# Infinite number of degrees of freedom

In a system with a finite number of degrees of freedom $$\eta_i$$, $$i=1,\ldots, N$$ , the partition function depends on the N external fields that may couple linearly to the $$\eta_i$$ in the Hamiltonian $$Z[H_i]= Tr \exp \left[ -\beta(\mathscr H - \sum_i H_i \eta_i \right]$$

In a system with infinite number of degrees of freedom, the partition function becomes a functional of $$H(\mathbf r)$$: $$Z[H(\mathbf r)]= Tr \exp \left[ -\beta(\mathscr H - \int d^d \mathbf r H(\mathbf r) \eta(\mathbf r) \right]$$

What is meant by infinite number of degrees of freedom? Initially, there was a finite number (N) of axes in the system, and $$\eta_i$$ was constant throughout a given direction. I understand that for the nonhomogeneous case, $$\eta$$ has to depend on $$\mathbf r$$. But to do this, the author says that is due to an infinite number of degrees of freedom. In the continuum limit is the number of axes is infinity?