# A Canonical Approach to the Einstein-Hilbert Action in Two Spacetime Dimensions

###### Abstract

The canonical structure of the Einstein-Hilbert Lagrange density is examined in two spacetime dimensions, using the metric density and symmetric affine connection as dynamical variables. The Hamiltonian reduces to a linear combination of three first class constraints with a local algebra. The first class constraints are used to find a generator of gauge transformations that has a closed off-shell algebra and which leaves the Lagrangian and invariant. These transformations are distinct from diffeomorphism invariance, and are gauge transformations characterized by a symmetric matrix .

###### pacs:

11.10.EfThe canonical structure of the -dimensional Einstein-Hilbert (EH) action has been examined for some time Dirac1958 ; ADM . The two dimensional (2D) version of this action merits attention for the insight it can provide, even though when expressed solely in terms of the metric tensor the Lagrangian reduces to a total derivative and there are consequently no physical degrees of freedom. There has been interest in analyzing the structure of this theory, despite its topological nature Teitelboim ; Labeotida ; Montano .

As dynamical variables, we select the metric density and symmetric affine connection , as was done originally by Einstein Ein1925 (though this is often called the Palatini approach Pal ). We do not parameterize in a way that singles out the dynamics on a particular spatial surface, as was done in Dirac1958 ; ADM . Upon using these variables in 2D, we show that it is particularly easy to apply the Dirac constraint formalism Dirac1964 to analyze the canonical structure of the classical action . Without having to even partially fix a gauge, we find that the Hamiltonian reduces to a linear combination of three secondary first class constraints. Unlike the constraints in the Dirac-ADM approach Dirac1958 ; ADM for Teitelboim , these constraints obey an algebra with field independent structure constants; it is a local algebra. A local algebraic structure has also been found in dilaton gravity but with field-dependent structure constants GrumKumVas2002 . A model of 2-dimensional gravity with an gauge symmetry appears in Cham , though this model also involves an auxiliary scalar field.

From the full set of first class constraints (both primary and secondary), a generator of gauge transformations involving three local gauge parameters can be constructed using the approach of Castellani Cast1982 . The generator obeys a closed algebra, even off-shell, and results in a gauge transformation that leaves , and the equations of motion invariant. The gauge transformation is distinct from the usual diffeomorphism transformation.

The EH action in the first order formulation is

(1) |

The Lagrange density of (1) is polynomial of order three in the components of and components of which are all treated as being independent Ein1925 . This formulation is well suited to a canonical analysis of because second order derivatives do not appear at the outset in the action. There are special features associated with the first order formalism when Gegenberg ; Lindstrom ; Deser88 ; Deser95 . In this case, the metric obeys the constraint and the equation of motion for does not uniquely fix it to be the Christoffel symbols .

We introduce generalized momenta conjugate to all independent variables

(2) |

Each of these equations constitutes a primary constraint

(3) |

The total Hamiltonian is

(4) |

where and are undetermined multipliers and is the usual canonical Hamiltonian given by

(5) |

(Latin indices are spatial.) The fundamental Poisson brackets (PB) for canonical variables are

(6) |

where .

Time independence of the primary constraints of (3) can either lead to new (secondary) constraints or fixing of some the multipliers in (4). The rank of the matrix constructed from the PB of primary constraint

(7) |

(where ) corresponds to the number of multipliers that are determined and the independent eigenvectors with zero eigenvalue produce the secondary constraints. The only non-zero PB among the primary constraints is

(8) |

From the matrix of (7) one can now find the secondary constraints. These secondary constraints must also be time independent; this may imply further tertiary constraints.

In the case the canonical analysis simplifies considerably as in this instance is linear in variables whose associated momenta are first class primary constraints. (For the dependance on such variables is non-linear, complicating the analysis considerably by leading to tertiary constraints.) We have found that for , the Dirac analysis shows that there are only primary and secondary constraints, and of these, six are first class and six second class. (When combined with six gauge conditions, this serves to eliminate all eighteen canonical degrees of freedom associated with , and their conjugate momenta).

After having made this analysis, second class constraints of the form can be used to explicitly eliminate the degrees of freedom associated with and by setting and equal to in the Hamiltonian and remaining constraints Dirac1964 ; this simplifies our analysis of gravity in two or more dimensions.

Equation (3) gives nine primary constraints

(9) |

(10) |

(11) |

The matrix (7) for the primary constraints (9-11) has rank six; hence there are six second class constraints (those of (9,10) which we group into three pairs of the special form and (the last pair could also be taken to be )). Elimination of these constraints by setting and converts the Hamiltonian (5) and the remaining constraints (11) into

(12) |

where

(13) |

and, after some rearrangement of terms,

(14) |

The phase space now consists of only ’s and their corresponding momenta ’s with the standard fundamental PB (6). As the time derivative of the primary constraints (13) must vanish we obtain three secondary constraints

(15) |

(16) |

(17) |

This converts the Hamiltonian (14) into linear combination of secondary constraints.

All primary constraints have a zero PB among themselves and with the secondary constraints . The only non-zero PB among the constraints are local:

(18) |

Upon replacing the classical PB by (quantum commutator), this becomes the Lie algebra of .

The approach of Castellani Cast1982 can be used to find the form of the gauge transformations implied by the six first class constraints. The generator is found by first setting and then examining The functions are found by requiring that vanish when the primary constraints vanish. The generator of the gauge transformation is given by , which in our case leads to the following expression

(19) |

The PB of these generators have an algebra that closes off shell

(20) |

where , , and the only non-zero structure functions are , , .

