# Path Integral for curved spacetime

#### friend

Does anyone know what the Feynman Path Integral would look like in a space that has a curved geometry? I'm NOT talking about expressing the path integral in curvilinear coordinates that merely parameterize the cartesian coordinates of flat space. I'm talking about a space with curvature, like in general relativity, etc. Thanks.

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#### Haelfix

Actually it looks exactly like the path integral you normally see, with two subleties.

One, the normalization is different. So Z(0) = <out, 0|0, in> =<0|0> = 1 no longer applies.

Two, You need to include a source function J( ), that in general will not vanish, even if you insist (it will reappear upon renormalization).

The hard part though are intepretational issues, as well as the renormalization/regularization subleties for curved space. That requires a full textbooks treatment though and is lengthy..

#### friend

Actually it looks exactly like the path integral you normally see, with two subleties.

One, the normalization is different. So Z(0) = <out, 0|0, in> =<0|0> = 1 no longer applies.

Two, You need to include a source function J( ), that in general will not vanish, even if you insist (it will reappear upon renormalization).

The hard part though are intepretational issues, as well as the renormalization/regularization subleties for curved space. That requires a full textbooks treatment though and is lengthy..
Is there a book that you know of that treats this subject in a complete and modern way? Does this book develop the subject from scratch, or does it generalize on the flat spacetime version? Thanks.

#### friend

Birrell and Davies is pretty good, from what I hear:

https://www.amazon.com/dp/0521278589/?tag=pfamazon01-20

It actually looks pretty affordable.
Yes, thank you. But the book you refer to by Birrel and Davies assumes a "working knowledge" of QFT in flat spacetime, which I do not have. I've read up on QFT a couple of time a couple years ago, but I never really worked with it.

So I'm also considering another book by Stephen A. Fulling, entitled, "Aspects of Quantum Field Theory in Curved Spacetime". See more info at:

https://www.amazon.com/dp/0521377684/?tag=pfamazon01-20

It start with the Path Integral, and it is geared more for mathematicians without knowledge of QFT. Does anyone have an opinion on this book or its author? Thanks.

#### George Jones

Staff Emeritus
Gold Member
I don't have Fulling with me, but, if I remember correctly (I'll check tomorrow or Monday), Fulling doesn't treat path integrals at all. Fulling is a math text, and I don't think there is (what mathematicians would call) a mathematically rigorous general formulation of path integrals (What is the measure?) in Minkowski spacetime, let alone curved spacetime.

Maybe you should have a look at https://www.amazon.com/dp/0521868343/?tag=pfamazon01-20. Read the review by smallphi on this page.

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#### BenTheMan

Yes, thank you. But the book you refer to by Birrel and Davies assumes a "working knowledge" of QFT in flat spacetime, which I do not have. I've read up on QFT a couple of time a couple years ago, but I never really worked with it.
Ahh. Well...I would suggest that you read up on QFT again, and this time work with it ;)

If you want a free book, one is here:

http://www.physics.ucsb.edu/~mark/qft.html

If you want to pay for a book, try Zee, QFT in a nutshell''.

Any book you buy about QFT in a curved space-time is going to assume that you already are pretty good at regular QFT. You gotta wak before you can run.

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