Path Integral for curved spacetime

In summary: 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.htmlIf you want to pay for a book, try Zee, ``QFT in a nutshell''.
  • #1
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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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  • #2
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..
 
  • #3
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.
 
  • #5
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.
 
  • #6
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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  • #8
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.
 

What is the path integral for curved spacetime?

The path integral for curved spacetime is a mathematical tool used in quantum field theory and general relativity to calculate the probability amplitude of a quantum particle traveling from one point in curved spacetime to another. It takes into account the effects of gravity on the particle's path.

How is the path integral for curved spacetime different from the flat spacetime case?

In flat spacetime, the path integral can be calculated using the principle of least action, where the path with the smallest action is the most likely one. However, in curved spacetime, the action is not well-defined and the path integral must take into account the curvature of spacetime.

What is the significance of the path integral for curved spacetime in quantum gravity?

The path integral for curved spacetime is a crucial tool in the study of quantum gravity, which aims to reconcile the theories of general relativity and quantum mechanics. It allows for the calculation of quantum effects in a gravitational field, which is essential for understanding the behavior of matter and energy at the smallest scales.

What are the challenges in using the path integral for curved spacetime?

One of the main challenges in using the path integral for curved spacetime is the mathematical complexity involved. The calculations can become very difficult, especially in cases where the curvature of spacetime is large. Another challenge is the lack of a complete, unified theory of quantum gravity, which makes the interpretation of the path integral results difficult.

How does the path integral for curved spacetime relate to other areas of physics?

The path integral for curved spacetime is closely related to other areas of physics, such as quantum field theory, general relativity, and cosmology. It provides a bridge between these theories and allows for the study of quantum effects in curved spacetime, which is important for understanding the behavior of matter and energy in the universe.

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