If we compute the gauge transformation of fields by taking their PB with the generator , then the Lagrangian is invariant under these transformations only when the constraints themselves vanish and the fields are on shell as in the Dirac-ADM formulation of gravity Cast1982 . However, by a contact transformation that correponds to a slight change in the choice of dynamical variables in the 2D EH action, it is possible to find gauge transformations that leave the Lagrangian invariant even off the constraint surface and when fields are off shell. We make a linear change of variables, suggested by (14),

(21) |

so that the Lagrangian becomes

(22) |

with

We can now repeat the Dirac procedure starting from (22). Introducing momenta for the variables (21) with the non-vanishing PB we obtain nine primary constraints, those of (9), and in addition,

(23) |

(24) |

As before, after the elimination of the primary second class constraints (9) and (24) we obtain the total Hamiltonian in reduced phase space with the three primary constraints (23),

(25) |

where

(26) |

The time derivatives of the remaining primary constraints vanish if we have the secondary constraints

(27) |

(28) |

(29) |

whose algebra is

(30) |

This algebra is identical to (18) and ensures that the time derivatives of all secondary constraints weakly vanish. The gauge generator becomes

(31) |

This too satisfies the algebra (20). It can be shown that now as for Yang-Mills theory Cast1982 . Consequently, the variables (21) lead to a local algebra of constraints with field independent structure constants and a closed algebra of generators off the constraint surface and off shell that generates transformations which leave invariant off shell. We thus have a truly canonical formulation of the 2D EH action.

gives rise to the transformations

(32) |

(33) |

Using these transformations it is straightforward calculation to demonstrate that under (33) the 2D action (1) is invariant, . This is an exact result as in Yang-Mills theory (not just on the constraint surface or on shell as in the case of generators of general coordinate transformations in gravity, derived from the Dirac-ADM constraint analysis of the EH action Cast1982 ). The transformation of (33) is distinct from a diffeomorphism, which is immediately apparent as it is characterized by three rather than two parameters. Indeed, (33) can be rewritten in a way that resembles the transformations appearing in Deser88 if we use the antisymmetric tensor and affine covariant derivatives :

(34) |

(35) |

where and .

We note that as in dimensions, , and hence in 2D, there is the extra condition

(36) |

However, with the generator (31), it is easy to show that and hence our formalism is consistent with (36). Supplementing the action with a term leads to a pair of primary constraints where and are momenta conjugate to and respectively. There then follows the secondary constraint . The constraint can be associated with the gauge condition ; and are second class constraints. Thus, there is no net change in the number of degrees of freedom in the model as a result of imposing the condition - there are still zero degrees of freedom left after all constraints are applied. (In some models of two dimensional gravity, there are a negative number of degrees of freedom Martinec ; Polchinski .)

If instead of using as a dynamical field, we were to use the metric , then it would not be necessary to impose (36). As will be reported elsewhere, the use of in place of results in there being seven first class constraints and gauge generator involving five independent functions, though the all canonical properties (such as having a local algebra and possessing off shell invariance, etc.) are the same as in the formalism developed above.

Application of the procedure employed in this letter to the EH action in dimensions is being considered; it is expected that the simple canonical properties present in ordinary gauge theories could be preserved in these dimensions as well. This would provide an interesting alternative approach to quantum gravity.

This work is supported in part by funds provided by NSERC.

D.G.C.McKeon would like to thank Perimeter Institute for hospitality while part of this work was completed and R. and D. MacKenzie for helpful advice.

## References

- (1) P.A.M.Dirac, Proc.Roy.Soc. A246, 333 (1958).
- (2) R.Arnowitt, S.Deser and C.W.Misner, Phys.Rev. 116, 1322 (1959).
- (3) C.Teitelboim, Phys.Lett, B126, 41 (1983).
- (4) J.Labastida, M.Pernici and E.Witten, Nucl.Phys. B110, 611 (1988).
- (5) D.Montano and J.Sonnenschein, Nucl.Phys. B313, 258 (1989).
- (6) A.Einstein, Sitzungsber.preuss.Akad.Wiss., phys.-math. K1, 414 (1925) and The complete collection of scientific papers (Nauka, Moskow, 1966), v.2, p.171.
- (7) A.Palatini, Rendiconti del Circolo Mathetico di Palermo, 43, 203 (1919) and in Cosmology and Gravitation, edited by P.G.Bergmann and V. De Sabbata (Plenum press, New York, 1979), p.477; M.Ferraris and M.Francaviglia, Gen.Rel. and Grav. 14, 243 (1982).
- (8) P.A.M.Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964).
- (9) D.Grumiller, W.Kummer, and D.V.Vassilevich, Phys.Rep. 369, 327 (2002).
- (10) A.H.Chamseddine, Phys.Lett. B256, 379 (1991).
- (11) L.Castellani, Ann.Phys. 142, 357 (1982).
- (12) J.Gegenberg, P.F.Kelly, R.B.Mann and D.Vincent, Phys.Rev. D37, 3463 (1987).
- (13) U.Lindström and M.Rocek, Class.Quant.Grav. 4, L79 (1987).
- (14) S.Deser, J.McCarthy and Z.Yang, Phys.Lett. B222, 61 (1988).
- (15) S.Deser, gr-qc 9512022.
- (16) E.Martinec, Phys.Rev. D30, 1198 (1984).
- (17) J.Polchinski, Nucl.Phys. B324, 123 (1989).