The following content has been provided by the University of Erlangen-Nürnberg.

Thank you very much for the invitation to Erlangen.

Actually, it's the first time that I'm in Erlangen.

So it's nice to see that there's no very active group in mathematical physics.

Now I want to talk on quantum field theory on curved spacetime.

So that's the general theme of these lectures, quantum field theory on curved spacetimes.

And I would like to start with some introductory remarks.

Now, as you very well know, we have two very beautiful fundamental theories,

which is, on the one hand, quantum field theory, in particular in the form of the standard model of elementary particle physics.

And this theory describes what we see in experiments to a very high degree.

So you all heard that recently one missing part in the standard model, the X particle, was discovered.

And actually all attempts to find effects which are not explained by the standard model failed up to now.

So it seems to be a very well established fundamental physical theory.

There's another fundamental physical theory, namely theory of general relativity.

And although this theory is very well established, of course, it's not so easy to make measurements,

because the effects are small, but nevertheless, in all cases where one has measured effects, there are an agreement with general relativity.

Now there is one big unsolved problem in physics, namely how these two theories can be united in such a way that you have one theory which contains both sub-theories.

And this is a very big problem, I think, which has attacked since several decades, and the progress is slow, and I think we don't know how far we are from a solution.

Now there are certain attempts to look for theory of quantum gravity.

For instance, when you start from the theory of elementary particle physics, then people prefer a way of construction of quantum gravity which is near to elementary particle physics,

and one attempt in this direction is string theory.

If one starts from general relativity and wants to quantize this theory, then another approach is more natural, and this is what is called loop quantum gravity.

There are also other approaches which are less well known, but I think it's fair to say that up to now, one cannot decide whether any of these approaches really approximates quantum gravity.

Now what I want to describe is something which is less ambitious, namely I want to extend the framework of quantum field theory in such a way that at least one aspect of general relativity is taken into account,

namely that space-time has a non-trivial geometry, it's not necessarily a flat space.

So let me perhaps indicate this by such a drawing.

So this would be quantum field theory on curved space-time.

So there are some aspects of general relativity which are taken into account.

There are the dynamical aspects of general relativity which are not taken into account because this is a theory where space-time is given, has a non-trivial structure,

but the structure is not directly influenced by the quantum fields.

But the theory should be formulated in such a way that it covers the full usual quantum field theory on Minkowski space.

So it's called quantum field theory on curved space-time, but it means generic space-time, so flat space-times are included.

So that's what I want to describe.

Now, this is not a very new approach, and actually the treatment of quantum field theory on curved space-time was very popular in the early 70s,

and the most remarkable result of this time was Hawking's prediction that black holes evaporate due to quantum effects, actually quantum field theoretical effects.

But already this analysis showed that it's not so easy to transfer the notions which we are used to in quantum field theory on Minkowski space to the curved situation.

Actually, the phenomenon of Hawking radiation just relies on the fact that there is no unique concept of a vacuum and no unique concept of particles on curved space-time.

So let me draw this kind of conclusion which one can draw after some thinking from this analysis.

The first crucial observation is there is no vacuum.

So the old attempt to physics, one considers a vacuum just as the name says, where you remove everything, all particles are removed, so you have nothing that remains, this is a vacuum.

And this idea is in conflict with the basic principles of quantum physics.

Actually, we all know this very well, if you look at the harmonic oscillator, the ground set of the harmonic oscillator has fluctuations.

And the same phenomenon occurs for quantum field theory.

So in quantum field theory, we have quantum fluctuations as a necessary consequence of the loss of quantum field theory.

So this can be illustrated in the example of a free field on Minkowski space.

So let's take a free scalar field on Minkowski space.

And we have the equal time commutation relations, that's the time zero of x.

And here we take the time derivative of the field at time zero at position y.

And this is equal to i times the delta function of x minus y.

These are the canonical commutation relations.