Continuous Limit
The Toda chain equations of motion, in the continuous limit in which the distance between neighbors goes to zero, become the KdV equation. Here the index labeling the particle in the chain becomes the new spacial coordinate.
In contrast, the Toda field theory is achieved by introducing a new spacial coordinate which is independent of the chain index label. This is done in a relativistically invariant way, so that time and space are treated on equal grounds. This means that the Toda field theory is not a continuous limit of the Toda chain.
Read more about this topic: Toda Oscillator
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