Lagrangian constraints and pre-symplectic double forms

It is interesting to apply the pre-symplectic formalism described in Section 4.5 to the specific models defined in the preceding Sections. We consider first the Einstein-Cartan theory of Section 5.1. The normal equations (6.5) do not contain the ``velocities'' namely the structure coefficients, and therefore coincide with the primary constraints. By sustituting them into eq. (4.79), we obtain

$$\theta' = k \epsilon_{ikjl} \,\hat d \omega^{[ik]} \wedge \omega^j \... ...^l - k_1 \epsilon_{ikjl} \hat d (\omega^{[ik]} \wedge \omega^j \wedge \omega^l)$$

$$+ 3^{-1} k_2 \epsilon_{ikjn} g_{lm} \hat d (\omega^{[ik]} \wedge \omega^{[jl]} \wedge \omega^{[mn]}) + 2 k_5 \chi^k \hat d \eta_k.$$ (6.62)

Since we have $k_5 = 0$ and the other coefficients $k, k_1, k_2$ are constant, we obtain immediately the pre-symplectic double form

$$\Omega' = - \hat d \theta = k \epsilon_{ikjl} \,\hat d \ome... ...\,\hat d \omega^{[ik]} \wedge \hat d \omega^j \wedge \omega^l.$$ (6.63)

One must keep in mind that only the restriction of this form to the 3-dimensional surface $\Sigma$ contributes to the pre-symplectic form $\Omega(\Sigma)$.

In the scalar-tensor theory of Sections 6.1 and 6.2 the coefficients are given by eqs. (6.25) and (6.30) and the normal equations contain the structure coefficients through the fields $\phi$ and $\chi_k$. In order to obtain the constraint equations, we have to express them as functions of the ``canonical momenta'' $\sigma_{\alpha}$, for instance (for $m \neq 1, 2$) by means of the equations

$$\epsilon^{mnik} i(A_m) i(A_n) \sigma_{ik} = 24 (m-2)(m-1)^{-1} \phi^{m-1},$$ (6.64)

$$\epsilon^{mnik} i(A_m) i(A_n) \sigma_i = 12 \alpha \phi^{m-2} \chi^k,$$ (6.65)

that follow from eq. (6.5). Remember that the vector fields $A_{\alpha}$ are uniquely determined by the 1-forms $\omega^{\alpha}$.

In this way we obtain the pre-symplectic double form

$$\Omega' = 2^{-2} \epsilon_{ikjl} \hat d (\phi^{m - 1}) \wedge \hat d... ...epsilon_{ikjl} \, \hat d \omega^{[ik]} \wedge \hat d (\omega^j \wedge \omega^l)$$

$$- 2^{-2} (m-1)^{-1} \epsilon_{ikjl} \hat d (\phi^{m - 1})\wedge \hat d (\omega^{[ik]} \wedge \omega^j \wedge \omega^l)$$

$$- 2 \alpha \hat d (\phi^{m - 2} \chi^k) \wedge \hat d \eta_k. \quad$$ (6.66)

If we introduce the new variables $\Phi$ and $\tilde \omega^{ik}$ by means of eqs. (6.32) and (6.38), after some calculations, we get

$$\Omega' = 2^{-2} \epsilon_{ikjl} \, \hat d \tilde \omega^{[i... ...omega^l) - 2 \alpha \hat d (\Phi \chi^k) \wedge \hat d \eta_k.$$ (6.67)

Through this change of variables (that depends on $m$) we have obtained a pre-symplectic form that, as the field equations, does not contain the parameter $m$. This means that $m$ should also be absent in an hypothetic quantized theory, as soon as a parameter $\ell$ is not introduced.

If $n > 10$, namely if internal gauge fields are present, one has to take into account the primary constraint following from eq. (5.12), namely

$$i(A_{[ik]}) \sigma_a = i(A_b) \sigma_a = 0.$$ (6.68)

The contribution of the internal gauge fields to the pre-symplectic double form is formally unchanged, namely

$$\Omega' = \hat d \omega^a \wedge \hat d \sigma_a,$$ (6.69)

but it is defined in the submanifold $\Gamma'$. It may be difficult to find the connection with the primary constraint of the usual formalism in the spacetime $\mathcal{M}$. It is based on a different choice of the unconstrained phase space $\Gamma$, but what is relevant is the physical phase space.

If there is a scalar field $\Xi$ of the kind described in Section 5.3, from eq. (5.28) we obtain the primary constraints

$$i(A_{[ik]}) \pi^S = i(A_b) \pi^S = 0$$ (6.70)

and the contribution to the pre-symplectic double form maintains the form

$$\Omega' = \hat d \Xi^T \wedge \hat d \pi.$$ (6.71)

Finally, we have to consider the Dirac fields. Eq. (5.37) does not contain the ``velocities`` and therefore provides directly the primary constraint. The contribution to the pre-symplectic double form is

$$\Omega' = - i \hat d \Psi^T C \gamma^i \wedge \hat d \Psi \eta_i.$$ (6.72)

A new feature is that the fields $\Psi$ are anticommuting and this expression does not vanish because the matrices $C \gamma^i$, according to eq. (3.15), are symmetric.