The ${\varvec{SL}}(2,\mathbb {R})$ totally constrained model: three quantization approaches

Abstract

We provide a detailed comparison of the different approaches available for the quantization of a totally constrained system with a constraint algebra generating the non-compact $SL(2,\mathbb {R})$ group. In particular, we consider three schemes: the Refined Algebraic Quantization, the Master Constraint Programme and the Uniform Discretizations approach. For the latter, we provide a quantum description where we identify semiclassical sectors of the kinematical Hilbert space. We study the quantum dynamics of the system in order to show that it is compatible with the classical continuum evolution. Among these quantization approaches, the Uniform Discretizations provides the simpler description in agreement with the classical theory of this particular model, and it is expected to give new insights about the quantum dynamics of more realistic totally constrained models such as canonical general relativity.

Appendices

Appendix 1: Physical states: normalizable solutions

In this appendix, we will describe the spectral resolution of $\hat{H}$ adopting the treatment of Ref. [22]. Essentially, one starts with a representation of the positive and negative discrete series of $sl(2,\mathbb {R})$. Each representation is associated with the corresponding Hilbert spaces of holomorphic and anti-holomorphic functions on the open unit disc in $\mathbb {C}$, respectively, endowed with the scalar product, in both cases,

$$\begin{aligned} \langle f,h\rangle _l =\frac{l-1}{\pi }\int \limits _{D} f(z) \overline{h(z)} (1-|z|^2)^{l-2} dx\,dy \end{aligned}$$

(91)

where $D$ is the unit disc and $dx dy$ is the Lebesgue measure on $\mathbb {C}$. If $l=1$ one simply considers the limit $l\rightarrow 1$ in the previous expression.

For the positive series, an orthonormal basis is given by

$$\begin{aligned} f^{l}_n:=\Big [\mu _{l}(n)\Big ]^{-\frac{1}{2}}z^{n}\quad (n \in \mathbb {N})\quad { \mathrm {with}}\quad \mu _{l}(n)=\frac{\varGamma (n+1)\varGamma (l)}{\varGamma (l+n)} , \end{aligned}$$

(92)

while for the negative series, the corresponding basis is given by the complex conjugated of $f^{l}_n$. There exists also a unitary map between the polarized basis $\{|k_+,k_-,k'_+,k'_-\rangle \}$ and the basis provided by $f^{l}_n\otimes (f^{l'}_{n'})^*$, given by

$$\begin{aligned} U: f^{|j|+1}_n\otimes \left( f^{|j'|+1}_{n'}\right) ^* \mapsto&(-1)^{n'}|k_+,k_-,k'_+,k'_-\rangle \quad \mathrm {where} \nonumber \\&2 n=k_++k_--|j|\, , \quad j=k_+-k_- \, , \nonumber \\&2 n'=k'_++k'_--|j'|\, , \quad j'=-k'_++k'_- . \end{aligned}$$

(93)

In this representation the Master Constraint is a differential operator [22], whose eigenfunctions are of the form

$$\begin{aligned} f_{k,j,j'}(z_1,\overline{z}_2,t)=f_{k,j,j'}(z_1\overline{z}_2,t)\,z_1^{\frac{1}{2}(k-|j|+|j'|)}, \end{aligned}$$

(94)

where the solutions that are regular at $z=0$, with $z:=z_1\overline{z}_2$, are

$$\begin{aligned}&f_{k,j,j'}(z,t)=(1-z)^{1-t-\frac{1}{2}(|j|+|j'|+2)}\nonumber \\&\quad \times F\Big (1-t+\frac{1}{2}(-|j|+|j'|),1-t+\frac{1}{2}k,1+\frac{1}{2}(k-|j|+|j'|);z\Big ), \end{aligned}$$

(95)

for $k-|j|+|j'|\ge 0$, and

$$\begin{aligned}&f_{k,j,j'}(z,t)=(1-z)^{1-t-\frac{1}{2}(|j|+|j'|+2)} z^{\frac{1}{2}(-k+|j|-|j'|)} \nonumber \\&\quad \times F\Big (1-t-\frac{1}{2}k,1-t+\frac{1}{2}(|j|-|j'|),1+\frac{1}{2}(-k+|j|-|j'|);z\Big ), \end{aligned}$$

(96)

for $k-|j|+|j'|\le 0$, being $t=\frac{1}{2}(1+\sqrt{1-\lambda +2k^2}),\, \mathrm {Re}(t) \ge \frac{1}{2}$ and $F(a,b,c;z)$ the hypergeometric function [33].

