A note on the Holst action, the time gauge, and the Barbero–Immirzi parameter

Abstract

In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero–Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero–Immirzi parameter as a potential way to investigate its fate and relevance in the quantum theory.

Appendices

Appendix A: Dirac bracket in the time gauge

An alternative to using the explicit resolution of second class constraints $S^i\approx 0$ (2.20b) and $\Psi ^{ab}\approx 0$ (2.26) that leads to the Levi–Civita connection (2.27), is to work with the Dirac bracket. Its computation in the time gauge could be particularly interesting to establish a contact with the covariant formulation mainly developed by Alexandrov [4–7]. Furthermore, by using the Dirac bracket instead of the explicit resolution $\omega ^i_a=\Gamma ^i_a(e)$, we can somehow keep at hand all the components of the connection, i.e. the rotational and the boost part, without singling out the rotational part by solving it in terms of the tetrad field.

To compute the bracket, it is convenient to use the freedom in choosing the parametrization of the second class constraints in order to replace the 3 constraints $S^i$ and the 6 constraints $\Psi ^{ab}$ by the 9 equivalent constraints

$$\begin{aligned} T^i_a\equiv \varepsilon ^{bc}_{~~a}(\partial _be^i_c-\varepsilon ^i_{~jk}\omega _b^je_c^k)\approx 0. \end{aligned}$$

(5.1)

The set of second class constraints is then given by $\pi ^a_i\approx 0$ and $T^i_a\approx 0$, and we can compute the Dirac matrix along with its inverse:

$$\begin{aligned} \Delta ^{ij}_{ab}&\equiv \big \lbrace \pi ^a_i,T^j_b\big \rbrace =-\varepsilon ^{ij}_{~~k} \varepsilon _{ab}^{~~c}e^k_c=\frac{2}{\sqrt{\det (E)}}E^b_{[i}E^a_{j]}, \nonumber \\ (\Delta ^{-1})^{jk}_{bc}&= \frac{1}{2\det (e)}(2e^j_ce^k_b-e^k_ce^j_b). \end{aligned}$$

(5.2)

Note that it is much simpler to compute the Dirac bracket with the constraints $T^i_a\approx 0$ instead of using the original set.

The Dirac bracket between any two phase space functions $f$ and $g$ is defined as

$$\begin{aligned} \big \lbrace f,g\big \rbrace _\text{ D }=\big \lbrace f,g\big \rbrace -\big \lbrace f,\pi ^a_i\big \rbrace (\Delta ^{-1})^{ij}_{ab}\big \lbrace T^j_b,g\big \rbrace -\big \lbrace f,T^i_a\big \rbrace (\Delta ^{-1})^{ij}_{ab}\big \lbrace \pi _j^b,g\big \rbrace . \end{aligned}$$

(5.3)

Among all the canonical variables $E^a_i,\,A^i_a,\,\omega ^i_a$, and $\pi ^a_i$, the only non-trivial Dirac brackets are given by

$$\begin{aligned} \big \lbrace E^a_i,A^j_b\big \rbrace _\text{ D }=-\gamma \delta ^a_b\delta ^j_i, \qquad \big \lbrace \omega ^i_a,A^j_b\big \rbrace _\text{ D }=(\Delta ^{-1})^{ik}_{ac}\big \lbrace T^k_c,A^j_b\big \rbrace . \end{aligned}$$

(5.4)

A relevant question that we do not address here is wether this formalism allows to define new $\mathfrak su (2)$ connections with interesting properties. This should be ultimately compared with the non-commutative shifted connection of the Lorentz-covariant formulation.

Appendix B: Symmetry reduction of the four-dimensional Holst action

In this appendix, we show that the three-dimensional Holst action (3.1) can be obtained from a symmetry reduction of the four-dimensional Holst action. This result strongly supports the idea that our three-dimensional action is the lower-dimensional analogue of the Holst action.

The Holst action is given by (2.8):

$$\begin{aligned} S\equiv S_\text{ HP }\!+\!\gamma ^{-1}S_\text{ H }\!=\!\frac{1}{4}\int \limits _\mathcal{M }\mathrm d ^4x\, \varepsilon ^{\mu \nu \rho \sigma }\left( \frac{1}{2}\varepsilon _{IJKL}e_\mu ^Ie_\nu ^J F_{\rho \sigma }^{KL}\!+\!\gamma ^{-1}\delta _{IJKL}e_\mu ^Ie_\nu ^JF_{\rho \sigma }^{IJ}\right) . \nonumber \\ \end{aligned}$$

(6.1)

