Some of the postulates made in deriving quantum mechanics are assumptions about the mathematical procedures to be used, and not statements about what is "really" going on. For example, the quantum wave function (a way to represent the state vector) is defined as a complex number whose squared magnitude gives the probability per unit volume for finding a particle in an infinitesimal volume centered at a particular point in space. Is this a law of physics? I would not think so, unless the wave function is itself an element of reality, which is by no means an accepted fact. Rather, this is the definition of a mathematical quantity in the model that we will use to calculate probabilities.[1] Just as we introduce space and time to describe motion, we introduce the wave function to describe a quantum state. Both are ingredients in the models that we use to describe observations. The test, as always, is whether they do so or not.

Still, it would be nice to have some reason beyond simply that "it works" to hypothesize the connection between the wave function and probability. The interference effects that mark the difference between quantum and classical particle physics arise in the model from that source. Indeed, since classical waves exhibit interference effects, the common explanation used in elementary discussions is that a "duality" exists in which particles are sometime waves and waves sometimes particles.

However, this duality is not present in the basic model, nor
does it properly describe observations. Even in an experiment such as
double-slit interference,
whether done with light or electrons, sufficiently sensitive

detectors will pick up individual, localized particles. In the case of light, which we normally think of as waves, the particles are photons. What is observed is a statistical distribution, or intensity pattern, of particles that can be mathematically described in terms of waves. It is hard to see what it is that is doing the waving.

So, the wave function is used to calculate probabilities. As
I have noted, at a more advanced level, the quantum state is
represented by a state vector in an abstract vector space. This
vector space is assumed to be "linear" in the same way that our
familiar three-dimensional space is linear. In the latter case, we can always
write a vector as the linear combination of three vectors, called *eigenvectors, *each pointing along a
particular coordinate axis. The same is assumed for state vectors, with the
generalization applying to any number of dimensions.

Now, we have seen that gauge invariance requires that the state vector be independent of the coordinate system. Thus it can be written as the superposition of any set of eigenvectors, each set corresponding to a different coordinate system. This can be expressed as the following principle:

*Superposition
principle*: The state vector is a linear combination of eigenvectors
independent of the coordinate system defining the eigenvectors.

The superposition principle is what gives us quantum interference and the so-called entanglement of quantum states. Each eigenvector represents a possible state of the system, and so quantum states are themselves superpositions, or coherent mixtures, of other states.

This would not be the case for nonlinear vectors. So, once again we find that a basic principle of physics, the superposition principle, is needed to guarantee point-of-view invariance, as is the linearity of the vector space.

While the mathematical formalism of quantum mechanics has
proved immensely successful in making predictions to compare with experiment,
no consensus has yet been achieved on what quantum mechanics "really
means," that is, what it implies about the nature of reality. Numerous
philosophical interpretations can be found in the literature, all of which are
consistent with the data; any that were inconsistent would not be published. No
test has been found for distinguishing between these interpretations (see my *The Unconscious Quantum* for further
discussion).[2]