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Suppose we have a integral measure $[\mathrm dx]$, and suppose we make the change of variables $$x=y+\sum_n c_nx_n, \tag{2.6}$$ where $x_n$ is a certain orthonormal basis. Why we should have

$$[\mathrm dx] \propto \prod_n \mathrm dc_n~?\tag{2.8}$$

This is done in Coleman's "Aspects of symmetry" page 269 (he takes $y$ to be the classical path).

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  • $\begingroup$ -1. Unclear what you are asking. It seems to me this is a purely mathematical question anyway. $\endgroup$
    – sammy gerbil
    Commented Mar 1, 2017 at 3:40
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    $\begingroup$ As far as Im concerned, the question is clear enough, and is not a purely mathematical question, because path-integrals are usually ill-defined from the mathematical POV, and it's physicists task to interpret what they mean. $\endgroup$
    – AccidentalFourierTransform
    Commented Mar 1, 2017 at 13:51

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In short eq. (2.8) is how Coleman defines the path integral measure. A virtual path $t \mapsto x(t)$ [satisfying Dirichlet boundary conditions eq. (2.7b)] are parametrized via a sequence $(c_n)_{n\in\mathbb{N}_0}$, cf. eq. (2.6). The elements $c_n$ of the sequence $(c_n)_{n\in\mathbb{N}_0}$ are the defining integration variables of the path integral. Coleman's construction is also reviewed in this related Phys.SE post.

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