Let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a $D$-dimensional model spacetime $\mathcal{M}_D$. Now consider the classical Lagrangian density, $\mathcal{L}(\phi^{(i)}, \partial_\mu \phi^{(i)}, x^\mu)$. We apply the following infintesimal fixed-boundary transformation to $\mathcal{M}_D$. \begin{align*} x \to \widetilde{x}^\mu &\equiv x^\mu + \delta x^\mu (x), \tag{1} \\ \text{such that, }\ \delta x^\mu\Big{|}_{\partial\Omega}&=0, \tag{2} \\ \text{and the fields transform as: }\ \phi^{(i)}(x) &\to \widetilde{\phi}^{(i)}(\widetilde{x}) \equiv \phi^{(i)} (x) + \delta\phi^{(i)} (x). \tag{3} \\ \end{align*}

According to my calculations, up to first order in the variation, the Lagrangian density is given by: $$ \boxed{ \delta \mathcal{L} = \partial_\mu \Big( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\partial_\nu \phi^{(i)} \delta x^\nu + \mathcal{L} \delta x^\mu \Big) - \mathcal{L} \partial_\mu (\delta x^\mu) }\tag{4} $$

Therefore, the conserved current is $$ \boxed{ J^{\mu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\partial_\nu \phi^{(i)} \delta x^\nu + \mathcal{L} \delta x^\mu - F^\mu } \tag{5}$$ where $F^\mu$ is some arbitrary field that vanishes on $ \partial \Omega$.

However, most textbooks ignore the second and the third terms in the above expression. Compare, for example, with Peskin and Schroeder (p.18) which sets:

$$ J^{\mu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - F^\mu. \tag{6} $$

For another example, Schweber (p. 208) ignores all terms but the first in the variation of the Lagrangian density, and writes:

$$ \delta \mathcal{L} = \partial_\mu \Big( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} \Big).\tag{7} $$

So what is going on here? Am I missing something? We seem to have set the same assumptions, but get different results. Am I wrong, or are they?

Let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a $D$-dimensional model spacetime $\mathcal{M}_D$. Now consider the classical Lagrangian density, $\mathcal{L}(\phi^{(i)}, \partial_\mu \phi^{(i)}, x^\mu)$. We apply the following infintesimal fixed-boundary transformation to $\mathcal{M}_D$. \begin{align*} x \to \widetilde{x}^\mu &\equiv x^\mu + \delta x^\mu (x), \tag{1} \\ \text{such that, }\ \delta x^\mu\Big{|}_{\partial\Omega}&=0, \tag{2} \\ \text{and the fields transform as: }\ \phi^{(i)}(x) &\to \widetilde{\phi}^{(i)}(\widetilde{x}) \equiv \phi^{(i)} (x) + \delta\phi^{(i)} (x). \tag{3} \\ \end{align*}

According to my calculations, up to first order in the variation, the Lagrangian density is given by: $$ \boxed{ \delta \mathcal{L} = \partial_\mu \Big( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\partial_\nu \phi^{(i)} \delta x^\nu + \mathcal{L} \delta x^\mu \Big) - \mathcal{L} \partial_\mu (\delta x^\mu) }\tag{4} $$

Therefore, the conserved current is $$ \boxed{ J^{\mu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\partial_\nu \phi^{(i)} \delta x^\nu + \mathcal{L} \delta x^\mu - F^\mu } \tag{5}$$ where $F^\mu$ is some arbitrary field that vanishes on $ \partial \Omega$.

However, most textbooks ignore the second and the third terms in the above expression. Compare, for example, with Peskin and Schroeder (p.18) which sets:

$$ J^{\mu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - F^\mu. \tag{6} $$

For another example, Schweber (p. 208) ignores all terms but the first in the variation of the Lagrangian density, and writes:

$$ \delta \mathcal{L} = \partial_\mu \Big( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} \Big).\tag{7} $$

So what is going on here? Am I missing something? We seem to have set the same assumptions, but get different results. Am I wrong, or are they?