In order not to reduce the internal gauge group $\text{ SO }(4)$, we only perform a space-time reduction. This can be done by assuming that the four-dimensional space-time has the topology $\mathcal{M }_4=\mathcal{M }_3\times \mathbb{I }$ where $\mathcal{M }_3$ is a three-dimensional space-time, and $\mathbb{I }$ is a spacelike segment with coordinates $x^3$. In this way, we single out the third spatial component $\mu =3$. Let us now impose the conditions

$$\begin{aligned} \partial _3=0,\qquad \qquad \omega _3^{IJ}=0. \end{aligned}$$

(6.2)

The first condition means that the fields do not depend on the third spatial direction $x^3$. The second one means that the parallel transport along the real line $\mathbb{I }$ is trivial. Therefore, the covariant derivative of the fields along the third direction vanishes.

Using the conditions (6.2), a direct calculation shows that the four-dimensional Holst action reduces to

$$\begin{aligned} S_\text{ red }=-\int \limits _\mathbb{I }\mathrm d x^3\int \limits _{\mathcal{M }_3}\mathrm d ^3x\,\varepsilon ^{\mu \nu \rho }\left( \frac{1}{2} \varepsilon _{IJKL}e_3^Ie^J_\mu F^{KL}_{\nu \rho }+\gamma ^{-1}e_3^Ie^J_\mu F^{IJ}_{\nu \rho }\right) , \end{aligned}$$

(6.3)

where now $\mu ,\nu ,\rho$ are three-dimensional space-time indices, and $\mathrm d ^3x=\mathrm d x^0\mathrm d x^1\mathrm d x^2$ is the volume form of $\mathcal{M }_3$. Apart from a global multiplicative factor that is not relevant at all, we recover the three-dimensional action (3.1) with the Barbero–Immirzi parameter, provided that we set $x^I\equiv e_3^I$.

Appendix C: Details on the $\varvec{\gamma }$-dependent three-dimensional gauge

In this appendix, we give some intermediate expressions that were used to derive (3.18). It is possible to write (3.14) as the sum of a canonical term, a term involving $\omega _0$, a term involving $K_0$, and finally a term involving $e_0$.

1.1 The canonical term

It is easy to see that the canonical term is given by

$$\begin{aligned}&-2\varepsilon ^{ab}\left[ e_a\cdot \partial _0(\omega _b+\gamma ^{-1}K_b)+(x\,{\scriptstyle {\times }}\,e_a)\cdot \partial _0(K_b+\gamma ^{-1}\omega _b)\right] \nonumber \\&\quad =\widetilde{E}^a_i\partial _0(\omega ^i_a+\gamma ^{-1}K^i_a)+X^a_i \partial _0(K^i_a+\gamma ^{-1}\omega ^i_a)\nonumber \\&\quad =E^a_i\partial _0\tilde{A}^i_a. \end{aligned}$$

(7.1)

1.2 The constraints $\varvec{G_i}$ and $\varvec{S_i}$

The Gauss constraint $G_i$ and the second class constraint $S_i$ are given by the terms in (3.14) containing the variables $\omega _0$ and $K_0$. The term involving $\omega _0$ is

$$\begin{aligned}&2\varepsilon ^{ab}\left[ (x\,{\scriptstyle {\times }}\,e_a\!+\!\gamma ^{-1}e_a)\cdot (\omega _0\,{\scriptstyle {\times }}\,K_b)\!+\!(e_a\!+\!\gamma ^{-1}x\,{\scriptstyle {\times }}\,e_a)\cdot (\partial _b\omega _0\!+\!\omega _0\,{\scriptstyle {\times }}\,\omega _b)\right] \nonumber \\&\quad =\omega _0\cdot \left[ \partial _a(\widetilde{E}^a\!+\!\gamma ^{-1}X^a)\!+\!(X^a\!+\!\gamma ^{-1} \widetilde{E}^a)\,{\scriptstyle {\times }}\,K_a\!+\!(\widetilde{E}^a\!+\!\gamma ^{-1}X^a)\,{\scriptstyle {\times }}\,\omega _a\right] . \end{aligned}$$

(7.2)

The term involving $K_0$ is

$$\begin{aligned}&2\varepsilon ^{ab}\left[ (x\,{\scriptstyle {\times }}\,e_a\!+\!\gamma ^{-1}e_a)\cdot (\partial _bK_0\!+\!K_0\,{\scriptstyle {\times }}\,\omega _b)\!+\!(e_a\!+\!\gamma ^{-1}x\,{\scriptstyle {\times }}\,e_a)\cdot (K_0\,{\scriptstyle {\times }}\,K_b)\right] \nonumber \\&\quad =K_0\cdot \left[ \partial _a(X^a\!+\!\gamma ^{-1}\widetilde{E}^a)\!+\!(X^a\!+\!\gamma ^{-1} \widetilde{E}^a)\,{\scriptstyle {\times }}\,\omega _a\!+\!(\widetilde{E}^a\!+\!\gamma ^{-1}X^a)\,{\scriptstyle {\times }}\,K_a\right] . \end{aligned}$$

