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In quantum mechanics, we have the famous uncertainty relation $$\Delta x \Delta p\geq \frac{\hbar}{2}$$ which is a result of the canonical commutation relation $[x,p] = i \hbar$.

Is there are similar relation in quantum field theory? In particular, is there are relation of the form $$\Delta \phi \Delta \Pi \neq 0 ,$$ where $\Pi$ denotes the conjugate momentum? This seems likely since the canonical commutation relation in QFT reads $[\phi(t,x),\Pi(t,y)] = i \hbar \delta (x-y)$. Moreover, if such a relation exists, how is it commonly interpreted?

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  • $\begingroup$ You might consider the EM field. $\endgroup$
    – Cosmas Zachos
    Nov 24, 2019 at 15:35
  • $\begingroup$ Linked and also. Read up on Heitler's "The quantum theory of radiation", Ch II sec 8. It is a long and glorious story, started by Jordan and Pauli in 1928, and crucially applied by Bohr and Rosenfeld to prove the inevitability of quantization of fields. $\endgroup$
    – Cosmas Zachos
    Nov 25, 2019 at 19:35

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QFT is sometimes introduced as a collection of oscillatory modes "in a box", each being a harmonic oscillator. With a finite number of modes (corresponding to a momentum cut-off) this is just a multidimensional harmonic oscillator so there is an uncertainty relation like we have in 3-space. This idea can be extended to infinite dimensions using some higher math.

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