Some General Results in Non-Some General Results in Non-covariant Gaugescovariant Gauges

S.D. JoglekarS.D. Joglekar

IIT KanpurIIT Kanpur

Talk given at THEP-I, held at IIT Roorkee from 16/3/05—Talk given at THEP-I, held at IIT Roorkee from 16/3/05—20/3/05 20/3/05

Plan of TalkPlan of Talk1.1. PreliminaryPreliminary2.2. Brief Statement of The Problem and approaches towards Brief Statement of The Problem and approaches towards

solving itsolving it3.3. Importance of The Boundary Term and the FFBRS solutionImportance of The Boundary Term and the FFBRS solution4.4. Properties of the naïve path-integral: IRGT WT-identities & Properties of the naïve path-integral: IRGT WT-identities &

Danger inherent in imposing a prescription by handDanger inherent in imposing a prescription by hand5.5. A General Approach to the study of non-covariant gauges: A General Approach to the study of non-covariant gauges:

Exact Exact BRST WT-identities and its new featuresBRST WT-identities and its new features6.6. A simple illustration of how these unusual results workA simple illustration of how these unusual results work7.7. Interpretation of ResultsInterpretation of Results8.8. Additional restrictions placed by IRGT on RenormalizationAdditional restrictions placed by IRGT on Renormalization9.9. A study of the additional restrictionsA study of the additional restrictions

ReferencesReferences: : 1. S. D. Joglekar, Euro. Phys. Journal-direct C12 1. S. D. Joglekar, Euro. Phys. Journal-direct C12 3, 3, 1-18 (2001). 1-18 (2001).

hep-th/0106264hep-th/01062642. S. D. Joglekar, Mod. Phys. Letts. A 2. S. D. Joglekar, Mod. Phys. Letts. A 17, 17, 2581-2596 (2002). hep-th/02050452581-2596 (2002). hep-th/02050453.3. S. D. Joglekar, Mod. Phys. Letts. A S. D. Joglekar, Mod. Phys. Letts. A 18, 18, 843 (2003). hep-th/0209073843 (2003). hep-th/0209073 Background:Background: 1. Bassetto et al: 1. Bassetto et al: 2. Leibbrandt 2. Leibbrandt

PreliminaryPreliminary Non-covariant gauges [Axial, temporal, light-cone, Non-covariant gauges [Axial, temporal, light-cone,

Coulomb, planar etc] have been widely used in Coulomb, planar etc] have been widely used in standard model calculations as well as in formal standard model calculations as well as in formal arguments in gauge theories and string theories.arguments in gauge theories and string theories.

For example, For example, axial and light-cone gauges are used widely in QCD axial and light-cone gauges are used widely in QCD

calculations because of its formal freedom from calculations because of its formal freedom from ghosts. ghosts.

Axial gauge is also useful in the treatment of the Axial gauge is also useful in the treatment of the Chern-Simon theories.Chern-Simon theories.

Light-cone gauge: useful in N=4 supersymmetric Y-M Light-cone gauge: useful in N=4 supersymmetric Y-M theory.theory.

Coulomb gauge: in the discussion of confinementCoulomb gauge: in the discussion of confinement Light-cone gauge: Cancellation of anomalies in Light-cone gauge: Cancellation of anomalies in

superstring theoriessuperstring theories

Problems with non-covariant Problems with non-covariant gaugesgauges

In axial gauges, it arises as the problem of In axial gauges, it arises as the problem of spurious singularities in the propagator:spurious singularities in the propagator:

In Coulomb gauges, it arises as the problem In Coulomb gauges, it arises as the problem of ill-defined energy-integrals as the of ill-defined energy-integrals as the propagator for the time-like component is : propagator for the time-like component is :

is poorly damped as is poorly damped as kk0 0 ∞∞ compared to the compared to the Lorentz gauges.Lorentz gauges.

Problem is intimately related with the residual Problem is intimately related with the residual gauge transformation.gauge transformation.

