Some General Results in Noncovariant Gauges
S.D. Joglekar
IIT Kanpur
Talk given at THEP-I, held at IIT Roorkee from 16/3/05—20/3/05
Plan of Talk
1.
2.
3.
4.
5.
6.
7.
8.
9.
Preliminary
Brief Statement of The Problem and approaches towards
solving it
Importance of The Boundary Term and the FFBRS solution
Properties of the naïve path-integral: IRGT WT-identities &
Danger inherent in imposing a prescription by hand
A General Approach to the study of non-covariant gauges:
Exact BRST WT-identities and its new features
A simple illustration of how these unusual results work
Interpretation of Results
Additional restrictions placed by IRGT on Renormalization
A study of the additional restrictions
References:
1. S. D. Joglekar, Euro. Phys. Journal-direct C12 3, 1-18 (2001). hep-th/0106264
2. S. D. Joglekar, Mod. Phys. Letts. A 17, 2581-2596 (2002). hep-th/0205045
3.
S. D. Joglekar, Mod. Phys. Letts. A 18, 843 (2003). hep-th/0209073
Background:
1. Bassetto et al:
2. Leibbrandt
Preliminary








Non-covariant gauges [Axial, temporal, light-cone, Coulomb,
planar etc] have been widely used in standard model
calculations as well as in formal arguments in gauge theories
and string theories.
For example,
axial and light-cone gauges are used widely in QCD
calculations because of its formal freedom from ghosts.
Axial gauge is also useful in the treatment of the Chern-Simon
theories.
Light-cone gauge: useful in N=4 supersymmetric Y-M theory.
Coulomb gauge: in the discussion of confinement
Light-cone gauge: Cancellation of anomalies in superstring
theories
Problems with non-covariant
gauges

In axial gauges, it arises as the problem of spurious
singularities in the propagator:
D

k 
i

 k 2  i 






In Coulomb gauges, it arises as the problem of illdefined energy-integrals as the propagator for the
time-like component is :
D00


k   k 
k  k 


 g  
2
k
.

 k . 


k  
i
0
~
k
0
| k |2
is poorly damped as k0  ∞ compared to the
Lorentz gauges.
Problem is intimately related with the residual gauge
transformation.
Various Approaches: a sketch
1. Prescriptions:
Based on the general hope that a simple solution should work

Simplicity in momentum space leading to an ease of calculation.
Examples:
CPV: 1/.k  ½ {1/(.k+i 1/(.k-i}
LM: 1/.k  * .k/(.k * .k+i); 0 ; *  0,  

Possibility of Wick rotation

Finally, agreement/ disagreement with the Lorentz gauge results only
known after a calculation.
2. Attempts at derivations via canonical quantization

Derivations not unambiguous

Exist arguments for variety of conflicting prescriptions

Gauge independence of results not obvious.

3. Attempts via interpolating gauges


Construct a gauge that has a variable parameter a which connects
Lorentz and a non-covariant gauge
Gauge-independence proves to be trickier than expected [SDJ: EPJ
C01]
A Non-trivial Problem
Importance of the boundary
term






A prescription such as CPV or LM amounts to giving the boundary
condition for the unphysical degrees of freedom. E.g.
L-M requires causal BC for all degrees of freedom
CPV 0  0  amounts to requiring ½(C+A) for unphysical d.f.
A natural question: How do you know that the chosen BC will
produce a result compatible with the Lorentz gauges?
Prompted us (SDJ, Misra, Bandhu) to devise an independent
path-integral formalism that takes into account the boundary
term carefully.
The approach uses Lorentz gauge path-integral together with the
BC term:
1

  A A  cc 
2

and performs a field-transformation to construct an axial-gauge
path-integral together with a transformed BC term that tells one
how the axial gauge poles should be treated.
FFBRS transformation approach
W   D A D c D c O  A e



1

iS L eff [ A, c , c ] d 4 x  A2  cc   i d n x{ J ( x ) A ( x )+ ( x ) c ( x )+c ( x )  ( x )}
2




One then constructs a field transformation in the gauge-ghost
sector (a field-dependent BRS-type) that converts the path-integral
from the Lorentz gauge to axial gauge. [SDJ,Mandal, Bandhu]
In this process, the term:   d 4 x  1 A2  cc  transforms to another
2

term
 O '  A, c, c 
which decides how the unphysical poles of the axial (or any other
gauge) are to be treated.
Procedure is very general and has worked for axial and Coulomb
gauges:
Study of the naïve path-integral


