Constraints and conserved charges for modified massive and massless Abelian 1-form and 2-form theories: a brief review

Abstract

We demonstrate that the generators for the local, continuous and infinitesimal classical gauge symmetry transformations in the cases of (1) the Stückelberg-modified massive Abelian 1-form and 2-form theories, and (2) the massless Abelian 1-form and 2-form free theories owe their origin to the first-class constraints (on these theories). We discuss the appearance of these constraints at the quantum level, within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, through the physicality criteria w.r.t. the conserved and nilpotent (anti-)BRST charges. One of the highlights of our present investigation is the derivation of the nilpotent versions of the (anti-)BRST charges (from the standard non-nilpotent Noether (anti-)BRST charges) which lead to the appearance of the operator forms of the first-class constraints through the physicality criteria at the quantum level in the context of the modified massive and massless Abelian 2-form theories. We also comment on (1) the existence of the Curci-Ferrari (CF) type restrictions on the Abelian 2-form theories (with and without mass), (2) the modifications in the Stückelberg-technique for the massive 2D Abelian 1-form and 4D Abelian 2-form theories and their consequences, and (3) the off-shell nilpotent version of the conserved co-BRST charge and its role in the physicality criteria for the Stückelberg-modified 4D massive Abelian 2-form theory.

Appendices

Appendix A: on the derivation of (85) in covariant notation

In Sect. 6.1, we have derived the well-defined form of the 2D Lagrangian density for the modified Proca theory with the gauge-fixing term in a non-covariant manner because we have focused on: $F_{01} = \partial _0\, A_1 - \partial _1\, A_0$ and applied the modified form of Stückelberg-technique [cf. Eq. 79]. The purpose of our present Appendix is to derive (85) in a covariant fashion as we have derived in the context of the modified massive Abelian 2-form theory (see, e.g., [50]). We begin with the Lagrangian density of the 2D Proca theory (74) and focus on the individual terms. It turns out that, under (79), the kinetic term of (74) transforms to

$$\begin{aligned} -\, \frac{1}{4}\,F^{\mu \nu }\, F_{\mu \nu } \longrightarrow -\, \frac{1}{4}\,F^{\mu \nu }\, F_{\mu \nu } \mp \frac{1}{2\,m}\, F^{\mu \nu }\,\Sigma _{\mu \nu } - \, \frac{1}{4\,m^2}\, \Sigma ^{\mu \nu }\,\Sigma _{\mu \nu }, \end{aligned}$$

(A.1)

where the new notation $\Sigma _{\mu \nu }$ is an antisymmetric $(\Sigma _{\mu \nu } = -\, \Sigma _{\nu \mu } )$ tensor that is defined in terms of the derivatives on the pseudo-scalar field $(\tilde{\phi })$ as follows

$$\begin{aligned} \Sigma _{\mu \nu } = \big (\varepsilon _{\mu \rho }\,\partial _\nu - \varepsilon _{\nu \rho }\,\partial _\mu \big )\, \big (\partial ^\rho \,\tilde{\phi }\big ), \end{aligned}$$

(A.2)

where $\varepsilon _{\mu \nu }$ is the antisymmetric ($\varepsilon _{\mu \nu } = -\, \varepsilon _{\nu \mu }$) 2D Levi-Civita tensor and $\tilde{\phi }$ is the pseudo-scalar field (that is present in the modified form of the Stückelberg-technique [cf. Eq. 79]). For the 2D Proca theory, it is clear that

$$\begin{aligned} -\, \frac{1}{4}\,F_{\mu \nu }\, F^{\mu \nu } = \frac{1}{2}\, E^2 \equiv \frac{1}{2}\, \big (-\, \varepsilon ^{\mu \nu }\, \partial _\mu \, A_\nu \big )\, \big (-\, \varepsilon ^{\alpha \beta }\, \partial _\alpha \, A_\beta \big ), \end{aligned}$$

