# Area spectrum in Lorentz covariant loop gravity

###### Abstract

We use the manifestly Lorentz covariant canonical formalism to
evaluate eigenvalues of the area operator acting on Wilson lines.
To this end we modify the standard definition of the loop states
to make it applicable to the present case of non-commutative
connections. The area operator is diagonalized by using the
usual shift ambiguity in definition of the connection. The eigenvalues
are then expressed through quadratic Casimir operators. No dependence
on the Immirzi parameter appears.

PACS numbers: 04.20.Fy, 04.60.-m

## I Introduction

Quantization of gravity is an extremely hard and interesting problem which remains unsolved so far. During last years a number of approaches have achieved a definite progress in treating various aspects of quantum gravity. The most elaborated and popular line of research is string theory which includes perturbative gravity in its spectrum and unifies it with other interactions. An alternative (or, perhaps, complementary) approach is the loop quantum gravity [loop] (for review, see [Rov-dif]). This program relies on the Dirac canonical quantization. It is explicitly nonperturbative and background independent so realizing the basic principles of general relativity. During the previous decade this approach has got rigorous mathematical foundations [cyl] and has led to interesting qualitative predictions about quantum spacetime.

These predictions originate from remarkable results obtained in the framework of loop quantum gravity, which are calculations of the volume and area spectra [area]-[volume]. It appeared, however, that the area spectrum depends on the so called Immirzi parameter [imir]. It parametrises a canonical transformation [barb] which introduces a new connection field. The reason for this dependence is that this transformation cannot be realized unitarily in the Hilbert space of quantum theory [Rov-Tim]. In the language of quantum field theory this means presence of a quantum anomaly. There exist two different types of the quantum anomalies. The first type of the anomalies appear when a symmetry of the classical action cannot be preserved by quantization due to divergences or other quantum effects. Chiral and conformal anomalies belong to this type. Their presence indicates emergence of a new physics. The most celebrated example is the chiral anomaly in QCD which has been used for description of the low energy hadron physics since late 60’s. Rather naturally, it has been suggested [Rov-Tim] that the anomaly in the mentioned canonical transformation belongs to this type and, consequently, the Immirzi parameter is a new fundamental constant.

One cannot however exclude the second possibility. An anomaly could appear if a symmetry is involuntary broken by the choice of a particular quantization scheme. If this is the case, the remedy can be in applying another quantization scheme which explicitly preserves as much important symmetries as possible. This is the rout we take in the present paper by applying the manifestly Lorentz covariant quantization of [SA] to calculation of the area spectrum.

There are already some evidences that the Immirzi parameter dependence may disappear in a more symmetric quantization scheme. In the paper [SA] the path integral quantization scheme of [AV] has been extended to arbitrary values of the Immirzi parameter. It has been demonstrated, that the Immirzi parameter dependence does not appear in the path integral. We should stress, that in principle the path integral formalism is capable to see non-perturbative effects (as e.g. the virtual black hole formation [KuVa]). Another important result was obtained recently by Samuel [Sam] who demonstrated that the Barbero connection is not a Lorentz connection.

Recently, the importance for the theory to be Lorentz-covariant has been also recognized in spin foam models [sfLor] which represent the modern development of loop quantum gravity [sf]. However, the Lorentz covariance has been introduced there without any reference to the canonical quantization. It is an important task to develop a Lorentz covariant formulation “from the first principles”.

In this paper we apply the Lorentz covariant canonical quantization developed in [SA] to loop quantum gravity. We re-derive the spectrum of the area operator in the new framework. To this end we construct the Wilson line operator with true Lorentz connection. Since the Dirac brackets of the connections are non-zero, there is not connection representation. However, by choosing an appropriate vacuum state we are able to construct the quantum states corresponding to the Wilson lines which behave in a very similar way to the ordinary loop states. However, the area operator is not necessarily diagonal on these states. To diagonalize this operator we use the usual ambiguity in the connection: any connection can be shifted by a vector and will still remain a proper connection. It appears, that the shift is uniquely defined by the requirements that it vanishes on the constraint surface and that the area operator is diagonal on the Wilson line states. This new connection obeys a remarkably simple bracket algebra. Eigenvalues of the area operator are then calculated. They do not depend on the Immirzi parameter.

The paper is organized as follows. In the next section we summarize the covariant canonical formulation of [SA]. In sec. III we discuss the choice of the connection variables to be used in the Wilson line states. Area spectrum is calculated in sec. IV. Section V is devoted to discussion of the results, problems and future perspectives. Appendices are intended to list various definitions and useful properties.

We use the following notations for indices. The indices from the middle of the alphabet label the space coordinates. The latin indices from the beginning of the alphabet are the indices, whereas the capital letters from the end of the alphabet are the indices.

## Ii -covariant canonical formulation

In this section we review the covariant formalism developed in [SA]. It is a canonical formulation of general relativity based on the generalized Hilbert–Palatini action suggested by Holst [Holst]

(1) |

Here the star operator is defined as , and is the curvature of the spin-connection . A decomposition of the fields reads:

(2) | |||

Here is the inverse of . The field describes deviation of the normal to the spacelike hypersurface from the time direction.

Let us introduce matrix fields carrying one Lorentz index

which form multiplets in the adjoint representation of so(3,1). In Appendix A we present the relations between the triad multiplets and introduce the numerical matrices and (LABEL:p-q), (LABEL:pb-q) appearing in the formulas below. In terms of these fields the decomposed action can be represented in the form:

(4) | |||||

where are structure constants, . The indices are raised and lowered with the help of the Killing form

(5) |

The limit gives Ashtekar gravity. Even though the Hamiltonian constraint in (4) has apparently a pole at one can demonstrate [SA] that this limit is non-singular.

The canonical variables of the model are and , and are first class constraints obeying the algebra presented in Appendix C. We call them the Gauss law, diffeomorphism and Hamiltonian constraints respectively. There are also two sets of the second class constraints. They are represented by symmetric fields .

(6) | |||||

(7) | |||||

(8) |

Symmetrization is taken with the weight . Antisymmetrization includes no weight.

The existence of the second class constraints gives rise to the Dirac bracket [Dirac]

(9) |

where . The matrix of commutators of the second class constraints can be found in Appendix B. Both and are triangular. Due to this when one of the functions in (9), or is a first class constraint, the Dirac bracket coincides with the ordinary one (except for the case when and depends on the connection). In particular, this gives

(10) | |||||

Finally, the Dirac brackets of the canonical variables have the form: