# Alternative quantization of quantum electrodynamics?

A Quantum field theory is determined, if a Hilbert space Basis with Operators acting on it (such that one element of an Hilbert space is also an element of the same Hilbert space if an Operator acting on it) and a Hamiltonian and commutator relations exists. Now I want to know whether Quantum electrodynamics can be reformulated in Terms of cobordisms instead of field distributions in spacetime.

Quantum electrodynamics has the Hamiltonian:

$$H = \int d^3x [\psi^\dagger \gamma_0(\gamma_i (\partial^i+ieA^i) +m_0) \psi + \frac{1}{2} (E^2+(rot A)^2)].$$

The commutator Relations for vector potential $A^i$ and electric fields $E^i$ are

$$[E^j(x),A^i(x')] = -i\delta_{ij} \delta(x-x')$$

for Fermion fields $\psi$ anticommutators are given by

$$\{\bar{\psi}(x),\psi(x')\} = -i\delta(x-x').$$

All other commutator Relations are zero. In ordinary Approach to Quantum electrodynamics one has a vacuum state $|0>$, where the Action of a field operator on it gives the corresponding field function and a creation or Annihilation of a field on the vacuum.

Is it possible to assume the vacuum state on a fixed time $t=t_0$ as a collection of 3-dimensional boundary, where the Hamiltonian Operator $H$ defines how this 3-dimensional boundary evolves to another 3-dimensional boundary at $t=t_0+dt$? Can I assume field Operators as cobordisms on such states? Maybe I can create a theory with same hamiltonian and commutator relation, but where I use boundaries as objects and cobordisms as morphisms?

I would say yes, because this can be achieved by a functor mapping form Hilbert space category to cobordism category.

Another question: Will I obtain different scattering amplitudes if I treat Quantum electrodynamics with cobordisms instead of ordinary well-known Hilbert space Framework?

Maybe also yes, because quantization is different.

The problem is that QED is neither topological nor conformal - the partition function, which would be the functor mapping a cobordism to an amplitude in this approach, has no well-defined value on a morphism in the category of cobordisms, since the objects of this category are customarily diffeomorphism (or homeomorphism) classes of manifolds and a morphisms $\Sigma$ between two such classes $M_1,M_2$ is the equivalence class of a cobordism $\sigma$ with $\partial\sigma = M_1 \sqcup M_2$. The partition function only gives a well-defined value $Z(\sigma)$ if $Z(\sigma) = Z(\sigma')$ for all other $\sigma'$ in this equivalence class, but this is essentially the definition of a topological quantum field theory, which QED is not.