![the curved space the curved space](https://nanohub.org/app/site/resources/2013/09/19336/slides/009.01.jpg)
To solve for the second relation,, we repeat this process, instead rewriting the left-hand side of (12) in terms of. Note that had we taken the Klein-Gordon product with, we’d simply have obtained. (6)), and using the fact that the modes are orthonormal (7), we then have
![the curved space the curved space](https://cdn-images-1.medium.com/max/1600/1*UDMTLq8c9-vG2vd3cIwtyQ.png)
Taking the Klein-Gordon product with (recall this is how we isolate the coefficients, cf. Equating the two mode expansions (1) and (3) for the same field then leads to the following relationship between the creation/annihilation operators:įor example, the first of these is obtained by expressing everything in terms of : Where the second equality in the last line follows from (8) similarly for. Taking the inner product of different sets of modes then allows one to isolate the Bogolyubov coefficients: The modes are orthonormal with respect to this inner product, and therefore provide a convenient basis for the solution space:Īnd similarly for, hence the funny split in the normalization we mentioned below (1) (for our notation/normalization conventions, see part 1). The idea behind this choice of inner product is that it allows the easy extraction of the Fourier coefficients from (1) as follows:Īnd similarly for commuting the two yields the standard commutation relation for the creation/annihilation operators above. Where the integral is performed over any Cauchy slice for the present case of 1d flat space. These can be determined by appealing to the normalization condition imposed by the Klein-Gordon inner product, (Note that unlike, we’re working directly in the continuum, so the coefficients should be interpreted as continuous functions rather than discrete matrices). These are the advertised Bogolyubov transformations, and the matrices, are called Bogolyubov coefficients. Since both sets of modes are complete, they must be linear combinations of one another, i.e., Which represent excitations with respect to some other vacuum state, i.e. Hence we could equally-well quantize the field with a different set of modes: Indeed, the spirit of general relativity is that there are no privileged coordinate systems. Another way to say this is that the vacuum state is not invariant under general diffeomorphisms, because these have the potential to mix positive- and negative-frequency modes. In general spacetimes however, there will be no Killing vectors with respect to which we can define positive-frequency modes, so this decomposition depends on the observer’s frame of reference. That is, the modes are of positive frequency with respect to some time parameter, by which we mean they’re eigenfunctions of the operator with eigenvalue : Note however that this statement depends on the existence of a global timelike Killing vector. Typically, we choose the positive solution to the Klein-Gordon equation,, so that the operator has an interpretation as the creation operator for a positive-frequency mode traveling forwards in time. (The derivation of this commutation relation, as well as the reason for splitting the normalization in this particular way when identifying the modes, will be discussed below). Where, , and the creation/annihilation operators satisfy the standard equal-time commutation relation, which in our convention is, with the vacuum state such that.
#The curved space free
Consider quantizing the free scalar field in -dimensional Minkowski (i.e., flat) spacetime. In a nutshell, a Bogolyubov transformation relates mode decompositions of the field in different states. In this post, I want to elucidate this issue by introducing a very important tool: the Bogolyubov (also transliterated as “Bogoliubov”) transformation. In addition to challenging our intuition, this fact lies at the heart of some of the most vexing paradoxes in physics, firewalls chief among them. That’s not to say it isn’t locally useful, but without specifying the details of the mode decomposition and the trajectory/background of the detector/observer, it’s simply not meaningful.
![the curved space the curved space](http://milesmathis.com/curve.jpg)
One of the most important lessons of QFT in curved space is that the notion of a particle is an observer-dependent concept.