22 . .

k k k kiD k g

kk i k

000 02

~| |

iD k k

k

Various Approaches: a Various Approaches: a sketchsketch1. 1. Prescriptions:Prescriptions: Based on the general hope that a simple solution should Based on the general hope that a simple solution should

work work Simplicity in momentum space leading to an ease of Simplicity in momentum space leading to an ease of

calculation.calculation.ExamplesExamples:: CPV: 1/CPV: 1/k k ½ {1/( ½ {1/(k+k+ii1/(1/(k-k-iiLM: 1/LM: 1/k k k/(k/(k k kk+i+i); ); Possibility of Wick rotationPossibility of Wick rotation Finally, agreement/ disagreement with the Lorentz gauge Finally, agreement/ disagreement with the Lorentz gauge

results only known after a calculation.results only known after a calculation.2.2. Attempts at derivations via canonical quantization Attempts at derivations via canonical quantization Derivations not unambiguousDerivations not unambiguous Exist arguments for variety of conflicting prescriptionsExist arguments for variety of conflicting prescriptions Gauge independence of results not obvious.Gauge independence of results not obvious.3. Attempts via interpolating gauges3. Attempts via interpolating gauges ConstructConstruct a gauge that has a variable parameter a gauge that has a variable parameter which which

connects Lorentz and a non-covariant gauge connects Lorentz and a non-covariant gauge Gauge-independence proves to be trickier than expected Gauge-independence proves to be trickier than expected

[SDJ: EPJ C01][SDJ: EPJ C01]

A Non-trivial ProblemA Non-trivial Problem

Importance of the Importance of the boundary termboundary term

A prescription such as CPV or LM amounts to giving the A prescription such as CPV or LM amounts to giving the boundary condition for the unphysical degrees of freedom. boundary condition for the unphysical degrees of freedom. E.g.E.g.

L-M requires causal BC for L-M requires causal BC for all all degrees of freedom degrees of freedom CPV CPV amounts to requiring ½(C+A) for unphysical amounts to requiring ½(C+A) for unphysical

d.f.d.f. A natural question: How do you know that the chosen BC will A natural question: How do you know that the chosen BC will

produce a result compatible with the Lorentz gauges?produce a result compatible with the Lorentz gauges? Prompted us (SDJ, Misra, Bandhu) to devise an independent Prompted us (SDJ, Misra, Bandhu) to devise an independent

path-integral formalism that takes into account the boundary path-integral formalism that takes into account the boundary term carefully.term carefully.

The approach uses Lorentz gauge path-integral The approach uses Lorentz gauge path-integral together together with with the the BC term: BC term:

and performs a field-transformation to construct an axial-and performs a field-transformation to construct an axial-gauge path-integral gauge path-integral together with together with a a transformedtransformed BC term BC term that tells one how the axial gauge poles should be treated.that tells one how the axial gauge poles should be treated.

1

2A A cc

FFBRS transformation FFBRS transformation approachapproach

One then constructs a field transformation in the gauge-One then constructs a field transformation in the gauge-ghost sector (a field-dependent BRS-type) that converts ghost sector (a field-dependent BRS-type) that converts the path-integral from the Lorentz gauge to axial gauge. the path-integral from the Lorentz gauge to axial gauge. [SDJ,Mandal, Bandhu][SDJ,Mandal, Bandhu]

In this process, the In this process, the term: transforms to term: transforms to another term another term

which decides how the unphysical poles of the axial (or any which decides how the unphysical poles of the axial (or any other gauge) are to be treated.other gauge) are to be treated.

Procedure is very general and has worked for axial and Procedure is very general and has worked for axial and Coulomb gauges:Coulomb gauges:

4 21

[ , , ] d { ( ) ( )+ ( ) ( )+ ( ) ( )}2

L neffiS A c c x A cc i d x J x A x x c x c x x

W D ADcDc O A e

4 21d

2x A cc

' , ,O A c c

Study of the naïve path-Study of the naïve path-integralintegral The comparison of this approach with other The comparison of this approach with other

attempts in the literature evoked many questions attempts in the literature evoked many questions that lead to this general approach and work on that lead to this general approach and work on interpolating gauges to be discussed below. We interpolating gauges to be discussed below. We wish to look at the general problem of non-covariant wish to look at the general problem of non-covariant gauges and problems associated with them in a gauges and problems associated with them in a general setting suggested by our [SDJ, Misra] earlier general setting suggested by our [SDJ, Misra] earlier work.work.

We consider first consider the general path-integral We consider first consider the general path-integral of the type:of the type:

[ , , ] { ( ) ( )+ ( ) ( )+ ( ) ( )}neffiS A c c i d x J x A x x c x c x x

W D ADcDc e

and study its properties.

