Square-Integrable Functions in Curved Spacetime

[stevendaryl]

In non-relativistic quantum mechanics, an important set of functions are the normalized square-integrable ones. Those are functions on $\mathcal{R}^3$ such that

$\int |\Psi(x,y,z)|^2 dx dy dz = 1$

I'm just curious as to whether there is some analogous concept for curved spacetime. One complication in curved spacetime is that an integral over "all space" requires a choice of a way to divide spacetime into spatial slices. Is the above condition on $\Psi$ independent of how one slices up spacetime?

 

[ChrisVer]

I also think you can use the invariant volume element in curved spacetimes. Since it's invariant it must have some certain form, which is generally
$dV= \sqrt{-g} d^{D}x$
with $g=detg$
I think it comes out of the Jacobian

 

[WannabeNewton]

What you have written down isn't even a Lorentz scalar in Minkowski space-time. Different time-like congruences of inertial observers determine different foliations of space-time into space-like hypersurfaces; two different foliations (amounting to two different families of inertial observers) will be related through Lorentz boosts. Geometrically, the foliations correspond to planes of simultaneity relative to a given family of inertial observers and Lorentz boosts will tilt the planes of simultaneity by angles related to the rapidity when going from one family of inertial observers to another.

Quantities that are invariant under such Lorentz boosts are of course Lorentz scalars. For example if we have a charged fluid with charge 4-current density $j^{\mu}$ then $\partial^{\mu}j_{\mu} = 0$ implies that $Q = \int_{\Sigma} \rho d^{3}x$ is a Lorentz invariant (here $\Sigma$ is a single plane of simultaneity relative to a given family of inertial observers).

In relativistic QM, the Lorentz invariant total probability* is given by $\int \rho d^{3}x = \int i[(\partial_{0}\varphi)\varphi^{\dagger} - (\partial_{0}\varphi^{\dagger})\varphi]d^{3}x = 1$. This value of unity for the total probability is preserved in the same manner the total charge $Q$ is due to the conservation of the probability 4-current density $j^{\mu} = i[(\partial^{\mu}\varphi )\varphi^{\dagger} - (\partial^{\mu}\varphi^{\dagger} )\varphi]$.

As you noted, what you have written down for the total probability is only valid in non-relativistic QM which uses Galilean relativity as a meta-theory of space-time, not special relativity. As such you can't even use that if you want a Lorentz invariant total probability let alone one that is valid for all space-time foliations, not just those determined by families of inertial observers in Minkowski space-time.

*of course once we go to QFT, the concepts of probability 4-current density and total probability are replaced, for obvious reasons, by those of charge 4-current density (operators) and total charge (operators) as per $j^{\mu} \rightarrow qj^{\mu}$.

 

[bcrowell]

The classification of functions in terms of continuity and integrability, on a manifold, is discussed in excruciating detail in Hawking and Ellis. For a discussion of volume integration in GR, see ch. 2 of the free online version of Carroll, http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll2.html

 

[sardar]

Thank you for your question. The concept of square-integrable functions in curved spacetime is indeed an important one in the study of quantum mechanics in general relativity. In curved spacetime, the metric is no longer constant and may vary from point to point, leading to a non-Euclidean geometry. This means that the definition of "volume" and "distance" may also vary, making the concept of a simple integral over all space more complicated.

To address this, one approach is to use the concept of a measure, which is a way of assigning a numerical value to a set of points in a space. In the case of curved spacetime, a measure can be defined using the metric tensor, which takes into account the varying geometry of spacetime.

Using this measure, we can then define a square-integrable function as one whose squared magnitude can be integrated over a region of curved spacetime, such that the integral converges to a finite value. This condition is independent of how one divides up the spacetime into spatial slices, as the measure takes into account the varying geometry.

In summary, the concept of square-integrable functions in curved spacetime does exist and is crucial in the study of quantum mechanics in general relativity. It is defined using a measure that takes into account the varying geometry of spacetime, making it independent of the choice of spatial slices.