Conventions on the Fourier transform
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You will recall there are various conventions adopted with regard to factors of \(2\pi\) in Fourier transforms. These notes use a single convention throughout, which is also the one used by Peskin & Schroeder: wavevectors are angular (so that \(\exp\left(i\vb{k}\cdot\vb{x}\right)\) has wavelength \(2\pi/\left|\vb{k}\right|\)), and every factor of \(2\pi\) travels with the momentum measure. Recall that we work in \(d\) spatial and \(D=d+1\) spacetime dimensions.
Infinite space
\[\begin{aligned} \tilde{f}\left(\vb{k}\right)= & \int\text{d}^{d}\vb{x}\exp\left(-i\vb{k}\cdot\vb{x}\right)f\left(\vb{x}\right)\\ f\left(\vb{x}\right)= & \int\frac{\text{d}^{d}\vb{k}}{\left(2\pi\right)^{d}}\exp\left(i\vb{k}\cdot\vb{x}\right)\tilde{f}\left(\vb{k}\right) \end{aligned}\]
with the completeness relation
\[\int\text{d}^{d}\vb{x}\exp\left(i\left(\vb{k}-\vb{q}\right)\cdot\vb{x}\right)=\left(2\pi\right)^{d}\delta^{d}\left(\vb{k}-\vb{q}\right).\]
(If you prefer the convention with \(\exp\left(\pm2\pi i\vb{k}\cdot\vb{x}\right)\) and no \(2\pi\)s in the measure, it is recovered by rescaling \(\vb{k}\rightarrow2\pi\vb{k}\). Mixing the two conventions is a common source of stray factors of \(2\pi\): for example the dispersion \(\omega=c\left|\vb{k}\right|\) becomes \(\omega=2\pi c\left|\vb{k}\right|\).)
Since we use the \(\left(+---\right)\) signature, spacetime Fourier transforms must have the opposite overall signs in the exponents, so that the spatial parts agree with the above:
\[\begin{aligned} \tilde{f}\left(k\right)= & \int\text{d}^{D}x\exp\left(ik_{\mu}x^{\mu}\right)f\left(x\right)\\ f\left(x\right)= & \int\frac{\text{d}^{D}k}{\left(2\pi\right)^{D}}\exp\left(-ik_{\mu}x^{\mu}\right)\tilde{f}\left(k\right) \end{aligned}\]
where \(k_{\mu}x^{\mu}=\omega t-\vb{k}\cdot\vb{x}\).
A finite box
In a box of side \(L\) and volume \(V=L^{d}\) with periodic boundary conditions, the allowed wavevectors are discrete, \(\vb{k}=2\pi\vb{n}/L\) with \(\vb{n}\in\mathbb{Z}^{d}\) (an integer number of whole wavelengths fits across the box). Then
\[\begin{aligned} \tilde{f}_{\vb{k}}= & \int_{V}\text{d}^{d}\vb{x}\exp\left(-i\vb{k}\cdot\vb{x}\right)f\left(\vb{x}\right)\\ f\left(\vb{x}\right)= & \frac{1}{V}\sum_{\vb{k}}\exp\left(i\vb{k}\cdot\vb{x}\right)\tilde{f}_{\vb{k}} \end{aligned}\]
with
\[\int_{V}\text{d}^{d}\vb{x}\exp\left(i\left(\vb{k}-\vb{q}\right)\cdot\vb{x}\right)=V\delta_{\vb{k}\vb{q}}.\]
A lattice
On a periodic chain of \(N\) sites at positions \(an\) (lattice constant \(a\)), the allowed wavevectors are \(k=2\pi j/Na\) for integer \(j\), and wavevectors differing by \(2\pi/a\) are equivalent. We take one representative of each, in the Brillouin zone \(-\pi/a<k\le\pi/a\), which contains exactly \(N\) allowed values. For operators we use the unitary (symmetric) normalisation
\[\begin{aligned} \tilde{f}_{k} & =\frac{1}{\sqrt{N}}\sum^{N}_{n=1}\exp\left(-ikan\right)f_{n}\\ f_{n} & =\frac{1}{\sqrt{N}}\sum_{k\in\text{BZ}}\exp\left(ikan\right)\tilde{f}_{k} \end{aligned}\]
with
\[\frac{1}{N}\sum^{N}_{n=1}\exp\left(i\left(k-q\right)an\right)=\delta_{kq}.\]
Because this transformation is unitary it preserves canonical commutation relations, which is why we use it for operators. For the same reason, field operators in a box are expanded as \(\hat{\varphi}_{\vb{x}}=V^{-1/2}\sum_{\vb{k}}\exp\left(i\vb{k}\cdot\vb{x}\right)\hat{\Phi}_{\vb{k}}\).
