We are now ready to go back to curved spacetime in the IN/OUT formalism!

We have

Indeed,

where when we get particles from gravity! The particle density reads

For instance, in a Friedmann-Robertson-Walker metric (FRW universe) we obtain

where is the scale factor and

is the energy of the mode k. The Heisenberg Uncertainty Principle (HUP) implies (heuristically) that

An interesting consequence of all this crazy stuff is that with massless particles ( ) and a de Sitter (dS) spacetime we get virtual particles produced by this effect or alternatively is finite!

The IN/OUT asymptotically stationary phase implies gravitational particle creation! The Bogoliubov coefficients are understood in the following sense:

contains “particles” (quasiparticles to be more precise) that are related to a set of normal modes. In fact, the complete set of Bogoliubov coefficients satisfy certain relations:

for bosons and

for fermions. The quantum analogue of this coefficients are the scattering problem over a potential barrier (they can be understood as “tunneling amplitudes” somehow).

In flat FRW universes, we get

The Klein-Gordon field has solutions (in curved spacetime!):

with

normalization. The time dependent simple harmonic oscillator with variable satisfy

and where is the minimal coupling case, and determines the conformal coupling case in D=4. Furthermore,

and thus

The particle number uncertainty due to the particle creation can be also calculated:

It has a minimum when

and

so

Fluctuations in the particle number are ALWAYS non zero if there is particle creation! The proof is easy to understand:

There is a nice relationship between the “squeezed” states and the vacuum in this formalism:

If is a squeezed state respect to , then

The conformal coupling is special: massless case implies that there is no particle creation (indeed, conformal field theories has no notion of particle theirselves!).

When we have a smooth expansion, in the massive case , then is the conformal case. Moreover, the solutions to the EOM can be easily found by common procedures of differential equations. Thus, we have

with

has two solutions with frequencies

and !

A well known example in the literature is when corresponds to the case

when

Thus,

and it corresponds to the adiabatic expansion case.

With a step function expansion (), we get

and it corresponds to the “sudden approximation”. The modes read then, in this “sudden” case,

and

if . The particle number, the critical frequency and the variation of correspond respectively to

After all these rudimentary facts, we can understand two of the most celebrated phenomena in curved spacetime QFT:

1) The Unruh effect (or the relativity of vacuum in curved spacetime QFT).

2) The Hawking effect (particle creation in the surrounding of black holes).

These two effects will be covered in my next posts from a very elementary viewpoint!

See you in the next TSOR posts!