The Hamilton–Jacobi analysis for higher-order Maxwell–Chern–Simons gauge theory

Abstract

By using the Hamilton–Jacobi [HJ] framework, the higher-order Maxwell–Chern–Simons theory is analyzed. The complete set of HJ Hamiltonians and a generalized HJ differential are reported, from which all symmetries of the theory are identified. In addition, we complete our study by performing the higher-order Gitman–Lyakhovich–Tyutin [GLT] framework and compare the results of both formalisms.

Appendix: Gauge transformations

1.1 HJ formalism

We start by calculating the characteristic equations from the fundamental differential, which will reveal the symmetries of the theory. Using (17), we find them to be

$$\begin{aligned} \nonumber d\xi _{0}= & {} v_{0}dt - d\sigma ^{2}, \\ \nonumber d\xi _{i}= & {} v_{i}dt + \partial _{i}d\sigma ^{3}, \\ \nonumber d\pi ^{0}= & {} \left[ \frac{1}{2}{\partial _{i}}v^{i} - \frac{1}{2}\nabla ^{2}\xi _{0} + \frac{3\theta }{4}\epsilon ^{i j}{\partial _{i}}\xi _{j} - \nabla ^{2}{\tilde{\pi }}^{0} + \frac{1}{4m}\epsilon ^{i j}\nabla ^{2}{\partial _{i}}\xi _{j} + \frac{1}{2}{\partial _{i}}\pi ^{i} \right] dt, \\ \nonumber d\pi ^{i}= & {} \left[ -\partial _{j}F^{i j}- \frac{\theta }{2}\epsilon ^{i j}v_{j} - \nabla ^{2}{\tilde{\pi }}^{i} \right] dt - \frac{1}{2m}\epsilon ^{i j}\partial _{j}d\sigma ^{1} + \left[ \frac{\theta }{2}\epsilon ^{i j}\partial _{j} + \frac{1}{2m}\epsilon ^{i j}\nabla ^{2}\partial _{j}\right] d\sigma ^{3}, \\ \nonumber dv_{0}= & {} \nabla ^{2}\xi _{0}dt + d\sigma ^{1}, \\ \nonumber dv_{i}= & {} \left[ \frac{1}{2} \nabla ^{2}\xi _{i}+ \frac{1}{2}\partial _{i}v_{0} -m\epsilon _{i j}v^{j} +m \epsilon _{i j}\partial ^{j}\xi _{0} + \frac{\theta m}{2}\xi _{i} + m\epsilon _{i j}\pi ^{j}\right] dt - \partial _{i}d\sigma ^{2}, \\ \nonumber d{\tilde{\pi }}^{0}= & {} -\pi ^{0}dt, \\ d{\tilde{\pi }}^{i}= & {} \left[ \frac{1}{2}v^{i} - \frac{1}{2}\partial ^{i}\xi _{0} + \frac{\theta }{4}\epsilon ^{i j}\xi _{j} - \frac{1}{4m}\epsilon ^{i j}\partial _{j}v_{0} + \frac{1}{4m}\epsilon ^{i j}\nabla ^{2}\xi _{j} - \frac{1}{2}\pi ^{i} \right] dt. \end{aligned}$$

(41)

The evolution of the dynamical variables with respect to our parameters $\sigma ^{i}$ is understood as canonical transformations, with the corresponding Hamiltonians $\Gamma ^{i}$ as generators [24, 25]. Due to Frobenius’ theorem [25], the transformation with respect to one of these parameters is independent of the evolution along the others. To relate these canonical transformations to the gauge ones, we set $dt=0$ [15], obtaining

$$\begin{aligned} \delta \xi _{0}= & {} -\delta \sigma ^{2}, \nonumber \\ \delta \xi _{i}= & {} \partial _{i} \delta \sigma ^{3}, \nonumber \\ \delta \pi ^{0}= & {} 0, \nonumber \\ \delta \pi ^{i}= & {} -\frac{1}{2m} \epsilon ^{i j} \partial _{j} \delta \sigma ^{1} + \left[ \frac{\theta }{2} \epsilon ^{i j} \partial _{j}+\frac{1}{2m} \epsilon ^{i j} \nabla ^{2} \partial _{j}\right] \delta \sigma ^{3}, \nonumber \\ \delta v_{0}= & {} \delta \sigma ^{1}, \nonumber \\ \delta v_{i}= & {} -\partial _{i} \delta \sigma ^{2}, \nonumber \\ \delta {\tilde{\pi }}^{0}= & {} 0, \nonumber \\ \delta {\tilde{\pi }}^{i}= & {} 0. \end{aligned}$$