Finally, we can use the map $U$ in (93) to transfer these results to the original kinematical Hilbert space ${\fancyscript{L}}^2(\mathbb {R}^4)$. To this end we rewrite (94) into a power series in $z_1$ and $\overline{z}_2$ using the definition of the hypergeometric function

$$\begin{aligned} F(a,b,c\,;z)=\frac{\varGamma (c)}{\varGamma (a)\varGamma (b)}\sum _{n=0}\frac{\varGamma (a+n)\varGamma (b+n)}{\varGamma (c+n)\varGamma (1+n)}z^n, \end{aligned}$$

(97)

and

$$\begin{aligned} (1-z)^{1-d}=\sum _{n=0}\frac{\varGamma (d+n-1)}{\varGamma (d-1)\varGamma (n+1)}z^n. \end{aligned}$$

(98)

For $k-|j|+|j'|\ge 0$ we obtain

$$\begin{aligned} f(t;k,j,j')&=U\Big (f_{k,j,j'}(z_1,\overline{z}_2,t)\Big )\nonumber \\&=\sum _{m=0} a_m \,|k_+(m),k_-(m),k'_+(m),k'_-(m)\rangle , \end{aligned}$$

(99)

where

$$\begin{aligned} k_+(m)&=m+\frac{1}{2}(k+j+|j'|),\quad k_-(m)=m+\frac{1}{2}(k-j+|j'|), \nonumber \\ k'_+(m)&=m+\frac{1}{2}(|j'|-j'),\quad k'_-(m)=m+\frac{1}{2}(|j'|+j'), \end{aligned}$$

(100)

and

$$\begin{aligned} a_m&=(-1)^m \left[ \mu _{(|j|+1)}\Big (m+\frac{1}{2}(k-|j|+|j'|) \Big )\right] ^{\frac{1}{2}} \left[ \mu _{(|j'|+1)}(m )\right] ^{\frac{1}{2}} \,\, \nonumber \\&\quad \times \frac{\varGamma \Big (1+\frac{1}{2}(k-|j|+|j'|) \Big )}{\varGamma \Big (1-t+\frac{1}{2}(-|j|+|j'|) \Big )\varGamma \Big (1-t+\frac{1}{2}k \Big )} \,\,\nonumber \\&\quad \times \,\sum _{l=0}^m \frac{\varGamma \Big (1-t+\frac{1}{2}(-|j|+|j'|)+l \Big ) \varGamma \Big (1-t+\frac{1}{2}k+l \Big )}{\varGamma \Big (1+\frac{1}{2}(k-|j|+|j'|)+l \Big )\varGamma \Big (1+l \Big )}\nonumber \\&\quad \times \frac{\varGamma \Big (t+\frac{1}{2}(|j|+|j'|) +(m-l)\Big )}{\varGamma \Big (m-l+1\Big )\varGamma \Big (t+\frac{1}{2}(|j|+|j'|) \Big )} . \end{aligned}$$

(101)

It is worth comment that replacing $k$ with $-k$, switching $|j|$ and $|j'|$ and multiplying with $(-1)^{\frac{1}{2}(-k+|j|-|j'|)}$, we obtain the coefficient $a_m$ for the solution corresponding to $k-|j|+|j'|\le 0$.

