Re: Energy I don't think energy can be negative in the sense that you mean. Binding energy is sort of negative in that it contributes negatively to the mass of a composite, i.e. the mass of an atom, for example, is less than the mass of its constituents.
Re: Energy -- Can energy be negative? It depends on your definition of energy. Potential energy can be negative, and often is. Any attractive force has a negative potential energy. If you take an electron and positron together, the total energy shrinks as you move them closer together, because the coulomb potential becomes more negative. A bound system will have the sum of potential energy and kinetic energy less than zero. However, the total energy, including the individual masses, is still positive. The total energy of any isolated system should be positive or else these systems could spontaneously appear anywhere and the vacuum would not be stable.
Re: Energy -- Can energy be negative? This is a complicated question. Typically we talk about energy as the ability to perform work. The amount of energy something contains depends on many different parameters of the system in question. For example, a container of hot liquid can be hooked up to a heat engine and the heat can be pumped to another reservoir and power the engine in the process, resulting in work. The amount of energy depends on the amount of liquid, the temperature, etc. There is also energy that is accessible by the annihilation of particles and antiparticles. In this case the energy is the sum of the mass of the two particles plus any energy they had in their velocities relative to each other. In the first example, once the two reservoirs of liquid reached equilibrium with each other and the engine, no more energy was available. The engine could perform no more work. Can we say that the energy is negative in any way? Sure, we can say that to get each reservoir back to their initial temperatures would require X amount of energy, and thus the system contains negative energy with respect to our end goal. Does this system have negative mass? Only if you compare the mass of the system before and after. If we measured the mass of the system we would ALWAYS measure a positive amount of mass. The 2nd example is similar. We can say that it would require energy to get 2 photons to interact and create an electron and positron. Again, we can say that a particular way the system can be set up would require an input of energy and thus it would have "negative" energy. But, like the 1st example any measurement of mass would always be positive.
Re: Energy -- Can energy be negative? A note on the equation E=MC^2. The equation is useless if you do not use it correctly. Even if the energy of a system is negative in some way it will NOT have negative mass, as you have neglected to include the actual mass of the system, which is always positive.
Re: Energy -- Can energy be negative? When you filter out all the nonsense, the question is basically an open one. It'd be very nifty and useful if energy can turn negative. There are some very interesting consequences to that in General Relativity. Traversible wormholes and warp drives are way easier with negative energy. And by "easier", I mean we don't know if these things are even possible without the negative energy. But there is no direct indication that such a thing is possible. In QM, only relative energy matters. It's really only in context of GR that negative energy makes sense, and even there simply having energy lower than zero-point might qualify. Short answer, nobody really knows, but it hasn't been observed.
Re: Energy -- Can energy be negative? There is an arbitrary setting of the zero of the energy scale. In some contexts, one can calculate a zero point energy of the vacuum. Efforts to actually calculate it in QFT usually give infinity, which is just meaningless. On the other hand, cosmology suggests that the vacuum has negative energy. Only differences in energy are meaningful, so it's probably most convenient to regard a vacuum as being at zero energy, and any thing that isn't vacuum has positive energy. The important energy quantity is the difference between an object's energy and the background vacuum. In the case of the Casimir effect, the background vacuum plays a real role.
Re: Energy -- Can energy be negative? If you consider the free space (with vacuum) as a state of zero energy, you can get volumes with negative energy (casimir effect). However, you need some material to get those areas, therefore the total energy is positive again.