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III) BVClassical BV formulation. Let us mention for completeness that the gauge transformation $\delta$ can be encoded as a BRST transformation, cf. e.g. Ref. 2 and this Phys.SE post. Roughly speaking, the Grassmann-even gauge parameter $\eta$ is then replaced by a Grassmann-odd Faddeev-Popov (FP) ghost $C$. (Actually, the gauge parameter $\eta$ will more precisely be replaced with the combination $e^{1-r}C$, where $r\in\mathbb{R}$ is a power, to be more general, cf. eq. (16) below.) To minimize the appearances of time derivatives, instead of using the Lagrangian (5), it becomes a bit simpler to start from the Hamiltonian Lagrangian

IV) Quantum master equation. The odd Laplacian reads

$$ \Delta~=~(-1)^{|\alpha|}\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ \delta(\tau\!-\!\tau^{\prime})\frac{\delta_L}{\delta\phi^{\alpha}(\tau)} \frac{\delta_L}{\delta\phi^{\ast}_{\alpha}(\tau^{\prime})}. \tag{19} $$$$ \Delta~=~(-1)^{|\alpha|}\int\! \mathrm{d}\tau~ \frac{\delta_L}{\delta\phi^{\alpha}(\tau)} \frac{\delta_L}{\delta\phi^{\ast}_{\alpha}(\tau)} ~=~(-1)^{|\alpha|}\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ \delta(\tau\!-\!\tau^{\prime})\frac{\delta_L}{\delta\phi^{\alpha}(\tau)} \frac{\delta_L}{\delta\phi^{\ast}_{\alpha}(\tau^{\prime})} \tag{19} $$

is a singular object, which strictly speaking needs to be regularized. We calculate formally

$$ \Delta S_{BV}~\stackrel{(13)+(19)}{=}~ 2(n\!-\!r)\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}C(\tau)~ \delta(\tau\!-\!\tau^{\prime}) \frac{d}{d\tau}\delta(\tau\!-\!\tau^{\prime}) $$ $$+r \iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}\dot{C}(\tau)~\delta(\tau\!-\!\tau^{\prime})^2~\neq~0, \tag{20} $$ where $n$ is the target space (TS) dimension. This showshows that the BV action (13) does not satisfy the quantum master equation; only the classical master equation. We will discuss appropriate modifications of the BV action (13) in Section VII.

V) BFVClassical BFV formulation. We identify $p_e\approx\epsilon B$ with the canonical momentum of the einbein $e$, and we identify the antifield $e^{\ast}\equiv \bar{P}$ with the FP ghost momentum. Introduce an ultra-local Poisson bracket $\{\cdot,\cdot\}_{PB}$ with the following canonical pairs

VII) BVQuantum BV formulation revisited. Let us putEqs. (20), (22) & (34) suggest that we should put $r=0$, so let us do this from now on. Using the eom for $x^{\mu}$ and $p_{\mu}$ inInspired by the BRSTBFV-BRST transformations (24), we rewrite themodify the BV Lagrangian (13) tointo

$^1$ We ignore temporal boundary terms. Effectively this means that we impose pertinent boundary conditions, and limit gauge symmetry to the bulk.

III) BV formulation. Let us mention for completeness that the gauge transformation $\delta$ can be encoded as a BRST transformation, cf. e.g. Ref. 2 and this Phys.SE post. Roughly speaking, the Grassmann-even gauge parameter $\eta$ is then replaced by a Grassmann-odd Faddeev-Popov (FP) ghost $C$. (Actually, the gauge parameter $\eta$ will more precisely be replaced with the combination $e^{1-r}C$, where $r\in\mathbb{R}$ is a power, to be more general, cf. eq. (16) below.) To minimize the appearances of time derivatives, instead of using the Lagrangian (5), it becomes a bit simpler to start from the Hamiltonian Lagrangian

IV) Quantum master equation. The odd Laplacian reads

$$ \Delta~=~(-1)^{|\alpha|}\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ \delta(\tau\!-\!\tau^{\prime})\frac{\delta_L}{\delta\phi^{\alpha}(\tau)} \frac{\delta_L}{\delta\phi^{\ast}_{\alpha}(\tau^{\prime})}. \tag{19} $$

We calculate

$$ \Delta S_{BV}~\stackrel{(13)+(19)}{=}~ 2(n\!-\!r)\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}C(\tau)~ \delta(\tau\!-\!\tau^{\prime}) \frac{d}{d\tau}\delta(\tau\!-\!\tau^{\prime}) $$ $$+r \iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}\dot{C}(\tau)~\delta(\tau\!-\!\tau^{\prime})^2~\neq~0, \tag{20} $$ where $n$ is the target space (TS) dimension. This show that the BV action (13) does not satisfy the quantum master equation; only the classical master equation. We will discuss appropriate modifications of the BV action (13) in Section VII.

V) BFV formulation. We identify $p_e\approx\epsilon B$ with the canonical momentum of the einbein $e$, and we identify the antifield $e^{\ast}\equiv \bar{P}$ with the FP ghost momentum. Introduce an ultra-local Poisson bracket $\{\cdot,\cdot\}_{PB}$ with the following canonical pairs

VII) BV formulation revisited. Let us put $r=0$ from now on. Using the eom for $x^{\mu}$ and $p_{\mu}$ in the BRST transformations (24), we rewrite the BV Lagrangian (13) to

$^1$ We ignore temporal boundary terms.

III) Classical BV formulation. Let us mention for completeness that the gauge transformation $\delta$ can be encoded as a BRST transformation, cf. e.g. Ref. 2 and this Phys.SE post. Roughly speaking, the Grassmann-even gauge parameter $\eta$ is then replaced by a Grassmann-odd Faddeev-Popov (FP) ghost $C$. (Actually, the gauge parameter $\eta$ will more precisely be replaced with the combination $e^{1-r}C$, where $r\in\mathbb{R}$ is a power, to be more general, cf. eq. (16) below.) To minimize the appearances of time derivatives, instead of using the Lagrangian (5), it becomes a bit simpler to start from the Hamiltonian Lagrangian

IV) Quantum master equation. The odd Laplacian

$$ \Delta~=~(-1)^{|\alpha|}\int\! \mathrm{d}\tau~ \frac{\delta_L}{\delta\phi^{\alpha}(\tau)} \frac{\delta_L}{\delta\phi^{\ast}_{\alpha}(\tau)} ~=~(-1)^{|\alpha|}\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ \delta(\tau\!-\!\tau^{\prime})\frac{\delta_L}{\delta\phi^{\alpha}(\tau)} \frac{\delta_L}{\delta\phi^{\ast}_{\alpha}(\tau^{\prime})} \tag{19} $$

is a singular object, which strictly speaking needs to be regularized. We calculate formally

$$ \Delta S_{BV}~\stackrel{(13)+(19)}{=}~ 2(n\!-\!r)\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}C(\tau)~ \delta(\tau\!-\!\tau^{\prime}) \frac{d}{d\tau}\delta(\tau\!-\!\tau^{\prime}) $$ $$+r \iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}\dot{C}(\tau)~\delta(\tau\!-\!\tau^{\prime})^2~\neq~0, \tag{20} $$ where $n$ is the target space (TS) dimension. This shows that the BV action (13) does not satisfy the quantum master equation; only the classical master equation. We will discuss appropriate modifications of the BV action (13) in Section VII.

V) Classical BFV formulation. We identify $p_e\approx\epsilon B$ with the canonical momentum of the einbein $e$, and we identify the antifield $e^{\ast}\equiv \bar{P}$ with the FP ghost momentum. Introduce an ultra-local Poisson bracket $\{\cdot,\cdot\}_{PB}$ with the following canonical pairs

VII) Quantum BV formulation. Eqs. (20), (22) & (34) suggest that we should put $r=0$, so let us do this from now on. Inspired by the BFV-BRST transformations (24), we modify the BV Lagrangian (13) into

$^1$ We ignore boundary terms. Effectively this means that we impose pertinent boundary conditions, and limit gauge symmetry to the bulk.

11 Added explanation
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$$\tag{1} e~=~e^{\prime} \frac{d\tau^{\prime}}{d\tau}$$$$ e~=~e^{\prime} \frac{d\tau^{\prime}}{d\tau}\tag{1} $$

$$\tag{2} \tau\longrightarrow \tau^{\prime}=f(\tau).$$$$ \tau\longrightarrow \tau^{\prime}=f(\tau).\tag{2} $$

$$\tag{3} X^{\mu}~=~X^{\prime \mu}$$$$ x^{\mu}~=~x^{\prime \mu}\tag{3} $$

$$\tag{4} \dot{X}^{\mu}~=~\dot{X}^{\prime \mu}\frac{d\tau^{\prime}}{d\tau}.$$$$ \dot{x}^{\mu}~=~\dot{x}^{\prime \mu}\frac{d\tau^{\prime}}{d\tau}.\tag{4}$$

$$\tag{5} S~=~\int \! \mathrm{d}\tau ~L , \qquad L~:=~\frac{\dot{X}^2}{2e}-\frac{e m^2}{2},$$$$ S~=~\int \! \mathrm{d}\tau ~L , \qquad L~:=~\frac{\dot{x}^2}{2e}-\frac{e m^2}{2},\tag{5}$$

$$ \tag{6} Y ~=~\eta \frac{d}{d\tau}~\in~ \Gamma(TI) $$$$ Y ~=~\eta \frac{d}{d\tau}~\in~ \Gamma(TI)\tag{6} $$

$$ \tag{7} {\cal L}_Y X^{\mu}~=~Y[X^{\mu}]~=~\eta \frac{dX^{\mu}}{d\tau}, $$$$ {\cal L}_Y x^{\mu}~=~Y[x^{\mu}]~=~\eta \frac{dx^{\mu}}{d\tau},\tag{7} $$

$$ ({\cal L}_Ye)\mathrm{d}\tau~:=~{\cal L}_Y\omega ~=~\{\mathrm{d}, i_Y\}\omega~=~\mathrm{d}i_Y\omega $$ $$ \tag{8}~=~\mathrm{d}(i_Y\omega)~=~\mathrm{d}(\eta e) ~=~\mathrm{d}\tau\frac{d}{d\tau}(\eta e), $$$$ ~=~\mathrm{d}(i_Y\omega)~=~\mathrm{d}(\eta e) ~=~\mathrm{d}\tau\frac{d}{d\tau}(\eta e),\tag{8} $$

$$ \tag{9} {\cal L}_Ye ~\stackrel{(8)}{=}~\frac{d}{d\tau}(\eta e).$$$$ {\cal L}_Ye ~\stackrel{(8)}{=}~\frac{d}{d\tau}(\eta e).\tag{9} $$

Formula (6), (7) and (9) correspond to eq. (1.10) in Ref. 1 $$ \tag{1.10} \tau\to \tilde{\tau}=\tau-\eta, \qquad \delta X^{\mu}~=~\eta\frac{d X^{\mu}}{d\tau}, \qquad \delta e ~=~\frac{d}{d\tau}(\eta e), $$$$ \tag{1.10} \tau\to \tilde{\tau}=\tau-\eta, \qquad \delta x^{\mu}~=~\eta\frac{d x^{\mu}}{d\tau}, \qquad \delta e ~=~\frac{d}{d\tau}(\eta e), $$ respectively.

$$ \tag{11} L_H~:=~ p_{\mu} \dot{X}^{\mu} - H, \qquad H~:=~ eT, \qquad T~:=~\frac{1}{2}(p^2+m^2), \qquad p^2~:=~ g^{\mu\nu}(X)~p_{\mu} p_{\nu},$$$$L_H~:=~ p_{\mu} \dot{x}^{\mu} - H, \qquad H~:=~ eT, \qquad T~:=~\frac{1}{2}(p^2+m^2), \qquad p^2~:=~ g^{\mu\nu}(x)~p_{\mu} p_{\nu}, \tag{11} $$

$$ \tag{12} \phi^{\alpha} ~=~ \{ X^{\mu};~p_{\mu};~ e;~C;~ \bar{C};~B\} $$$$ \phi^{\alpha} ~=~ \{ x^{\mu};~p_{\mu};~ e;~C;~ \bar{C};~B\} \tag{12}$$

are positions $X^{\mu}$$x^{\mu}$; momenta $p_{\mu}$; einbein $e$; FP ghost $C$; FP antighost $\bar{C}$; and Lautrup-Nakanishi (LN) Lagrange multiplier $B$, respectively. They are WL tensors of contravariant orders $0$; $0$; $-1$; $r$; $1$; and $1$, respectively. Each field $\phi^{\alpha}$ has a corresponding antifield $\phi^{\ast}_{\alpha}$ of opposite Grassmann parity. The corresponding BV action$^1$

$$ S_{BV}~=~\int \! \mathrm{d}\tau ~L_{BV} , $$ $$\tag{13} L_{BV}~=~L_H +\left(X^{\ast}_{\mu} \dot{X}^{\mu}+p_{\ast}^{\mu} \dot{p}_{\mu} +r C^{\ast}\dot{C} \right)e^{r-1}C +\underbrace{e^r C\dot{e}^{\ast}}_{\sim~e^{\ast}\frac{d}{d\tau}( e^r C)} + B \bar{C}^{\ast}, $$$$ L_{BV}~=~L_H +\left(x^{\ast}_{\mu} \dot{x}^{\mu}+p_{\ast}^{\mu} \dot{p}_{\mu} +r C^{\ast}\dot{C} \right)e^{r-1}C +\underbrace{e^r C\dot{e}^{\ast}}_{\sim~e^{\ast}\frac{d}{d\tau}( e^r C)} + B \bar{C}^{\ast},\tag{13} $$

$$ \tag{14} (S_{BV},S_{BV})~=~0, $$$$ (S_{BV},S_{BV})~=~0, \tag{14}$$

$$ \tag{15} (\phi^{\alpha}(\tau),\phi^{\ast}_{\beta}(\tau^{\prime})) ~=~\delta^{\alpha}_{\beta}~\delta(\tau\!-\!\tau^{\prime}).$$$$ (\phi^{\alpha}(\tau),\phi^{\ast}_{\beta}(\tau^{\prime})) ~=~\delta^{\alpha}_{\beta}~\delta(\tau\!-\!\tau^{\prime}). \tag{15}$$

$${\bf s}X^{\mu}~=~e^{r-1} C \dot{X}^{\mu}, \qquad {\bf s}p_{\mu}~=~e^{r-1} C \dot{p}_{\mu}, \qquad {\bf s}e~=~ \frac{d}{d\tau}( e^r C), $$$${\bf s}x^{\mu}~=~e^{r-1} C \dot{x}^{\mu}, \qquad {\bf s}p_{\mu}~=~e^{r-1} C \dot{p}_{\mu}, \qquad {\bf s}e~=~ \frac{d}{d\tau}( e^r C), $$ $$ \tag{16} {\bf s}C~=~ re^{r-1} C\dot{C},\qquad {\bf s}\bar{C}~=~ - B,\qquad {\bf s}B ~=~0, $$$$ {\bf s}C~=~ re^{r-1} C\dot{C},\qquad {\bf s}\bar{C}~=~ - B,\qquad {\bf s}B ~=~0, \tag{16} $$

