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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?

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In physics, infinite number of degrees of freedom means the state or configuration of a system cannot be given completely by finite number of variables, but requires infinite number of variables. These do not need to correspond to any physical space axes.

Often the infinite number of variables is due to working with field. Field is function of position, and since there is infinite number of positions, the field has infinite number of degrees of freedom.

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