Recently I have seen that the Einstein-Hilbert action (eventually with the Gibbon-Hawkings boundary term) is not all in the description of spacetime. There can exist also various additional terms in the action that give information about the topology of the spacetime manifold. These terms are:
- A Pontryagin term ($S_{Pont} = \int_\Omega \epsilon_{abcd} R^{ab} \wedge R^{cd}$ with curvature tensor $R^{ab}$ and the spacetime manifold $\Omega$)
- An Euler term that gives the Euler characteristic
- A Nieh-Yan term related to the spacetime torsion
A complete description of gravity is given by
$S = S_{EinstHilb} + S_{Pont} + S_{Euler} + S_{Nieh-Yan} + S_{gaugefixing} + S_{matter}$
(last term is dependent on the choice of gauge) and the partition function has the form
$Z = \int D[\phi] \int D[e_I^a] \int D[\omega_{IJ}^a] e^{iS}$
with matter fields $\phi$, tetrad $e_I^a$ and spin connection $\omega_{IJ}^a$. I know that the partition function will be UV divergent in the case when Einstein-Hilbert term (for classical General Relativity) is present. To resolve this trouble I assume that the Einstein-Hilbert action will not be quantized such that I will have a semiclassical theory; gravitational fields can be assumed to be classical that obey the equation
$R_{IJ}-\frac{1}{2}R g_{IJ} = 8 \pi G T_{IJ}$.
Gauge fixing I neglect also for simplification.
But all other topological terms I can quantize, since these are simply topological invariants; only numbers. I have seen e.g. in String theory that an expansion over all possible topological stuctures can be performed. Now I decompose gravitational fields in form of the following:
$e_I^a = e_I^a|_0 + e'_I^a$
$\omega_{IJ}^a = \omega_{IJ}^a|_0 + \omega'_{IJ}^a$
Here, the subscript 0 denotes the field that has no contribution to gravity, but induces nonzero topological invariants, depending on the manifold structure that I have. Primed fields are fields obtained by classical gravity, these do not change topological terms. Finally, I have the following path integral:
$Z = \sum_{Topologies}g_1^{n_{Pont}}g_2^{n_{Euler}}g_3^{n_{Nieh-Yan}} \int D[\phi] \int D[e'_I^a] \int D[\omega'_{IJ}^a] \delta(\delta_{e^a,\omega^a}S_{EinstHilb})e^{iS_{matter}}|_{n_{Pont},n_{Euler},n_{Nieh-Yan}}$.
The couplings $g_1,g_2,g_3$ are couplings corresponding to the topological structure and $n_{Pont},n_{Euler},n_{Nieh-Yan}$ characterizes the manifold topology. Is this path integral correct? Due to coordinate and Lorentz invariance I can fix gauge such that the location of topological features does not affect path integral; is a aum over topologies sufficient or I must integrate over fields to represent different topologies?
Now if I transform spacetime manifold integrals into momentum space I will get e.g.
$\int_{\Omega} d^4x \psi^*(x) \psi(x) = \int_{\mathbb{R}}d^4x 1_\Omega(x) \psi^*(x) \psi(x) = \frac{1}{(2 \pi)^12} \int d^4x \int d^4K 1_\Omega^{fouriertransformed}(K) e^{iKx} \int d^4k e^{-ikx} \phi^*(k) \int d^4k' e^{ik'x} \phi(k') = \frac{1}{(2 \pi)^8} \int d^4k \int d^4k' \int d^4 K \delta(k-k'+K) 1_\Omega^{fouriertransformed}(K) \phi^*(k) \phi(k') $
meaning that there will be excess of energy and momentum due to eventual boundaries of the spacetime manifold $\Omega$; this excess occurs if the distribution $1_\Omega$ is somewhere zero because of internal boundaries in spacetime.
Next Question: This energy-momentum excesses represent Heisenberg's uncertainty, where some spacetime regions that are "cut out" of spacetime define a given length and time interval. A minimum energy and momentum uncertainty arises. What would we observe if spacetime would have microscopic holes? Would we observe that some particles seem to pop out from nothingless (physically, energy is borrowed to create particle-antiparticle pairs) that will vanish a shorter time later near the holes of spacetime?
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