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BRST Inner Product Spaces and the Gribov Obstruction

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BRST Inner Product Spaces and the Gribov Obstruction PITHA – 98/27 BRST Inner Product Spaces and the Gribov Obstruction Norbert DUCHTING¨ 1∗, Sergei V. SHABANOV2†‡, Thomas STROBL1§ 1 Institut f¨ur Theoretische Physik, RWTH Aachen D-52056 Aachen, Germany 2 Department of Mathematics, University of Florida, Gainesville, FL 32611–8105, USA Abstract A global extension of the Batalin–Marnelius proposal for a BRST inner product to gauge theories with topologically nontrivial gauge orbits is discussed. It is shown that their (appropriately adapted) method is applicable to a large class of mechanical models with a semisimple gauge group in the adjoint and fundamental representa- tion. This includes cases where the Faddeev–Popov method fails. arXiv:hep-th/9808151v1 25 Aug 1998 Simple models are found also, however, which do not allow for a well– defined global extension of the Batalin–Marnelius inner product due to a Gribov obstruction. Reasons for the partial success and failure are worked out and possible ways to circumvent the problem are briefly discussed. ∗email: [email protected] †email: [email protected]fl.edu ‡On leave from the Laboratory of Theoretical Physics, JINR, Dubna, Russia §email: [email protected] 1 Introduction and Overview Canonical quantization of gauge theories leads, in general, to an ill–defined scalar product for physical states. In the Dirac approach [1] physical states are selected by the quantum constraints. Assuming that the theory under consideration is of the Yang–Mills type with its gauge group acting on some configuration space, the total Hilbert space may be realized by square in- tegrable functions on the configuration space and the quantum constraints imply gauge invariance of these wave functions. Thus the physical wave func- tions must be constant along the orbits traversed by gauge transformations in the configuration space. Consequently, the norm of the physical states is infinite, if the gauge orbits are noncompact or if the number of nonphysical degrees of freedom is infinite as in gauge field theories. The problem is similar to that which occurs also in the path integral quantization of gauge theories where the integral over gauge field configura- tions diverges because of the gauge invariance of the action. In their seminal work [2] Faddeev and Popov proposed a solution based on the idea that in order for the path integral measure to be finite, only one representative of each gauge orbit should be taken into account. They provided a systematic way of implementing gauge conditions in the integral so that, by inclusion of an appropriate additional contribution to the measure, namely the Faddeev– Popov (FP) determinant, the resulting integral becomes independent of the choice of gauge and effectively ranges over the physical degrees of freedom only. However, if the gauge orbits possess a nontrivial topology, as often happens in physically interesting theories, a good choice of gauge turns out to be impossible. This deficiency is known as the Gribov obstruction [3, 4, 5]. It can be illustrated with simple mechanical models [6] that ignoring global deficiencies of a particular gauge can result in explicitly wrong predictions of the corresponding path integral quantization. 1 The FP path integral measure specifies uniquely the measure in the scalar product. In fact, the norm of the Dirac states can be made finite by reducing the initial measure to the gauge fixing surface and inserting the corresponding FP determinant to maintain the gauge invariance of the physical amplitudes. The Gribov obstruction will again be present, if the gauge orbits have a non- trivial topology. So, the FP approach would, in general, lead to an ill–defined scalar product. In many of these cases, one may, however, further modify the resulting FP inner product so as to obtain a finally reasonable norm for physical states. In the simplest case this is effected, e.g., by restricting the domain of integration along the gauge fixing surface to its modular domain (i.e. to a region which contains no points that are still gauge equivalent to others on that surface, their so–called Gribov copies). More recent suggestions to handle the norm regularization problem within the Dirac approach include a redefinition of the scalar product along the lines suggested in [7, 8] or the transition to the coherent state representation for constrained systems as performed in [9, 10]. In gauge field theories and especially in their path integral formulation, an explicit Lorentz invariance of the quantum theory is desired, which is not available in the Dirac Hamiltonian approach. The Lorentz invariance can, however, be achieved within the (nonminimal) BRST quantization program (see, e.g., [11]). BRST quantization is based