$$ \tag{17} \psi ~:=~\int \! \mathrm{d}\tau~\bar{C}\left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right), $$$$ \psi ~:=~\int \! \mathrm{d}\tau~\bar{C}\left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right), \tag{17} $$ where $\xi,\epsilon\in\mathbb{R}$ are gauge-fixing parameters. Moreover, $\chi(e)=(e\!-\!e_0)\chi^{\prime}$ is a gauge-fixing condition (which we will assume is affine in $e$, so that the derivative $\chi^{\prime}$ is constant). The gauge-fixed Lagrangian becomes

$$ \tag{18} L_{\rm gf}~=~ \left. L_{BV} \right|_{\phi^{\ast}~=~\frac{\delta \psi}{\delta \phi}}~=~ L_H + \overbrace{\underbrace{\left(\chi^{\prime}\bar{C}-\epsilon\dot{\bar{C}}\right) \frac{d}{d\tau}(e^r C)}_{ \sim~ \bar{C} \left(\frac{\chi^{\prime}}{2}+\epsilon\frac{d}{d\tau}\right)\frac{d}{d\tau}(e^r C) + e^r C\left(\frac{\chi^{\prime}}{2}-\epsilon\frac{d}{d\tau}\right)\frac{d}{d\tau}\bar{C} }}^{\text{Faddeev-Popov term}} + \overbrace{B \left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right)}^{\text{gauge-fixing term}} , $$$$ L_{\rm gf}~=~ \left. L_{BV} \right|_{\phi^{\ast}~=~\frac{\delta \psi}{\delta \phi}}~=~ L_H + \overbrace{\underbrace{\left(\chi^{\prime}\bar{C}-\epsilon\dot{\bar{C}}\right) \frac{d}{d\tau}(e^r C)}_{ \sim~ \bar{C} \left(\frac{\chi^{\prime}}{2}+\epsilon\frac{d}{d\tau}\right)\frac{d}{d\tau}(e^r C) + e^r C\left(\frac{\chi^{\prime}}{2}-\epsilon\frac{d}{d\tau}\right)\frac{d}{d\tau}\bar{C} }}^{\text{Faddeev-Popov term}} + \overbrace{B \left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right)}^{\text{gauge-fixing term}} , \tag{18} $$

IV) Quantum master equation. The odd Laplacian reads

$$ \Delta~=~(-1)^{|\alpha|}\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ \delta(\tau\!-\!\tau^{\prime})\frac{\delta_L}{\delta\phi^{\alpha}(\tau)} \frac{\delta_L}{\delta\phi^{\ast}_{\alpha}(\tau^{\prime})}. \tag{19} $$

We calculate

$$ \Delta S_{BV}~\stackrel{(13)+(19)}{=}~ 2(n\!-\!r)\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}C(\tau)~ \delta(\tau\!-\!\tau^{\prime}) \frac{d}{d\tau}\delta(\tau\!-\!\tau^{\prime}) $$ $$+r \iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}\dot{C}(\tau)~\delta(\tau\!-\!\tau^{\prime})^2~\neq~0, \tag{20} $$ where $n$ is the target space (TS) dimension. This show that the BV action (13) does not satisfy the quantum master equation; only the classical master equation. We will discuss appropriate modifications of the BV action (13) in Section VII.

V) BFV formulation. We identify $p_e\approx\epsilon B$ with the canonical momentum of the einbein $e$, and we identify the antifield $e^{\ast}\equiv \bar{P}$ with the FP ghost momentum. Introduce an ultra-local Poisson bracket $\{\cdot,\cdot\}_{PB}$ with the following canonical pairs

$$ \{X^{\mu}(\tau), p_{\nu}(\tau^{\prime})\}_{PB} ~=~\delta^{\mu}_{\nu}~\delta(\tau\!-\!\tau^{\prime}), \qquad \{e(\tau)^rC(\tau), \bar{P}(\tau^{\prime})\}_{PB} ~=~\delta(\tau\!-\!\tau^{\prime}), $$$$ \{x^{\mu}(\tau), p_{\nu}(\tau^{\prime})\}_{PB} ~=~\delta^{\mu}_{\nu}~\delta(\tau\!-\!\tau^{\prime}), \qquad \{e(\tau)^rC(\tau), \bar{P}(\tau^{\prime})\}_{PB} ~=~\delta(\tau\!-\!\tau^{\prime}), $$ $$ \tag{19} \{e(\tau), B (\tau^{\prime})\}_{PB} ~=~\frac{1}{\epsilon}\delta(\tau\!-\!\tau^{\prime}), \qquad \{\bar{C}(\tau), P(\tau^{\prime})\}_{PB} ~=~\frac{1}{\epsilon}\delta(\tau\!-\!\tau^{\prime}). $$$$ \{e(\tau), B (\tau^{\prime})\}_{PB} ~=~\frac{1}{\epsilon}\delta(\tau\!-\!\tau^{\prime}), \qquad \{\bar{C}(\tau), P(\tau^{\prime})\}_{PB} ~=~\frac{1}{\epsilon}\delta(\tau\!-\!\tau^{\prime}).\tag{21} $$

$$\tag{20} \{C(\tau), \bar{P}(\tau^{\prime})\}_{PB}~=~e(\tau)^{-r}\delta(\tau\!-\!\tau^{\prime}), \qquad \{ B (\tau), C(\tau^{\prime})\}_{PB}~=~\frac{r}{\epsilon} \frac{C(\tau)}{e(\tau)} \delta(\tau\!-\!\tau^{\prime}), $$$$ \{C(\tau), \bar{P}(\tau^{\prime})\}_{PB} ~=~e(\tau)^{-r}\delta(\tau\!-\!\tau^{\prime}), \qquad \{ B (\tau), C(\tau^{\prime})\}_{PB} ~=~\frac{r}{\epsilon} \frac{C(\tau)}{e(\tau)} \delta(\tau\!-\!\tau^{\prime}), \tag{22} $$

$$ \tag{21} \{e(\tau)^r C(\tau), B(\tau^{\prime})\}_{PB}~=~0. $$$$ \{e(\tau)^r C(\tau), B(\tau^{\prime})\}_{PB}~=~0. \tag{23} $$

$${\bf s}X^{\mu}~=~e^r C g^{\mu\nu}(X)p_{\nu}~\approx~ e^{r-1} C \dot{X}^{\mu}, \qquad {\bf s}e~=~P~\approx~ \frac{d}{d\tau}( e^r C) , $$$${\bf s}x^{\mu}~=~e^r C g^{\mu\nu}(x)p_{\nu} ~\approx~ e^{r-1} C \dot{x}^{\mu}, \qquad {\bf s}p_{\mu} ~=~ -\frac{1}{2}e^r C \partial_{\mu}g^{\nu\lambda}(x)~p_{\nu}p_{\lambda} ~\approx~ e^{r-1} C \dot{p}_{\mu}, $$ $$ \tag{22} {\bf s}C~=~r\frac{C}{e}P ~\approx~ re^{r-1} C\dot{C},\qquad {\bf s}\bar{C}~=~ - B,\qquad {\bf s} B ~=~0, $$$${\bf s}e~=~P~\approx~ \frac{d}{d\tau}( e^r C) , \qquad {\bf s}C~=~r\frac{C}{e}P ~\approx~ re^{r-1} C\dot{C},\qquad {\bf s}\bar{C}~=~ - B,\qquad {\bf s} B ~=~0, \tag{24} $$

