The first thing taught anywhere, in any class of the world about probabilities is that "probability is always positive." Never question this. Only this time at NIBM I felt like, "but why?" Why not negative? And then I questioned myself as to what could it mean to say something had a negative chance of happening? I believe that the possible explanation that struck me was an indication that I'd begun to think like a banker. It was the idea of negative money. Bankers are rich because of development of this very idea. Most of this money is not real, it exists only on paper. It may happen to probabilities as well.
Further research was gratifying. There were others who had thought along the same lines. They were physicists. Problems of physics brought physicists to the necessity to use not only classical probability but also negative probability. Negative probabilities emerged in physics in 1930s when Dirac(1930) and Heisenberg(1931) introduced probability distributions with negative values within the context of quantum theory. However, both physicists missed its significance and possibility to take negative values, using this distribution as an approximation to the full quantum description of a system such as the atom. Wigner(1932) came to the conclusion that quantum corrections often lead to negative probabilities while he was supplanting the wave function from Schrodinger's equation with a probability distribution in phase space. To do this, he introduced a function, which looked like a conventional probability distribution and has later been better known as the Wigner quasi-probability distribution because in contrast to conventional probability distributions, it took negative values, which could not be eliminated or made non-negative. Dirac(1942) not only supported Wigner’s approach but also introduced the physical concept of negative energy.
Further research was gratifying. There were others who had thought along the same lines. They were physicists. Problems of physics brought physicists to the necessity to use not only classical probability but also negative probability. Negative probabilities emerged in physics in 1930s when Dirac(1930) and Heisenberg(1931) introduced probability distributions with negative values within the context of quantum theory. However, both physicists missed its significance and possibility to take negative values, using this distribution as an approximation to the full quantum description of a system such as the atom. Wigner(1932) came to the conclusion that quantum corrections often lead to negative probabilities while he was supplanting the wave function from Schrodinger's equation with a probability distribution in phase space. To do this, he introduced a function, which looked like a conventional probability distribution and has later been better known as the Wigner quasi-probability distribution because in contrast to conventional probability distributions, it took negative values, which could not be eliminated or made non-negative. Dirac(1942) not only supported Wigner’s approach but also introduced the physical concept of negative energy.
He wrote:
“Negative energies and probabilities should not be considered as nonsense.
They are well-defined concepts mathematically, like a negative of money."
Richard Feynman emphasised that even if the final answer of a calculation must be positive, negative numbers are often allowed to appear in the intermediate steps and this can happen with probabilities.
Interpretations to Negative Probabilities
