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In QFT, we use normal ordering to eliminate infinity from hamiltonian. In path integral formulation of QFT though, since what we integrate over is "classical field configuration", instead of operators, normal ordering does not seem to appear.

However, since normal ordering can be understood as renormalization, it seems that path integral should also see signs of this renormalization. Where does this normal ordering effect appear in path integral?

Qmechanic
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Brion Brion
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    related: Why path integral approach may suffer from operator ordering problem?. – AccidentalFourierTransform Apr 06 '18 at 16:01
  • Normal ordering is a property that can be assigned only to free theories or nearly free theories. What is more fundamental is time-ordering which the path integral keeps track of pretty well. In most quantum field theory calculations we are interested in computing time-ordered correlators. In perturbation theory, normal ordering is introduced only as a tool to calculate the time-ordered correlators. The path integral removes the need for this intermediate step. – Prahar Apr 06 '18 at 16:32
  • That is correct @Brion Brion, in the path integral normal ordering appears as renormalisation. The precise manner in which this appears in the path integral is explained in great detail in: https://arxiv.org/abs/1512.02604. Briefly, normal ordering is associated to subtracting self-contractions from local interactions of the action. IF the theory is renormalisable, these self-contractions can be absorbed into renormalisations of the various bare couplings. See the reference for all the glory details ... – Wakabaloola Apr 06 '18 at 20:11
  • @Prahar: the path integral does NOT remove the need for normal ordering (because the path integral does not automatically make your theory finite); if you renormalise couplings and absorb the infinities you are doing normal ordering (possibly without realising it). (Just to be clear, not all infinities are associated to normal ordering, so the bare couplings and bare fields subtract additional infinities also.) – Wakabaloola Apr 06 '18 at 20:13
  • One more comment: version 1 of http://arxiv.org/abs/1512.02604 (which you can find in the linked page) actually contains more details, and a whole section (Sec.3) on conventional normal ordering in the path integral. The main focus of the paper is on a generalisation of normal ordering whereby one subtracts all self-contractions using the full connected renormalised Green function at coincident points (so not just the free 2-pt function at coincident points as in conventional normal ordering), and this ensures you are doing perturbation around the exact quantum vacuum of your theory. – Wakabaloola Apr 06 '18 at 20:33
  • Related post by OP: https://physics.stackexchange.com/q/398154/2451 – Qmechanic Apr 10 '18 at 15:24
  • Another related post: https://physics.stackexchange.com/questions/564924/perturbative-expansion-and-self-contractions-in-functional-integral/565056#565056 – Weather Report Jul 11 '20 at 12:19

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