which should be compared with eq. (16). Here the $\approx$ symbol means equality modulo eqs. of motion. The BRST transformation (2224) is generated by

$$\tag{23} \mathbb{Q}~:=~ \int \! \mathrm{d}\tau ~Q, \qquad \{\mathbb{Q}, \mathbb{Q}\}_{PB}~=~0,$$$$ \mathbb{Q}~:=~ \int \! \mathrm{d}\tau ~Q, \qquad \{\mathbb{Q}, \mathbb{Q}\}_{PB}~=~0,\tag{25}$$

$$\tag{24} -Q~:=~T e^r C + \epsilon B P ~\approx~ T e^r C + \epsilon B \frac{d}{d\tau}( e^r C)$$$$ -Q~:=~T e^r C + \epsilon B P ~\approx~ T e^r C + \epsilon B \frac{d}{d\tau}( e^r C)\tag{26}$$

$$ \tag{25} S_{BFV} ~=~ \int \! \mathrm{d}\tau~\left(\dot{X}^{\mu}p_{\mu}+e^r C\dot{\bar{P}} \right) -\left\{ \psi, \mathbb{Q} \right\}_{PB} ~=~ \int \! \mathrm{d}\tau ~L_{BFV} , $$$$ S_{BFV} ~=~ \int \! \mathrm{d}\tau~\left(\dot{x}^{\mu}p_{\mu}+e^r C\dot{\bar{P}} \right) -\left\{ \psi, \mathbb{Q} \right\}_{PB} ~=~ \int \! \mathrm{d}\tau ~L_{BFV} , \tag{27} $$

$$\tag{26} \psi ~:=~\int \! \mathrm{d}\tau \left(\bar{C} \left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right) -\bar{P}e\right), $$$$ \psi ~:=~\int \! \mathrm{d}\tau \left(\bar{C} \left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right) -\bar{P}e\right),\tag{28} $$

and where the BFV Lagrangian reads$^1$$^2$

$$ L_{BFV}~=~\left(p_{\mu}\dot{X}^{\mu}+ e^r C\dot{\bar{P}} \right) + \epsilon\left( B \dot{e} + \bar{C} \dot{P}\right) + \left(-eT +\bar{C}\chi^{\prime} P +B \left(\frac{\xi}{2}B+\chi(e)\right) -\bar{P}P \right)$$$$ L_{BFV}~=~\left(p_{\mu}\dot{x}^{\mu}+ e^r C\dot{\bar{P}} \right) + \epsilon\left( B \dot{e} + \bar{C} \dot{P}\right) + \left(-eT +\bar{C}\chi^{\prime} P +B \left(\frac{\xi}{2}B+\chi(e)\right) -\bar{P}P \right)$$ $$\tag{27} ~\sim~ L_H+ \underbrace{\epsilon\left( B \dot{e} + \bar{C} \dot{P}\right)}_{\text{kinetic term}}+ \bar{P}\left( \frac{d}{d\tau}( e^r C)-P\right) + \underbrace{\bar{C} \chi^{\prime} P}_{\text{FP term}} + \underbrace{B \left(\frac{\xi}{2}B+\chi(e)\right)}_{\text{gauge-fixing term}} .$$$$ ~\sim~ L_H+ \underbrace{\epsilon\left( B \dot{e} + \bar{C} \dot{P}\right)}_{\text{kinetic term}}+ \bar{P}\left( \frac{d}{d\tau}( e^r C)-P\right) + \underbrace{\bar{C} \chi^{\prime} P}_{\text{FP term}} + \underbrace{B \left(\frac{\xi}{2}B+\chi(e)\right)}_{\text{gauge-fixing term}} \tag{29} .$$

VVI) Dirac bracket. Let us integrate out the two FP momenta $P$ and $\bar{P}$. Then the BFV Lagrangian (2729) becomes the gauge-fixed Lagrangian (18) from Section III. The corresponding two 2nd class constraints

$$\tag{28} \Theta~:=~ P - \frac{d}{d\tau}( e^r C)~\approx~0, \qquad \bar{\Theta}~:=~ \bar{P} - \chi^{\prime} \bar{C}+\epsilon\dot{\bar{C}}~\approx~0, $$$$ \Theta~:=~ P - \frac{d}{d\tau}( e^r C)~\approx~0, \qquad \bar{\Theta}~:=~ \bar{P} - \chi^{\prime} \bar{C}+\epsilon\dot{\bar{C}}~\approx~0,\tag{30} $$

$$\tag{29} \Delta(\tau,\tau^{\prime} ) ~:=~ \{\Theta(\tau), \bar{\Theta}(\tau^{\prime}) \}_{PB} ~=~ -\left(\frac{\chi^{\prime}}{\epsilon}+2\frac{d}{d\tau} \right) \delta(\tau\!-\!\tau^{\prime}), $$$$ \Delta(\tau,\tau^{\prime} ) ~:=~ \{\Theta(\tau), \bar{\Theta}(\tau^{\prime}) \}_{PB} ~=~ -\left(\frac{\chi^{\prime}}{\epsilon}+2\frac{d}{d\tau} \right) \delta(\tau\!-\!\tau^{\prime}),\tag{31} $$

$$\tag{30} \Delta^{-1}(\tau,\tau^{\prime} ) ~=~ - \frac{1}{4} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}). $$$$ \Delta^{-1}(\tau,\tau^{\prime} ) ~=~ - \frac{1}{4} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}).\tag{32} $$

$$\tag{31} \{e(\tau)^rC(\tau), \bar{C}(\tau^{\prime})\}_{DB} ~=~ \frac{1}{4\epsilon} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}). $$$$ \{e(\tau)^rC(\tau), \bar{C}(\tau^{\prime})\}_{DB} ~=~ \frac{1}{4\epsilon} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}).\tag{33} $$

Alternatively, the Poisson structure (3133) could be deduced from the FP term in the gauge-fixed Lagrangian (18).

$$ \{C(\tau), \bar{C}(\tau^{\prime})\}_{DB}~=~\frac{e(\tau)^{-r}}{4\epsilon} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}) , $$$$ \{C(\tau), \bar{C}(\tau^{\prime})\}_{DB} ~=~\frac{e(\tau)^{-r}}{4\epsilon} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}) , $$ $$ \tag{32} \{ B (\tau), C(\tau^{\prime})\}_{DB}~=~\frac{r}{\epsilon} \frac{C(\tau)}{e(\tau)} \delta(\tau\!-\!\tau^{\prime}), $$$$ \{ B (\tau), C(\tau^{\prime})\}_{DB} ~=~\frac{r}{\epsilon} \frac{C(\tau)}{e(\tau)} \delta(\tau\!-\!\tau^{\prime}), \tag{34}$$

$$ \tag{33} \{e(\tau)^r C(\tau), B(\tau^{\prime})\}_{DB}~=~0. $$$$ \{e(\tau)^r C(\tau), B(\tau^{\prime})\}_{DB}~=~0.\tag{35} $$

VII) BV formulation revisited. Let us put $r=0$ from now on. Using the eom for $x^{\mu}$ and $p_{\mu}$ in the BRST transformations (24), we rewrite the BV Lagrangian (13) to

$$ \tilde{L}_{BV}~=~L_H +x^{\ast}_{\mu} g^{\mu\nu}(x)p_{\nu}C -\frac{1}{2}p_{\ast}^{\mu} \partial_{\mu}g^{\nu\lambda}(x)~p_{\nu}p_{\lambda} C +e^{\ast}\dot{C} + B \bar{C}^{\ast}. \tag{36} $$

One may show that the quantum master equation is now satisfied$^1$

$$ (\tilde{S}_{BV}, \tilde{S}_{BV})~=~0~=~\Delta\tilde{S}_{BV}. \tag{37} $$

The modification (36) does not alter the gauge-fixed Lagrangian (18) apart from putting $r=0$.

