Quantum Field Theory in Curved Spacetime?

[asimov42]

Hi all,

I have a question about the formulation of quantum field theories in curved spacetime. I'm still learning, and so I might not articulate this very well, but I'm wondering:

If a region of spacetime can warp and curve, dynamically changing its shape in response to changes in energy density, then I assume a quantum field defined over the same region would also have to expand / contract? Can the field be thought of as something 'physical', and if so, in the case of expansion, where does the extra 'quantity' of field (for lack of a better description) come from? Or is it better to think of the field as being 'stretched' and 'compressed'?

Thanks all.

 

[Fredrik]

Think of it as a function that has spacetime as its domain of definition.

 

[richard14]

question?
i am a new student in physics, but is it true that the big bang breaks one of the more
imortants laws of physics the conservation of momentum.Since a singularity a creative point of energy exploted in space in order to be done there were suppose to be two
particles, a particle and its nemesis to create a big explosion this is called annihilation
but were do the pair antipair came from, and if the singularity was pure energy were did it come from E=MC2

 

[Fredrik]

question?
i am a new student in physics, but is it true that the big bang breaks one of the more
imortants laws of physics the conservation of momentum.Since a singularity a creative point of energy exploted in space in order to be done there were suppose to be two
particles, a particle and its nemesis to create a big explosion this is called annihilation
but were do the pair antipair came from, and if the singularity was pure energy were did it come from E=MC2

The right place to ask that question is in the relativity forum, in a new thread, but there isn't much discussion going on here anyway, so I don't think I'll ruin this thread by answering your question.

The "big bang theory" (the simplest version of it anyway) is just a part of the FLRW solutions of Einstein's equation. They define a coordinate system that assigns a time coordinate t>0 to every event in spacetime. (No event has t=0 or t<0). The "big bang" is the limit $t\rightarrow 0$. When you take that limit, some variables go to infinity (e.g. density and pressue) and others go to zero (e.g. the distance between two arbitrary points in "space").

So the big bang isn't an explosion. It's not a point in spacetime. It's not "a creative point of energy" or "pure energy". It's just a name for the funny stuff that happens when you take a certain limit.

 

[asimov42]

But should the field be thought of as something 'physical'? Or simply as a mathematical representation for some underlying phenomenon?

 

[Peter Morgan]

But should the field be thought of as something 'physical'? Or simply as a mathematical representation for some underlying phenomenon?

You should think of the quantum field in curved spacetime in the same way as you think of quantum field theory in Minkowski space -- whatever that is. It's an algebra of operators on a curved spacetime background instead of on a flat spacetime background. The background is non-dynamical, so we can effectively take the background as a given spacetime, otherwise you would be talking about quantum gravity and all hopes of understanding are in the future.

Insofar as quantum field theory in Minkowski space is carried through in terms of solutions of the Klein-Gordon and other classical free fields on a flat Minkowski space (i.e. in terms of orthogonal modes of the free field, effectively Fourier analysis), quantum field theory in a curved spacetime is carried out in terms of solutions of the Klein-Gordon and other classical free fields on whatever curved spacetime we have chosen (ie. in terms of orthogonal modes of the free field in the curved spacetime). Wald's book, "Quantum field theory in curved spacetime and black hole thermodynamics", is a pretty good presentation.

 

[asimov42]

Hi Peter,

Thanks - that helps a lot. Just for my own clarification (again as a complete novice), if we say that the background is non-dynamical, we mean that spacetime has a fixed structure (i.e. a fixed, invariant curvature at each point)?

 

[Peter Morgan]

Hi Peter,

Thanks - that helps a lot. Just for my own clarification (again as a complete novice), if we say that the background is non-dynamical, we mean that spacetime has a fixed structure (i.e. a fixed, invariant curvature at each point)?

If you understood my comment, you may not be too much of a novice to try reading Wald, right? It doesn't much matter how you fix the geometric structure, but it has to be enough that "the causal behavior of the curved spacetime [is] sufficiently well-behaved that the space of solutions to the classical field equations have the same basic structure as in Minkowski space. As we shall see in section 4.1, the condition of global hyperbolicity ensures that this is the case."[Wald, from the first para of Ch.4] If you need the details, you really have to read Wald or something similar.