1.1 Normalizable eigenfunctions of $\hat{H}$:

Let us focus on the normalizable eigenfunctions of $\hat{H}$ fulfilling the condition $2t<|k|<\lambda ^H_\mathrm{discr}$. More precisely, we will start with those such that $k-|j|+|j'|\ge 0$. Since $|j|-|j'|=\pm 2t$ and $|k|>2t$, i.e. $k\pm 2t>0$, the only possibility is that of $k>0$. If we substitute this in (95) we get

$$\begin{aligned} f_{\pm }(z)=(1-z)^{-t\mp t-|j'|}F\left( 1-t\mp t,1-t+\frac{1}{2}k,1\mp t+\frac{1}{2}k;z\right) , \end{aligned}$$

(102)

where

$$\begin{aligned}&f_{+}(z)=(1-z)^{-2t-|j'|}F\left( 1-2t,1-t+\frac{1}{2}k,1- t+\frac{1}{2}k;z\right) \nonumber \\&\quad =(1-z)^{-|j'|-1}=\sum _{l=0}\frac{\varGamma \left( |j'|+l+1\right) }{\varGamma \left( |j'|+1\right) \varGamma \left( l+1\right) }z^{l}, \end{aligned}$$

(103)

and

$$\begin{aligned} f_{-}(z)&=(1-z)^{-|j'|}F\left( 1,1-t+\frac{1}{2}k,1+ t+\frac{1}{2}k;z\right) \nonumber \\&=\frac{\varGamma \left( 1+t+\frac{1}{2}k\right) }{\varGamma \left( 1-t+\frac{1}{2}k\right) }\sum _{n,l=0}\frac{\varGamma \left( 1-t+\frac{1}{2}k+n\right) }{\varGamma \left( 1+t+\frac{1}{2}k+n\right) }\frac{\varGamma \left( |j'|+l\right) }{\varGamma \left( |j'|\right) \varGamma \left( l+1\right) }z^{n+l}. \end{aligned}$$

(104)

In each case, the corresponding eigenfunction (94) is

$$\begin{aligned} f_{+}(z_1,\bar{z}_2)&=f_{+}(z_1\bar{z}_2)z_1^{\frac{1}{2}k-t}=\sum _{l=0}\frac{\varGamma \left( |j'|+l+1\right) }{\varGamma \left( |j'|+1\right) \varGamma \left( l+1\right) }{\bar{z}}_2^{l}\,z_1^{l-t+\frac{1}{2}k}, \end{aligned}$$

(105)

and

$$\begin{aligned} f_{-}(z_1,\bar{z}_2)&=f_{-}(z_1\bar{z}_2)z_1^{\frac{1}{2}k+t}=\frac{\varGamma \left( 1+t+\frac{1}{2}k\right) }{\varGamma \left( 1-t+\frac{1}{2}k\right) }\nonumber \\&\quad \times \sum _{n,l=0}\frac{\varGamma \left( 1-t+\frac{1}{2}k+n\right) }{\varGamma \left( 1+t+\frac{1}{2}k+n\right) }\frac{\varGamma \left( |j'|+l\right) }{\varGamma \left( |j'|\right) \varGamma \left( l+1\right) }\bar{z}_2^{n+l}\,z_1^{n+l+t+\frac{1}{2}k}. \end{aligned}$$

(106)

Let us now consider the case in which $k-|j|+|j'|\le 0$. Again, $|j|-|j'|=\pm 2t$ and $|k|>2t$. Since $k\pm 2t<0$, we conclude that $k<0$. Implementing all this in (96)

$$\begin{aligned}&f_{\pm }(z)=(1-z)^{-t\mp t-|j'|}z^{\pm t-\frac{1}{2}k}\times F\left( 1-t-\frac{1}{2}k,1-t\mp t,1\mp t-\frac{1}{2}k;z\right) \nonumber \\&\quad =(1-z)^{-t\mp t-|j'|}z^{\pm t+\frac{1}{2}|k|}F\left( 1-t+\frac{1}{2}|k|,1-t\mp t,1\mp t+\frac{1}{2}|k|;z\right) , \end{aligned}$$

(107)

or more specifically

$$\begin{aligned} f_{+}(z)&=(1-z)^{-2t-|j'|}z^{t+\frac{1}{2}|k|}F\left( 1-t+\frac{1}{2}|k|,1,1+ t+\frac{1}{2}|k|;z\right) \nonumber \\&=\frac{\varGamma \left( 1+t+\frac{1}{2}|k|\right) }{\varGamma \left( 1-t+\frac{1}{2}|k|\right) }\nonumber \\&\quad \times \sum _{n,l=0}\frac{\varGamma \left( 1-t+\frac{1}{2}|k|+n\right) }{\varGamma \left( 1+t+\frac{1}{2}|k|+n\right) }\frac{\varGamma \left( 2t +|j'|+l\right) }{\varGamma \left( 2t+|j'|\right) \varGamma \left( l+1\right) } z^{n+l+t+\frac{1}{2}|k|}, \end{aligned}$$

(108)

and

$$\begin{aligned}&f_{-}(z)=(1-z)^{-|j'|}z^{-t+\frac{1}{2}|k|}F\left( 1-t+\frac{1}{2}|k|,1-2t,1- t+\frac{1}{2}|k|;z\right) \nonumber \\&\quad =(1-z)^{2t-|j'|-1}z^{-t+\frac{1}{2}|k|}=\sum _{l=0}\frac{\varGamma \left( |j'|-2t+l+1\right) }{\varGamma \left( |j'|-2t+1\right) \varGamma \left( l+1\right) }z^{l+\frac{1}{2}|k|-t}. \end{aligned}$$

(109)

The corresponding eigenfunctions—see (94)—are

$$\begin{aligned}&f_{+}(z_1,\bar{z}_2)=f_{+}(z_1\bar{z}_2)z_1^{-\frac{1}{2}|k|-t}=\frac{\varGamma \left( 1+t+\frac{1}{2}|k|\right) }{\varGamma \left( 1-t+\frac{1}{2}|k|\right) }\nonumber \\&\quad \times \sum _{n,l=0}\frac{\varGamma \left( 1-t+\frac{1}{2}|k|+n\right) }{\varGamma \left( 1+t+\frac{1}{2}|k|+n\right) }\frac{\varGamma \left( 2t+|j'|+l\right) }{\varGamma \left( 2t+|j'|\right) \varGamma \left( l+1\right) }z_1^{n+l}\,\bar{z}_2^{n+l+t+\frac{1}{2}|k|}, \end{aligned}$$

(110)

and

$$\begin{aligned} f_{-}(z_1,\bar{z}_2)&=f_{-}(z_1\bar{z}_2)z_1^{-\frac{1}{2}|k|+t}\nonumber \\&=\sum _{l=0}\frac{\varGamma \left( |j'|-2t+l+1\right) }{\varGamma \left( |j'|-2t+1\right) \varGamma \left( l+1\right) }z_1^{l}\,{\bar{z}}_2^{l-t+\frac{1}{2}|k|}. \end{aligned}$$

(111)

The last step in our calculations consists of applying the unitary transformation (93) to the previous functions, and normalize them. The resulting eigenfunctions now read

$$\begin{aligned} f_\pm (t,k,j,j') = \sum _{m=0} a_{\pm ,m} |k_+(m),k_-(m),k'_+(m),k'_-(m)\rangle , \end{aligned}$$

(112)

with the coefficients $a_m$ given by

• (a) $k-|j|+|j'|\ge 0$ and $|j|-|j'|=2t$,

  $$\begin{aligned} a_{+,m}&=(-1)^{m}\left[ \mu _{(|j'|+2t+1)}\left( m-t+\frac{1}{2}k\right) \right] ^{\frac{1}{2}} \left[ \mu _{(|j'|+1)}(m )\right] ^{\frac{1}{2}}\nonumber \\&\quad \times \frac{\varGamma \left( |j'|+m+1\right) }{\varGamma \left( |j'|+1\right) \varGamma \left( m+1\right) }. \end{aligned}$$

  (113)

