What does $\int d^3 x $ mean?
I found this in a lecture on quantum field theory, and it was not explained.
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11$d^3x$ a short hand for $dx_1 dx_2 dx_3$, the volume element of a 3 dimension integral. – achille hui Aug 29 '13 at 19:13
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possible duplicate of Meaning of $\int\mathop{}!\mathrm{d}^4x$ – Did Aug 29 '13 at 20:10
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In physics notation, it means a triple integral over the whole 3-dimensional space. An equation like: $$ \int d^3x\,f(x) $$
actually means: $$ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f(x,y,z)\,dx\,dy\,dz. $$
For example, a 3-dimensional Fourier transform can be written as:
$$ \tilde{f}(p) = \int d^3x\,f(x)\,e^{-ip\cdot x}, $$
which means: $$ \tilde{f}(p_1, p_2,p_3) = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f(x,y,z)\,e^{-ip_1x-ip_2y-ip_3z}\,dx\,dy\,dz. $$
It is a little unclear at first, but as you can see, it shortens the formulas a lot.
geodude
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