The comparison of this approach with other attempts in
the literature evoked many questions that lead to this
general approach and work on interpolating gauges to be
discussed below. We wish to look at the general problem of
non-covariant gauges and problems associated with them in
a general setting suggested by our [SDJ, Misra] earlier
work.
We consider first consider the general path-integral of the
type:

iSeff [ A,c ,c ]i d n x{ J ( x ) A( x )+ ( x ) c ( x )+c ( x ) ( x )}
W   D ADc Dc e
and study its properties.
Some General Remarks





We recall that the path-integral (without any further
modifications) is often used in the formal manipulations and
for WT identities. The latter are important while discussing
renormalization.
We should recall that the path-integral has a residual gauge
invariance, (without the term); and this results in the
problem associated with the unphysical poles.
It is generally not recognized, how badly behaved is the pathintegral, without the term.
We then introduce a general form of corrective measure We
shall study some properties of the path-integral with this
general corrective measure.
We shall demonstrate several unusual features of both the
path-integrals.
IRGT: Infinitesimal Residual Gauge
Transformations

We generalize residual gauge transformation to BRS space
so as to leave the entire effective action invariant:
  x    Da   x  , and an analogous transformation
on matter fields
Here,  is an infinitesimal parameter. This leaves the
gauge-fixing term invariant.
• Sgh is invariant under the above, combined with “local
vector” transformations on ghosts:
 ca ( x)   gf a c  ( x)  ( x);
 c a ( x)   gf a c  ( x)  ( x)
And analogous transformations on matter fields.
These are NOT special cases of the BRS transformations
IRGT (CONTD)


We also require that the above transformations do not
alter the boundary conditions on the path-integrals
These transformations lead to relations between
Green’s functions. These relations will be called the
IRGT WT-identities.
EXAMPLES (contd)
Consequences of IRGT

iSeff [ A,c ,c , ]i d n x{ J ( x ) A( x )+K†  † K  ( x ) c ( x )+c ( x ) ( x )}
W [ J , K , K*]   D e

Theorem I: The Minkowski space Lagrangian path-integral of
(i) for "type-R gauges" leads to physically unacceptable
results from the IRGT WT-identities:



d a


a
0   d x  J 0  x, t   igJ   x, t  f
 .W  J , K , K *

dt

J
x
,
t







3




 *

a
a
  d x  gK i  x, t  Tij

gK
x
,
t
T


 .W  J , K , K *
j
ij
*

K
x
,
t

K
x
,
t

 

j 
i 

3
•The propagator for the scalar field vanishes D(x,t;y,t’)=0 for
x y, t t’ [Differentiate w.r.t. K* (x,t) and K(y,t’) ]
•Evidently this conclusion incompatible with the corresponding
one for the Lorentz gauges.
Consequences of IRGT (contd)
•
D(x,t;y,t’)=0 for x y, t t’ ; [Differentiate w.r.t. J
(x,t) and J (y,t’) ]
• Differentiate w.r.t. J0 (y,t’)  d/dt (t-t’) =0
 The path-integral manipulations along the lines
of derivation of WT-identities lead to absurd
results
 It becomes apparent that the role of any corrective
measure is very important and not peripheral.
 We shall later see how some of these results get
corrected.
We next study properties of a general path-integral in which
a general corrective measure is introduced.
We consider
W   D ADc Dc e

iSeff [ A,c,c ] O A,c ,c i d n x{ J ( x ) A( x )+ ( x ) c ( x )+c ( x ) ( x )}
Earlier instances of “risky” situations (i)
involving  (ii) “Dangerous” consequences