(42)

In HJ, to find the gauge transformations it is necessary to see the specific conditions in which (42) acts into the Lagrangian. Thus, the Lagrangian (1) becomes invariant under these transformations if $\delta L =0$. This will result in relations between the parameters $\sigma ^{2}, \sigma ^3$. The variation in the Lagrangian is

$$\begin{aligned} \delta L = \int dt\;d^{2}x\; \left[ \frac{\partial {\mathcal {L}}}{\partial A_{\mu }}\delta A_{\mu } + \frac{\partial {\mathcal {L}}}{\partial (\partial _{\nu }A_{\mu })}\delta (\partial _{\nu }A_{\mu }) + \frac{\partial {\mathcal {L}}}{\partial (\partial _{\nu }\partial ^{\mu }A_{\mu })}\delta (\partial _{\nu }\partial ^{\mu }A_{\mu })\right] ; \end{aligned}$$

here we use $A_{\mu }$ instead of $\xi _{\mu }$ to more easily compare both formalisms. This, up to a total time derivative, is found to be

$$\begin{aligned} \delta L = \int dt\;d^{2}x\; \left[ \theta \epsilon ^{\sigma \nu \lambda }\partial _{\nu } A_{\lambda } + \partial _{\rho }F^{\rho \sigma } - \frac{1}{2m}\epsilon ^{\sigma \rho \mu }\left( \partial _{0}\partial ^{0}\partial _{\rho }A_{\mu }\right) + \frac{1}{m}\epsilon ^{\sigma \nu \lambda }\nabla ^{2}\partial _{\nu } A_{\lambda }\right] \delta A_{\sigma } = 0. \nonumber \\ \end{aligned}$$

(43)

We can combine the first and second equations in (42) to write the variation in $A_{\sigma }$ as

$$\begin{aligned} \delta A_{\sigma } = - \delta _{\sigma }^{0}\delta \sigma ^{2} + \delta _{\sigma }^{i} \partial _{i} \delta \sigma ^{3}; \end{aligned}$$

(44)

thus, by using (44) into (43) the variation in the action takes the form

$$\begin{aligned} \delta L = -\int dt\;d^{2}x\;\left( \theta \epsilon ^{i j}\partial _{i} A_{j} + \partial _{i}F^{i 0} - \frac{1}{2m}\epsilon ^{i j}\partial _{i}{\ddot{A}}_{j} + \frac{1}{m}\epsilon ^{i j}\nabla ^{2}\partial _{i} A_{j}\right) \left( \delta \sigma ^{2} + \partial _{0} \delta \sigma ^{3}\right) = 0.\nonumber \\ \end{aligned}$$

(45)

The theory will be invariant under (42) if the parameters $\sigma ^{i}$ obey

$$\begin{aligned} \delta \sigma ^{2} = -\partial _{0} \delta \sigma ^{3}; \end{aligned}$$

(46)

hence, from (44) the gauge transformations are given by

$$\begin{aligned} \delta A_{\mu } = \partial _{\mu } \delta \sigma ^{3}. \end{aligned}$$

(47)

Additionally, since $v_{\mu } = {\dot{A}}_{\mu }$, it can be seen that $\delta \sigma ^{1} = \partial _{0}\partial _{0} \delta \sigma ^{3}$.

1.2 GLT formalism

In this section, we use Castellani’s procedure [22, 26, 27] to obtain the gauge transformations. We start this calculation with the Hamiltonian (21), the constraints given in (32), and the Dirac brackets (36). First, we define the gauge generator as

$$\begin{aligned} G=\int \epsilon _{a} \gamma ^{a} d^{2}x, \end{aligned}$$

(48)

where $\epsilon _{a}$ are the gauge parameters and $a=1,2,3$. This generates infinitesimal gauge transformations on phase space variables, say F, through

$$\begin{aligned} \delta F=\int \delta \epsilon _{a}(y)\left\{ F(x), \gamma ^{a}(y)\right\} _{D} d^{2} y. \end{aligned}$$

(49)

In particular, the generator obeys the following equation, called the master equation,

$$\begin{aligned} \frac{\partial }{\partial t} G+\left\{ G, {\mathcal {H}}_{T}\right\} _{D}=0. \end{aligned}$$