EDIT: Condition (2) is unnecessary, as it was never used in the derivation of the current. Please ignore its presence in the above text.

Let us consider N independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a D-dimensional model spacetime $\mathcal{M}_D$. Now consider the classical Lagrangian density, $\mathcal{L}(\phi^{(i)}, \partial_\mu \phi^{(i)}, x^\mu)$. We apply the following infintesimal fixed-boundary transformation to $\mathcal{M}_D$. \begin{align*} x \to \widetilde{x}^\mu &\equiv x^\mu + \delta x^\mu (x), \\ \text{such that, }\ \delta x^\mu\Big{|}_{\partial\Omega}&=0, \\ \text{and the fields transform as: }\ \phi^{(i)}(x) &\to \widetilde{\phi}^{(i)}(\widetilde{x}) \equiv \phi^{(i)} (x) + \delta\phi^{(i)} (x). \\ \end{align*}

According to my calculations, up to first order in the variation, the Lagrangian density is given by: $$ \boxed{ \delta \mathcal{L} = \partial_\mu \Big( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\partial_\nu \phi^{(i)} \delta x^\nu + \mathcal{L} \delta x^\mu \Big) - \mathcal{L} \partial_\mu (\delta x^\mu) }$$

Therefore, the conserved current is $$ \boxed{ J^{\mu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\partial_\nu \phi^{(i)} \delta x^\nu + \mathcal{L} \delta x^\mu - F^\mu }$$ where $F^\mu$ is some arbitrary field that vanishes on $ \partial \Omega$.

However, most textbooks ignore the second and the third terms in the above expression. Compare, for example, with Peskin and Schroeder (p.18) which sets:

$$ J^{\mu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - F^\mu $$

For another example, Schweber (p. 208) ignores all terms but the first in the variation of the Lagrangian density, and writes:

$$ \delta \mathcal{L} = \partial_\mu \Big( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} \Big)$$

So what is going on here? Am I missing something? We seem to have set the same assumptions, but get different results. Am I wrong, or are they?

Let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a $D$-dimensional model spacetime $\mathcal{M}_D$. Now consider the classical Lagrangian density, $\mathcal{L}(\phi^{(i)}, \partial_\mu \phi^{(i)}, x^\mu)$. We apply the following infintesimal fixed-boundary transformation to $\mathcal{M}_D$. \begin{align*} x \to \widetilde{x}^\mu &\equiv x^\mu + \delta x^\mu (x), \tag{1} \\ \text{such that, }\ \delta x^\mu\Big{|}_{\partial\Omega}&=0, \tag{2} \\ \text{and the fields transform as: }\ \phi^{(i)}(x) &\to \widetilde{\phi}^{(i)}(\widetilde{x}) \equiv \phi^{(i)} (x) + \delta\phi^{(i)} (x). \tag{3} \\ \end{align*}

According to my calculations, up to first order in the variation, the Lagrangian density is given by: $$ \boxed{ \delta \mathcal{L} = \partial_\mu \Big( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\partial_\nu \phi^{(i)} \delta x^\nu + \mathcal{L} \delta x^\mu \Big) - \mathcal{L} \partial_\mu (\delta x^\mu) }\tag{4} $$

Therefore, the conserved current is $$ \boxed{ J^{\mu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\partial_\nu \phi^{(i)} \delta x^\nu + \mathcal{L} \delta x^\mu - F^\mu } \tag{5}$$ where $F^\mu$ is some arbitrary field that vanishes on $ \partial \Omega$.

However, most textbooks ignore the second and the third terms in the above expression. Compare, for example, with Peskin and Schroeder (p.18) which sets:

$$ J^{\mu} = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} - F^\mu. \tag{6} $$

For another example, Schweber (p. 208) ignores all terms but the first in the variation of the Lagrangian density, and writes:

$$ \delta \mathcal{L} = \partial_\mu \Big( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^{(i)} )}\delta\phi^{(i)} \Big).\tag{7} $$

So what is going on here? Am I missing something? We seem to have set the same assumptions, but get different results. Am I wrong, or are they?