References:

$^1$ We ignore temporal boundary terms.

$^2$ The $\epsilon$-dependence in the BFV action (2527) comes only from the gauge-fixing fermion (2628). The $\epsilon$-dependence can be removed via redefinition

$$\tag{34} \epsilon B~\longrightarrow~ B, \qquad \epsilon\bar{C}~\longrightarrow~ \bar{C}, \qquad \frac{\chi}{\epsilon} ~\longrightarrow~ \chi, \qquad \frac{\xi}{\epsilon^2} ~\longrightarrow~ \xi . $$$$ \epsilon B~\longrightarrow~ B, \qquad \epsilon\bar{C}~\longrightarrow~ \bar{C}, \qquad \frac{\chi}{\epsilon} ~\longrightarrow~ \chi, \qquad \frac{\xi}{\epsilon^2} ~\longrightarrow~ \xi .\tag{38} $$

In the limit $\epsilon\to 0$, the infinities on the rhs. of the Poisson brackets (1921) should be interpreted as zero, i.e. the corresponding canonical variables become decoupled.

$$\tag{1} e~=~e^{\prime} \frac{d\tau^{\prime}}{d\tau}$$

$$\tag{2} \tau\longrightarrow \tau^{\prime}=f(\tau).$$

$$\tag{3} X^{\mu}~=~X^{\prime \mu}$$

$$\tag{4} \dot{X}^{\mu}~=~\dot{X}^{\prime \mu}\frac{d\tau^{\prime}}{d\tau}.$$

$$\tag{5} S~=~\int \! \mathrm{d}\tau ~L , \qquad L~:=~\frac{\dot{X}^2}{2e}-\frac{e m^2}{2},$$

$$ \tag{6} Y ~=~\eta \frac{d}{d\tau}~\in~ \Gamma(TI) $$

$$ \tag{7} {\cal L}_Y X^{\mu}~=~Y[X^{\mu}]~=~\eta \frac{dX^{\mu}}{d\tau}, $$

$$ ({\cal L}_Ye)\mathrm{d}\tau~:=~{\cal L}_Y\omega ~=~\{\mathrm{d}, i_Y\}\omega~=~\mathrm{d}i_Y\omega $$ $$ \tag{8}~=~\mathrm{d}(i_Y\omega)~=~\mathrm{d}(\eta e) ~=~\mathrm{d}\tau\frac{d}{d\tau}(\eta e), $$

$$ \tag{9} {\cal L}_Ye ~\stackrel{(8)}{=}~\frac{d}{d\tau}(\eta e).$$

Formula (6), (7) and (9) correspond to eq. (1.10) in Ref. 1 $$ \tag{1.10} \tau\to \tilde{\tau}=\tau-\eta, \qquad \delta X^{\mu}~=~\eta\frac{d X^{\mu}}{d\tau}, \qquad \delta e ~=~\frac{d}{d\tau}(\eta e), $$ respectively.

$$ \tag{11} L_H~:=~ p_{\mu} \dot{X}^{\mu} - H, \qquad H~:=~ eT, \qquad T~:=~\frac{1}{2}(p^2+m^2), \qquad p^2~:=~ g^{\mu\nu}(X)~p_{\mu} p_{\nu},$$

$$ \tag{12} \phi^{\alpha} ~=~ \{ X^{\mu};~p_{\mu};~ e;~C;~ \bar{C};~B\} $$

are positions $X^{\mu}$; momenta $p_{\mu}$; einbein $e$; FP ghost $C$; FP antighost $\bar{C}$; and Lautrup-Nakanishi (LN) Lagrange multiplier $B$, respectively. They are WL tensors of contravariant orders $0$; $0$; $-1$; $r$; $1$; and $1$, respectively. Each field $\phi^{\alpha}$ has a corresponding antifield $\phi^{\ast}_{\alpha}$ of opposite Grassmann parity. The BV action

$$ S_{BV}~=~\int \! \mathrm{d}\tau ~L_{BV} , $$ $$\tag{13} L_{BV}~=~L_H +\left(X^{\ast}_{\mu} \dot{X}^{\mu}+p_{\ast}^{\mu} \dot{p}_{\mu} +r C^{\ast}\dot{C} \right)e^{r-1}C +\underbrace{e^r C\dot{e}^{\ast}}_{\sim~e^{\ast}\frac{d}{d\tau}( e^r C)} + B \bar{C}^{\ast}, $$

$$ \tag{14} (S_{BV},S_{BV})~=~0, $$

$$ \tag{15} (\phi^{\alpha}(\tau),\phi^{\ast}_{\beta}(\tau^{\prime})) ~=~\delta^{\alpha}_{\beta}~\delta(\tau\!-\!\tau^{\prime}).$$

$${\bf s}X^{\mu}~=~e^{r-1} C \dot{X}^{\mu}, \qquad {\bf s}p_{\mu}~=~e^{r-1} C \dot{p}_{\mu}, \qquad {\bf s}e~=~ \frac{d}{d\tau}( e^r C), $$ $$ \tag{16} {\bf s}C~=~ re^{r-1} C\dot{C},\qquad {\bf s}\bar{C}~=~ - B,\qquad {\bf s}B ~=~0, $$

$$ \tag{17} \psi ~:=~\int \! \mathrm{d}\tau~\bar{C}\left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right), $$ where $\xi,\epsilon\in\mathbb{R}$ are gauge-fixing parameters. Moreover, $\chi(e)=(e\!-\!e_0)\chi^{\prime}$ is a gauge-fixing condition (which we will assume is affine in $e$, so that the derivative $\chi^{\prime}$ is constant). The gauge-fixed Lagrangian becomes

$$ \tag{18} L_{\rm gf}~=~ \left. L_{BV} \right|_{\phi^{\ast}~=~\frac{\delta \psi}{\delta \phi}}~=~ L_H + \overbrace{\underbrace{\left(\chi^{\prime}\bar{C}-\epsilon\dot{\bar{C}}\right) \frac{d}{d\tau}(e^r C)}_{ \sim~ \bar{C} \left(\frac{\chi^{\prime}}{2}+\epsilon\frac{d}{d\tau}\right)\frac{d}{d\tau}(e^r C) + e^r C\left(\frac{\chi^{\prime}}{2}-\epsilon\frac{d}{d\tau}\right)\frac{d}{d\tau}\bar{C} }}^{\text{Faddeev-Popov term}} + \overbrace{B \left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right)}^{\text{gauge-fixing term}} , $$