The Wald is quite readable, and you would very likely learn some interesting stuff about QFT in flat spacetime.

 

[Naty1]

To express, I hope, the same thoughts another way,

The background is non-dynamical, so we can effectively take the background as a given spacetime, otherwise you would be talking about quantum gravity and all hopes of understanding are in the future.

this can also be illustrated by noting QFT is a background dependent theory...this means we pick a fixed geometrical background and work with it...like string theory(s). In contrast, general relativity (GR) has a flexible changing "background independent" formulation...via the curvature tensor, no preselected, fixed geometry is independently chosen, instead the background changes dynamically, in the case of GR according to mass, energy and pressure changes...a quantum gravity formulation that IS background independent is loop quantum gravity, but that has not yet been reconciled with GR, and is I think still incomplete...

So

If a region of spacetime can warp and curve, dynamically changing its shape in response to changes in energy density, then I assume a quantum field defined over the same region would also have to expand / contract?

the immediate answer is no; that formulation is still being sought and if formulated might combine quantum mechanics and GR...stay tuned! Still plenty of opportunities for Nobel Prizes!

One clue that things are still a bit confused are all these formulations... if we REALLY knew what we were talking about, there would be one agreed upon formulation...

"we know much, we understand little"

 

[Haelfix]

The curved spacetime is assumed to first order to be nondynamical. EG you are not computing graviton interactions and/or backreactions (although you could in principle).

The approximation is pretty good unless you get energies close to the Planck scale. Beyond that, you have to do quantum gravity.

The main problem is calculational. You lose many global symmetries working in curved spacetimes, and it makes things really technically challenging (which is why there are only about 3-4 famous results coming from this formalism).

In general, the exact form of the metric is usually left blank until late in the calculation (eg its pretty general, with only a few limitations: eg non hyperbolic, or metrics with bad asymptotics cannot be dealt with easily).

The good news is that the causal structure of the classical theory is very constraining on the quantum picture. The bad news is spin structure is really hard to deal with and a lot of ones intuition about acceptable quantum observables ceases to be mathematically well defined..

 

[asimov42]

To express, I hope, the same thoughts another way,

this can also be illustrated by noting QFT is a background dependent theory...this means we pick a fixed geometrical background and work with it...like string theory(s). In contrast, general relativity (GR) has a flexible changing "background independent" formulation...via the curvature tensor, no preselected, fixed geometry is independently chosen, instead the background changes dynamically, in the case of GR according to mass, energy and pressure changes...a quantum gravity formulation that IS background independent is loop quantum gravity, but that has not yet been reconciled with GR, and is I think still incomplete...

Are classical field theories also considered to be background dependent? I.e. formulated over a selected fixed geometrical background? This might be a naive question and may not make sense to ask - I admit, I'm somewhat foggy on this :-)

Thanks very much everyone - I appreciate the help.

 

[atyy]

Are classical field theories also considered to be background dependent? I.e. formulated over a selected fixed geometrical background? This might be a naive question and may not make sense to ask - I admit, I'm somewhat foggy on this :-)

Thanks very much everyone - I appreciate the help.

In special relativity, the flat Minkowski metric is fixed and Maxwell's equations are solved.

In general relativity, the metric is not fixed and Maxwell's equations are solved simultaneously with the Einstein field equations, so electromagnetic fields cause spacetime curvature.

However, if the curvature produced by the electromagnetic fields is small, it is an excellent approximation to ignore them when solving for the metric. For example, in the solar deflection of light, the usual presentation is to fix the metric as that caused by the sun alone, then determine how light propagates on that metric.

 

[Naty1]

The approximation is pretty good unless you get energies close to the Planck scale. Beyond that, you have to do quantum gravity.

Good point.

Loop quantum gravity uses a dynamical background but I don't know how far formulations have been developed. Lee Smolin works that area and in THE TROUBLE WITH PHYSICS disccusess problems and opportunities with different avenues of approach.

He works at the PERIMETER INSTITUTE ...They have some interesting stuff available on line..

 

[Phrak]

aty. Re: maxwell on curved spacetime. Are you expressing fields with exterior derivatives or covariant derivatives? just curious