Fall 2006 QG Seminar Errata

John Baez

Here's a list of errors we've found in the Fall 2006 Quantum Gravity Seminar notes. If you discover any more errors, please let me know, and we'll eventually try to correct them, or at least add them to this list.

Quantization and Cohomology

• Week 2 - On page 14 (the last page), "reparameterization" should be "parametrization".
• Week 6 - On page 33 (the first page), in the last line of the calculation,
  -E(t₁) + E(t₀)
  should be
  -E(t₁ - t₀)
  This is the energy E times the difference in times t₁ - t₀. The energy E is just a fixed number, not a function of time.
• Week 6 - On page 34 (the second page), I accidentally switched to conventions where ω = dα. We've been using ω = -dα, which is annoying, but in line with standard usage.
• Week 6 - On page 36 (the fourth page), in the definition of the set Y, I started writing the coordinates on T*(X × R) in this order:
  (q,p,t,p₀)
  instead of the order I'd been using since page 34:
  (q,t,p,p₀)
  It doesn't really matter, but ultimately it's best to use
  (q,t,p,p₀)
  since this lumps all the "position" coordinates together - both position in space and position in time - and eventually we'll study spacetimes that aren't of the form X × R, so we won't even be able to separate the space and time coordinates if we want to!

Classical versus Quantum Computation

• Week 0 - Toby Bartels pointed out some errors in terminology and also made some other nice points:

  John wrote:

  Some of you were wondering about which order to evaluate the expression in exercise 1.1. Now that I look at it, I see there's no ambiguity: Selinger packs this expression with enough parentheses to make it unambiguous. Just do stuff inside parentheses first!

  Also, something of the form λ f.λx.t is unambiguous, since λf.λx makes no sense; so it's λf.(λx.t). As for λz.z + 1, this is ambiguous if you can add a function and a number, since that is what (λz.z) + 1 does; but Selinger probably meant λz.(z + 1).

  While I'm here, I'll make a few comments based on Derek's notes.

  While you're contrasting classical logic with quantum logic here, there is a (different) contrast of classical with constructive logic (also called "intuitionistic" logic). I see the lambda calculus a lot in the metamathematics of constructive mathematics (which also involves a lot of Cartesian closed categories). So while the lambda calculus is classical as opposed to quantum, it's quite happily constructive as well.

  Interestingly, the "easy proof" of existence of noncomputable functions is nonconstructive, while the "better proof" is constructive. Indeed, in the Russian (Markov's) school of constructive mathematics, it is accepted as a fact (axiomatic) that every total function from the set of natural numbers to itself is recursive/computable! (Thus, there is a set --the set of Turing machines that never halt-- with an injection to a countable set --the set of Turing machines-- and a surjection to an uncountable set --the set of total functions. This is consistent with, say, Aczel's constructive set theory CZF.) However, all constructivists (well, except finitists) must agree that the partial function given on the top of page 3 is uncomputable. (Using classical logic, you could turn this into a total function, and many classical mathematicians would do so, but it's not necessary.)

  Also (this is less philosophical, more along the lines of errata), I've never seen the term "α-reduction", only "α-equivalence"; this is because there is no preferred direction (unlike with β-reduction, which simpifies (λx.fx)a to fa, for example, but would rather not expand fa to (λx.fx)a). Even "η-reduction" is viewed with some suspicion, because the correct direction seems to a matter of dispute. I see that Selinger's notes give a direction to it, but it's not the direction that you might expect!

• Week 1 - In the split exact sequence on page 9, the groups labelled F(A) and F(B) should be called F(X) and F(A), respectively, for consistency with previous notation.

• Week 6 - The theorem at the bottom of the fourth page of these notes is missing a key assumption, namely that A ≠ B. If A = B the result fails, since given a non-identity automorphism in a category C, we can't ever find an equivalence F: C → C' sending this to an identity automorphism.

  Luckily, in the free cartesian closed category on one object X equipped with an isomorphism α : X ≅ hom(X,X), we have X ≠ hom(X,X). (I should say even more about this, but I won't now.)

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