IV) BFV formulation. We identify $p_e\approx\epsilon B$ with the canonical momentum of the einbein $e$, and we identify the antifield $e^{\ast}\equiv \bar{P}$ with the FP ghost momentum. Introduce an ultra-local Poisson bracket $\{\cdot,\cdot\}_{PB}$ with the following canonical pairs

$$ \{X^{\mu}(\tau), p_{\nu}(\tau^{\prime})\}_{PB} ~=~\delta^{\mu}_{\nu}~\delta(\tau\!-\!\tau^{\prime}), \qquad \{e(\tau)^rC(\tau), \bar{P}(\tau^{\prime})\}_{PB} ~=~\delta(\tau\!-\!\tau^{\prime}), $$ $$ \tag{19} \{e(\tau), B (\tau^{\prime})\}_{PB} ~=~\frac{1}{\epsilon}\delta(\tau\!-\!\tau^{\prime}), \qquad \{\bar{C}(\tau), P(\tau^{\prime})\}_{PB} ~=~\frac{1}{\epsilon}\delta(\tau\!-\!\tau^{\prime}). $$

$$\tag{20} \{C(\tau), \bar{P}(\tau^{\prime})\}_{PB}~=~e(\tau)^{-r}\delta(\tau\!-\!\tau^{\prime}), \qquad \{ B (\tau), C(\tau^{\prime})\}_{PB}~=~\frac{r}{\epsilon} \frac{C(\tau)}{e(\tau)} \delta(\tau\!-\!\tau^{\prime}), $$

$$ \tag{21} \{e(\tau)^r C(\tau), B(\tau^{\prime})\}_{PB}~=~0. $$

$${\bf s}X^{\mu}~=~e^r C g^{\mu\nu}(X)p_{\nu}~\approx~ e^{r-1} C \dot{X}^{\mu}, \qquad {\bf s}e~=~P~\approx~ \frac{d}{d\tau}( e^r C) , $$ $$ \tag{22} {\bf s}C~=~r\frac{C}{e}P ~\approx~ re^{r-1} C\dot{C},\qquad {\bf s}\bar{C}~=~ - B,\qquad {\bf s} B ~=~0, $$

which should be compared with eq. (16). Here the $\approx$ symbol means equality modulo eqs. of motion. The BRST transformation (22) is generated by

$$\tag{23} \mathbb{Q}~:=~ \int \! \mathrm{d}\tau ~Q, \qquad \{\mathbb{Q}, \mathbb{Q}\}_{PB}~=~0,$$

$$\tag{24} -Q~:=~T e^r C + \epsilon B P ~\approx~ T e^r C + \epsilon B \frac{d}{d\tau}( e^r C)$$

$$ \tag{25} S_{BFV} ~=~ \int \! \mathrm{d}\tau~\left(\dot{X}^{\mu}p_{\mu}+e^r C\dot{\bar{P}} \right) -\left\{ \psi, \mathbb{Q} \right\}_{PB} ~=~ \int \! \mathrm{d}\tau ~L_{BFV} , $$

$$\tag{26} \psi ~:=~\int \! \mathrm{d}\tau \left(\bar{C} \left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right) -\bar{P}e\right), $$

and where the BFV Lagrangian reads$^1$

$$ L_{BFV}~=~\left(p_{\mu}\dot{X}^{\mu}+ e^r C\dot{\bar{P}} \right) + \epsilon\left( B \dot{e} + \bar{C} \dot{P}\right) + \left(-eT +\bar{C}\chi^{\prime} P +B \left(\frac{\xi}{2}B+\chi(e)\right) -\bar{P}P \right)$$ $$\tag{27} ~\sim~ L_H+ \underbrace{\epsilon\left( B \dot{e} + \bar{C} \dot{P}\right)}_{\text{kinetic term}}+ \bar{P}\left( \frac{d}{d\tau}( e^r C)-P\right) + \underbrace{\bar{C} \chi^{\prime} P}_{\text{FP term}} + \underbrace{B \left(\frac{\xi}{2}B+\chi(e)\right)}_{\text{gauge-fixing term}} .$$

V) Dirac bracket. Let us integrate out the two FP momenta $P$ and $\bar{P}$. Then the BFV Lagrangian (27) becomes the gauge-fixed Lagrangian (18) from Section III. The corresponding two 2nd class constraints

$$\tag{28} \Theta~:=~ P - \frac{d}{d\tau}( e^r C)~\approx~0, \qquad \bar{\Theta}~:=~ \bar{P} - \chi^{\prime} \bar{C}+\epsilon\dot{\bar{C}}~\approx~0, $$

$$\tag{29} \Delta(\tau,\tau^{\prime} ) ~:=~ \{\Theta(\tau), \bar{\Theta}(\tau^{\prime}) \}_{PB} ~=~ -\left(\frac{\chi^{\prime}}{\epsilon}+2\frac{d}{d\tau} \right) \delta(\tau\!-\!\tau^{\prime}), $$

$$\tag{30} \Delta^{-1}(\tau,\tau^{\prime} ) ~=~ - \frac{1}{4} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}). $$

$$\tag{31} \{e(\tau)^rC(\tau), \bar{C}(\tau^{\prime})\}_{DB} ~=~ \frac{1}{4\epsilon} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}). $$

Alternatively, the Poisson structure (31) could be deduced from the FP term in the gauge-fixed Lagrangian (18).

$$ \{C(\tau), \bar{C}(\tau^{\prime})\}_{DB}~=~\frac{e(\tau)^{-r}}{4\epsilon} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}) , $$ $$ \tag{32} \{ B (\tau), C(\tau^{\prime})\}_{DB}~=~\frac{r}{\epsilon} \frac{C(\tau)}{e(\tau)} \delta(\tau\!-\!\tau^{\prime}), $$

$$ \tag{33} \{e(\tau)^r C(\tau), B(\tau^{\prime})\}_{DB}~=~0. $$

References:

$^1$ The $\epsilon$-dependence in the BFV action (25) comes only from the gauge-fixing fermion (26). The $\epsilon$-dependence can be removed via redefinition

$$\tag{34} \epsilon B~\longrightarrow~ B, \qquad \epsilon\bar{C}~\longrightarrow~ \bar{C}, \qquad \frac{\chi}{\epsilon} ~\longrightarrow~ \chi, \qquad \frac{\xi}{\epsilon^2} ~\longrightarrow~ \xi . $$

In the limit $\epsilon\to 0$, the infinities on the rhs. of the Poisson brackets (19) should be interpreted as zero, i.e. the corresponding canonical variables become decoupled.

$$ e~=~e^{\prime} \frac{d\tau^{\prime}}{d\tau}\tag{1} $$

$$ \tau\longrightarrow \tau^{\prime}=f(\tau).\tag{2} $$

$$ x^{\mu}~=~x^{\prime \mu}\tag{3} $$

$$ \dot{x}^{\mu}~=~\dot{x}^{\prime \mu}\frac{d\tau^{\prime}}{d\tau}.\tag{4}$$

$$ S~=~\int \! \mathrm{d}\tau ~L , \qquad L~:=~\frac{\dot{x}^2}{2e}-\frac{e m^2}{2},\tag{5}$$

$$ Y ~=~\eta \frac{d}{d\tau}~\in~ \Gamma(TI)\tag{6} $$

$$ {\cal L}_Y x^{\mu}~=~Y[x^{\mu}]~=~\eta \frac{dx^{\mu}}{d\tau},\tag{7} $$

$$ ({\cal L}_Ye)\mathrm{d}\tau~:=~{\cal L}_Y\omega ~=~\{\mathrm{d}, i_Y\}\omega~=~\mathrm{d}i_Y\omega $$ $$ ~=~\mathrm{d}(i_Y\omega)~=~\mathrm{d}(\eta e) ~=~\mathrm{d}\tau\frac{d}{d\tau}(\eta e),\tag{8} $$

$$ {\cal L}_Ye ~\stackrel{(8)}{=}~\frac{d}{d\tau}(\eta e).\tag{9} $$

Formula (6), (7) and (9) correspond to eq. (1.10) in Ref. 1 $$ \tag{1.10} \tau\to \tilde{\tau}=\tau-\eta, \qquad \delta x^{\mu}~=~\eta\frac{d x^{\mu}}{d\tau}, \qquad \delta e ~=~\frac{d}{d\tau}(\eta e), $$ respectively.

