The Role of Classical Variables

The traditional way to construct a quantum theory is to posit some classical con­figuration space (such as the space of all possible positions of a set of particles). A quantum state is then a wave function, which assigns a complex number to every possible configuration, such that (ultimately) the square of that number will give the probability of observing the system in that configuration.

This gives us a representation of H, but the Hilbert space itself is simply a vector space with a norm (a way of taking the dot product between two vectors). That gives us very little structure to work with: all Hilbert spaces of the same finite dimensionality are isomorphic, as are infinite-dimensional ones that are separable (possessing a countable dense subset, which implies a countable orthonormal basis). We may therefore ask, once H is constructed, is there any remnant of the original classical configuration space left in the theory?

The answer is “not fundamentally, no.” A given representation might be useful for purposes of intuition or calculational convenience, but it is not necessary for the fundamental definition of the theory. Representations are very far from unique, even if we limit our attention to representations corresponding to sensible physical theories.

One lesson of dualities in quantum field theories is that a single quantum theory can be thought of as describing completely different classical variables. The fun­damental nature of the “stuff” being described by a theory can change under such dualities, as in that between the sine-Gordon boson in 1 + 1 dimensions the theory of a massive Thirring fermion [2]. Even the dimensionality of space can change, as is well- appreciated in the context of the AdS/CFT correspondence, where a single quantum theory can be interpreted as either a conformal field theory in a fixed d-dimensional Minkowski background or a gravitational theory in a dynamical (d + 1 )-dimensional spacetime with asymptotically anti-de Sitter boundary conditions [3].

The lesson we draw from this is that Nature at its most fundamental is simply described by a vector in Hilbert space. Classical concepts must emerge from this structure in an appropriate limit. The problem is that Hilbert space is relatively featureless; given that Hilbert spaces of fixed finite or countable dimension D are all isomorphic, it is a challenge to see precisely how a rich classical world is supposed to emerge.

Ultimately, all we have to work with is the Hamiltonian and the specific vector describing the universe. In the absence of any preferred basis, the Hamiltonian is fixed by its spectrum, the list of energy eigenvalues:

and the state is specified by its components in the energy eigenbasis,

The question becomes, how do we go from such austere lists of numbers to the fullness of the world around us?

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