I am currently learning about QFT in curved spacetime using these notes.

Given a subspace $S_p$ of positive frequency solutions to the Klein-Gordon equation, we defined the corresponding annihilation operators $a(f) := (f, \Phi)$, where $\Phi$ is our quantum field and $(.,.)$ is the Klein-Gordon "inner product":

$$(f,g) := i \displaystyle\int_{\Sigma} d^3x \sqrt{h}n^a(\bar{f}\nabla_a\beta - \beta\nabla_a\bar{f}).$$

I wanted to find the commutation relations for these annihilation operators, and I found: $$ [a(f), a(g)] = (f, \bar{g}).$$

In the case that our spacetime is stationary with timelike Killing vector $K$, and we take $S_p$ to be the space of eigenfunctions of $\mathcal{L}_K$ with negative imaginary eigenvalue, then $\bar{f}$ is a negative frequency solution. By the antihermiticity of $\mathcal{L}_K$ it follows that positive and negative frequency solutions are orthogonal and hence $[a(f), a(g)] = 0$

However for an arbitrary choice of $S_p$, do we necessarily have that positive and negative frequency solutions are always orthogonal? If not, does this mean that $[a(f), a(g)]$ will not be equal to $0$ in those cases, or have I done the above calculation wrong?


1 Answer 1


By "an arbitrary choice of $S_p$" I assume you mean "if we don't stick to the space of positive-frequency solutions induced by the Killing vector field". If so, one will impose "by hand" that such a result holds. When looking for candidates for a space $S_p$ of "generalized positive-frequency" solutions to the Klein--Gordon equation, one asks that

  1. the Klein--Gordon "inner product" should be positive definite on $S_p$;
  2. $S_p$ and its conjugate (i.e., the negative-frequency solutions) should span the space of complexified solutions to the Klein--Gordon equation;
  3. positive-frequency and negative-frequency should be orthogonal.

This is pretty much the conditions you need not to run into any trouble. I guess one could say that, among other things, the first one allows you to get a Hilbert space in the end of the day, the second one ensures you are not forgetting anyone, and the third one guarantees you'll be able to decompose stuff in positive and negative frequency (it would be odd if a negative-frequency solution had a component along a positive-frequency direction in the solution space, so to speak). From a quick look at the notes you linked, it seems they mention the first two axioms, but not the third.

I guess it is worth mentioning that, usually, one would specify $S_p$ not explicitly, but instead by choosing a "well-behaved" real inner product $\mu \colon S \times S \to \mathbb{R}$, which is then used to define $S_p$, and by means of this approach orthogonality will also follow. So, in summary, we necessarily have that positive and negative frequency solutions are always orthogonal essentially because we require that.

In case you would like some further reading, Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics covers these issues really well. I think you'd be interested particularly in the discussion on pages 27--30 (where the axioms I listed are provided and so is the general idea of specifying $S_p$ by means of a real inner product $\mu$) and afterwards on pages 41--43, which show the equivalence between such freedoms. In particular, what I meant by "well-behaved" inner product is that Eq. (3.2.16) should hold.


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