(7.3)

Introducing the index $\alpha =\{1,2\}$ to write $i=\{1,2,3\}=\{\alpha ,3\}$, we can express the constraints imposed by the multipliers $\omega _0$ and $K_0$ as

$$\begin{aligned}&\left\{ \begin{array}{l} \partial _a(\widetilde{E}^a+\gamma ^{-1}X^a)+(X^a+\gamma ^{-1}\widetilde{E}^a)\,{\scriptstyle {\times }}\,K_a+(\widetilde{E}^a+\gamma ^{-1}X^a)\,{\scriptstyle {\times }}\,\omega _a\approx 0 \\ \partial _a(X^a+\gamma ^{-1}\widetilde{E}^a)+(X^a+\gamma ^{-1} \widetilde{E}^a)\,{\scriptstyle {\times }}\,\omega _a+(\widetilde{E}^a+\gamma ^{-1}X^a)\,{\scriptstyle {\times }}\,K_a\approx 0 \end{array}\right. \qquad \end{aligned}$$

(7.4)

$$\begin{aligned}&\Leftrightarrow \left\{ \begin{array}{l} \partial _a\widetilde{E}^a_\alpha +\gamma ^{-1}\varepsilon ^{\alpha \beta } \widetilde{E}^a_\beta K^3_a-\varepsilon ^{\alpha \beta }X^a_3K^\beta _a+\varepsilon ^{\alpha \beta } \widetilde{E}^a_\beta \omega ^3_a-\gamma ^{-1}\varepsilon ^{ \alpha \beta }X^a_3\omega ^\beta _a\approx 0\\ \gamma ^{-1}\partial _aX^a_3+\gamma ^{-1}\varepsilon ^{\alpha \beta }\widetilde{E}_\alpha K^\beta _a+\varepsilon ^{\alpha \beta }\widetilde{E}^a_\alpha \omega ^\beta _a\approx 0\\ \gamma ^{-1}\partial _a\widetilde{E}^a_\alpha \!+\!\gamma ^{-1}\varepsilon ^{\alpha \beta } \widetilde{E}^a_\beta \omega ^3_a\!-\!\varepsilon ^{\alpha \beta }X^a_3\omega ^\beta _ a\!+\!\varepsilon ^{\alpha \beta }\widetilde{E}^a_\beta K^3_a\!-\!\gamma ^{-1}\varepsilon ^{\alpha \beta }X^a_3K^\beta _a\!\approx \!0\\ \partial _aX^a_3+\gamma ^{-1}\varepsilon ^{\alpha \beta }\widetilde{E}_\alpha \omega ^\beta _a+\varepsilon ^{\alpha \beta }\widetilde{E}^a_\alpha K^\beta _a\approx 0 \end{array}\right. \qquad \end{aligned}$$

(7.5)

$$\begin{aligned}&\Leftrightarrow \left\{ \begin{array}{l} \partial _a\widetilde{E}^a_\alpha +\varepsilon ^{\alpha \beta }(B^3_a \widetilde{E}^a_\beta -B^\beta _aX^a_3)\approx 0\\ \partial _aX^a_3+\gamma \varepsilon ^{\alpha \beta }\widetilde{E}^a_\alpha \tilde{A}^\beta _a \approx 0\\ \partial _a\widetilde{E}^a_\alpha +\gamma (\tilde{A}^3_a \widetilde{E}^a_\beta -\tilde{A}^\beta _aX^a_3)\approx 0\\ \partial _aX^a_3+\varepsilon ^{\alpha \beta }\widetilde{E}^a_\alpha B^\beta _a\approx 0, \end{array}\right. \end{aligned}$$

(7.6)

where we have introduced the variable

$$\begin{aligned} B^i_a=(B^\alpha _a,B^3_a)\equiv (K^\alpha _a+\gamma ^{-1}\omega ^\alpha _a, \omega ^3_a+\gamma ^{-1}K^3_a). \end{aligned}$$

(7.7)

Now, one can see that the second and third equations of the system (7.6) are equivalent to

$$\begin{aligned} G\equiv \partial _aE^a-\gamma \tilde{A}_a\,{\scriptstyle {\times }}\,E^a\approx 0, \end{aligned}$$

(7.8)

while the first and fourth equations are equivalent to

$$\begin{aligned} \partial _aE^a-B_a\,{\scriptstyle {\times }}\,E^a\approx 0. \end{aligned}$$

(7.9)

Finally, combining (7.8) and (7.9) and using the definition

$$\begin{aligned} \Omega ^i_a\equiv \frac{\gamma }{\gamma +1}(\tilde{A}^i_a+B^i_a), \end{aligned}$$

(7.10)

we obtain the constraint $S\equiv \partial _aE^a-\Omega _a\,{\scriptstyle {\times }}\,E^a\approx 0$. Keeping track of the multipliers $\omega _0$ and $K_0$ during the above steps, it is easy to obtain the expressions (3.19).

1.3 The constraints $\widetilde{\varvec{H}}^{\varvec{a}}$ and $\varvec{C}$

The term involving $e_0$ is

$$\begin{aligned}&\varepsilon ^{ab}e_0\cdot \Big \lbrace \partial _a(\omega _b+\gamma ^{-1}K_b)-\partial _b( \omega _a+\gamma ^{-1}K_a)\nonumber \\&\qquad +\omega _b\,{\scriptstyle {\times }}\,\omega _a-K_a\,{\scriptstyle {\times }}\,K_b-\gamma ^{-1}K_a\,{\scriptstyle {\times }}\,\omega _b+\gamma ^{-1}K_b\,{\scriptstyle {\times }}\,\omega _a\nonumber \\&\qquad +\Big (\partial _a(K_b+\gamma ^{-1}\omega _b)-\partial _b(K_a+\gamma ^{-1} \omega _a)-K_a\,{\scriptstyle {\times }}\,\omega _b \nonumber \\&\qquad +K_b\,{\scriptstyle {\times }}\,\omega _a+\gamma ^{-1}\omega _b\,{\scriptstyle {\times }}\,\omega _a-\gamma ^{-1}K_a\,{\scriptstyle {\times }}\,K_b\Big )\,{\scriptstyle {\times }}\,x\Big \rbrace \nonumber \\&\quad \equiv \varepsilon ^{ab}e_0\cdot H_{ab}. \end{aligned}$$

(7.11)

The components $i=\{\alpha ,3\}$ of $H^i_{ab}$ are given by

$$\begin{aligned} H^\alpha _{ab}&= \partial _a(\omega ^\alpha _b+\gamma ^{-1}K^\alpha _b)-\partial _b( \omega ^\alpha _a+\gamma ^{-1}K^\alpha _a)\nonumber \\&\quad +\varepsilon ^{\alpha \beta }\Big (\omega ^\beta _b\omega ^3_a-\omega ^3_b \omega ^\beta _a-K^\beta _aK^3_b+K^3_aK^\beta _b \nonumber \\&\quad +\gamma ^{-1}\left[ K^3_a\omega ^\beta _b-K^\beta _a\omega ^3_b+K^\beta _b \omega ^3_a-K^3_b\omega ^\beta _a\right] \Big )\nonumber \\&\quad -\varepsilon ^{\alpha \beta }x^\beta \Big \{\partial _a(K^3_b+\gamma ^{-1} \omega ^3_b)-\partial _b(K^3_a+\gamma ^{-1}\omega ^3_a)\nonumber \\&\quad +\varepsilon _{\gamma \delta }\Big (K^\gamma _b\omega ^\delta _a-K^\gamma _a \omega ^\delta _b+\gamma ^{-1}\omega ^\gamma _b \omega ^\delta _a-\gamma ^{-1}K^\gamma _aK^\delta _b\Big )\Big \}\nonumber \\&\equiv M_{ab}^\alpha -\varepsilon ^{\alpha \beta }x^\beta M_{ab}^3, \end{aligned}$$

(7.12)

and

$$\begin{aligned} H^3_{ab}&= \partial _a(\omega ^3_b+\gamma ^{-1}K^3_b)-\partial _b( \omega ^3_a+\gamma ^{-1}K^3_a) \nonumber \\&\quad +\varepsilon _{\alpha \beta }\left( \omega ^\alpha _b \omega ^\beta _a-K^\alpha _aK^\beta _b-\gamma ^{-1}K^\alpha _a \omega ^\beta _b+\gamma ^{-1}K^\alpha _b\omega ^\beta _a\right) \nonumber \\&\quad -\varepsilon _{\alpha \beta }x^\alpha \Big \{\partial _a(K^\beta _b+ \gamma ^{-1}\omega ^\beta _b)-\partial _b(K^\beta _a+\gamma ^{-1}\omega ^\beta _a)\nonumber \\&\quad +\varepsilon _{\beta \delta }\Big (-K^\delta _a\omega ^3_b+K^3_a\omega ^\delta _b+K^\delta _b \omega ^3_a-K^3_b\omega ^\delta _a \nonumber \\&\quad +\gamma ^{-1}\left[ \omega ^\delta _b\omega ^3_a-\omega ^3_b \omega ^\delta _a-K^\delta _aK^3_b+K^3_aK^\delta _b\right] \Big )\Big \}\nonumber \\&\equiv N_{ab}^3-\varepsilon _{\alpha \beta }x^\alpha N_{ab}^\beta . \end{aligned}$$

