Some General Results in Non-covariant Gauges

1. Some General Results in Non-covariant Gauges S.D. Joglekar IIT Kanpur Talk given at THEP-I, held at IIT Roorkee from 16/3/05—20/3/05

2. Plan of Talk • 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 • S. D. Joglekar, Mod. Phys. Letts. A 18, 843 (2003). hep-th/0209073 Background: 1. Bassetto et al: 2. Leibbrandt

3. 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

4. Problems with non-covariant gauges • In axial gauges, it arises as the problem of spurious singularities in the propagator: • In Coulomb gauges, it arises as the problem of ill-defined energy-integrals as the propagator for the time-like component is : • is poorly damped as k0 ∞ compared to the Lorentz gauges. • Problem is intimately related with the residual gauge transformation.

5. 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/h.k  ½ {1/(h.k+ie) +1/(h.k-ie)} LM: 1/h.k  h* .k/(h.k h* .k+ie); h2=0 ; h* = (h0, - h) • 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 • Constructa 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]

6. A Non-trivial Problem

7. 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 (h0 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: 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.

8. FFBRS transformation approach • 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 e-term: transforms to another term 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:

9. 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: • and study its properties.

10. Some General Remarks • We recall that the path-integral (without any furthermodifications)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 e-term); and this results in the problem associated with the unphysical poles. • It is generally not recognized, how badly behaved is the path-integral, without the e-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.

11. IRGT: Infinitesimal Residual Gauge Transformations • We generalize residual gauge transformation to BRS space so as to leave the entire effective action invariant: • Here, s is an infinitesimal parameter. This leaves the gauge-fixing term invariant. • Sgh is invariant under the above, combined with “local vector” transformations on ghosts: And analogous transformations on matter fields. These are NOT special cases of the BRS transformations

12. 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.

13. EXAMPLES (contd)

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

15. Consequences of IRGT (contd) • Dmn(x,t;y,t’)=0 for x y, t t’ ; [Differentiate w.r.t. Jm(x,t) and Jn (y,t’) ] • Differentiate w.r.t. J0 (y,t’)  d/dt d(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.

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

17. Earlier instances of “risky” situations (i) involving e (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 ein 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.

18. A 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 e-term and propagator with a prescription. The latter necessarily contains some small parameter, which we define in terms of e. Thus the inverses of these will differ by a term ~ e. • Even when this is not possible, we shall see that analogous conclusions should be expected.

19. 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 e-term. • Consequence: BRST WT-identities can receive contributions from the e-term as e 0.

20. An illustration

24. 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 e-term • The e-dependent WT identity contains consequences of both: • BRST invariance of Seff • The specific eO-term • The e-term must contribute (as e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

25. 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: • The result is: Theorem III: The local quadratic e-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 e-term then leads to a term in the IRGT WT-identity of depending on And is

26. Additional consequences due to IRGT • In particular, it leads to an identity: • It is the corrected version of the absurd relation d/dt d(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.

27. 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 With, We perform the following IRGT (with qa =qa(x0,x1,x2))

28. Additional considerations in renormalization (contd) • Under IRGT, Seffis invariant. This leads to the IRGT WT-identity: The above IRGT identity is over and above the formal BRS identity. We note that the last term can have a finite limit as e 0. Without it we would again land with absurd relations.

29. 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: 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 generalquadratic form in A:

30. 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 eO 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 DO ~  ∂mAm. This leads to the IRGT WT-identity: Each term in the above is multiplicatively renormalizable. But the multiplicative scales do not agree.

31. Renormalization of e-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 + eO. • This possibility has been analyzed partially under certain assumptions and restrictions on O have been spelt out.

32. 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: • Temporal gauge:A0iA4; axial gauge: h.A h.A • But not light-cone gauge: A0-A3 iA4--A3 • The underlying question is whether there will exist an Euclidean formulation (with no e-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.

33. A comment on Wick Rotation [contd] • An analysis along the present lines seems to indicate that for these non-covariant gauges, this is unlikely.

34. 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 e-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.

35. AN ANALOGY WITH SYMMETRIES AND ANOMALY

37. FAQ1: A Criticism: Fault of PI-formulation? • It is sometimes believed that the path-integral is a ill-defined object and illegitimate operations of path-integrals 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 e-term is needed in an essential way for its validity.

38. FAQ2: Divergence structure and prescription

39. FAQ2:(Contd) • ILMhas no divergences at all.

40. FAQ3: FFBRS approach

41. FAQ3: FFBRS approach (contd.)

42. FAQ4: Absurd conclusion from PI with residual gauge symmetry