# Massive spin-1 field and Proca Lagrangian

In his book Quantum Field Theory and the Standard Model, Matthew D. Schwartz derives the Lagrangian for the massive spin 1 field (section 8.2.2). In eq. (8.23) he finds this to be \begin{align} \mathcal L&=\frac{1}{2}A_\mu\square A^\mu-\frac{1}{2}A_\mu\partial^\mu\partial_\nu A^\nu+\frac{1}{2}m^2A_\mu A^\mu,\tag{8.23} \end{align} where $$\square = \partial_\mu\partial^\mu$$. In the very same equation, he equates this to the Proca Lagrangian \begin{align} \mathcal L=\mathcal L_\mathrm{Proca}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^2A_\mu A^\mu,\tag{8.23} \end{align} where $$F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$$.

I fail to understand however, how the first Lagrangian can be rewritten to this Proca Lagrangian. My attempt was to rewrite the first term of the Proca Lagrangian into something that resembles the first two terms of the first Lagrangian above. It involves the product rule \begin{align} -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}&=-\frac{1}{4}(2\partial_\mu A_\nu\partial^\mu A^\nu-2\partial_\mu A_\nu\partial^\nu A^\mu)\\ &=-\frac{1}{4}(2\partial_\mu[A_\nu\partial^\mu A^\nu]-2A_\nu\partial_\mu\partial^\mu A^\nu-2\partial_\mu[A_\nu\partial^\nu A^\mu]+2A_\nu\partial_\mu\partial^\nu A^\mu)\\ &=\frac{1}{2}A_\mu\square A^\mu-\frac{1}{2}A_\mu\partial^\mu\partial_\nu A^\nu+\frac{1}{2}\partial_\mu(A_\nu\partial^\nu A^\mu)-\frac{1}{2}\partial_\mu(A_\nu\partial^\mu A^\nu), \end{align} having applied some relabelling in the second term of the final expression. The first two terms in this final expression are the first two terms in the Lagrangian, but then I'm stuck with the final two terms. Could someone explain to me what I'm missing here?

Also, the equation of motion for the Proca Lagrangian are \begin{align} (\square+m^2)A_\mu=0\tag{8.18}\\ \partial_\mu A^\mu=0. \end{align} Substituting this in the first Lagrangian would make it vanish. How does that make sense?

• So the question is about total derivative terms? Commented May 1, 2020 at 15:15
• Indeed, as well as why the Lagrangian seems to vanish under the equations of motion. Commented May 1, 2020 at 15:45
• Commented May 2, 2020 at 18:31
• @The One : Is your field $A$ complex or real ? If your field $A$ would be complex, your formula would be wrong. Do you agree ? en.wikipedia.org/wiki/Proca_action Commented Jan 10, 2021 at 12:29
• @Mathieu Krisztian The fields A_\mu under question are real. Commented Nov 28, 2021 at 21:30

This somewhat sloppy statement means that the two Lagrangian expressions are the same up to a total derivative. Such a total derivative does not contribute to the action $$A=\int d^4x \cal L$$, for suitable conditions at the edge of the integration domain, and therefore is considered to be without physical consequence.
• Using different non-gravitational Lagrangians differing by a total time derivative does not imply working with distinct theories or with different distributions of energy and momentum in GR. The energy-momentum tensor $T_{\mu\nu}$ referred to in Einstein's equations is not necessarily the Noether charge, but instead it is the accepted "real" distribution of energy and momentum, which is unique and the same for all the various possible equivalent choices of Lagrangians and canonical energy-momentum tensors. Commented Mar 19 at 22:47