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My understanding is this: In the canonical approach to QFT there is an ambiguity in the order in which we write operators when calculating matrix elements. The different choices just correspond to different vacuum energies, which we are free to ignore since we only measure energy differences. So we are free to choose any ordering prescription we like - normal ordering is the special choice where all the creation operators are to the left of annihalation operators.

My questions are:

1. I heard a claim that normal ordering, defined as placing all the creation operators to the left of the annihalation operators, fails to remove the infinity from all diagrams. In particular the diagram in [tex] \phi^4 [/tex] theory which I will now try to describe:

Draw a straight line between 2 points x and y. Now bring a circle down so it makes contact with the line in only one place, forming a 4-vertex with the line. Now take 2 points on that circle, and draw 2 separate lines connecting those 2 points.

Is this true? And if it is, couldn't we just define a different ordering in which all the divergences go away?

2. Can normal ordering be used to remove the infinities in any theory? I heard a claim that it can fail in interacting theories, and that is when we bring in the heavy artillery of renormalisation.

3. Is normal ordering equivalent to a particular renormalisation scheme?