Non-covariant gauge literature has several specific instances
where it has been observed that it may be risky to ignore
terms with  in the numerator:
Works of Cheng and Tsai ~ 1986-7
Works of Andrasi and Taylor~ 1988-92
Landshoff and P. van Niewenhuizen
In addition, a specific instance of a dangerous consequence
from path-integral having residual gauge invariance has
appeared in a work by Baulieu and Zwanziger in 1999.
The present derivation gives a coherent approach and a
rational for such possibilities. In addition, it brings out a
number of further results and the full IRGT identity which
leads to such possibilities.
A general formulation
W   D ADc Dc e






iSeff [ A,c ,c ] O A,c ,c i d n x{ J ( x ) A( x )+ ( x ) c ( x )+c ( x ) ( x )}
where, O[A,c,c-] is some operator, which necessarily breaks the
residual gauge-invariance, and study its properties.
The operator O  A, c, c 
is supposed to take care of the
unphysical poles in some way, not necessarily the correct one.
Various prescriptions are special cases of this form.
To see this, consider the propagator without the -term and
propagator with a prescription. The latter necessarily contains
some small parameter, which we define in terms of . Thus the
inverses of these will differ by a term ~ .
Even when this is not possible, we shall see that analogous
conclusions should be expected.
IRGT v/s BRST







The IRGT are not a special case of BRS transformations.
A natural question: Are these IRGT identities something
extra? Something spurious?
Answer: No, the correct versions of these are contained in the
correct BRST WT-identities.
Theorem: The exact IRGT WT-identities are derivable from
the exact BRST identities.
Then, can we forget the IRGT identities, being a part of
BRST?
We have already seen the crucial fact about IRGT, that their
mathematical validity depends crucially on the presence of the
term.
Consequence: BRST WT-identities can receive
contributions from the term as  0.
An illustration
J2   
2
2
dx
dx
x
exp

ax

J
x

J
x


x
 1 2 2  1 1 1 2 2 2 }
I  J1 , J 2 ,  
Interpretation of Results
A non-covariant gauge is defined by two things:
 An BRST invariant action
 A prescription to deal with the singularities, or an -term
• The -dependent WT identity contains consequences of
both:
 BRST invariance of Seff
 The specific O-term
•The -term must contribute (as  0) to avoid absurd
relations exhibited earlier. Renormalization must deal with
both of these.
•The additional IRGT identities must also be dealt with
under renormalization
Additional consequences due to
IRGT


These relations crucially depend on what we take for the
operator O.
We shall illustrate this with a specific form of O and for the
Coulomb gauge:
4
1

O  i  d x  A2  cc 
2

The result is:
Theorem III: The local quadratic -term implies additional
constraints on Green's functions that are derivable from IRGT for
the type-R gauges.
Now we shall illustrate the proof for the Coulomb gauge. The term then leads to a term in the IRGT WT-identity of depending
on
  D a  i. Aa

And is
i  d x  D a  i  d x
3
3
 a
A
t
Additional consequences due to
IRGT

In particular, it leads to an identity:
• It is the corrected version of the absurd relation d/dt (t-t’) 0
 It puts a constraint on the un-renormalized propagator to all
orders:
• In particular, this gives non-trivial information about the exact
propagator at p = 0, p0  0.
Additional considerations in
renormalization: a partial analysis


How do these additional equations affect the renormalization of gauge
theories ?
To study this, we consider for simplicity, the axial gauge A3 =0 and
the following path-integral
W   D A Dc Dc e
With,

iSeff [ A,c ,c ] O A,c ,c i d n x{ J ( x ) A( x )+ ( x ) c ( x )+c ( x ) ( x )}
Seff  S0   d 4 x ba A3a  S gh
We perform the following IRGT (with a ax0,x1,x2))
Additional considerations in
renormalization (contd)

Under IRGT, Seff is invariant. This leads to the IRGT WTidentity:
4
a
a


d
x
J
(
x
)
D

i

D
O
A

( x )}
 0;






where
4
4
4


d
xO

d
xO

d
x
D
O
A

( x)






W  J 
 a
a 
0   dx3   J  ( x)W  J   igf J  ( x) 
 i DOa  A
 J  ( x)
  
The above IRGT identity is over and above the formal BRS
identity. We note that the last term can have a finite limit as 
0. Without it we would again land with absurd relations.



