Gauge symmetries of a theory

[ubugnu]

Homework Statement

I want to derive Gauge symmetries of the following gauge theory:

$$S=\int\;dt L=\int d^4 x \;\epsilon^{\mu\nu\rho\sigma} B_{\mu\nu\;IJ} F_{\mu\nu}^{\;\;IJ}$$

Where $$B$$ is an antisymmetric tensor of rank two and $$F$$ is the curvature of a connection $$A$$ i.e: $$F=dA+A\wedge A$$, $$\mu,\nu...$$ are space-time indices and $$I,J...$$ are Lie Algebra indices (internal indices) I would like to find its symmetries.

Homework Equations

I rewrite the Lagrangian by splitting time and space indices $$\{\mu,\nu...=0..3\}\equiv \{0; i,j,...=1..3\}$$ I find:

$$L = \int d^3 x\;(P^i_{\;IJ}\dot{A}_i+B_i^{\,IJ}\Pi^i_{\,IJ}+A_0^{\;IJ}\Pi_{IJ})$$

Where $$\dot{A}_i = \partial_0 A_i$$, $$P^i_{\;IJ} = 2\epsilon^{ijk}B_{jk\,IJ}$$ is hence the conjugate momentum of $$A_i^{\,IJ}$$

$$B_i^{\,IJ}$$ and $$A_0^{\;IJ}$$ being Lagrange multipliers we obtain respectively two primary and two secondary constraints:

$$\Phi_{IJ} = P^0_{\;IJ} \approx0$$

$$\Phi_{\;\;IJ}^{\mu\nu} = P^{\mu\nu}_{\;\;IJ} \approx0$$

$$\Pi^i_{\,IJ} = 2\epsilon^{ijk}F_{jk\,IJ} \approx0$$

$$\Pi_{IJ}=(D_i P^i)_{IJ} \approx0$$

Where $$P^0_{\;IJ}$$ are the conjugate momentums of $$A_0^{\,IJ}$$ and $$P^{\mu\nu}_{\;\;IJ}$$ those of $$B_{\mu\nu}^{\;\;IJ}$$. Making these constraints constant in time produces no further constraints.

Whiche gives us a general constraint:

$$\Phi = \int d^3 x \;(\epsilon^{IJ}P^0_{\,IJ}+\epsilon_{\mu\nu}^{IJ}\;P^{\mu\nu}_{\;\;IJ}+\eta^{IJ}\Pi_{IJ}+\eta_i^{IJ}\Pi^i_{\;IJ})$$

The Attempt at a Solution

Each quantity $$F$$ have thus a Gauge transformation $$\delta F = \{F,\Phi\}$$ where $$\{...\}$$ denotes the Poisson bracket.

Knowing that this theory have the following Gauge symmetry:

$$\delta A = D\omega$$

$$\delta B = [B,\omega]$$

Where $$\omega$$ is a 0-form, I would like to retrieve these transformations using the relation below. (where $$\Phi$$ is considered as the generator of the Gauge symmetry) but my problem is that I don't know how to proceed, I already did this with a Yang-Mills theory and it worked... but for this theory it seems to le intractable! Someone to guide me?

 

[mark726]

Thank you for your question. To find the gauge symmetries of this gauge theory, we can follow a similar approach as in the case of a Yang-Mills theory. We start by writing down the Lagrangian in terms of the field variables and their conjugate momenta, as you have done. Then, we can use the primary and secondary constraints to derive the gauge transformations for the fields.

For the gauge field A, we have the constraint \Phi_{IJ} = P^0_{\;IJ} \approx0, which implies that A_0^{\;IJ} is a Lagrange multiplier. This means that A_0^{\;IJ} is not a dynamical field and can be set to zero without changing the equations of motion. Therefore, the gauge transformation for A is simply \delta A_i = D_i \omega, where \omega is a 0-form.

For the field B, we have the constraint \Pi^i_{\,IJ} = 2\epsilon^{ijk}F_{jk\,IJ} \approx0, which implies that B_{ij}^{\;\;IJ} is also a Lagrange multiplier. Therefore, the gauge transformation for B is \delta B_{ij}^{\;\;IJ} = [B_{ij}, \omega], where \omega is again a 0-form.

Using these transformations, we can check that the Lagrangian remains invariant under these gauge transformations. Furthermore, we can show that these gauge transformations generate the correct transformations for the field strengths F and G. This can be done by using the Poisson bracket relation \{F_{\mu\nu}, \Phi\} = \partial_\mu \omega_\nu - \partial_\nu \omega_\mu + [A_\mu, \omega_\nu] - [A_\nu, \omega_\mu] and similar for G. This will give us the desired gauge symmetries for this gauge theory.

I hope this helps you in your derivation. If you have any further questions, please don't hesitate to ask.