Some General Remarks Some General Remarks

We recall that the path-integral (without any furtherWe recall that the path-integral (without any further modifications)modifications) is often used in the formal is often used in the formal manipulations and for WT identities. The latter are manipulations and for WT identities. The latter are important while discussing renormalization.important while discussing renormalization.

We should recall that the path-integral has a residual We should recall that the path-integral has a residual gauge invariance, (without the gauge invariance, (without the term); and this term); and this results in the problem associated with the unphysical results in the problem associated with the unphysical poles.poles.

It is generally not recognized, how badly behaved is It is generally not recognized, how badly behaved is the path-integral, without the the path-integral, without the term.term.

We then introduce a general form of corrective measure We shall study some properties of the path-We shall study some properties of the path-integral integral with this general corrective measurewith this general corrective measure..

We shall demonstrate several unusual features of both the path-integrals.

IRGT: IRGT: Infinitesimal Residual Gauge Infinitesimal Residual Gauge TransformationsTransformations

We generalize residual gauge transformation to BRS We generalize residual gauge transformation to BRS space so as to leave the entire effective action invariant:space so as to leave the entire effective action invariant:

, and an analogous transformation

on matter fields

x D x

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:

( ) ( ) ( );

( ) ( ) ( )

c x gf c x x

c x gf c x x

And analogous transformations on matter fields.

These are NOT special cases of the BRS transformations

IRGT IRGT (CONTD) (CONTD)

We also require that the above transformations do not We also require that the above transformations do not alter the boundary conditions on the path-integralsalter the boundary conditions on the path-integrals

These transformations lead to relations between These transformations lead to relations between Green’s functions. These relations will be called the Green’s functions. These relations will be called the IRGT WT-identities.IRGT WT-identities.

EXAMPLES EXAMPLES (contd) (contd)

Consequences of IRGTConsequences of IRGT

Theorem I: The Minkowski space Lagrangian path-Theorem I: The Minkowski space Lagrangian path-integral of (i) for "type-R gauges" leads to physically integral of (i) for "type-R gauges" leads to physically unacceptable results from the IRGT WT-identities:unacceptable results from the IRGT WT-identities:

30

3 **

0 d , , . , , *,

d , , . , , *, ,i ij j ij

j i

dJ t igJ t f W J K K

dt J t

gK t T gK t T W J K KK t K t

x x xx

x x xx x

•The propagator for the scalar field vanishes (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.

† †[ , , , ] { ( ) ( )+K ( ) ( )+ ( ) ( )}[ , , *]

neffiS A c c i d x J x A x K x c x c x x

W J K K D e

Consequences of IRGT Consequences of IRGT (contd)(contd)

• (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 The path-integral manipulations along the lines path-integral manipulations along the lines

of derivation of WT-identities lead to absurd of derivation of WT-identities lead to absurd resultsresults

It becomes apparent that the role of any corrective It becomes apparent that the role of any corrective measure is very important and not peripheral.measure is very important and not peripheral.

We shall later see how some of these results get We shall later see how some of these results get corrected.corrected.

We next study properties of a general path-integral We next study properties of a general path-integral in which a general corrective measure is introduced.in which a general corrective measure is introduced.We considerWe consider

[ , , ] , , { ( ) ( )+ ( ) ( )+ ( ) ( )}neffiS A c c O A c c i d x J x A x x c x c x x

W D ADcDc e

Earlier instances of “risky” situations (i) Earlier instances of “risky” situations (i) involving involving (ii)(ii)“Dangerous” “Dangerous” consequencesconsequences Non-covariant gauge literature has several specific Non-covariant gauge literature has several specific

instances where it has been observed that it may be instances where it has been observed that it may be risky to ignore terms with risky to ignore terms with in the numerator:in the numerator:

Works of Cheng and Tsai ~ 1986-7Works of Cheng and Tsai ~ 1986-7 Works of Andrasi and Taylor~ 1988-92Works of Andrasi and Taylor~ 1988-92 Landshoff and P. van NiewenhuizenLandshoff and P. van Niewenhuizen

In addition, a specific instance of a dangerous In addition, a specific instance of a dangerous consequence from path-integral having residual gauge consequence from path-integral having residual gauge invariance has appeared in a work by Baulieu and invariance has appeared in a work by Baulieu and Zwanziger in 1999.Zwanziger in 1999.