$$L_H~:=~ p_{\mu} \dot{x}^{\mu} - H, \qquad H~:=~ eT, \qquad T~:=~\frac{1}{2}(p^2+m^2), \qquad p^2~:=~ g^{\mu\nu}(x)~p_{\mu} p_{\nu}, \tag{11} $$

$$ \phi^{\alpha} ~=~ \{ x^{\mu};~p_{\mu};~ e;~C;~ \bar{C};~B\} \tag{12}$$

are positions $x^{\mu}$; momenta $p_{\mu}$; einbein $e$; FP ghost $C$; FP antighost $\bar{C}$; and Lautrup-Nakanishi (LN) Lagrange multiplier $B$, respectively. They are WL tensors of contravariant orders $0$; $0$; $-1$; $r$; $1$; and $1$, respectively. Each field $\phi^{\alpha}$ has a corresponding antifield $\phi^{\ast}_{\alpha}$ of opposite Grassmann parity. The corresponding BV action$^1$

$$ S_{BV}~=~\int \! \mathrm{d}\tau ~L_{BV} , $$ $$ L_{BV}~=~L_H +\left(x^{\ast}_{\mu} \dot{x}^{\mu}+p_{\ast}^{\mu} \dot{p}_{\mu} +r C^{\ast}\dot{C} \right)e^{r-1}C +\underbrace{e^r C\dot{e}^{\ast}}_{\sim~e^{\ast}\frac{d}{d\tau}( e^r C)} + B \bar{C}^{\ast},\tag{13} $$

$$ (S_{BV},S_{BV})~=~0, \tag{14}$$

$$ (\phi^{\alpha}(\tau),\phi^{\ast}_{\beta}(\tau^{\prime})) ~=~\delta^{\alpha}_{\beta}~\delta(\tau\!-\!\tau^{\prime}). \tag{15}$$

$${\bf s}x^{\mu}~=~e^{r-1} C \dot{x}^{\mu}, \qquad {\bf s}p_{\mu}~=~e^{r-1} C \dot{p}_{\mu}, \qquad {\bf s}e~=~ \frac{d}{d\tau}( e^r C), $$ $$ {\bf s}C~=~ re^{r-1} C\dot{C},\qquad {\bf s}\bar{C}~=~ - B,\qquad {\bf s}B ~=~0, \tag{16} $$

$$ \psi ~:=~\int \! \mathrm{d}\tau~\bar{C}\left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right), \tag{17} $$ where $\xi,\epsilon\in\mathbb{R}$ are gauge-fixing parameters. Moreover, $\chi(e)=(e\!-\!e_0)\chi^{\prime}$ is a gauge-fixing condition (which we will assume is affine in $e$, so that the derivative $\chi^{\prime}$ is constant). The gauge-fixed Lagrangian becomes

$$ L_{\rm gf}~=~ \left. L_{BV} \right|_{\phi^{\ast}~=~\frac{\delta \psi}{\delta \phi}}~=~ L_H + \overbrace{\underbrace{\left(\chi^{\prime}\bar{C}-\epsilon\dot{\bar{C}}\right) \frac{d}{d\tau}(e^r C)}_{ \sim~ \bar{C} \left(\frac{\chi^{\prime}}{2}+\epsilon\frac{d}{d\tau}\right)\frac{d}{d\tau}(e^r C) + e^r C\left(\frac{\chi^{\prime}}{2}-\epsilon\frac{d}{d\tau}\right)\frac{d}{d\tau}\bar{C} }}^{\text{Faddeev-Popov term}} + \overbrace{B \left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right)}^{\text{gauge-fixing term}} , \tag{18} $$

IV) Quantum master equation. The odd Laplacian reads

$$ \Delta~=~(-1)^{|\alpha|}\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ \delta(\tau\!-\!\tau^{\prime})\frac{\delta_L}{\delta\phi^{\alpha}(\tau)} \frac{\delta_L}{\delta\phi^{\ast}_{\alpha}(\tau^{\prime})}. \tag{19} $$

We calculate

$$ \Delta S_{BV}~\stackrel{(13)+(19)}{=}~ 2(n\!-\!r)\iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}C(\tau)~ \delta(\tau\!-\!\tau^{\prime}) \frac{d}{d\tau}\delta(\tau\!-\!\tau^{\prime}) $$ $$+r \iint\! \mathrm{d}\tau~\mathrm{d}\tau^{\prime}~ e(\tau)^{r-1}\dot{C}(\tau)~\delta(\tau\!-\!\tau^{\prime})^2~\neq~0, \tag{20} $$ where $n$ is the target space (TS) dimension. This show that the BV action (13) does not satisfy the quantum master equation; only the classical master equation. We will discuss appropriate modifications of the BV action (13) in Section VII.

V) BFV formulation. We identify $p_e\approx\epsilon B$ with the canonical momentum of the einbein $e$, and we identify the antifield $e^{\ast}\equiv \bar{P}$ with the FP ghost momentum. Introduce an ultra-local Poisson bracket $\{\cdot,\cdot\}_{PB}$ with the following canonical pairs

$$ \{x^{\mu}(\tau), p_{\nu}(\tau^{\prime})\}_{PB} ~=~\delta^{\mu}_{\nu}~\delta(\tau\!-\!\tau^{\prime}), \qquad \{e(\tau)^rC(\tau), \bar{P}(\tau^{\prime})\}_{PB} ~=~\delta(\tau\!-\!\tau^{\prime}), $$ $$ \{e(\tau), B (\tau^{\prime})\}_{PB} ~=~\frac{1}{\epsilon}\delta(\tau\!-\!\tau^{\prime}), \qquad \{\bar{C}(\tau), P(\tau^{\prime})\}_{PB} ~=~\frac{1}{\epsilon}\delta(\tau\!-\!\tau^{\prime}).\tag{21} $$

$$ \{C(\tau), \bar{P}(\tau^{\prime})\}_{PB} ~=~e(\tau)^{-r}\delta(\tau\!-\!\tau^{\prime}), \qquad \{ B (\tau), C(\tau^{\prime})\}_{PB} ~=~\frac{r}{\epsilon} \frac{C(\tau)}{e(\tau)} \delta(\tau\!-\!\tau^{\prime}), \tag{22} $$

$$ \{e(\tau)^r C(\tau), B(\tau^{\prime})\}_{PB}~=~0. \tag{23} $$

$${\bf s}x^{\mu}~=~e^r C g^{\mu\nu}(x)p_{\nu} ~\approx~ e^{r-1} C \dot{x}^{\mu}, \qquad {\bf s}p_{\mu} ~=~ -\frac{1}{2}e^r C \partial_{\mu}g^{\nu\lambda}(x)~p_{\nu}p_{\lambda} ~\approx~ e^{r-1} C \dot{p}_{\mu}, $$ $${\bf s}e~=~P~\approx~ \frac{d}{d\tau}( e^r C) , \qquad {\bf s}C~=~r\frac{C}{e}P ~\approx~ re^{r-1} C\dot{C},\qquad {\bf s}\bar{C}~=~ - B,\qquad {\bf s} B ~=~0, \tag{24} $$

which should be compared with eq. (16). Here the $\approx$ symbol means equality modulo eqs. of motion. The BRST transformation (24) is generated by

$$ \mathbb{Q}~:=~ \int \! \mathrm{d}\tau ~Q, \qquad \{\mathbb{Q}, \mathbb{Q}\}_{PB}~=~0,\tag{25}$$

$$ -Q~:=~T e^r C + \epsilon B P ~\approx~ T e^r C + \epsilon B \frac{d}{d\tau}( e^r C)\tag{26}$$

$$ S_{BFV} ~=~ \int \! \mathrm{d}\tau~\left(\dot{x}^{\mu}p_{\mu}+e^r C\dot{\bar{P}} \right) -\left\{ \psi, \mathbb{Q} \right\}_{PB} ~=~ \int \! \mathrm{d}\tau ~L_{BFV} , \tag{27} $$

