Functional Integration pp 327-361 | Cite as

# Gauge Theory without Ghosts

## 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## 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

- Q
_{α} Fiber generators (pointwise linearly independent vertical vector fields)

*c*^{γ}_{αβ}Structure constants of

*G**S*Classical action functional

*F*ΦFrame bundle over Φ

- Γ
_{γ}^{i}_{jk} Pullback to Φ of Riemannian connection on

*F*Φ- ω
^{α}_{i} Connection 1-form on Φ (defined uniquely by γ and the Q

_{α})- Π
^{i}_{j} Horizontal projection operator

*I*^{A},*K*^{α}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*- Γ
^{A}_{BC} Pullback to Φ/

*G*of Riemannian connection on*F*(Φ/*G*)- Γ
^{i}_{jk} Vilkovisky’s connection (extension of Γ

^{ A }_{ BC }to Φ)- φ
^{i} Arbitrary local field variables

- ϕ
^{a} Gaussian normal fields

- Γ
Effective action

- μ[φ
_{*}, ϕ] Measure for the functional integral defining Γ

- Σ
Γ —

*S*(loop contribution to Γ)*A*^{α}_{μ}Gauge potentials for Yang—Mills theory

- γ
_{αβ} Cartan—Killing metric

*F*^{α}_{μν}Yang—Mills curvature 2-form

*f*^{α}_{βγ}Structure constants of the Yang—Mills Lie group

*f*_{α}Generators of the adjoint representation: (

*f*^{β}_{αγ})*Q*^{α}_{μβ′}Fiber generators for the Yang—Mills field

- γ
_{α}^{μ}_{β′}^{ν′} Ultralocal invariant metric for the Yang—Mills field

- Π
^{α}_{μβ}^{ν′} 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 Γ

_{γ}^{ i }_{ jk }- ;
Denotes covariant functional differentiation based on Vilkovisky’s connection Γ

^{ i }_{ jk }

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## References

- [1]R. P. Feynman,
*Acta Physica Polonica***24**, 697 (1963). The work presented in this paper was carried out in 1962.MathSciNetGoogle Scholar - [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]The question of Lorentz invariance arises because Feynman was assuming spacetime to be asymptotically Minkowskian.Google Scholar
- [4]B. DeWitt,
*Dynamical Theory of Groups and Fields*(Gordon and Breach, 1965).Google Scholar - [5]The linear independence of the
**Q**_{α}and the invertibility and ultralocality of γ*i, j*guarantee that F_{αβ}will be an invertible operator, typically differential or pseudodifferential.Google Scholar - [6]G. A. Vilkovisky in
*Quantum Theory of Gravity*, éd. S. M. Christensen (Adam Hilger, 1984).Google Scholar - [7]
- [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]J. C. Collins,
*Renormalization*(Cambridge University Press, 1984).Google Scholar - [10]In the case of the adjoint representation, generated by the
*f*_{α}of Eq. (59) these equations reduce to Eqs. (58) together with γ^{αβ}*f*_{α}*f*_{β}= −λl_{r}.Google Scholar - [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