Minkowski space

At the beginning of the twentieth century, Albert Einstein replaced the aether theory by relativity, but a twenty-first century aether is still puzzling physicists today. This modern aether is the quantum vacuum. The aether was thought to be an all-penetrating mysterious substance that carries light through space like air carries sound. Take away all light, and the aether would still be there, defining a universal frame of reference. Now, according to quantum field theory, the state of absolute darkness, the vacuum state, is still a physical state filling space throughout, similar to the aether. There is an important difference though: one does not notice motion at uniform speed relative to the quantum vacuum, but, as we describe in this chapter, during acceleration the vacuum glows, although slightly, causing friction. Furthermore, as Stephen Hawking predicted in 1974 (Hawking, 1974), the quantum vacuum should also cause black holes to evaporate, because at the event horizon particles are created from nothing, at the expense of the black hole's mass. In this chapter we also describe this creation of radiation at horizons. None of these fascinating phenomema have been observed in astrophysics yet, but they can be demonstrated in laboratory analogues (Philbin et al., 2008a).

The quantum vacuum may also account for the dark energy that, according to astronomical data, constitutes the lion's share of the energy of the Universe (Hogan, 2007).

Wigner representation

In classical optics (Born and Wolf, 1999), the state of the electromagnetic oscillator is perfectly described by the statistics of the classical amplitude α. The amplitude may be completely fixed (then the field is coherent), or αmay fluctuate (then the field is partially coherent or incoherent). In classical optics as well as in classical mechanics, we can characterize the statistics of the complex amplitude?or, equivalently, the statistics of the components position qand momentum pintroducing a phase space distribution W(q, p). (As explained in Section 3.1, the real and the imaginary part of the complex amplitude αcan be regarded as the position and the momentum of the electromagnetic oscillator.) The distribution W(q, p) quantifies the probability of finding a particular pair of qand pvalues in their simultaneous measurement. Knowing the phase space probability distribution, all statistical quantities of the electromagnetic oscillator can be predicted by calculation. In this sense the phase space distribution describes the state in classical physics.

All this is much more subtle in quantum mechanics. First of all, Heisenberg's uncertainty principle prevents us from observing position momentum simultaneously andprecisely. So it seems there is no point in thinking about quantum phase space. But wait! In quantum mechanics we cannot directly observe quantum states either. Nevertheless, we are legitimately entitled to use the concept of states as if they were existing entities (whatever they are). We use their properties to predict the statistics of observations.