From the box to infinite space
1 Canonical Field Quantization works in a finite box (and on a lattice), where everything is discrete and finite. From 2 Path Integral Field Quantization onwards we take the infinite-volume limit \(V\rightarrow\infty\). The two descriptions are related by the following dictionary:
| finite box, volume \(V\) | infinite volume |
|---|---|
| \(\sum_{\vb{k}}\) | \(V\int\text{d}^{d}\vb{k}/\left(2\pi\right)^{d}\) |
| \(V\delta_{\vb{k}\vb{q}}\) | \(\left(2\pi\right)^{d}\delta^{d}\left(\vb{k}-\vb{q}\right)\) |
| \(V\) | \(\left(2\pi\right)^{d}\delta^{d}\left(\vb{0}\right)\) |
| \(\sqrt{V}\hat{a}_{\vb{k}}\), with \(\left[\hat{a}_{\vb{k}},\hat{a}^{\dagger}_{\vb{q}}\right]=\delta_{\vb{k}\vb{q}}\) | \(\hat{a}\left(\vb{k}\right)\), with \(\left[\hat{a}\left(\vb{k}\right),\hat{a}^{\dagger}\left(\vb{q}\right)\right]=\left(2\pi\right)^{d}\delta^{d}\left(\vb{k}-\vb{q}\right)\) |
| \(\sqrt{V}|\vb{k}\rangle\), with \(\langle\vb{k}|\vb{q}\rangle=\delta_{\vb{k}\vb{q}}\) | \(|\vb{k}\rangle\), with \(\langle\vb{k}|\vb{q}\rangle=\left(2\pi\right)^{d}\delta^{d}\left(\vb{k}-\vb{q}\right)\) |
The third line is worth dwelling on: a Dirac delta evaluated at zero is not a mysterious ‘uncountable’ infinity, it is simply the volume of space. Infinities proportional to \(V\) are infrared (IR) divergences, and are harmless whenever we ask about densities. Infinities which arise instead from arbitrarily large \(\left|\vb{k}\right|\) are ultraviolet (UV) divergences, and have a different physical origin: they appear when we forget that our description stops being valid at short distances (for example, at the lattice scale).
Other conventions at a glance
- Metric signature \(\left(+---\right)\), natural units \(\hbar=c=1\).
- Relativistically normalised states: \(|k\rangle=\sqrt{2E_{\vb{k}}}|\vb{k}\rangle\), so that \(\langle k|q\rangle=2E_{\vb{k}}\left(2\pi\right)^{d}\delta^{d}\left(\vb{k}-\vb{q}\right)\).
- Free real scalar field: \(\hat{\varphi}_{x}=\int\frac{\text{d}^{d}\vb{k}}{\left(2\pi\right)^{d}}\frac{1}{\sqrt{2\omega_{\vb{k}}}}\left(\hat{a}\left(\vb{k}\right)\exp\left(-ik\cdot x\right)+\hat{a}^{\dagger}\left(\vb{k}\right)\exp\left(ik\cdot x\right)\right)\).
- Green’s functions are defined with a factor of \(-i\), \(\left(\partial^{2}+m^{2}\right)G\left(x-y\right)=-i\delta^{D}\left(x-y\right)\), so that \(\tilde{G}\left(p\right)=i/\left(p^{2}-m^{2}+i0^{+}\right)\) and \(G_{xy}=\langle\Omega|\mathcal{T}\hat{\varphi}_{x}\hat{\varphi}_{y}|\Omega\rangle\).
- \(\varphi^{4}\) theory has \(S_{\text{int}}=-\frac{\lambda}{4!}\int\text{d}^{D}x\varphi^{4}_{x}\) with \(\lambda>0\), so that the potential is bounded below, and each vertex contributes \(-i\lambda\).