The present derivation gives a coherent approach and The present derivation gives a coherent approach and a rational for such possibilities. In addition, it brings a rational for such possibilities. In addition, it brings out a number of further results and the full IRGT out a number of further results and the full IRGT identity which leads to such possibilities.identity which leads to such possibilities.

A general formulationA general formulation

where, O[A,c,c-] is some operator, which necessarily breaks the residual gauge-invariance, and study its properties.

The operator 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 mall parameter, which we define in terms of define in terms of . Thus the inverses of these will . Thus the inverses of these will differ by a term ~ differ by a term ~

Even when this is not possible, we shall see that Even when this is not possible, we shall see that analogous conclusions should be expected. analogous conclusions should be expected.

[ , , ] , , { ( ) ( )+ ( ) ( )+ ( ) ( )}neffiS A c c O A c c i d x J x A x x c x c x x

W D ADcDc e

, ,O A c c

IRGT v/s BRSTIRGT v/s BRST The IRGT are The IRGT are notnot a special case of BRS a special case of BRS

transformations.transformations. A natural question: Are these IRGT identities A natural question: Are these IRGT identities

something extra? Something spurious?something extra? Something spurious? Answer: No, the Answer: No, the correct correct versions of these are versions of these are

contained incontained in the the correct correct BRST WT-identities.BRST WT-identities. Theorem: The Theorem: The exactexact IRGT WT-identities are derivable IRGT WT-identities are derivable

from the from the exactexact BRST identities. BRST identities. Then, can we forget the IRGT identities, being a part Then, can we forget the IRGT identities, being a part

of BRST?of BRST? We have already seen the crucial fact about IRGT, that We have already seen the crucial fact about IRGT, that

their mathematical validity depends crucially on the their mathematical validity depends crucially on the presence of the presence of the term.term.

ConsequenceConsequence: : BRST WT-identities can receive BRST WT-identities can receive contributions from the contributions from the term as term as

An illustrationAn illustration

2 21 2 2 1 1 1 2 2 2

21 2

exp

, ,

dx dx x ax J x J x xJ

I J J

Interpretation of Interpretation of ResultsResultsA non-covariant gauge is defined by A non-covariant gauge is defined by two two things: things: An BRST invariant actionAn BRST invariant action A prescription to deal with the singularities, or an A prescription to deal with the singularities, or an -term-term

• The The -dependent WT identity contains -dependent WT identity contains consequences of consequences of both:both: BRST invariance of BRST invariance of SeffSeff The specific The specific O-termO-term

•The The -term must contribute (as -term must contribute (as 0) to avoid 0) to avoid absurd relations exhibited earlier. Renormalization absurd relations exhibited earlier. Renormalization must deal with both of these. must deal with both of these.

•The additional IRGT identities must also be dealt The additional IRGT identities must also be dealt with under renormalizationwith under renormalization

Additional consequences due to Additional consequences due to IRGTIRGT

These relations crucially depend on what we take for the These relations crucially depend on what we take for the operator operator O.O.

We shall illustrate this with a specific form of We shall illustrate this with a specific form of O O and for and for the Coulomb gauge:the Coulomb gauge:

The result is:The result is:

Theorem III: The local quadratic Theorem III: The local quadratic -term implies additional -term implies additional constraints on Green's functions that are derivable from constraints on Green's functions that are derivable from

IRGT for the type-R gaugesIRGT for the type-R gauges..Now we shall illustrate the proof for the Coulomb gauge. Now we shall illustrate the proof for the Coulomb gauge.

The The -term then leads to a term in the IRGT WT-identity -term then leads to a term in the IRGT WT-identity of depending onof depending on

4 21

2O i d x A cc

.i A

And is 3 3

i d i d At

x x

Additional consequences due to Additional consequences due to IRGTIRGT

In particular, it leads to an identity:In particular, it leads to an identity:

• It is the corrected version of the absurd relation d/dt (t-t’)

t 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 Additional considerations in renormalization: a partial renormalization: a partial analysisanalysis How do these additional equations affect the renormalization How do these additional equations affect the renormalization

of gauge theories ?of gauge theories ? To study this, we consider for simplicity, the axial gauge ATo study this, we consider for simplicity, the axial gauge A33

=0 and the following path-integral=0 and the following path-integral

[ , , ] , , { ( ) ( )+ ( ) ( )+ ( ) ( )}neffiS A c c O A c c i d x J x A x x c x c x x

W D ADcDc e

With, 40 3deff ghS S xb A S

We perform the following IRGT (with x0,x1,x2))

Additional considerations in Additional considerations in renormalization renormalization (contd)(contd) Under IRGT, Under IRGT, SSeffeff is invariant. This leads to the IRGT WT-is invariant. This leads to the IRGT WT-

identity:identity:

4

4 4 4

d ( ) ( ) 0;

d d d ( )

x J x D i O A x

where

xO xO x O A x

30 ( ) ( )( )

W Jdx J x W J igf J x i O 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 Without it we would again land with absurd relations.