$$ \psi ~:=~\int \! \mathrm{d}\tau \left(\bar{C} \left(\frac{\xi}{2}B +\chi(e) +\epsilon \dot{e}\right) -\bar{P}e\right),\tag{28} $$

and where the BFV Lagrangian reads$^2$

$$ L_{BFV}~=~\left(p_{\mu}\dot{x}^{\mu}+ e^r C\dot{\bar{P}} \right) + \epsilon\left( B \dot{e} + \bar{C} \dot{P}\right) + \left(-eT +\bar{C}\chi^{\prime} P +B \left(\frac{\xi}{2}B+\chi(e)\right) -\bar{P}P \right)$$ $$ ~\sim~ L_H+ \underbrace{\epsilon\left( B \dot{e} + \bar{C} \dot{P}\right)}_{\text{kinetic term}}+ \bar{P}\left( \frac{d}{d\tau}( e^r C)-P\right) + \underbrace{\bar{C} \chi^{\prime} P}_{\text{FP term}} + \underbrace{B \left(\frac{\xi}{2}B+\chi(e)\right)}_{\text{gauge-fixing term}} \tag{29} .$$

VI) Dirac bracket. Let us integrate out the two FP momenta $P$ and $\bar{P}$. Then the BFV Lagrangian (29) becomes the gauge-fixed Lagrangian (18) from Section III. The corresponding two 2nd class constraints

$$ \Theta~:=~ P - \frac{d}{d\tau}( e^r C)~\approx~0, \qquad \bar{\Theta}~:=~ \bar{P} - \chi^{\prime} \bar{C}+\epsilon\dot{\bar{C}}~\approx~0,\tag{30} $$

$$ \Delta(\tau,\tau^{\prime} ) ~:=~ \{\Theta(\tau), \bar{\Theta}(\tau^{\prime}) \}_{PB} ~=~ -\left(\frac{\chi^{\prime}}{\epsilon}+2\frac{d}{d\tau} \right) \delta(\tau\!-\!\tau^{\prime}),\tag{31} $$

$$ \Delta^{-1}(\tau,\tau^{\prime} ) ~=~ - \frac{1}{4} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}).\tag{32} $$

$$ \{e(\tau)^rC(\tau), \bar{C}(\tau^{\prime})\}_{DB} ~=~ \frac{1}{4\epsilon} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}).\tag{33} $$

Alternatively, the Poisson structure (33) could be deduced from the FP term in the gauge-fixed Lagrangian (18).

$$ \{C(\tau), \bar{C}(\tau^{\prime})\}_{DB} ~=~\frac{e(\tau)^{-r}}{4\epsilon} \exp\left[\frac{(\tau^{\prime}-\tau)\chi^{\prime}}{2\epsilon}\right] {\rm sgn}(\tau\!-\!\tau^{\prime}) , $$ $$ \{ B (\tau), C(\tau^{\prime})\}_{DB} ~=~\frac{r}{\epsilon} \frac{C(\tau)}{e(\tau)} \delta(\tau\!-\!\tau^{\prime}), \tag{34}$$

$$ \{e(\tau)^r C(\tau), B(\tau^{\prime})\}_{DB}~=~0.\tag{35} $$

VII) BV formulation revisited. Let us put $r=0$ from now on. Using the eom for $x^{\mu}$ and $p_{\mu}$ in the BRST transformations (24), we rewrite the BV Lagrangian (13) to

$$ \tilde{L}_{BV}~=~L_H +x^{\ast}_{\mu} g^{\mu\nu}(x)p_{\nu}C -\frac{1}{2}p_{\ast}^{\mu} \partial_{\mu}g^{\nu\lambda}(x)~p_{\nu}p_{\lambda} C +e^{\ast}\dot{C} + B \bar{C}^{\ast}. \tag{36} $$

One may show that the quantum master equation is now satisfied$^1$

$$ (\tilde{S}_{BV}, \tilde{S}_{BV})~=~0~=~\Delta\tilde{S}_{BV}. \tag{37} $$

The modification (36) does not alter the gauge-fixed Lagrangian (18) apart from putting $r=0$.

References:

$^1$ We ignore temporal boundary terms.

$^2$ The $\epsilon$-dependence in the BFV action (27) comes only from the gauge-fixing fermion (28). The $\epsilon$-dependence can be removed via redefinition

$$ \epsilon B~\longrightarrow~ B, \qquad \epsilon\bar{C}~\longrightarrow~ \bar{C}, \qquad \frac{\chi}{\epsilon} ~\longrightarrow~ \chi, \qquad \frac{\xi}{\epsilon^2} ~\longrightarrow~ \xi .\tag{38} $$

In the limit $\epsilon\to 0$, the infinities on the rhs. of the Poisson brackets (21) should be interpreted as zero, i.e. the corresponding canonical variables become decoupled.

10 replaced http://physics.stackexchange.com/ with https://physics.stackexchange.com/
source | link

should be invariant under reparametrizations (2). See also thisthis related Phys.SE post.

III) BV formulation. Let us mention for completeness that the gauge transformation $\delta$ can be encoded as a BRST transformation, cf. e.g. Ref. 2 and thisthis Phys.SE post. Roughly speaking, the Grassmann-even gauge parameter $\eta$ is then replaced by a Grassmann-odd Faddeev-Popov (FP) ghost $C$. (Actually, the gauge parameter $\eta$ will more precisely be replaced with the combination $e^{1-r}C$, where $r\in\mathbb{R}$ is a power, to be more general, cf. eq. (16) below.) To minimize the appearances of time derivatives, instead of using the Lagrangian (5), it becomes a bit simpler to start from the Hamiltonian Lagrangian

cf. e.g. thisthis Phys.SE post. Here we will use the Batalin-Vilkovisky (BV) formalism, cf. Ref. 3. The fields

should be invariant under reparametrizations (2). See also this related Phys.SE post.

III) BV formulation. Let us mention for completeness that the gauge transformation $\delta$ can be encoded as a BRST transformation, cf. e.g. Ref. 2 and this Phys.SE post. Roughly speaking, the Grassmann-even gauge parameter $\eta$ is then replaced by a Grassmann-odd Faddeev-Popov (FP) ghost $C$. (Actually, the gauge parameter $\eta$ will more precisely be replaced with the combination $e^{1-r}C$, where $r\in\mathbb{R}$ is a power, to be more general, cf. eq. (16) below.) To minimize the appearances of time derivatives, instead of using the Lagrangian (5), it becomes a bit simpler to start from the Hamiltonian Lagrangian

cf. e.g. this Phys.SE post. Here we will use the Batalin-Vilkovisky (BV) formalism, cf. Ref. 3. The fields

should be invariant under reparametrizations (2). See also this related Phys.SE post.

III) BV formulation. Let us mention for completeness that the gauge transformation $\delta$ can be encoded as a BRST transformation, cf. e.g. Ref. 2 and this Phys.SE post. Roughly speaking, the Grassmann-even gauge parameter $\eta$ is then replaced by a Grassmann-odd Faddeev-Popov (FP) ghost $C$. (Actually, the gauge parameter $\eta$ will more precisely be replaced with the combination $e^{1-r}C$, where $r\in\mathbb{R}$ is a power, to be more general, cf. eq. (16) below.) To minimize the appearances of time derivatives, instead of using the Lagrangian (5), it becomes a bit simpler to start from the Hamiltonian Lagrangian

cf. e.g. this Phys.SE post. Here we will use the Batalin-Vilkovisky (BV) formalism, cf. Ref. 3. The fields

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