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I am reading a book on quantum field theory, while I have never been trained as a physicist. I found a big gap in language and have trouble understanding what physicists mean by "quantum field".

If I understand correctly, after quantizing twice (field and operator). quantum field should be an operator valued distribution. Am I right? I would appreciate it if some mathematician who are familiar with physics could kindly explain "quantum field" in terms of mathematics.

Thank you in advance.

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