Tuesday, October 03, 2006

Coleman and cosmological constant

It's fun to use Sidney Coleman's notes for Quantum Field Theory I. Of course, basic quantum field theory has been more or less a closed subject for some time. In some cases, his discussion of the renormalization group is not entirely modern and he couldn't point to extensions of the material that were found in the last decades, but otherwise, the whole material is still a classic piece.

Sometimes you think that you have encountered an error. For example, I thought that the following statements were wrong. Imagine that you have a unitary operator "U" representing translations such that it conjugates the quantum field operators in the following way:
  • U . O(x) . U^{-1} = O(x+a)
Then the question is how does the same translation operator "U" conjugate the position(s) "q" of individual particle(s) in a single-particle limit i.e. what is
  • U . q . U^{-1} = ?
You could guess immediately: it must be "q+a". You just shift everything by "a". Coleman emphasized it was "q-a" instead. I needed twice 5 minutes, Lisa Randall's authority who remembered this subtlety, and a 4-hour break in the middle to realize that he was right. ;-)




A simple argument of mine is as follows: the conjugation of "O(x)" changes the point associated with "O(x)" - namely the point "x" - to "x+a" after you conjugate it. What is the point associated with "q"? It is "x=0". Why? For example, particles localized at "x=0" have a priviliged, vanishing expectation value of "q".

If you conjugate it, the priviliged point of the operator "x=0" must be shifted to "x = 0+a = a". What is the operator resembling "q" whose characteristic position is "x=a"? Yes, it is "q-a" because it is "q-a" whose expectation value vanishes for a particle localized at "x=a". ;-) The opposite sign can also be interpreted as coming from active vs. passive transformations.

But one comment that seems obsolete is priceless and I exploded with a laughter when I was reading it. After Coleman derives the divergent normal ordering constant and says that it doesn't matter in non-gravitational theories (so far so good), he says:
  • Adding a term "lambda g_{mu nu}" to the stress energy tensor is equivalent to changing the cosmological constant, a term introduced by Einstein and repudiated by him 10 years later. No astronomer has ever observed a nonzero cosmological constant. The theory is eventually going to be applied to strong interactions, maybe even to some grand unified theory. Strong interactions have energy typically of order 1 GeV and a characteristic length of 1 fermi = 10^{-13} cm. With a cosmic energy density of order 10^{39} GeV/cm^3, the Universe would be about 1 km long according to Einstein's equations. You couldn't even get to MIT without coming back where you started. We won't talk about why the cosmological constant is zero in this course. They don't explain it in any course given at Harvard because nobody knows why it is zero. ;-)
That was a happy world before the astronomers ate the poisoned apple to see that the constant is actually nonzero and, indeed, 120 orders of magnitude lower than expected. Coleman's was the holy world without the devils of the anthropic principle. Despite the unholy effect of the discovery of an accelerating universe in 1998 had on theoretical physics, the people who did it have a chance to grab the Nobel prize tomorrow in the morning. Good luck to them. ;-)

Incidentally, NASA announced today that they plan a more careful research of supernovae to refine out knowledge about the dark energy.

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