(A.3)

where $E = \partial _0\, A_1 - \partial _1\, A_0$ has been expressed in its covariant form as: $E = -\, \varepsilon ^{\mu \nu }\, \partial _\mu \, A_\nu$. The second term of (A.1) is a term that contains higher derivatives (e.g., three derivatives for our 2D massive Abelian 1-form theory). To get rid of one higher derivative, we see that the second term can be expressed as

$$\begin{aligned} \pm \frac{1}{m}\,F^{\mu \nu }\, \big (\varepsilon _{\nu \lambda }\, \partial _\mu \, \partial ^\lambda \, \tilde{\phi }\big ) \equiv \mp \frac{1}{m}\, \big (\partial _\mu \, F^{\mu \nu } \big ) \, \varepsilon _{\nu \lambda } \partial ^\lambda \, \tilde{\phi }, \end{aligned}$$

(A.4)

where we have dropped a total spacetime derivative and used the antisymmetric properties of $F^{\mu \nu }$ and $\Sigma _{\mu \nu }$. Using the on-shell condition: $\partial _\mu \, F^{\mu \nu } + m^2\, A^\nu = 0$ (which is equivalent to $(\Box + m^2 )\, A_\mu = 0$ provided we take into account the subsidiary condition: $\partial \cdot A = 0$ for $m^2 \ne 0$), we can re-express (A.4) as:

$$\begin{aligned} \pm m\, A^\nu \, \varepsilon _{\nu \lambda }\, \partial ^\lambda \, \tilde{\phi }= \mp m \, (\varepsilon ^{\nu \lambda }\, \partial _\lambda \, A_\nu )\, \tilde{\phi }\equiv \mp m\, (- \, \varepsilon ^{\mu \nu }\, \partial _\mu \, A_\nu )\, \tilde{\phi }. \end{aligned}$$

(A.5)

To derive (A.5), we have used the on-shell conditions: $\partial _\mu \, F^{\mu \nu } + m^2\ A^\nu = 0$ and $(\Box + m^2 )\, A_\mu = 0$ in an ad-hoc and arbitrary fashion. However, we shall see that the appropriately defined Lagrangian density (with an appropriate gauge-fixing term) for the modified 2D Proca theory will produce these equivalent equations as the EL-EoMs.

At this stage, let us focus on the third term of (A.1) which contains four derivatives. We can write explicitly this term as:

$$\begin{aligned} - \, \frac{1}{4\,m^2}\, \Sigma ^{\mu \nu }\,\Sigma _{\mu \nu } = - \, \frac{1}{4\,m^2}\,\Big [\big (\varepsilon ^{\nu \lambda }\,\partial ^\mu - \varepsilon ^{\mu \lambda }\,\partial ^\nu \big ) \Big ] \big (\partial _\lambda \,\tilde{\phi }\big )\, \Big [ \big (\varepsilon _{\nu \rho }\,\partial _\mu - \varepsilon _{\mu \rho }\,\partial _\nu \big )\Big ] \big (\partial ^\rho \,\tilde{\phi }\big ), \end{aligned}$$

(A.6)

which will lead to four terms. However, it turns out that two of them are total spacetime derivatives which can be automatically ignored as they are a part of the Lagrangian density. Only two terms contribute equally which can be added together to produce:

$$\begin{aligned} - \, \frac{1}{2\,m^2}\, \big (\varepsilon ^{\nu \lambda }\,\partial ^\mu \partial _\lambda \,\tilde{\phi }\big ) \, \big (\varepsilon _{\nu \rho }\,\partial _\mu \partial ^\rho \,\tilde{\phi }\big ) \equiv \frac{1}{2\,m^2}\, \big (\Box \, \partial _\lambda \,\tilde{\phi }\big )\, \big (\partial ^\lambda \,\tilde{\phi }\big ), \end{aligned}$$

