghost
number
is
only
conserved
modulo
two.
lThis
work was
supported
by
the
Director,
Office
of
Energy
Research
Office
of
High
Energy
and
Nuclear
Physics,
Division
of
High Energy
Physics
of
the
U.S.
Department
of
Energy
under
Contract
DE-AC03-
76SF00098.
2
Participating
Guest
at
Lawrence
Berkeley
Laboratory.
,(
- 2 -
INTRODUCTION
The
extended
BRS
equations
[1-4]
which
govern
the
unitarity
[1,5]
and
renormalizability
[1,2,6,7]
of
Yang-Mills
theory
are
Sp(2)
symmetric
in
the
ghost
antighost
fields
[8,9,10].
On
the
other
hand,
the
fami-
liar
Faddeev
POPJv
gauges
[11]
break
this
symmetry and
the
antighost
plays
no
role
[12]
in
the
classification
of
anomalies
[13,14,15].
The
purpose
of
this
note
is
to
analyze
this
situation
in
some
de-
tail,
and
to
emphasize
the
geometrical
interpretation
of
the
formalism
In
the
first
section,
we
shall
study
the
Sp(2»)Q
BRS
semi-direct
algebra
and
its
eventual
decontraction
to
OSp(I/2).
In
section
2,
we
discuss
the
Curci
Ferrari
gauges
[2]
which
allow
for
a
controlled
breaking
of
the
ghost-antighost
symmetry
[7]
and
re-
late
them
to
the
one
parameter
family
of
parallel
transports
that
Cartan
[16]
has
defined"'on
a
Lie
group.
In
section
3,
we
generalize
these
gauges
by
allowing
the
creation
of
ghost
pairs.
The
theory
remains
unitary
and
renormalizable.
Nevertheless,
these
gauges
offer
the
possibility
of
occurence,
in
per-
turbation
theory,
of
anomalies
with
ghost
number 3
considered
recently
by Faddeev [17] and Zumino
[18].
In
the
last
section
we
show
that
the
Osp(I/2)
group
decontraction
discussed
in
the
first
section
leads
to
the
massive
Curci-Ferrari
gauges
[2]
and
we
explain
algebraically
why
these
gauges
break
unitarity.
This
work
is
a complement
to
our
earlier
detailed
study
with

- 3 -
Laurent
Baulieu
of
the
renormalizability
of
the
extended
BRS
invariance
entitled
"The
principle
of
BRS
symmetry" [7] and
we
shall
use
the
same
notations.
It
can
however be
read
independently.
1.
THE
Sp(2)
>4
BRS
ALGEBRA
Let
A:
denote
the
Yang-Mills
field.
ca
the
scalar
anticommuting
Faddeev
Popov
ghost.
and
~a
the
antighost.
all
valued
in
the
adjoint
representation
of
the
Lie
group
G.
The
extended
BRS
equations
[2.3,4,7]
can
be
written:
s'\
-
DIJc
S,\
-
Di
sc
- -
t
[c.c]
sc
- -
[e.e]
se
+
sc
+
[c.e]
- 0
Let
us
define
a composite
connection
form
A - A dx
V
+ c + e
IJ
and a
composite
differential
operator:
a - dxiJa + s + J
IJ
The
BRS
equations
(1.1)
can be
rewritten
as
Maurer
Cartan
equations
[ 19 ]
. where
" - F
F
..
d A +
F - d A +
[A.A]
[A.A]
- t
(a
A - a A +
[A
.A
])
dxIJdx
V
IJ
v v
IJ
IJ
v
",-
(l.l
)
(1.
2a)
(1.
2b)
(1.
3)
- 4 -
To
define
se
in
equ.
(1.1).
one
introduces
an
auxiliary
field
b
a
of
dimension 2
[5.6]
such
that
[20]:
se
- b -
[e.c]
sc-
-b
-
[e.c]
Using
(1.1).
one
may
verify
that
s2
'\
-
(ss
+
ss)
AIJ
-
s2,\
- 0
s
2
c -
s2e
- 0
Imposing
s2 -
ss
+
ss
-
s2
- 0
on
all
fields
yields
the
variation
of
b
[20]:
sb
- - 1
[c.b]
- 1
[[c.c].e]
2 8
sb
= - t
[e,b]
+ i
[[e,e],c]
(
1.4)
(1.
5)
(1.6)
0.7)
These
well
known
equations
are
explicitly
symmetric
under
the
exchange
of
c and
e,
sand
s and
the
reversal
of
the
sign
of
b.
It
is
often
more
convenient
[4-7]
to
define
the
variable
b'
- b - t
[e.c]
and
rewrite
the
auxiliary
equations
1.4
-
1.7
as
se
-
b'
BC
-
b'
-
[c,e]
(1.
4a)
sb'
- 0
sb'
[e.b'
]
(1.
7
a)
However.
the
ce
symmetry
of
the
equations
is
less
manifest.
....
-

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The authors study the ghost antighost symmetry of the extended BRS equations, discuss the geometrical interpretation of the formalism and define a new class of gauges in which the ghost number is only conserved modulo two. LThis work was supported by the Director, Office of Energy Research Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U. S. Department of Energy under Contract DE-AC0376SF00098.