Quantum wave equations in curved space

1. Dec 23, 2012

FunkyDwarf

Hi,

I've been looking at the Klein Gordon equation, Maxwell's equation, and the Dirac equation in curved space and I was wondering if there is an underlying formalism regarding how to derive them from their flat space counterparts.

What I mean is, at the heart of the whole process for all spins is the switching from normal derivatives to covariant derivatives. However, in the Dirac case most people apply the tetrad formalism of transforming to a locally flat frame: why isn't this done in the other cases? (is it just that there is an easier way?)

Is it not possible to take the Dirac Lagrangian, switch round the derivatives accordingly as you would with the spin-0 Lagrangian, and derive the wave equations that way?

-FD

2. Dec 23, 2012

HallsofIvy

Staff Emeritus
No, the equations in general curved space cannot be derived directly from the flat space equations. That is because "flat space" is a "local" special case of the more general formulas and there are an infinite number of possible equations that will reduce to the flat space equations "locally".

3. Dec 23, 2012

bcrowell

Staff Emeritus
Could you amplify on this, maybe give an example? This is the opposite of what I'd thought, but maybe we're thinking of different things...?

4. Dec 23, 2012

Bill_K

Isn't it true that writing the curved space theory requires that one pick the coupling to gravity, and just as in QED where there is a "minimal coupling" in which additional terms like σμνFμν are not added to the Lagrangian. The minimal coupling to gravity says you first write the flat space theory in curvilinear coordinates, and then simply forget that space is flat.

5. Dec 23, 2012

FunkyDwarf

6. Dec 23, 2012

Bill_K

Because the Dirac Equation involves an internal symmetry group, and the tetrad formalism is needed to accommodate this. In flat space we can take the gamma matrices as constant, but in curved space they obey {γμν} = 2gμν and cannot possibly be constant. The covariant derivative therefore must include a term that compensates for the change of γμ from point to point, and this is what the tetrad formalism provides. The spinor indices on γ and the wavefunction ψ are relative to a local Lorentz frame. In flat space, one component of the wavefunction, for example, is "spin up", but in a curved spacetime, "spin up" is not globally defined. You can parallel transport a spin along a single curve, but along some other curve it will be different. The best one can do is to define spin up to mean projection on one of the tetrad basis vectors.

7. Dec 23, 2012

Naty1

8. Dec 24, 2012

andrien

9. Dec 24, 2012

haael

This is a similar problem to the transition from SR to GR.

The difference between special and general relativity is that the first one's vacuum is flat. The question is: how a "flat" equation generalizes into a "curved" one? The obvious fact is that all geometrical objects must remain tensors. This introduces a condition that a usual derivative changes into a covariant one.

$F_{,x} = 0$ (in flat spacetime)
$F_{;x} = 0$ (in curved spacetime)

But tha's not all. Suppose there is some measurement of curvature; let's call it R. Now in flat spacetime $R = 0$. So the equation may as well generalize to:

$F_{,x} = 0$ (in flat spacetime)
$F_{;x} = R$ (in curved spacetime?)

Instead of R there may be any power of R or in general any function of R that equals 0 for $R = 0$.

In general relativity we have a very strong rule - the equivalence principle - that tells us exactly where we should put R and which power of it. But is the equivalence principle applicable to the quantum world? This is basically trying to account the gravitational intetaction of quantum fields. This might not be easy.

10. Dec 24, 2012

Bill_K

You don't mean R by itself, since that would belong to a description of gravity alone without the field. You mean an interaction term, something like φR. This is what I was referring to above in the comment on minimal coupling:

11. Dec 24, 2012

George Jones

Staff Emeritus
I believe this is discussed in Birrell and Davies, but I don't have my copy with me, so I can't check.

12. Dec 24, 2012

bcrowell

Staff Emeritus
Is this what's known as semiclassical gravity? Is the stuff about minimal coupling and so on physically dealing with how the matter fields influence the gravitational field? If so, then in the limit where the curved spacetime is taken to be a fixed background, do you get something reasonable by taking, say, the Klein-Gordon equation and simply replacing the derivatives with covariant derivatives? E.g., I believe Hawking and Ellis discuss global hyperbolicity by talking about whether the Klein-Gordon equation has unique solutions on a given background, and I don't recall that they did anything fancier than simply replacing the derivatives with covariant derivatives.

13. Dec 24, 2012

vanhees71

Of course, we have no working quantum theory of gravitation yet. What, however, can be established to a certain extent is quantum field theory in a "classical" space-time background.

Usual relativistic quantum field theory (like the standard model of elementary particles) ignores gravity at all and uses flat Minkowski space as space time. In an analogous way, one can try to do quantum field theory in more general space times like a Robertson-Walker metric, describing the universe on the long-range scale.

This is highly non-trivial, particularly the definition of what a particle state might be. That's already non-trivial in usual Minkowski space, only using a non-inertial (i.e., accelerated) frame of reference. What's defined as "the vacuum" of free particles in an inertial frame comes out to describe a thermal system of many particles in a uniformly accelerated reference frame (Rindler coordinates of Minkoski space time).

To define even what the vacuum might be in a general curved space time is the more complicated. A naive definition leads to trouble with the definition of particle states. A good review about these issues is

Bryce S. DeWitt. Quantum field theory in curved spacetime. Phys. Rept., 19:295-357, 1975.
http://dx.doi.org/10.1016/0370-1573(75)90051-4 [Broken]

Last edited by a moderator: May 6, 2017
14. Dec 24, 2012

TrickyDicky

But isn't this trivializing geometric curvature?
One needs to decide whether gravity is more like a field or rather something more geometrical following the original spirit of GR; you seem to favor the former view, right?

15. Dec 24, 2012

George Jones

Staff Emeritus
Yes.
I think Hawking and Ellis was published in 1973, and written earlier, which was before this stuff was worked out.
Yes. I have used amazon.com to look at the stuff in Birrell and Davies (section 3.2). In the scalar field Lagrangian, they include a term $\xi R \phi^2$ ($\xi$ is a constant) that represents coupling between the scalar and gravitational fields. This introduces a term $\xi R \phi$ in the Klein-Gordon equation. They say that two values of $\xi$ "are of particular interest": 1) minimal coupling with $\xi = 0$; 2) conformal coupling with $\xi =3/2$. The latter is used when the massless scalar field in order to make things invariant under conformal transformation.