Why can't we see virtual photons? If photons are the carriers for the electromagnetic force, then why can we not see electromagnetic fields, given that photons are involved in this interaction?
 A: Virtual photons and generally virtual particles are mathematical constructs coming from the iconal representation of scattering amplitudes by Feynman diagrams. To make the point clear here is a virtual particle with an enormous mass with respect to the incoming and outgoing real particles. A Feynman diagram is a prescription for writing the mathematical formulae that have to be integrated over in order to have predictive numbers.


A free neutron will decay by emitting a W-, which produces an electron and an antineutrino.

The W- has a mass of order 80GeV. The neutron has a mass of order 1 GeV. Since energy conservation is a fundamental law, it is evident that this so called W is not a real particle  but represents a mathematical function, a propagator, that has to be integrated over when one wants to turn the Feynman diagram into the calculation of the lifetime.
Why is it labeled W-? because it carries  all the quantum numbers of W and the mass in the propagator is the mass of the W. It cannot be seen or measured and it is there for mathematical consistency and conservation of quantum numbers.
In a diagram where a virtual photon is involved, one also cannot see it because it is a mathematical functions to be integrated over, not a real free particle.

If photons are the carriers for the electromagnetic force, then why can we not see electromagnetic fields, given that photons are involved in this interaction?

Because photons when functioning as carriers of the electromagnetic force are virtual, are mathematical constructs, (like the W above which is the carrier of the weak force). They only exist in a mathematical formula which has to be integrated in order to get real numbers predicting a crossection for a scattering event,  or a lifetime for a decay.
A: If you had ask a slightly different question "Why can't we interact directly with virtual photons" I could say that, in the formulation of Feynman series as scattering off-shell segments of virtual photons and virtual electrically charged matter-antimatter pairs, any virtual particle is by definition not part of the asymptotic states
What is an asymptotic state? When we study a Feynman propagator, we usually think of them to behave as a numerical machine that takes some input physical state at time $t$, and return some output physical state at time $t+T$. In the case of a scattering interaction (the canonical case of study in QED), both input and output states are assumed to be states in the irreducible representations of the Poincare algebra. Usually, these are described as asymptotic states
In other words, if a 'virtual photon' interacted with the detector, we wouldn't call it virtual to begin with, but we still would account it as a real photon in the perturbation expansion
A: Hawking radiation starts with the creation of a pair of virtual photons, I believe. One is absorbed by the Black Hole and the other could be observed if it were not so faint.
My understanding is that in the general case a virtual pair re-combines (annihilates) within a time on the order of a Planck time and cannot be observed.
