Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.

The trick to get the Noether current consists in making the variation local: the standard argument, which doesn't convince me and for which I'd like a more formal explanation, is that, since the global symmetry is in force, the only term appearing in the variation will be proportional to derivatives of $\epsilon,$ and thus the involved current $J^\mu$ will be conserved on-shell:

$$ \delta S = \int \mathrm{d}^n x \ J^\mu \partial_\mu \epsilon .\tag{*}$$

This is stated, e.g., in Superstring Theory: Volume 1 by Green Schwarz Witten on page 69 and The Quantum Theory of Fields, Volume 1 by Weinberg on page 307.

In other words, why a term $ \int \mathrm{d}^n x \ K(x) \epsilon(x)$ is forbidden?

Taking from the answer below, I believe two nice references are

1. theorem 4.1
2. example 2.2.5

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.

The trick to get the Noether current consists in making the variation local: the standard argument, which doesn't convince me and for which I'd like a more formal explanation, is that, since the global symmetry is in force, the only term appearing in the variation will be proportional to derivatives of $\epsilon,$ and thus the involved current $J^\mu$ will be conserved on-shell:

$$ \delta S = \int \mathrm{d}^n x \ J^\mu \partial_\mu \epsilon .\tag{*}$$

This is stated, e.g., in Superstring Theory: Volume 1 by Green Schwarz Witten on page 69 and The Quantum Theory of Fields, Volume 1 by Weinberg on page 307.

In other words, why a term $$ \int \mathrm{d}^n x \ K(x) \ \epsilon(x)$$ is forbidden?

Taking from the answer below, I believe two nice references are

1. theorem 4.1
2. example 2.2.5

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.

The trick to get the Noether current consists in making the variation local: the standard argument, which does't convince me and for which I'd like a more formal explanation, is that, since the global symmetry is in force, the only term appearing in the variation will be proportional to derivatives of $\epsilon,$ and thus the involved current $J^\mu$ will be conserved on shell: $$ \delta S = \int \mathrm{d}^n x \ J^\mu \partial_\mu \epsilon $$

This is stated, e.g., in Superstring Theory: Volume 1 by Green Schwarz Witten on page 69 and The Quantum Theory of Fields, Volume 1 by Weinberg on page 307.

In other words, why a term $ \int \mathrm{d}^n x \ K(x) \epsilon(x)$ is forbidden?

Taking from the answer below, I believe two nice references are

1. theorem 4.1
2. example 2.2.5

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.

The trick to get the Noether current consists in making the variation local: the standard argument, which doesn't convince me and for which I'd like a more formal explanation, is that, since the global symmetry is in force, the only term appearing in the variation will be proportional to derivatives of $\epsilon,$ and thus the involved current $J^\mu$ will be conserved on-shell:

$$ \delta S = \int \mathrm{d}^n x \ J^\mu \partial_\mu \epsilon .\tag{*}$$

This is stated, e.g., in Superstring Theory: Volume 1 by Green Schwarz Witten on page 69 and The Quantum Theory of Fields, Volume 1 by Weinberg on page 307.

In other words, why a term $ \int \mathrm{d}^n x \ K(x) \epsilon(x)$ is forbidden?

Taking from the answer below, I believe two nice references are

1. theorem 4.1
2. example 2.2.5

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.

The trick to get the Noether current consists in making the variation local: the standard argument, which does't convince me and for which I'd like a more formal explanation, is that, since the global symmetry is in force, the only term appearing in the variation will be proportional to derivatives of $\epsilon,$ and thus the involved current $J^\mu$ will be conserved on shell: $$ \delta S = \int \mathrm{d}^n x \ J^\mu \partial_\mu \epsilon $$

This is stated, e.g., in Superstring Theory: Volume 1 by Green Schwarz Witten on page 69 and The Quantum Theory of Fields, Volume 1 by Weinberg on page 307.

In other words, why a term $ \int \mathrm{d}^n x \ K(x) \epsilon(x)$ is forbidden?

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.

The trick to get the Noether current consists in making the variation local: the standard argument, which does't convince me and for which I'd like a more formal explanation, is that, since the global symmetry is in force, the only term appearing in the variation will be proportional to derivatives of $\epsilon,$ and thus the involved current $J^\mu$ will be conserved on shell: $$ \delta S = \int \mathrm{d}^n x \ J^\mu \partial_\mu \epsilon $$

This is stated, e.g., in Superstring Theory: Volume 1 by Green Schwarz Witten on page 69 and The Quantum Theory of Fields, Volume 1 by Weinberg on page 307.

In other words, why a term $ \int \mathrm{d}^n x \ K(x) \epsilon(x)$ is forbidden?

Taking from the answer below, I believe two nice references are

1. theorem 4.1
2. example 2.2.5