Has Quantum Mechanics Been Fully Explored in Curved Spacetime?

[Schreiberdk]

Hi PF

I know that a goal of the current research in High Energy physics is to get a describtion of QFT in curved spacetime. I assume this has yet to be described fully, but how about plain quantum mechanics in curved spacetime? Has it yet been described?

Schreiber

 

[marcus]

You might be interested in this series of lectures given by an expert in this area, namely
QFT on (fixed) curved spacetime backgrounds
==quote from this year's QG school program==
Stefan Hollands - Exact QFT in curved backgrounds (lecture notes: 1, 2, 3, 4)

QFT on curved backgrounds is the formulation of QFT which does not require the symmetries of Minkowski or (Anti) de Sitter space times. From the mathematical point of view this is the highest development of QFT. It is also an important intermediate step beetwing the standard QFT and quantum gravity. As an approximation to quantum gravity it supplies some of the most potent intuitions of the field (holography, black hole entropy). The lecture will cover the recent results and successes in the exact construction of these quantum field theories.
==endquote==
Have a look at the school program. There are abstracts and links to lecture notes.
http://www.fuw.edu.pl/~kostecki/school3/

The Zakopane QG school is a one or two week school that has been held a few times to help grad students and others get into the main lines of QG research. The program this year consisted of 5 "core lecture series" plus many individual talks. QFT on curved background was considered one of the 5 core topics, so Hollands was asked to give a series of 4 lectures on it.

My impression is that QFT on (fixed) curved background is well developed but that what is NOT yet secure, and is the main work in progress, is backgroundless QFT----that means no prior fixed geometry. There the spacetime geometry would be fully interactive with the matter field.

 

[friend]

My impression is that QFT on (fixed) curved background is well developed but that what is NOT yet secure, and is the main work in progress, is backgroundless QFT----that means no prior fixed geometry. There the spacetime geometry would be fully interactive with the matter field.

Have they examined the difference between two small changes in curvature with QFT? Wouldn't that be enough to give some insight on how changes in curvature change QFT and more importantly how changes in QFT change curvature?

 

[marcus]

Have they examined the difference between two small changes in curvature with QFT? Wouldn't that be enough to give some insight on how changes in curvature change QFT and more importantly how changes in QFT change curvature?

Good idea! Someone else may have to answer your question, since I do not follow curved-QFT at all closely. I have not read of anyone doing what you say, but they could have been trying it. Actually Holland would be a good person to ask since he taught the Zakopane QG school course on this. He is at Cardiff U in the UK. Would you like his email to write him?
If you have any difficulty finding it please send me PM. (But I think you would have no trouble.)

 

[Bill_K]

The standard reference on the subject is "Quantum Fields in Curved Space" by Birrell and Davies.

 

[George Jones]

The standard reference on the subject is "Quantum Fields in Curved Space" by Birrell and Davies.

To this standard, I'll add

Introduction to Quantum Effects in Gravity by Mukhanov and and Winitzki (2009)
https://www.amazon.com/dp/0521868343/?tag=pfamazon01-20

Quantum Field Theory in Curved Spacetime by Parker and Toms.
https://www.amazon.com/dp/0521877873/?tag=pfamazon01-20

Mukhanov and Winitzki is meant as an "easy", pedagogical, and short introduction.

 

[martinbn]

If I understand your question correctly, you are asking about quantum mechanics on curved background, not about quantum field theory. I have asked that question but didn't get a satisfactory answer. I also couldn't find anything, but that may be simply because I didn't look enough.

The way I understand the question is the following. Consider a free nonrelativistic particle. The space of states can be taken $L^2(\mathbb R^3)$, well it should be the projective space and I am ignoring spin, the physical observables are curtain operators on that space and the evolution of the system is described by say the Schrödinger equation. So, at least naively one can try to replace $\mathbb R^3$ with his favorite manifold with a fixed measure. May be a Lie group or a homogeneous space would be of special interest. I expected to find such things at least because they would be mathematically interesting. The quantum behavior of a particle on the upper half plane (with the standard non-euclidean geometry) sounds interesting enough for someone to have tried it.

On the other hand there might be reasons why this is not done (if indeed it isn't). Curved background may be interesting in the case of gravity, hence relativistic setting, hence QFT. Or there might be reasons why things don't work out or why they are difficult.

In any case I like the question and would like to see some answers.

p.s. Sorry, how do I latex inline?

 

[qsa]

the simple answer is that QFT is QM+SR(special relativity) you cannot go directly to QM +GR(dynamic spacetime) like he said because that is QG (like LQG), and other QGs avoid the dynamic space time and concentrate on more viable problems. so an intermediate step is Dirac equation with a static curved spacetime,like a solution to a balckhole. there are many variations.(google).

example

https://www.physicsforums.com/showthread.php?t=185158