Comment on "New variables for 1+1 dimensional gravity"

TL;DR

This paper clarifies that the results by Gambini, Pullin, and Rastgoo are specific instances within a broader framework for canonical variables in dilaton gravity models, emphasizing the generality of the latter approach.

Contribution

It demonstrates that previous specific results are special cases of a more general canonical variable treatment for dilaton gravity.

Findings

Previous results are special cases of a general framework.

The general treatment encompasses earlier specific models.

Clarifies the scope of canonical variables in 1+1 dimensional gravity.

Abstract

The results reported by Gambini, Pullin and Rastgoo in (2010 Class. Quantum Grav. 27 025002) are special cases of a general treatment of canonical variables for dilaton gravity models published in (2009 Class. Quantum Grav. 26 035018).

Equations

$$S=-\frac{1}{2G}{\int }_M\left(\varphi \mathrm{d}\omega +\frac{1}{2}V(\varphi )ϵ+X_a\mathrm{D}{e}^a\right)$$

$$S=\frac{1}{2G}{\int }_M\left(e^a\wedge \mathrm{d}{X}_a+\omega \mathrm{d}\varphi -X_a{ϵ}^a{}_b\omega \wedge e^b-\frac{1}{2}V(\varphi )ϵ\right).$$

$$\{X^i(x),A_j(y)\}=2G{\delta }_j^i\delta (x-y),$$

$${\tilde{C}}^i=\frac{1}{2G}\left((X^i{)}^{\prime }+P^{ij}{A}_j\right)$$

$$P=\left(\begin{array}[]{ccc}0&-\frac{1}{2}V(\phi)&-X^{-}\\ \frac{1}{2}V(\phi)&0&X^{+}\\ X^{-}&-X^{+}&0\end{array}\right)\,.$$

$$X:=\sqrt{X^{+}{X}^{-}}\text{mbox}ande:=\sqrt{e_x^{+}{e}_x^{-}}$$

$$X^{\pm }=X\exp (\pm \beta )\text{mbox}and{e}_x^{\pm }=e\exp (\pm \alpha ).$$

$$Q^e=2X\cosh (\alpha -\beta )\text{mbox}and{Q}^{\alpha }=2eX\sinh (\alpha -\beta )$$

$$\{Q^e(x),e(y)\}=\{Q^{\alpha }(x),\alpha (y)\}=\{\varphi (x),{\omega }_x(y)\}=2G\delta (x-y).$$

$$X^{\pm }=\frac{e{Q}^e\mp Q^{\alpha }}{2e}\exp (\pm \alpha ).$$

$${\tilde{C}}^{\pm }=\frac{1}{2G}\left({\left(\frac{e{Q}^e\mp Q^{\alpha }}{2e}\right)}^{\prime }\pm \frac{e{Q}^e\mp Q^{\alpha }}{2e}({\omega }_x+{\alpha }^{\prime })\pm \frac{1}{2}V(\varphi )e\right)\exp (\pm \alpha )$$

$${\tilde{C}}^3=\frac{1}{2G}({\varphi }^{\prime }+Q^{\alpha }).$$

$$\{K_x(x),E^x(y)\}=2G\delta (x-y),\{K_{\phi }(x),E^{\phi }(y)\}=G\delta (x-y).$$

$$Q^e$$

$$Q^{\alpha }$$

$${\omega }_x$$

$$H[N]$$

Full text

Different sets of canonical variables for $1+1$-dimensional models of gravity without local physical degrees of freedom have been discussed in [1]. These variables are related to connection formulations as used in classical theories underlying loop quantum gravity. All models of this form are contained in the class of 2-dimensional dilaton models, or equivalently in the class of Poisson Sigma models [3, 4, 5]. Since these classes had already been formulated in terms of connection variables [2], there should be a strict relation between the different sets of variables. In this comment we work out the relationship.

We start with standard formulations of $1+1$-dimensional actions for a dyad $e^{a}$ with volume form $\epsilon$, a connection 1-form $\omega$ and a dilaton field $\phi$. For our purpose here, it suffices to consider torsion-free models, such that the condition ${\rm D}e^{a}=0$, using the covariant derivative given by $\omega$, is implemented by Lagrange multipliers $X_{a}$. The dilaton gravity action with potential $V(\phi)$ is then

[TABLE]

and takes, after integrating by parts, the form

[TABLE]

Here, $M$ is a $1+1$-dimensional manifold with coordinates $(t,x)$.