Additional considerations in renormalization: Additional considerations in renormalization: An analysis of IRGT identityAn analysis of IRGT identity

How do these additional considerations affect the How do these additional considerations affect the renormalization of gauge theories ?renormalization of gauge theories ?

To study this, it convenient to recall the usual set-up To study this, it convenient to recall the usual set-up of the axial gauges and their expected renormalization of the axial gauges and their expected renormalization and see if these extra relations agree with them.and see if these extra relations agree with them.

We associate with an axial gauge AWe associate with an axial gauge A33 =0, the following =0, the following freedom from ghosts,freedom from ghosts, A multiplicative renormalization scheme: A multiplicative renormalization scheme:

1/ 2 1/ 23 3 3 1; ;R R RA Z A A Z A g Z g

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 4d d d ( ) ( )xO A x yA x a x y A y

A “Spectator” prescription A “Spectator” prescription termterm It is usually believed that the prescription for treating It is usually believed that the prescription for treating

axial gauge poles is axial gauge poles is unaffected by renormalization. unaffected by renormalization. This amounts to assuming that the This amounts to assuming that the O O term remains term remains unchanged during renormalization process. We shall unchanged during renormalization process. We shall call this a “spectator prescription term”.call this a “spectator prescription term”.

A local A local O O is not compatible with these relations. To see is not compatible with these relations. To see this, recall this, recall ∂∂AA

This leads to the IRGT WT-This leads to the IRGT WT-identity:identity:

30 ( ) ( ) ( )

( )

W Jdx J x W J igf J x i A x

J x

Each term in the above is multiplicatively renormalizable. But the multiplicative scales do not agree.

Renormalization of Renormalization of termterm In the more general case, we have to entertain the In the more general case, we have to entertain the

possibility that the prescription term receives possibility that the prescription term receives renormalization.renormalization.

In fact, in the present picture this possibility becomes In fact, in the present picture this possibility becomes transparent and natural because we are looking at the transparent and natural because we are looking at the prescription as coming from just another term in the prescription as coming from just another term in the total action total action SSeffeff + + ..

This possibility has been analyzed partially under This possibility has been analyzed partially under certain assumptions and restrictions on certain assumptions and restrictions on O O have been have been spelt out.spelt out.

A comment on Wick A comment on Wick RotationRotation This comment applies to a subset of non-covariant gauges This comment applies to a subset of non-covariant gauges

which are such that the Wick-rotated gauge-function which are such that the Wick-rotated gauge-function F F EE[A] is either purely real or purely imaginary. Example:[A] is either purely real or purely imaginary. Example:

Coulomb gauge: Coulomb gauge: Temporal gauge:ATemporal gauge:A00 iAiA44; axial gauge: ; axial gauge: A A A A

But But not not light-cone gauge: A light-cone gauge: A00-A-A33 iAiA44--A--A3 3

The underlying question is whether there will exist an The underlying question is whether there will exist an Euclidean formulation (with Euclidean formulation (with no no termterm) from which the ) from which the correct correct Minkowski formulation Minkowski formulation withwith the prescription term the prescription term can be obtained from Wick-rotation.can be obtained from Wick-rotation.

In covariant gauges, we know that this is always possible.In covariant gauges, we know that this is always possible. This question is important, because often possibility of This question is important, because often possibility of

Wick rotation has been considered a Wick rotation has been considered a desirable desirable criterion for criterion for a prescription.a prescription.

A A

A comment on Wick Rotation A comment on Wick Rotation [contd][contd] An analysis along the present lines seems to indicate An analysis along the present lines seems to indicate

that for these non-covariant gauges, this is unlikely.that for these non-covariant gauges, this is unlikely.