• (b) $k-|j|+|j'|\ge 0$ and $|j|-|j'|=-2t$,

  $$\begin{aligned}&a_{-,m}=(-1)^m\left[ \mu _{(|j'|-2t+1)}\left( m+t+\frac{1}{2}k\right) \right] ^{\frac{1}{2}} \left[ \mu _{(|j'|+1)}(m )\right] ^{\frac{1}{2}} \nonumber \\&\quad \times \frac{\varGamma \left( 1+t+\frac{1}{2}k\right) }{\varGamma \left( 1-t+\frac{1}{2}k\right) }\sum _{l=0}^m\frac{\varGamma \left( 1-t+\frac{1}{2}k+l\right) }{\varGamma \left( 1+t+\frac{1}{2}k+l\right) }\frac{\varGamma \left( |j'|+m-l\right) }{\varGamma \left( |j'|\right) \varGamma \left( m-l+1\right) }. \end{aligned}$$

  (114)

• (c) $k-|j|+|j'|\le 0$ and $|j|-|j'|=2t$,

  $$\begin{aligned}&a_{+,m}=(-1)^m\left[ \mu _{(|j'|+2t+1)}\left( m\right) \right] ^{\frac{1}{2}} \left[ \mu _{(|j'|+1)}(m+t+\frac{1}{2}|k| )\right] ^{\frac{1}{2}} \nonumber \\&\times \frac{\varGamma \left( 1+t+\frac{1}{2}|k|\right) }{\varGamma \left( 1-t+\frac{1}{2}|k|\right) }\sum _{l=0}^{m}\frac{\varGamma \left( 1-t+\frac{1}{2}|k|+l\right) }{\varGamma \left( 1+t+\frac{1}{2}|k|+l\right) }\frac{\varGamma \left( 2t+|j'|+m-l\right) }{\varGamma \left( 2t+|j'|\right) \varGamma \left( m-l+1\right) }. \end{aligned}$$

  (115)

• (d) $k-|j|+|j'|\le 0$ and $|j|-|j'|=-2t$,

  $$\begin{aligned} a_{-,m}&=(-1)^m\left[ \mu _{(|j'|-2t+1)}\left( m\right) \right] ^{\frac{1}{2}} \left[ \mu _{(|j'|+1)}(m-t+\frac{1}{2}|k| )\right] ^{\frac{1}{2}}\nonumber \\&\quad \times \frac{\varGamma \left( |j'|-2t+m+1\right) }{\varGamma \left( |j'|-2t+1\right) \varGamma \left( m+1\right) }. \end{aligned}$$

  (116)

The normalized eigenfunctions are finally defined as

$$\begin{aligned} |j,j'\rangle _{t,k}:= \sum _{m=0}\tilde{a}_m |k_+(m),k_-(m),k'_+(m),k'_-(m)\rangle , \end{aligned}$$

(117)

with

$$\begin{aligned} \tilde{a}_m =a_m\left( \sum _{l=0}|a_l|^2\right) ^{-1}. \end{aligned}$$

(118)

The states belonging to the infrarred spectrum of $\hat{H}$ solve the three constraints $\hat{H}_\pm$ and $\hat{D}$ when quantum corrections of the Planck order are neglected.

1.2 Algebraic quantization and physical states

Given the solutions (94) to the Master Constraint, one can ask which is the relation between the states annihilated by $\hat{M}$ and the solutions (19) found within the Algebraic Quantization approach.

They can be easily computed by means of (94) just solving the equation $\hat{M} \varPsi =0$. In this case, we set $t=1$ and $k=0$ in (94). After applying the map (93), the resulting solutions are

$$\begin{aligned} \!\!f(t=1;k=0,j=m,j'=\varepsilon m)= \sum _{l=0} (-1)^l |k_+(l),k_-(l),k'_+(l),k'_-(l)\rangle , \end{aligned}$$

(119)

where $f(t=1;k=0,j=m,j'=\varepsilon m)=\varPsi _{m,\varepsilon }$. These states solve simultaneously the three constraints $\hat{H}_\pm$ and $\hat{D}$. Nevertheless, they do not belong to ${\fancyscript{H}}_\mathrm{kin}$. Hence additional considerations are necessary in order to endow them with Hilbert space structure.