(A.7)

where we have dropped a total spacetime derivative and have taken: $\varepsilon ^{\nu \lambda }\, \varepsilon _{\nu \rho } = -\, \delta ^\lambda _\rho$. To get rid of the higher derivative in the above equation, we take the help of the on-shell condition: $(\Box + m^2 )\, \tilde{\phi }= 0 \Longrightarrow (\Box + m^2 )\, \partial _\mu \, \tilde{\phi }= 0$. Finally, we obtain, from (A.7), the following

$$\begin{aligned} \frac{1}{2}\,\partial _\mu \, \tilde{\phi }\, \partial ^\mu \, \tilde{\phi }= -\, \frac{1}{2} \tilde{\phi }\, \Box \, \tilde{\phi }\;\; \equiv \;\;+ \frac{1}{2}\, m^2\,\tilde{\phi }^2, \end{aligned}$$

(A.8)

where we have used: $(\Box + m^2 )\, \tilde{\phi }= 0$ in an ad-hoc and arbitrary fashion. However, we shall show its sanctity from the appropriately defined 2D Lagrangian density.

Taking into account the equations (A.3), (A.5) and (A.8), we observe that (A.1) can be expressed in the language of the renormalizable terms for our 2D theory as:

$$\begin{aligned}{} & {} -\, \frac{1}{4}\,F^{\mu \nu }\, F_{\mu \nu } \longrightarrow \frac{1}{2}\, \Big [ (-\, \varepsilon ^{\mu \nu }\, \partial _\mu \, A_\nu )\, (-\, \varepsilon ^{\alpha \beta }\, \partial _\alpha \, A_\beta ) \mp 2\, m\, (- \, \varepsilon ^{\mu \nu }\, \partial _\mu \, A_\nu )\, \tilde{\phi }+ m^2\,\tilde{\phi }^2 \Big ] \nonumber \\{} & {} \quad = \frac{1}{2}\, \Big [ (-\, \varepsilon ^{\mu \nu }\, \partial _\mu \, A_\nu ) \mp m\, \tilde{\phi }\Big ]^2 \equiv \frac{1}{2}\,\Big [ E \mp m\, \tilde{\phi }\Big ]^2. \end{aligned}$$

(A.9)

Thus, we have obtained a well-defined transformations (A.9) from the transformations (A.1) as far as our modified 2D Proca theory is concerned. It should be noted that the latter contained the higher derivatives in the second and third terms. We now focus on the transformation of the mass term of (74) under the modified form of the Stückelberg-technique in (79), namely;

$$\begin{aligned}{} & {} \frac{m^2}{2}\, A_\mu \,A^\mu \longrightarrow \frac{m^2}{2}\, A_\mu \,A^\mu \mp m \,A_\mu \,\partial ^\mu \, \phi + \frac{1}{2}\, \partial _\mu \, \phi \, \partial ^\mu \, \phi - \frac{1}{2}\,\partial _\mu \,\tilde{\phi }\,\,\partial ^\mu \,\tilde{\phi }\mp m\, \varepsilon ^{\mu \rho } \, A_\mu \, \partial _\rho \tilde{\phi }\nonumber \\{} & {} \quad \equiv \frac{m^2}{2}\, A_\mu \,A^\mu \mp m \,A_\mu \,\partial ^\mu \, \phi + \frac{1}{2}\, \partial _\mu \, \phi \, \partial ^\mu \, \phi - \frac{1}{2}\,\partial _\mu \,\tilde{\phi }\,\,\partial ^\mu \,\tilde{\phi }\pm m\, (-\, \varepsilon ^{\mu \nu }\, \partial _\mu \, A_\nu ) \,\tilde{\phi }\end{aligned}$$

(A.10)

where we have dropped a total spacetime derivative term. Taking into account: $E = -\, \varepsilon ^{\mu \nu }\, \partial _\mu \, A_\nu$, we observe that the modified form of the 2D Proca Lagrangian density (74), along with a gauge-fixing term in the ’t Hooft gauge [i.e., $-\, (1/2)\, (\partial \cdot A \pm m\, \phi )^2$] is nothing but the sum of (A.9), (A.10) and the appropriate gauge-fixing term that turns out to be equal to (85) which we have derived in a different manner in Sect. 6.1. It is straightforward to check that (85) produces the on-shell conditions: $(\Box + m^2 )\, A_\mu = 0, \; (\Box + m^2 )\, \phi = 0$ and $(\Box + m^2 )\, \tilde{\phi }= 0$ which have been taken into considerations to get rid of the higher derivative terms (emerging out from the modified form of the Stückelberg-technique [79]).

