Functional derivative for the same function expressed in different coordinates

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This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf



On page 17, they took a functional derivative of Z[J] with respect to iJ to obtain an expression for G(0)(x1,x2). We're supposed to take derivatives with respect to J(x), but on page 17 the document took derivatives with respect to J(x'), where x0=ix0' (the subscript 0 indicates the first element of x; the other elements remain equivalent).



Is the results the same or did the document made a mistake?



Note: The definition of functional derivative the document is using is a delta function as the test function, as explained in section 4 of the following Wikipedia article: https://en.wikipedia.org/wiki/Functional_derivative#Using_the_delta_function_as_a_test_function






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    This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf



    On page 17, they took a functional derivative of Z[J] with respect to iJ to obtain an expression for G(0)(x1,x2). We're supposed to take derivatives with respect to J(x), but on page 17 the document took derivatives with respect to J(x'), where x0=ix0' (the subscript 0 indicates the first element of x; the other elements remain equivalent).



    Is the results the same or did the document made a mistake?



    Note: The definition of functional derivative the document is using is a delta function as the test function, as explained in section 4 of the following Wikipedia article: https://en.wikipedia.org/wiki/Functional_derivative#Using_the_delta_function_as_a_test_function






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      This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf



      On page 17, they took a functional derivative of Z[J] with respect to iJ to obtain an expression for G(0)(x1,x2). We're supposed to take derivatives with respect to J(x), but on page 17 the document took derivatives with respect to J(x'), where x0=ix0' (the subscript 0 indicates the first element of x; the other elements remain equivalent).



      Is the results the same or did the document made a mistake?



      Note: The definition of functional derivative the document is using is a delta function as the test function, as explained in section 4 of the following Wikipedia article: https://en.wikipedia.org/wiki/Functional_derivative#Using_the_delta_function_as_a_test_function






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      share|cite|improve this question













      This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf



      On page 17, they took a functional derivative of Z[J] with respect to iJ to obtain an expression for G(0)(x1,x2). We're supposed to take derivatives with respect to J(x), but on page 17 the document took derivatives with respect to J(x'), where x0=ix0' (the subscript 0 indicates the first element of x; the other elements remain equivalent).



      Is the results the same or did the document made a mistake?



      Note: The definition of functional derivative the document is using is a delta function as the test function, as explained in section 4 of the following Wikipedia article: https://en.wikipedia.org/wiki/Functional_derivative#Using_the_delta_function_as_a_test_function






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      asked Sep 1 at 7:10









      Taylor Tiger

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