My question is about quantum field theory, but I post it here because I want to understand the mathematics.
In the functional integral approach to quantum field theory, one considers the functional integral in Euclidean signature (i.e. partition function)
$$\mathcal{Z}=\int[\mathcal{D}\phi]e^{-S[\phi(x)]},$$
where $[\mathcal{D}\phi]\equiv\prod_{x}d\phi(x)$, $\phi\in C^{\infty}(M)$, and $S[\phi(x)]=\int_{M}dx\mathcal{L}(\phi,\nabla\phi)$ is the classical action.
My question is concerning the Wilsonian approach to renormalization group flow. As introduced in the Wikipedia page, one considers a UV-cutoff in momentum space $|\vec{p}|\leq\Lambda$.
For each field $\phi(x)$, naively one can expect its Fourier modes ranges in the entire momentum space, i.e. $|\vec{p}|\in[0,\infty)$, and one has the Fourier transformation
$$\phi(x)=\int\frac{d^{D}\vec{p}}{(2\pi)^{D}}e^{ip\cdot x}\hat{\phi}(p).\tag{1}$$
Imposing such a UV cutoff means we replace $\phi(x)\in C^{\infty}$ by a regularized field $\phi_{\Lambda}$:
$$\phi_{\Lambda}(x)\equiv\int_{|\vec{p}|\leq\Lambda}\frac{d^{D}\vec{p}}{(2\pi)^{D}}e^{ip\cdot x}\hat{\phi}(p).\tag{2}$$
I have to admit that there's a certain psychological barrier to feel comfortable about the above expression. Physicists are happy to call it a "Fourier transform" because humans are ignorant to the laws of physics in the ultraviolet domain. Correspondingly, this UV-cutoff in the momentum space means that the field in spacetime is put on a lattice of size larger than the scale $L=1/\Lambda$.
But this has never made any sense to me. If we consider a field on lattice, then, by period boundary condition, one must have
$$\phi(x)=\frac{1}{L}\sum_{n\in\mathbb{Z}}e^{ip_{n}\cdot x}\hat{\phi}_{n}\quad\mathrm{and}\quad\hat{\phi}_{n}=\frac{1}{2}\int_{-L}^{L}dxe^{-ip_{n}\cdot x}\phi(x),$$
where $p_{n}=2\pi n/L$. The UV-cutoff in momentum space does not appear in the above expressions at all.
So how can equation (2) makes any sense in mathematics?
Thanks Qiaochu Yuan very much for pointing out my mistake. I should've put the field on a lattice in position space. Then, I should have
$$\phi(X_{n})=\frac{1}{2}\int_{-\Lambda}^{\Lambda}dp\,e^{-iX_{n}\,\cdot\,p}\hat{\phi}(p)\quad\mathrm{and}\quad\hat{\phi}(p)=\frac{1}{\Lambda}\sum_{n\in\mathbb{Z}}e^{iX_{n}\,\cdot\,p}\phi(X_{n}),$$
where $X_{n}=\frac{2\pi n}{\Lambda}$, and the lattice size is of the scale $L\sim\frac{1}{\Lambda}$.
But then I have one more question. The above equations require the periodic condition of the Fourier modes of the field, i.e $$\hat{\phi}(p)=\hat{\phi}(p+\Lambda).\tag{3}$$
This still doesn't make sense in QFT because we are supposed to be ignorant of the laws of physics in the UV.
So how do we make sense of equations (2) and (3)?