on the extension of the original phase space by Lagrange multiplier and ghost sectors. For a set of first–class constraints a Ga one introduces the Lagrange multipliers y and their canonical momenta p a and adds canonical pairs ( a, ) and ( ¯a, ¯ ) of fermionic (Grassmann) y C Pa C Pa ghost and antighost variables, respectively. The extended phase space has a natural grading with respect to the ghost number operator N:[N, a] = C a, [N, ¯a] = ¯a, etc. Finally, the BRST charge Q is constructed. It is a C C −C 2 hermitian nilpotent operator (Q† = Q, Q2 = 0) such that, at least generically, the Dirac physical subspace formed by functions on the orbit space is iso- morphic to the subspace composed of elements of BRST–cohomology classes (usually at ghost number zero). That is, one looks for wave functions that are annihilated by Q, called BRST–closed, and identifies those differing by elements of the image of this operator (identification by BRST–exact states). Formally different representatives chosen from an equivalence class yield the same physical answers since s ( s + Q p )= s s , (1) h 1| | 2i | i h 1| 2i where we made use of Q s = 0 and the hermiticity of Q. However, it turns | 1i out that in practice the physical states (among others) often do not have a well–defined norm in the original (indefinite) Hilbert space. Typically, the norm is proportional to the meaningless factor 0. The infinity comes ∞· from the integration over the Lagrange multipliers, while zero results from the Berezin integral over the ghosts and antighosts (cf., e.g., [11] or Sec. 2 below, providing a simple illustration). So, such as in Dirac quantization, also in canonical BRST quantization there is a problem with the inner product [12], which, moreover, has not been resolved in generality up to present day. In this work we shall discuss an approach due to Batalin and Marnelius (B &M ) to this problem [13]. Their main idea is to not alter the original inner product, but to single out specifically chosen representatives in the BRST cohomology classes which then have a well–defined inner product among each other. They provide a scheme to construct a (hermitian) gauge fixing fermion Ψ and a space of so–called auxiliary states s , which, as we will | i0 see, resemble the physical (gauge invariant) states obtained in the ghost– free Dirac approach. The representatives of the cohomology which have a 3 well–defined inner product are then provided by the BRST singlets s := e[Q,Ψ]+ s . (2) | i | i0 More precisely, Batalin and Marnelius define the inner product of physical states, which are in a one–to–one correspondence with the states s , by | i0 means of s s′ = s exp (2[Q, Ψ] ) s′ , (3) h | i 0h | + | i0 which follows formally from the above representation of s and the analo- | i gous one for s′ when using (naive) hermiticity of Q and Ψ. Note that the | i auxiliary states s have a specific dependence on the ghost and nonphysical | i0 variables. So they are also called the ghost– and gauge–fixed states. The BRST transformed states (2) with a generic Ψ yield, at least up to global issues, the whole cohomology class represented by s . The conventional | i0 BRST inner product between quantum states is not always well–defined. The goal of Batalin and Marnelius was to develop a formalism for selecting a set of representatives s together with an adapted Ψ such that the resulting | i0 representatives s have a well–defined inner product with one another. The | i reason for introducing the states s on an intermediate level is that gener- | i0 ically they are much simpler than the states s , containing, e.g., no ghosts | i in an appropriate polarization and sometimes even coinciding literally with the states found in a Dirac quantization (cf. also the examples below in this paper). Arriving at formula (3) in this way, one implicitly has defined an inner product between the BRST cohomology classes (represented by s or s ). | i0 | i The B &M solution (2) of the BRST inner product cohomologies is local [13]. So the question of a global extendibility of their formalism arises. As follows from (3), the choice of gauge conditions — or, equivalently, the choice of the gauge fixing fermion Ψ — is an essential ingredient of the 4 B &M approach, such as it is in the FP procedure for defining an inner product between Dirac quantum states. In gauge theories with a nontrivial topology of the gauge orbits in the configuration space, there is, in general, a Gribov obstruction to a (globally well–defined) choice of a gauge condition, which can cause serious deficiencies of the FP inner product. Therefore one might expect an analogous problem within the B &M procedure. The aim of the paper is to investigate possible global obstructions to the B &M construction that might occur through a non–Euclidean gauge orbit space geometry [14], which has not yet been addressed.
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