Appendix B: on the CF-type restrictions

The purpose of this Appendix is to show (very concisely) that the Lagrangian density (109) respects the anti-BRST symmetry transformations, too, provided the symmetry consideration is discussed on a sub-manifold in the quantum Hilbert space of fields where the CF-type restrictions: $B_\mu - {\bar{B}}_\mu - \partial _\mu \, \phi = 0$ and $B + {\bar{B}} + m\, \phi = 0$ are satisfied. In this context, first of all, we list here the infinitesimal and continuous anti-BRST symmetry transformations ($s_{ab}$) from our earlier work (see, e.g., [40] for details).

$$\begin{aligned} s_{ab} B_{\mu \nu }= & {} - \,(\partial _\mu {\bar{C}}_\nu - \partial _\nu \bar{C}_\mu ), \quad s_{ab} {\bar{C}}_\mu = - \,\partial _\mu \bar{\beta }, \quad s_{ab} \phi _\mu = \partial _\mu {\bar{C}} - m\, {\bar{C}}_\mu , \quad s_{ab} \phi = \rho ,\nonumber \\ s_{ab} C_\mu= \, & {} {\bar{B}}_\mu , \quad s_{ab} \beta = - \,\lambda , \quad s_{ab} {\bar{C}} = -\, m\, \bar{\beta }, \quad s_{ab} C = {\bar{B}}, \quad s_{ab} B = - \, m\, \rho , \nonumber \\ s_{ab} B_\mu= \, & {} \partial _\mu \rho , \qquad s_{ab} [\rho , \, \lambda , \, \bar{\beta }, \, {\bar{B}}, \, \, {{\mathcal {B}}}, \, \bar{{\mathcal {B}}}, \, \tilde{\phi }, \, {\bar{B}}_\mu , \, {{\mathcal {B}}}_\mu , \, \tilde{\phi }_\mu , H_{\mu \nu \kappa }] = 0, \end{aligned}$$

(B.1)

which are found to be off-shell nilpotent $(s_{ab}^2 = 0)$ and absolutely anticommuting $(s_b\, s_{ab} + s_{ab}\, s_b = 0)$ with the off-shell nilpotent $(s_{b}^2 = 0)$ BRST symmetry transformations (110) provided the CF-type restriction: $B_\mu - {\bar{B}}_\mu - \partial _\mu \, \phi = 0$, $B + {\bar{B}} + m\, \phi = 0$ are invoked for the proof of the latter (see, e.g., [29, 30, 50]). Mathematically, this can be expressed as:

$$\begin{aligned} \{s_b, \, s_{ab}\}B_{\mu \nu }= & {} \partial _\mu (B_\nu - {\bar{B}}_\nu ) - \partial _\nu (B_\mu - {\bar{B}}_\mu ),\nonumber \\ \{s_b, \, s_{ab}\}\Phi _\mu= & {} \partial _\mu (B + {\bar{B}}) + m\, (B_\mu - {\bar{B}}_\mu ). \end{aligned}$$

(B.2)

In other words, we note that $\{ s_b, \, s_{ab} \} = 0$ in (B.2) provided we invoke the validity of the above CF-typer restrictions. It is straightforward to note that when we apply the anti-BRST symmetry transformations on ${{\mathcal {L}}}_{{{\mathcal {B}}}}$ [cf. Eq. 109], we obtain the following:

$$\begin{aligned} s_{ab} {{\mathcal {L}}}_{{{\mathcal {B}}}}= \, & {} \partial _\mu \bigg [-\, m \, \varepsilon ^{\mu \nu \eta \kappa } \tilde{\phi }_\nu \big (\partial _\eta {\bar{C}}_\kappa \big ) + B^{\mu \nu }\, \partial _\nu \, \rho + ({\bar{B}}^\mu + m\, \phi ^\mu )\, \rho - \lambda \, \big (\partial ^\mu \bar{\beta }\big ) \nonumber \\{} & {} - \big (\partial ^\mu {\bar{C}}^\nu - \partial ^\nu {\bar{C}}^\mu \big )\, B_\nu - \big (\partial ^\mu {\bar{C}} - m {\bar{C}}^\mu \big )\, B \bigg ] + \big [B_\mu - {\bar{B}}_\mu - \partial _\mu \varphi \big ] \big (\partial ^\mu \rho \big ) \nonumber \\{} & {} + \big (\partial ^\mu {\bar{C}}^\nu - \partial ^\nu {\bar{C}}^\mu \big )\, \partial _\mu \big [B_\nu - {\bar{B}}_\nu - \partial _\nu \varphi \big ] + m\, \big (\partial ^\mu {\bar{C}} - m {\bar{C}}^\mu \big )\, \big [B_\mu - {\bar{B}}_\mu - \partial _\mu \phi \big ] \nonumber \\{} & {} + m\, \big [B + {\bar{B}} + m \phi \big ] \rho + \big (\partial ^\mu {\bar{C}} - m {\bar{C}}^\mu \big )\,\partial _\mu \big [B + {\bar{B}} + m \phi \big ], ~~~~~~~~~~~~~~~~ \end{aligned}$$

(B.3)

A close look at the above expression shows that the Lagrangian density ${{\mathcal {L}}}_{{\mathcal {B}}}$ [cf. Eq. 109] respects the anti-BRST symmetry transformations, too, provided we take into account the validity of the CF-type restrictions: $B_\mu - \bar{B}_\mu - \partial _\mu \phi = 0$ and $B + {\bar{B}} + m\, \phi = 0$. In other words, if we impose the CF-type restrictions from outside, we find that ${{\mathcal {L}}}_{{{\mathcal {B}}}}$ also respects anti-BRST symmetry transformations because this Lagrangian density transforms to:

$$\begin{aligned} s_{ab} {{\mathcal {L}}}_{{{\mathcal {B}}}}= \, & {} \partial _\mu \bigg [-\, m \, \varepsilon ^{\mu \nu \eta \kappa } \tilde{\phi }_\nu \big (\partial _\eta {\bar{C}}_\kappa \big ) + B^{\mu \nu }\, \partial _\nu \, \rho + ({\bar{B}}^\mu + m\, \phi ^\mu )\, \rho - \lambda \, \big (\partial ^\mu \bar{\beta }\big ) \nonumber \\{} & {} - \big (\partial ^\mu {\bar{C}}^\nu - \partial ^\nu {\bar{C}}^\mu \big )\, B_\nu - \big (\partial ^\mu {\bar{C}} - m {\bar{C}}^\mu \big )\, B \bigg ]. \end{aligned}$$

(B.4)

Hence, the action integral: $S = \int d^4 x \, {{\mathcal {L}}}_{{{\mathcal {B}}}}$ respects $(s_{ab}\, S = 0)$ the anti-BRST symmetry transformations (B.1) on the sub-manifold of the quantum Hilbert space of fields where the CF-type restrictions (i.e., $B_\mu - {\bar{B}}_\mu - \partial _\mu \, \phi = 0$, $B + {\bar{B}} + m\, \phi = 0$) are satisfied.

We wrap-up this Appendix with the remark that for the massless Abelian 2-form theory, the Lagrangian density (109) reduces to the following form:

$$\begin{aligned} {{\mathcal {L}}}_{{\mathcal {B}}}^{(m = 0)}= \, & {} {{\mathcal {B}}}_\mu \,{{\mathcal {B}}}^\mu + {{\mathcal {B}}}^\mu \,\Big [\frac{1}{2}\,\varepsilon _{\mu \nu \lambda \xi }\,\partial ^\nu \,B^{\lambda \xi } - \partial _\mu \tilde{\phi }\Big ] + B^\mu \, \Big [ \partial ^\nu \,B_{\nu \mu } - \partial _\mu \,\phi \Big ]\, + B^\mu \,B_\mu \nonumber \\{} & {} + \partial _\mu \,\bar{\beta }\, \partial ^\mu \,\beta + (\partial _\mu \,{\bar{C}}_\nu - \partial _\nu \,\bar{C}_\mu )\,(\partial ^\mu \,C^\nu ) + \Big (\partial \cdot {\bar{C}} + \rho \Big )\,\lambda + \Big (\partial \cdot C - \lambda \Big )\,\rho , \end{aligned}$$