In Poisson Sigma models, one organizes the variables in new sets $X^{i}=(X^{-},X^{+},\phi)$, $A_{i}=(e_{x}^{+},e_{x}^{-},\omega_{x})$, and $\Lambda_{i}=(e_{t}^{+},e_{t}^{-},\omega_{t})$. A canonical analysis leads to Poisson brackets

[TABLE]

while the $\Lambda_{i}$ serve as Lagrange multipliers of first-class constraints

[TABLE]

with the Poisson tensor [3, 4, 5]

[TABLE]

The variables introduced in [2] are obtained by using the “absolute values”

[TABLE]

and boost parameters $\alpha$ and $\beta$ in

[TABLE]

They are related to the canonical variables

[TABLE]

with

[TABLE]

The inverse transformation is

[TABLE]

In canonical variables, the constraints are

[TABLE]

and

[TABLE]

For spherically symmetric gravity, corresponding to a specific dilaton potential $V(\phi)=-2/\sqrt{\phi}$ [5], one can compare the new canonical variables to those used in real connection formulations such as [6], denoted as $(K_{\varphi},E^{\varphi};K_{x},E^{x};\eta,P^{\eta})$ and with Poisson brackets

[TABLE]

(Note that there is no factor of two in the second equation because the pair $(K_{\varphi},E^{\varphi})$ represents two angular directions which were independent before symmetry reduction.) They are related to $(Q^{e},e;Q^{\alpha},\alpha;\phi,\omega_{x})$ by the canonical transformation

[TABLE]

(Unlike the pair $(K_{\varphi},E^{\varphi})$, the pair $(e,Q^{e})$ is defined with a factor of two in the Poisson bracket (9).) These relations have been derived in [2]; see Eq. (42) in this paper. Also in [2], the spherically symmetric Hamiltonian constraint has been obtained as

[TABLE]

For an arbitrary dilaton potential, this formulation generalizes the connection formulation of spherically symmetric canonical gravity to arbitrary $1+1$-dimensional dilaton models. Deriving just this kind of result was the aim of [1]. In fact, the definitions of [1] are nothing but the results of [2] with different names chosen for the new variables. The expression for $e$ in (14) is the same as (51) in [1], $Q^{e}$ in (14) is (57), and $\omega_{x}$ in (16) is (60).

There is a different formulation in [1] for the specific case of the CGHS model (constant dilaton potential). This model, as presented in [1] has a Hamiltonian constraint with only $K_{x}K_{\varphi}$ in its kinetic part but no contribution of $K_{\varphi}^{2}$, unlike what we have in (S0.Ex2). However, this formulation is not new either, even though it does not correspond to a connection formulation as defined in [2]. It rather amounts to using the original canonical variables (8) of dilaton models, along with $\omega_{x}$, and just renaming them as $\omega_{x}\equiv K_{x}$, $Q^{e}\equiv K_{\varphi}$ and $e\equiv E^{\varphi}$. Except for the new names, these variables, in particular $Q^{e}$, have been introduced in [2]; see Eq. (20) in this paper. Except for using the SU(1,1)-invariant $Q^{e}$, they correspond to a first-order formulation of Poisson Sigma models. The constraints (11) in these variables depend on $\omega_{x}$ in a linear fashion, and so does the Hamiltonian constraint obtained by a linear combination. In these variables, therefore, the Hamiltonian constraint has only one term $Q^{e}\omega_{x}=K_{x}K_{\varphi}$ but not contribution from $K_{\varphi}^{2}$ (which would amount to $(Q^{e})^{2}$, but there is no such term in (11)). The quadratic term is introduced in (S0.Ex2) by applying the canonical transformation (14) and (16) via the product $Q^{e}\omega_{x}$ in (11).

Even though [1] cites [2], it misses the close relationship between the results.

Acknowledgements

This work was supported in part by NSF grant PHY-1607414.