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I came across this in the lecture notes of quantum field theory by David Tong. Inside time ordering interactions aren’t taken to be normal ordered. Interaction hamiltonian should be normal ordered otherwise it is not well defined (due to ordering ambiguity and related singularities). Most standard QFT textbooks don’t address this issue. Am i missing something here or normal ordering was assumed?

Roy
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    Possible duplicates: https://physics.stackexchange.com/q/133426/2451 , https://physics.stackexchange.com/q/10804/2451 and links therein. – Qmechanic Nov 01 '20 at 22:02
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    This doesn’t answer my question. My question is whether we use normal ordered hamiltonian in s matrix or not and why books don’t take normal ordered hamiltonian? I am confused about this point. – Roy Nov 02 '20 at 02:15
  • Which textbooks? – Qmechanic Nov 02 '20 at 02:24
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    Schwarz’s and peskin’s book don’t mention this issue. So am i right or wrong? – Roy Nov 02 '20 at 02:34
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    Also Joe Polchinski in his string theory book says normal ordering is of little use in most interacting field theories , because these have additional divergences from interaction vertices approaching the composite operator or one another. Again i am confused because without normal ordering these composite operators aren’t well defined. – Roy Nov 02 '20 at 02:50
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    normal ordering can be undone by an appropriate choice of counterterms. since normal ordering does not remove all divergences in interacting field theories one needs further counterterms. so one can instead work with a different set of counterterms from the outset, one that removes all divergences from the outset with no reference to normal ordering. there is a normal ordering that is useful also in interacting theories (google: complete normal ordering), which serves to ensure you land on the quantum corrected vacuum, but here too there are further divergences and more counterterms are needed – Wakabaloola Nov 02 '20 at 05:34
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    an analogy might be useful: the difference between using normal ordering plus counterterms vs using only a different set of counterterms is like choosing to represent an integer using one partition over another. they both sum to the same number, and similarly you get the same result for the amplitudes and observables using either approach. – Wakabaloola Nov 02 '20 at 05:40
  • Thank you.but my original question is why textbooks don’t normal order hamiltonians in s matrix?otherwise they are not well defined.if they don’t use normal ordering then there should be self contractions in canonical approach and accordingly counterterms. but there isn’t any reference to these counterterms. – Roy Nov 02 '20 at 06:00
  • Is it because in first order in s matrix it does not matter whether we use normal ordered hamiltonian or not?other ambiguous terms simply don’t contribute to first order? – Roy Nov 02 '20 at 06:06
  • Normal order is discussed p. 100-103 in Schwartz and p. 88 + p. 116 in P&S. – Qmechanic Nov 02 '20 at 08:59
  • And in Peskin and Schroeder's "An Introduction to Quantum Field Theory" in section 4.3 titled "Wick's Theorem" – hft Oct 20 '21 at 20:27

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Here there should be normal ordering for individual interaction terms inside time ordering.

Roy
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  • Yes. The fact that they are normal ordered is manifested by the fact that you remove vacuum bubbles when computing the diagrams. – Connor Behan Oct 20 '21 at 20:24
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Most standard QFT textbooks don’t address this issue. Am i missing something here or normal ordering was assumed?

What? No. What could you possibly mean by "most standard QFT textbooks..."?

Check out the section titled "Wick's Theorem" in Peskin and Schroeder's textbook titled: "An Introduction to Quantum Field Theory" (first edition, copyright 1995).

In the first edition of the above-mentioned textbook, Wick's Theorem is discussed in section 4.3 on page 88.

hft
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