Gauge Theory without Ghosts

  • B. DeWitt
  • C. Molina-París
Part of the NATO ASI Series book series (NSSB, volume 361)

Abstract

A quantum effective action for gauge field theories is constructed that is gauge invariant and independent of the choice of gauge breaking terms in the functional integral that defines it. The loop expansion of this effective action leads to new Feynman rules, involving new vertex functions but without diagrams containing ghost lines. The new rules are given in full for the Yang—Mills field, both with and without coupling to fermions, and renormalization procedures are described. No BRST arguments are needed. Implications are briefly discussed.

Keywords

Gauge Theory Gauge Invariance Dimensional Regularization Vertex Function Counter Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Summary of Notation

Φ

Space of field histories (a principal fiber bundle)

G

Gauge group (the typical fiber)

Φ/G

Base space (the space in which real physics takes place)

γ

Ultralocal, gauge-invariant metric on Φ

Qα

Fiber generators (pointwise linearly independent vertical vector fields)

cγαβ

Structure constants of G

S

Classical action functional

FΦ

Frame bundle over Φ

Γγijk

Pullback to Φ of Riemannian connection on FΦ

ωαi

Connection 1-form on Φ (defined uniquely by γ and the Qα)

Πij

Horizontal projection operator

IA, Kα

Fiber-adapted coordinates

\(\hat{\mathfrak{F}}\hat{\mathfrak{G}}\)

Ghost operator and ghost propagator

g

Projection of γ onto Φ/G

F(Φ/G)

Frame bundle over Φ/G

ΓABC

Pullback to Φ/G of Riemannian connection on F(Φ/G)

Γijk

Vilkovisky’s connection (extension of Γ A BC to Φ)

φi

Arbitrary local field variables

ϕa

Gaussian normal fields

Γ

Effective action

μ[φ*, ϕ]

Measure for the functional integral defining Γ

Σ

Γ — S (loop contribution to Γ)

Aαμ

Gauge potentials for Yang—Mills theory

γαβ

Cartan—Killing metric

Fαμν

Yang—Mills curvature 2-form

fαβγ

Structure constants of the Yang—Mills Lie group

fα

Generators of the adjoint representation: (f β αγ)

Qαμβ′

Fiber generators for the Yang—Mills field

γαμβ′ν′

Ultralocal invariant metric for the Yang—Mills field

Παμβν′

Horizontal projection operator for the Yang—Mills field

Z

Yang—Mills field renormalization constant

Y

Renormalization constant for the three-pronged vertex

X

Renormalization constant for the four-pronged vertex

Ξ

Coefficient of the nonlocal counter term needed for mopping up trace residues coming from the divergent parts of graphs having four external lines

μ

Auxiliary mass for dimensional regularization

\(\tilde Z\)

Spinor field renormalization constant

\(\tilde \Xi \)

Coefficient of the nonlocal counter term needed to complete the renormalization of the fermion vertex

Denotes covariant functional differentiation based on the Riemannian connection Γγ i jk

;

Denotes covariant functional differentiation based on Vilkovisky’s connection Γ i jk

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References

  1. [1]
    R. P. Feynman, Acta Physica Polonica 24, 697 (1963). The work presented in this paper was carried out in 1962.MathSciNetGoogle Scholar
  2. [2]
    Throwing away the noncausal loops can be shown to be equivalent to throwing away nonvanishing contributions from arcs at infinity in the Wick rotation procedure. See B. DeWitt, Supermanifolds (Second Edition) (Cambridge University Press, 1992) chapter 6 (especially exercise 6.11, pp. 393-395) where rejection of the noncausal loops is also shown to be intimately connected to the choice of measure for the functional integral that defines the loop expansion.Google Scholar
  3. [3]
    The question of Lorentz invariance arises because Feynman was assuming spacetime to be asymptotically Minkowskian.Google Scholar
  4. [4]
    B. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, 1965).Google Scholar
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    The linear independence of the Q α and the invertibility and ultralocality of γi, j guarantee that Fαβ will be an invertible operator, typically differential or pseudodifferential.Google Scholar
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    G. A. Vilkovisky in Quantum Theory of Gravity, éd. S. M. Christensen (Adam Hilger, 1984).Google Scholar
  7. [7]
    This is not true in gravity theory, for which γij is not flat.Google Scholar
  8. [8]
    This would not be true if the invariant metric \(\gamma {\alpha ^\mu }{\beta '^{v'}}\) had not been chosen to be ultralocal. In fact the renormalization program almost certainly could not be carried out with any choice but (63).Google Scholar
  9. [9]
    J. C. Collins, Renormalization (Cambridge University Press, 1984).Google Scholar
  10. [10]
    In the case of the adjoint representation, generated by the f α of Eq. (59) these equations reduce to Eqs. (58) together with γαβ f α f β = −λlr.Google Scholar
  11. [11]
    If more than one Majorana field is present (a complex spinor field counts as two) each will contribute its own \( - \frac{2}{3}\left( {\frac{n}{r}} \right)\Lambda \)] term to expression (132). If too many are present the theory will cease to be asympotically free.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • B. DeWitt
    • 1
  • C. Molina-París
    • 1
  1. 1.Center for Relativity and Department of PhysicsUniversity of TexasAustinUSA

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