Quantum field theory in spacetime with different topologies

Question

Suppose I want to compute scattering amplitudes in a quantum field theory with the following action:

$$S = S_{\mathrm{matter}}+S_{\mathrm{topological}}.$$

Here, I have involved matter (without gravitational interactions) and some possibilities of spacetime topologies. After some searching in the web I have found that there can be three topological terms added dependent on the spacetime structure:

• A Pontryagin term
• An Euler term
• A Nieh-Yan term

First question: If I want to include these terms, do I have to include the gravitational term (the Einstein-Hilbert action) or can I construct a quantum field theory with only these topological terms?

Computing the partition function

$$Z = \sum_{\mathrm{topologies}} \int D[\phi] e^{iS}$$

with the matter fields $\phi$ leads to finite results; quantizing gravity is not yet known due to non-renormalizability. In string theory sum over topologies is known as the genus expansion; can I do a similar expansion over topologies for a 4-dimensional spacetime? Can I assume flat spacetime, but different topologies and different distributions of spacetime boundaries in a quantum field theory and if yes, how the partition function would look precisely?

I could also do QFT in classicaly curved spacetime, but quantization of fields become more difficult; therefore: Minkowski metric on regions $\Omega$ where quantum fields can be defined.

If I have an action with boundaries, i.e. $S_{matter} = \int_\Omega d^4x \mathcal{L}_{matter}$ and I make mode expansion in $k$-space for the fields $\phi$ then there will occur not simply delta distributions for energy-momentum conservation (it is $\delta(k_{incoming}^\mu - k_{outcoming}^\mu)$), there will occur instead $sinc(k_{incoming}^\mu - k_{outcoming}^\mu)$ if there is some boundary of spacetime. This implies that there will exist amplitudes, where energy and momentum is not conserved and due to symmetry of the sinc function with the argument, energy gain has same probability as energy loss; thus energy is still conserved in average. By taking the limit $\hbar \mapsto 0$, the sinc-function turns into delta function; that means that these excesses of energy and momentum are quantum effects.

The last question is: What would an observer see if spacetime has somewhere one or more boundaries? Will the observer see that particle-antiparticle pairs pop out for a while near these boundaries?

My main idea is that gravity terms can be neglected, but topological terms can be respected, such that one has an expansion for the scattering amplitude

$A = A_{matter,Pontryagin = 0} + g^1A_{matter,Pontryagin = 1} + g^2A_{matter,Pontryagin = 2} + \dots$

with the coupling to the Pontryagin term $g$ and where the first term in the expansion is for a spacetime without boundaries (energy-momentum strictly conserved in this contribution). I summarize the questions:

• Is gravity (dependence on spacetime geometry) neglectible; I assume that the magnitude of the energy-momentum tensor of the system is sufficiently small and that no external gravitational fields are present?
• How I define a plausible partition function if I want to do a "sum over (flat) spacetime topologies"?
• What would someone observe at spacetime boundaries?

Hints would be greatly appreciated!

Answer 1

Answer to the first question: It depends on the energy and length scale. If the energy density of the system is sufficiently small, I can neglect gravitational contributions. Formulating Einstein's field equations in dimensionless parameters (denoted by tilde) it holds:

$\frac{1}{L^2}(\tilde{R_{\mu \nu}} - \frac{1}{2}\tilde{R}g_{\mu \nu}) = -\frac{8 \pi GMc^2}{L^3c^4} \tilde{T_{\mu \nu}}$.

Here, $L$ is the characteristic length scale and $M$ the characteristic mass scale. One observes that the gravitational sources are proportional to $\frac{GM}{Lc^2}$; thus, for low masses and large lengths gravity can be neglected.

Answer to all other questions: The whole action is locally diffeomorphism invariant and does not depend on displacements $u_\mu$ (generator of diffeomorphisms) of spacetime. However, we will compute path integrals of the form

$Z = \int \mathcal{D}[u_\mu] \delta_{Gauge-fix}(u_\mu) \int \mathcal{D}[\phi] e^{iS}$

that in generally may contain sum over non-continuous functions of $u_\mu(x)$ (not a diffeomorphism; it changes topology of spacetime manifold!). This means also that we must split the integral over displacement into different topologies and into TRUE diffeomorphisms; hence:

$Z = \sum_{Topologies} \int \mathcal{D}[u_\mu]|_{diffeomorphic} \delta_{Gauge-fix}(u_\mu) \int \mathcal{D}[\phi] e^{iS_{matter}+iS_{top}}$.

The evaluation of the path integral over displacements is now possible and may lead to a gauge fixing factor $e^{iS_{gauge}}$ dependent on ghost fields $\xi$. The factor $e^{iS_{top}}$ is dependent on topology only. This term will give a number. Finally:

$Z = \sum_{Topologies} e^{iS_{top}|_{Topology}} \int \mathcal{D}[\xi]\int \mathcal{D}[\phi]e^{iS_{matter}+iS_{gauge}}$.

When dealing with Lagrangian densities integrated over manifolds with holes, one will obtain terms like $sinc(k_{in}-k_{out})$ instead of delta distributions after Fourier mode decompositions. One can show that this also goes to a delta distribution term for $\hbar \mapsto 0$. Such a term encodes simply Heisenberg's uncertainty principle.