ConclusionsConclusions We have constructed a generalization if residual gauge We have constructed a generalization if residual gauge

transformation to the BRS space and shown that the transformation to the BRS space and shown that the naïve path-integral leads to physically/mathematically naïve path-integral leads to physically/mathematically unacceptable results by performing IRGT.unacceptable results by performing IRGT.

This brings out the fact that even formal manipulations This brings out the fact that even formal manipulations require careful treatment.require careful treatment.

We constructed a general framework in which to study We constructed a general framework in which to study the non-covariant gauges.the non-covariant gauges.

We showed that the exact IRGT identities are We showed that the exact IRGT identities are contained in the contained in the exactexact BRST. BRST.

Unlike the covariant gauges, the Unlike the covariant gauges, the -term can contribute -term can contribute to the BRST WT-identities.to the BRST WT-identities.

The IRGT identities lead to additional consequences The IRGT identities lead to additional consequences that have to be taken into account while discussing that have to be taken into account while discussing renormalization.renormalization.

We have partially analyzed these conditions.We have partially analyzed these conditions.

AN ANALOGY WITH SYMMETRIES AN ANALOGY WITH SYMMETRIES AND ANOMALYAND ANOMALYSYMMETRIES AND SYMMETRIES AND ANOMALYANOMALY

NON-COVARIANT GAUGESNON-COVARIANT GAUGES

(1)(1) Classical SymmetryClassical Symmetry (1) BRST symmetry of action(1) BRST symmetry of action

(2) Quantum Theory embodying (2) Quantum Theory embodying the symmetry is ill-defined. the symmetry is ill-defined.

(2) Quantum Theory embodying (2) Quantum Theory embodying the symmetry is ill-defined.the symmetry is ill-defined.

(3) Quantum Theory has to be (3) Quantum Theory has to be regularizedregularized

(3) Path-integral has to be made (3) Path-integral has to be made well-defined by a prescription.well-defined by a prescription.

(4) A given regularization may (4) A given regularization may break symmetrybreak symmetry

(4) A prescription may not (4) A prescription may not preserve the BRST symmetrypreserve the BRST symmetry

(5) If a symmetry-preserving (5) If a symmetry-preserving regularization exists that regularization exists that symmetry may be preserved by symmetry may be preserved by Quantum TheoryQuantum Theory

(5) It is possible that here the path (5) It is possible that here the path -integral construction (SDJ,AM) -integral construction (SDJ,AM) that does not impose a that does not impose a prescription by hand may be prescription by hand may be compatible with BRScompatible with BRS

(6) If a symmetry preserving (6) If a symmetry preserving regularization does not exist, we regularization does not exist, we have an anomaly and a quantity have an anomaly and a quantity which ought to vanish, leads to which ought to vanish, leads to non-vanishing contribution. non-vanishing contribution.

(6) For a general prescription, the (6) For a general prescription, the limit limit 0 gives a non-vanishing 0 gives a non-vanishing contribution.contribution.

FAQ1: A Criticism: FAQ1: A Criticism: Fault of PI-formulation?Fault of PI-formulation? It is sometimes believed that the path-integral is a ill-It is sometimes believed that the path-integral is a ill-

defined object and illegitimate operations of path-defined object and illegitimate operations of path-integrals are often the cause of absurd results.integrals are often the cause of absurd results.

This can be resolved in a simple manner: We can This can be resolved in a simple manner: We can always think of the Field theory as defined in terms always think of the Field theory as defined in terms Feynman rules alone. We can then evaluate the Feynman rules alone. We can then evaluate the quantities on the left hand side and evaluate the quantities on the left hand side and evaluate the result. There is a one-to-one correspondence between result. There is a one-to-one correspondence between path-integral manipulations and Feynman path-integral manipulations and Feynman diagrammatic approach.diagrammatic approach.

These identities are valid even in tree approximation, These identities are valid even in tree approximation, wherewhere the the term is term is needed needed in an essential way for its in an essential way for its validity.validity.

FAQ2: Divergence structure FAQ2: Divergence structure and prescriptionand prescription

FAQ2FAQ2:(Contd):(Contd)

(1)ILM has no divergences at all.

FAQ3: FFBRS approachFAQ3: FFBRS approach

FAQ3: FFBRS approach FAQ3: FFBRS approach (contd.) (contd.)

FAQ4: Absurd conclusion from FAQ4: Absurd conclusion from PI with residual gauge PI with residual gauge symmetrysymmetry