Formulations of Quantum Mechanics

I said that quantum mechanics is a set of rules for predicting experiments, but even that needs to be qualified, because many different sets of rules work equally well. A variety of formulations have been developed for quantum mechanics. They are all mathematically equivalent, in that they make identical predictions for any experiment you can do. But they begin with very different postulates, and they are formulated in terms of very different mathematical objects.

For example, you have probably learned that the wavefunction plays a central role in quantum mechanics, but strictly speaking that is incorrect. It is not central to quantum mechanics, only to the Schrödinger formulation of quantum mechanics. On the other hand, the path integral formulation does not postulate anything resembling a wavefunction. Instead, it has a rule for computing transition probabilities by summing over all possible classical trajectories. Of course, if you start from the postulates of the path integral formulation, you can define a quantity corresponding to the wavefunction and rederive the Schrödinger formulation. They are mathematically equivalent, after all. But is the wavefunction a fundamental quantity, or just an arbitrarily defined abstraction? That depends on which formulation you start from.

The phase space formulation replaces the wavefunction with a different quantity called the Wigner function. Unlike the wavefunction, which is defined over the configuration space of the particles, the Wigner function is defined over the full phase space, both positions and momenta. You have probably been told that in quantum mechanics, it “doesn’t make sense” to talk about a particle having a specific position and a specific momentum at the same time. That there is no well-defined way to even ask the question. But really, that is only true of the Schrödinger formulation, not of quantum mechanics in general. In the phase space formulation, it is entirely reasonable to ask for the probability that a particle has a specific position and a specific momentum, both at the same time, and it gives you a well defined answer.

The two-state vector formulation actually has two wavefunction-like objects, one that evolves forward in time and another that evolves backward in time. The role played by these objects is very different from the familiar wavefunction, however, because they do not represent probability distributions. The two-state vector formulation is entirely deterministic. Given the values of the two state vectors, the result of any measurement is uniquely determined.

Since many sets of postulates all lead to identical predictions, and some of them do not involve a wavefunction, we cannot assume the wavefunction directly corresponds to any physical object. Unfortunately, many people do assume that. Worse, they often portray it as a natural or even inevitable conclusion. In reality, it is pure speculation that is not in any way justified by experiment. Whenever you hear someone assume the wavefunction is a physical object, you should immediately become suspicious of their conclusions.