I have a question regarding the implementation of constraint equations as delta functions in integrals. My confusion can best be illustrated with a quick example: Consider a Gaussian integral of the form $$ I = \int\text{d}x\,e^{-x^2}\,.$$ Let's say you want to implement the constraint $x=x_0$ on $x$. You can do this by multiplying the integrand with a delta function, yielding $$ I_{\text{constraint}} = \int\text{d}x\,\delta(x-x_0)e^{-x^2}\,.$$ The solution of this integral is of course $ e^{-x_0^2}\,,$ as it should be. However, I can for whatever reason also write the constraint equation as $2x=2x_0$ resulting in a different result $$ I'_{\text{constraint}} = \int\text{d}x\,\delta(2x-2x_0)e^{-x^2} = \frac{1}{2}I_{\text{constraint}}\,.$$ In this case, I know that the second result is wrong, but I don't exactly understand how to fix it. Even worse: Let's say there is way to fix it by somehow rescaling the integrand; what happens if I don't know which of the two constraint equation is the "right" one?
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$\begingroup$ This looks like a Mathematics question. Can you explain how this is related to physics specifically? $\endgroup$Vincent Thacker– Vincent Thacker2025-11-26 23:50:51 +00:00Commented 2 days ago
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$\begingroup$ Yes, this question is actually motivated from a more complicated problem where I try to implement physical conditions on fields in a path integral via Lagrange multipliers, i.e., the field must have certain values at certain points in time due to the underlying dynamics. $\endgroup$Physic_Student– Physic_Student2025-11-26 23:58:59 +00:00Commented 2 days ago
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3$\begingroup$ This might be useful: en.wikipedia.org/wiki/Dirac_delta_function#Properties In particular the scaling property of delta function: $\delta(ax) = \delta(x)/|a|$ $\endgroup$Karel– Karel2025-11-27 07:49:38 +00:00Commented 2 days ago
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1 Answer
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TL;DR: One needs additional information about the physical system at hand to determine the proper normalization of the constraint and delta function.
An important example in gauge theory: The role of the Faddeev-Popov determinant is partly to ensure that the path integral is invariant under reparametrizations of the gauge fixing conditions imposed via delta functions.