Path Integral for curved spacetime

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Discussion Overview

The discussion revolves around the Feynman Path Integral in the context of curved spacetime, particularly as it relates to general relativity. Participants explore the differences in formulation and interpretation of the path integral when applied to curved geometries, as opposed to flat spacetime. The conversation includes references to relevant literature and the prerequisites for understanding the subject.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Meta-discussion

Main Points Raised

  • One participant questions how the Feynman Path Integral is expressed in curved geometries, emphasizing that they are not referring to curvilinear coordinates in flat space.
  • Another participant asserts that the path integral retains a similar form but notes differences in normalization and the necessity of including a source function J( ).
  • Concerns are raised about interpretational issues and the complexities of renormalization in curved space, suggesting that these topics require extensive treatment.
  • Participants discuss the suitability of various texts, including Birrell and Davies, for understanding the path integral in curved spacetime, with some expressing concerns about the assumed prior knowledge of quantum field theory (QFT).
  • There is mention of Stephen A. Fulling's book, with one participant suggesting it may be more accessible for those without a strong background in QFT, although another participant doubts it covers path integrals adequately.
  • Additional recommendations for resources include a free online book and a suggestion to read Zee's "QFT in a nutshell," highlighting the expectation that readers should have a solid foundation in regular QFT before tackling curved spacetime topics.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of path integrals in curved spacetime, with no consensus on the adequacy of specific texts or the prerequisites needed for understanding the subject. The discussion remains unresolved regarding the best resources and the treatment of path integrals in the literature.

Contextual Notes

Some participants note limitations in the existing literature regarding the mathematical rigor of path integrals in curved spacetime and the assumptions required for understanding the material. There is also a recognition that a solid grasp of QFT is generally necessary to engage with the topic effectively.

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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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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..
 
Haelfix said:
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.
 
BenTheMan said:
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.
 
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.
 
Last edited by a moderator:
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 got to wak before you can run.
 

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