(B.5)

which transforms under the following anti-BRST symmetry transformations:

$$\begin{aligned}{} & {} s_{ab} B_{\mu \nu } = \, - \,(\partial _\mu {\bar{C}}_\nu - \partial _\nu \bar{C}_\mu ), \,\, s_{ab} {\bar{C}}_\mu = - \,\partial _\mu \bar{\beta }, \,\, s_{ab} C_\mu = {\bar{B}}_\mu , \quad s_{ab} \beta = - \,\lambda , \nonumber \\{} & {} s_{ab} C = {\bar{B}}, \quad s_{ab} \phi = \rho , \quad s_{ab} B_\mu = \partial _\mu \rho , \quad s_{ab} [ \rho , \, \lambda , \, \bar{\beta }, \,\tilde{\phi },\, {{\mathcal {B}}}_\mu , \, {\bar{B}}_\mu , \, H_{\mu \nu \kappa }] = 0, \end{aligned}$$

(B.6)

to the following total spacetime derivative plus terms that vanish due to CF-type restriction:

$$\begin{aligned} s_{ab}\, {{\mathcal {L}}}_{{\mathcal {B}}}^{(m = 0)}= \, & {} \partial _\mu \, \Big [ {\bar{B}}^\mu \rho + B^{\mu \nu }\, \partial _\nu \, \rho - \lambda \, \partial ^\mu \, \bar{\beta }- (\partial ^\mu \, {\bar{C}}^\nu - \partial ^\nu \, {\bar{C}}^\mu )\, B_\nu \Big ] \nonumber \\{} & {} + (B_\mu - {\bar{B}}_\mu - \partial _\mu \phi )\, (\partial ^\mu \, \rho ) + (\partial ^\mu \, {\bar{C}}^\nu - \partial ^\nu \, {\bar{C}}^\mu )\,\partial _\mu \, (B_\nu - {\bar{B}}_\nu - \partial _\nu \phi ) \Big ]. \end{aligned}$$

(B.7)

In other words, if we invoke the validity of the CF-type restrictions: $B_\mu - {\bar{B}}_\mu - \partial _\mu \phi = 0$, we obtain the following

$$\begin{aligned} s_{ab}\, {{\mathcal {L}}}_{{\mathcal {B}}}^{(m = 0)} = \partial _\mu \, \Big [ {\bar{B}}^\mu \rho + B^{\mu \nu }\, \partial _\nu \, \rho - \lambda \, \partial ^\mu \, \bar{\beta }- (\partial ^\mu \, {\bar{C}}^\nu - \partial ^\nu \, {\bar{C}}^\mu )\, B_\nu \Big ], \end{aligned}$$

(B.8)

which shows that the action integral: $S = \int d^4 x \, {{\mathcal {L}}}_{{{\mathcal {B}}}}^{(m = 0)}$ respects $(s_{ab} \, S = 0)$ the anti-BRST symmetry transformations (B.6). Thus, as far as the symmetry considerations are concerned, we have shown that the CF-type restrictions are to be respected if we wish to have the BRST and anti-BRST symmetry together for the Lagrangian density (109) for the modified massive as well as massless cases of the Abelian 2-form theories. The existence of the CF-type restriction(s) is as fundamental at the quantum level as the existence of the first-class constraints for a classical gauge theory when the latter is discussed within the framework of BRST formalism. The CF-type restriction(s) of the higher Abelian p-form ($p = 2, 3$) gauge theories are connected with the geometrical objects called gerbes [62, 63].