Do virtual particles emit radiation? When spontaneous electron / positron pairs annihilate, do they emit photons?
If so, does this not negate the first law of thermodynamics?
 A: The short answer is "no", the total mass-energy is zero so there's nothing to create when the meet again.
The longer answer is that it is vital to understand that virtual particles are not "real". It's just a name we give to lines on a diagram. Literally. We have those lines, and a name for them, because it makes certain aspects of QFT easier to understand.
For instance, it's easy to understand the interaction between two electrons by considering it to be mediated by photons - virtual photons. That helps us understand things like conservation of momentum or what have you. But that's not what's "really" happening, these interactions are just part of the physics of QFT.
There is great argument over this statement, and how to describe this entire concept, and you could do worse than to spend a few minutes reading some of the related questions here on SOP.
A: The quantum fields theory teaches us that the universe is a bit more complex than "real" vs. "unreal" or "exists" vs. "doesn't exist". The deeper knowledge we get, the more we challenge these very concepts. What exactly is "real"? What exactly does "exists" mean? And as we look closer and closer, none of it is what it seems. Is vacuum an empty space? Well, the answer is not exactly "yes" and it's not exactly "no". It's a bit, or rather a lot, more complex. Oh, and, by the way, the space itself may just be curved...
So let us try to make some intuitive sense of how "virtual" particles are different from "real" particles and along the way answer the OP's question on why exactly the annihilation of a virtual electron-positron pair does not result in the emission of photons.
Everyone knows now about the particle-wave dualism. Every particle has wave properties. This is why quantum mechanics is also known as wave mechanics. So what exactly is a wave? Well, oscillations, say, like a sine. A sine wave usually is a projection of some rotation. This is reflected in wave functions often having an exponent in the power of time with an imaginary unit that mathematically represents rotation in the complex plane. Closer to real life, we see the same idea in the fact that photons of the same spin are circularly polarized. So let's assume for a moment that a particle, while acting as a wave, essentially follows a "dance of rotation" and let's for the time being and the sake of simplicity just leave aside the specifics of what exactly rotates, whether it is its wave function or whatever.
Real particles move in time. What exactly does this mean? Two things:


*

*They have energy, because time and energy are two sides of the same nature (a Fourier pair or conjugates). Energy is what moves in time and time is what energy moves in. Energy cannot stay still in time, it must move, and only energy can move in time. Energy and time are two sides of the same coin, the "coin of existence". This fundamental symmetry of nature is most known by its famous consequence - the law of energy conservation: if time is uniform, energy stays the same.

*If the particle, while acting like a wave, "rotates", as  discussed above, and also moves forward in time, its "trajectory" looks like a spiral. Again, as mentioned above, this is not the actual trajectory of a particle, we are just trying to create an intuitive visualization that soon will reveal with clarity the difference between real and virtual particles. Moving on the "spiral", the particle makes a full rotation plus a step forward in time thus never closing the loops.
Now let's look at the energy of the photon. It is proportional to its frequency and therefore reverse proportional to the period of its oscillation or rotation. How about a virtual photon? The uncertainty principle tells us that it can exist for a period of time reverse proportional to its energy and therefore to its frequency. Put these two simple statements or equations together and you get a dramatic insight on the nature of virtual particles: the virtual photon exists exactly for one full period of its oscillation or rotation. So virtual particles do not move in time on a "spiral" like real particles. Instead, virtual particles complete the loop without moving forward in time.
So a photon (or any particle) is like a "rotating" wave. If it has energy, it moves forward in time, because this is what having energy means. And so it "moves" (in some sense) on a "spiral". And therefore it's "real". In contrast, if the photon does not have energy, it does not move forward in time, but stays in the same moment (whose exact length is defined by the uncertainty principle), because only energy can move in time. And so such a photon "moves" (in the same sense) in a circle and therefore is "virtual".
So here we are, "real" particles move in time and have energy while "virtual" particles "stay" in time and don't have energy (other than "borrowed" for a brief moment per the uncertainty principle). Otherwise they are the same particles with the same charge, spin, "color", interacting with the same class of particles, etc.
Now let's see if we can take a virtual particle and make it real. Can we? Yes, but only as long as we obey all applicable conservation laws. The easiest way to obey these laws is to create particles in pairs, like electron-positron. When we create both, we automatically obey the conservation of charge, lepton and electron quantum numbers, and can easily obey the momentum and angular momentum (spin) conservation this way. So what's left? Energy. Energy is the only thing that is not the opposite, but the same for a particle and anti-particle. For example, the charge of the electron is negative while positive for the positron, but energy is positive and the same for both. So, how do we change a virtual electron-positron pair to real? Simple, by giving them enough energy to come to existence and start moving in time. This is the essence of the production of the electron-positron pair by a high energy photon.
So, what happens when an electron and positron annihilate? Well, they disappear from existence. They were close to each other one moment and gone the next moment. This means that they stopped moving in time. Therefore this means that they became virtual. I've never heard this statement before, but it is true. When the electron and positron annihilate, they don't just disappear, they become virtual, stop moving in time, and disappear for us, because we keep moving in time while the pair remains in the past. And so it does disappear in this sense, but not without a dramatic consequence - the pair must release all its energy, because energy must keep moving forward in time. And so two photons are emitted (as one would recoil and thus violate the conservation of momentum).
To summarize, virtual electron-positron pairs exist in the moment of time (defined by the uncertainty principle), but don't move forward in time, because they don't have energy while only energy can move in time. Giving them enough energy via a photon allows them to start moving in time that we observe as a production of a real pair. When they "annihilate", they release this energy back as two photons, which keeps traveling forward in time. Having lost its energy, the pair stops moving in time and remains in the moment of its annihilation thus disappearing from existence.
If you've made it this far, the answer to the OP's question should become self evident. A virtual electron-positron pair does not have any energy to release as photons. In fact, it never really annihilates, it just stays in time while we are passing by and lose its sight in the mist of the past. When people say, "virtual particle pairs are constantly created and destroyed in a vacuum", this is not exactly correct. They are not created and destroyed. They simply "are", each in its own moment of time rotating in its own loop while we are passing by them in the fast train of time.
Finally I'd like to clarify that this explanation does not pretend to be "mathematically correct", so the specialists do not need to reply with impenetrable formulas to "disprove" everything I said. This explanation is only for the rest of us to get a glance at the types of things happening in the quantum world and hopefully realize that the concepts of time and energy, existence and non-existence, "real" and "virtual", or "matter" and "anti-matter" are not as simple as they once seemed, but still can be intuitively understood.