(50)

where ${\mathcal {H}}_{T} = {\mathcal {H}} + u_{a}\gamma ^{a}$ is the total Hamiltonian. From the algebra of the constraints and the canonical Hamiltonian ${\mathcal {H}}$, we can obtain the structure functions $V_{b}^{a}$, $C_{c}^{a b}$, given by

$$\begin{aligned} \left\{ {\mathcal {H}}, \gamma ^{a}(\mathrm {x})\right\} _{D}= & {} \int d^{2}y\; V_{b}^{a}(x, y) \gamma ^{b}(y), \end{aligned}$$

(51)

$$\begin{aligned} \left\{ \gamma ^{a}(x), \gamma ^{b}(y)\right\} _{D}= & {} \int d^{2}z\; C_{c}^{a b}(x, y, z) \gamma ^{c}(z). \end{aligned}$$

(52)

Using these, the master equation becomes

$$\begin{aligned} \frac{d \epsilon _{a}(x)}{d t}-\int d^{2}y\; \epsilon _{b}(y) V_{a}^{b}(x, y) - \int d^{2}y\; d^{2}z\; \epsilon _{b}(y) \gamma _{c}(z) C_{a}^{c b}(x, y, z) = 0. \end{aligned}$$

(53)

Since the only nonzero structure functions are

$$\begin{aligned} V_{2}^{1}= & {} -\delta ^{2}(x-y) \quad , \quad V_{3}^{2} = -\delta ^{2}(x-y), \end{aligned}$$

with all the $C_{c}^{a b}=0$. We obtain the following relations between the generators.

$$\begin{aligned} \epsilon _{1}= & {} \ddot{\epsilon }_{3}, \nonumber \\ \epsilon _{2}= & {} -{\dot{\epsilon }}_{3}. \end{aligned}$$

(54)

Therefore, the generator has only one parameter and can be written as

$$\begin{aligned} G=\int d^{2}x\; \left( \delta \ddot{\epsilon }_{3} \gamma ^{1} - \delta {\dot{\epsilon }}_{3} \gamma ^{2} + \delta \epsilon _{3} \gamma ^{3}\right) ; \end{aligned}$$

(55)

using (49) the gauge transformations of the variables are

$$\begin{aligned} \delta A_{0}= & {} \int \delta \epsilon _{2}(y)\left[ \delta ^{2}(x-y)\right] d^{2}y, \nonumber \\ \delta A_{i}= & {} \int \delta \epsilon _{3}(y)\left[ \frac{\partial }{\partial y^{i}} \delta ^{2}(x-y)\right] d^{2}y, \nonumber \\ \delta \pi ^{0}= & {} \int 0 d^{2}y, \nonumber \\ \delta \pi ^{i}= & {} \int \delta \epsilon _{1}(y)\left[ \frac{1}{2m}\epsilon ^{i j}\frac{\partial }{\partial x^{j}}\delta ^{2}(x-y)\right] \nonumber \\&+ \delta \epsilon _{3}(y)\left[ -\frac{\theta }{2}\epsilon ^{i j}\frac{\partial }{\partial x^{j}}\delta ^{2}(x-y) \right. \nonumber \\&\left. - \frac{1}{2m}\epsilon ^{i j}\nabla _{y}^{2}\frac{\partial }{\partial x^{j}} \delta ^{2}(x-y)\right] d^{2}y, \nonumber \\ \delta v_{0}= & {} \int \delta \epsilon _{1}(y)\left[ -\delta ^{2}(x-y)\right] d^{2}y, \nonumber \\ \delta v_{i}= & {} \int \delta \epsilon _{2}(y)\left[ \frac{\partial }{\partial x^{i}} \delta ^{2}(x-y)\right] d^{2}y, \nonumber \\ \delta {\tilde{\pi }}^{0}= & {} \int 0 d^{2}y, \nonumber \\ \delta {\tilde{\pi }}^{i}= & {} \int 0 d^{2}y, \end{aligned}$$

(56)

and by using (54) the following gauge transformations are found

$$\begin{aligned} \delta A_{\mu }= & {} -\partial _{\mu } \delta \epsilon _{3}, \nonumber \\ \delta \pi ^{\mu }= & {} \epsilon ^{0 \mu j}\left( - \frac{\theta }{2} + \frac{1}{2m} - \frac{1}{2m}\nabla ^{2}\right) \partial _{j} \delta \epsilon _{3}, \nonumber \\ \delta v_{\mu }= & {} -\partial _{\mu } \delta {\dot{\epsilon }}_{3}, \nonumber \\ \delta {\tilde{\pi }}^{\mu }= & {} 0. \end{aligned}$$

(57)

By identifying $\sigma ^{3} = - \epsilon _{3}$, both formalisms agree (see equations (47) and (42)).