(7.13)

Now, with (7.12) and (7.10), one can rewrite the components $M^\alpha _{ab}$ and $M^3_{ab}$ in terms of $\tilde{A}^i_a$ and $B^i_a$ to see that we have

$$\begin{aligned} M^i_{ab}&= \partial _a\tilde{A}^i_b-\partial _b\tilde{A}^i_a-\gamma \varepsilon ^i_{~jk}\tilde{A}^j_a\tilde{A}^k_b+\frac{\gamma }{\gamma ^2-1}\varepsilon ^i_{~jk}( \gamma \tilde{A}^j_a-B^j_a)(\gamma \tilde{A}^k_b-B^k_b)\nonumber \\&= \partial _a\tilde{A}^i_b-\partial _b\tilde{A}^i_a-\gamma \varepsilon ^i_{~jk}\tilde{A}^j_a\tilde{A}^k_b+(\gamma ^{-1}-\gamma ^{-3})\varepsilon ^i_{~jk}( \Omega ^j_a-\gamma \tilde{A}^j_a)(\Omega ^k_b-\gamma \tilde{A}^k_b). \nonumber \\ \end{aligned}$$

(7.14)

Therefore, we can write that

$$\begin{aligned} \widetilde{E}^a_\alpha H^\alpha _{12}=\widetilde{E}^a_\alpha M_{12}^\alpha +X^a_3M_{12}^3=E^a_iM_{12}^i\equiv \widetilde{H}^a. \end{aligned}$$

(7.15)

Similarly, using (7.13) and (7.10), one can see that

$$\begin{aligned} N^i_{ab}\!&= \!\partial _aB^i_b\!-\!\partial _bB^i_a\!+\!\frac{\gamma }{\gamma ^2-1} \varepsilon ^i_{~jk}(B^j_a\tilde{A}^k_b\!+\!\tilde{A}^j_aB^k_b\!-\!\gamma \tilde{A}^j_a\tilde{A}^k_b\!-\!\gamma B^j_aB^k_b)\nonumber \\&= \gamma ^{-1}\partial _a\tilde{A}^i_b\!-\!\gamma ^{-1}\partial _b \tilde{A}^i_a\!+\!(1\!-\!\gamma ^{-2})(\partial _a\Omega ^i_b\!-\!\partial _b \Omega ^i_a\!-\!\varepsilon ^i_{~jk}\Omega ^j_a\Omega ^k_b). \end{aligned}$$

(7.16)

Using the expression

$$\begin{aligned} E^1\,{\scriptstyle {\times }}\,E^2&= (\widetilde{E}^1_1,\widetilde{E}^1_2,X^1_3)\,{\scriptstyle {\times }}\,( \widetilde{E}^2_1,\widetilde{E}^2_2,X^2_3)\nonumber \\&= \big (\widetilde{E}^1_2X^2_3-\widetilde{E}^2_2X^1_3, \widetilde{E}^2_1X^1_3-\widetilde{E}^1_1X^2_3,\det (\widetilde{E})\big )\nonumber \\&= \det (\widetilde{E})(x_2,-x_1,1), \end{aligned}$$

(7.17)

we can now write that

$$\begin{aligned} H^3_{12}=N_{12}^3-\varepsilon _{\alpha \beta }x^\alpha N_{12}^\beta =\frac{(E^1 \,{\scriptstyle {\times }}\,E^2)^i}{\det (\widetilde{E})}N^i_{12}. \end{aligned}$$

(7.18)

Finally, using (7.15) and (7.18) together with the definitions $e^\alpha _0\equiv N^a\widetilde{E}^a_\alpha /2$ and $e^3_0\equiv N/2$, we arrive at the decomposition

$$\begin{aligned} 2e^i_0H^i_{12}=2e^\alpha _0H^\alpha _{12}+2e^3_0H^3_{12}=N^aE^a_iM_{12}^i+N\frac{(E^1 \,{\scriptstyle {\times }}\,E^2)^i}{\det (\widetilde{E})}N^i_{12}\equiv N^a\widetilde{H}_a+NC.\nonumber \\ \end{aligned}$$

(7.19)