Additional considerations in renormalization: An
analysis of IRGT identity





How do these additional considerations affect the
renormalization of gauge theories ?
To study this, it convenient to recall the usual set-up of the
axial gauges and their expected renormalization and see if
these extra relations agree with them.
We associate with an axial gauge A3 =0, the following
freedom from ghosts,
A multiplicative renormalization scheme:
A3  Z 1/ 2 A3R ; A  Z31/ 2 AR   0,,; g  Z1g R
We ask under what conditions on O are these relations
compatible with the above scheme?
We shall assume, without loss of generality, that O has a general
quadratic form in A:
4
4
4 a

a
d
xO
A

d
x
d
yA
(
x
)
a
x

y
A




 ( y)

  
A “Spectator” prescription term


It is usually believed that the prescription for treating axial
gauge poles is unaffected by renormalization. This amounts to
assuming that the O term remains unchanged during
renormalization process. We shall call this a “spectator
prescription term”.
A local O is not compatible with these relations. To see this,
recall D ~  ∂A. This leads to the IRGT WT-identity:

W  J 
 a
a 
0   dx3   J  ( x)W  J   igf J  ( x) 
 i

J
(
x
)

  
a
  A ( x)



Each term in the above is multiplicatively renormalizable. But the
multiplicative scales do not agree.
Renormalization of term



In the more general case, we have to entertain the possibility
that the prescription term receives renormalization.
In fact, in the present picture this possibility becomes
transparent and natural because we are looking at the
prescription as coming from just another term in the total
action Seff + .
This possibility has been analyzed partially under certain
assumptions and restrictions on O have been spelt out.
A comment on Wick Rotation







This comment applies to a subset of non-covariant gauges
which are such that the Wick-rotated gauge-function F E[A] is
either purely real or purely imaginary. Example:
Coulomb gauge:  A  A
Temporal gauge:A0 iA4; axial gauge: .A .A
But not light-cone gauge: A0-A3 iA4--A3
The underlying question is whether there will exist an Euclidean
formulation (with no term) from which the correct Minkowski
formulation with the prescription term can be obtained from
Wick-rotation.
In covariant gauges, we know that this is always possible.
This question is important, because often possibility of Wick
rotation has been considered a desirable criterion for a
prescription.
A comment on Wick Rotation [contd]

An analysis along the present lines seems to indicate that for
these non-covariant gauges, this is unlikely.
Conclusions







We have constructed a generalization if residual gauge
transformation to the BRS space and shown that the naïve
path-integral leads to physically/mathematically unacceptable
results by performing IRGT.
This brings out the fact that even formal manipulations require
careful treatment.
We constructed a general framework in which to study the
non-covariant gauges.
We showed that the exact IRGT identities are contained in the
exact BRST.
Unlike the covariant gauges, the -term can contribute to the
BRST WT-identities.
The IRGT identities lead to additional consequences that have
to be taken into account while discussing renormalization.
We have partially analyzed these conditions.
AN ANALOGY WITH SYMMETRIES AND
ANOMALY
SYMMETRIES AND ANOMALY
(1)
Classical Symmetry
NON-COVARIANT GAUGES
(1) BRST symmetry of action
(2) Quantum Theory embodying the
symmetry is ill-defined.
(2) Quantum Theory embodying the
symmetry is ill-defined.
(3) Quantum Theory has to be
regularized
(3) Path-integral has to be made welldefined by a prescription.
(4) A given regularization may break
symmetry
(4) A prescription may not preserve the
BRST symmetry
(5) If a symmetry-preserving
(5) It is possible that here the path regularization exists that symmetry may integral construction (SDJ,AM) that
be preserved by Quantum Theory
does not impose a prescription by hand
may be compatible with BRS
(6) If a symmetry preserving
regularization does not exist, we have
an anomaly and a quantity which ought
to vanish, leads to non-vanishing
contribution.
(6) For a general prescription, the limit
 0 gives a non-vanishing
contribution.
FAQ1: A Criticism:
Fault of PI-formulation?



It is sometimes believed that the path-integral is a illdefined object and illegitimate operations of pathintegrals are often the cause of absurd results.
This can be resolved in a simple manner: We can
always think of the Field theory as defined in terms
Feynman rules alone. We can then evaluate the
quantities on the left hand side and evaluate the
result. There is a one-to-one correspondence between
path-integral manipulations and Feynman
diagrammatic approach.
These identities are valid even in tree approximation,
where the term is needed in an essential way for its
validity.
FAQ2: Divergence structure and
prescription
FAQ2:(Contd)
(1) ILM has no divergences at all.
FAQ3: FFBRS approach
FAQ3: FFBRS approach
(contd.)
FAQ4: Absurd conclusion from PI
with residual gauge symmetry