Appendix 2: Constraint observable algebra

In this appendix we will study several properties of the quantum constraints $\hat{H}_+$ and $\hat{D}$ defined in Eq. (38). In particular, we are interested in the determination of their action on the eigenfunctions of $\hat{H}$. This space of states {$|q_3,p_3\rangle _{tk}$} or {$|x,k,q_3,p_3\rangle$} is characterized by the eigenvalue of the Casimir $\hat{\fancyscript{C}}$ which are labeled by $t$ or $x$, depending if it corresponds to the discrete or the continuous spectrum; $k$, which is the eigenvalue of the constraint $\hat{H}_-$; and $\lambda _H$, the eigenvalue of the Hamiltonian $\hat{H}$. For simplicity, we will restrict the study to {$|q_3,p_3\rangle _{tk}$}, but the very same conclusions are also valid for {$|x,k,q_3,p_3\rangle$}.

Since the quantum constraints fulfill the commutation relations (30), it seems natural to introduce the ladder operators

$$\begin{aligned} \hat{K}_\pm = \hat{H}_+\pm i\hat{D}. \end{aligned}$$

(120)

Their commutators with the constraint $\hat{H}_-$ can be straightforwardly deduced by means of the commutation relations (30), yielding

$$\begin{aligned}{}[\hat{H}_-,\hat{K}_\pm ]=\pm 2\hat{K}_\pm . \end{aligned}$$

(121)

Now, using these commutators, one can easily conclude that the operators $\hat{K}_\pm$ acting on states of the form $|q_3,p_3\rangle _{tk}$ shift the label $k$ in two units, that is,

$$\begin{aligned} H_-(\hat{K}_\pm |q_3,p_3\rangle _{tk})=(k\pm 2)\hat{K}_\pm |q_3,p_3\rangle _{tk}. \end{aligned}$$

(122)

Moreover, the square of the norms of $\hat{K}_\pm |q_3,p_3\rangle _{tk}$ fulfill

$$\begin{aligned} ||\hat{K}_\pm |q_3,p_3\rangle _{tk}||^2=(2\lambda _h-k^2\pm 2k)||\;|q_3,p_3\rangle _{tk}||^2. \end{aligned}$$

(123)

This result, together with the relations (38) and the reality of the coefficients $\tilde{a}_{m}$ in Eq. (117) for the normalized eigenfunctions $|q_3,p_3\rangle _{tk}$, allow us to conclude that the action of the operators $\hat{K}_\pm$ is given by

$$\begin{aligned} \hat{K}_\pm |q_3,p_3\rangle _{tk}=(-1)^{r_\pm (q_3,p_3,k,t)}\sqrt{2\lambda _h-k^2\pm 2k}|q_3,p_3\rangle _{t(k\pm 2 )}, \end{aligned}$$

(124)

with $r_\pm (q_3,p_3,k,t)$ some integers that can depend on the corresponding quantum numbers, such that $r_\pm (q_3,p_3,k,t)+r_\mp (q_3,p_3,k\pm 2,t)$ are even integers. It is worth commenting that the action of $\hat{K}_\pm$ on the eigenstates {$|q_3,p_3\rangle _{tk}$} with $k=\pm 2t$, respectively, is by annihilation. For the generalized eigenfunctions {$|x,k,q_3,p_3\rangle$}, it only happens for $k=\pm 1$ and $x=0$.

Now, a straightforward calculation allows us to conclude that

$$\begin{aligned} \hat{H}_+=\frac{1}{2}(\hat{K}_++\hat{K}_-),\quad \hat{D} = \frac{i}{2}(\hat{K}_--\hat{K}_+). \end{aligned}$$

(125)

Therefore, the action of the constraints on the solution space provided by condition (59) mixes states with labels $k\pm 2$.