What is the fundamental probabilistic interpretation of Quantum Fields?

In quantum mechanics, particles are described by wave functions, which describe probability amplitudes. In quantum field theory, particles are described by excitations of quantum fields. What is the analog of the quantum mechanical wave function? Is it a spectrum of field configurations (in analogy with QM wave functions' spectrum of particle observables), where each field configuration can be associated with a probability amplitude? Or is the field just essentially a superposition of infinitely many wave functions for each point along the field (as if you quantized a continuous mattress of infinitesimal coupled particles)?

Answer

(Lubos just posted an answer, but I think this is sufficiently orthogonal to post too). The usual wavefunction for a bosonic field is a complex number for each field configuration at all points of space:

$$\Psi(\phi(x))$$

This wavefuntion(al) obeys the Schrodinger equation with the field Hamiltonian, where the field momentum is a variational derivative operator acting on $\Psi$. This formulation is fine in principle, but it is not useful to work with this object directly under usual circumstances for the following reasons:

• You need to regulate the field theory for this wavefunctional to make mathematical sense. If you try to set up the theory in the continuum right from the beginning, to specify a wavefunction over each field configuration you need to work just as hard as to do a rigorous definition for the field theory. For example, just to normalize the wavefunction over all constant time slice field values, you need to do a path integral over all the constant time field configurations. This is a path integral in one dimension less, but the thing you are integrating is no longer a local action, so there is no gain in simplicity. Even after you normalize, the expectation value of operators in the wavefunctional is a field theory problem in itself, in one dimension less, but with a nonlocal action.


• Once you regulate on a lattice, the field wavefunctional is just an ordinary wavefunction of all the field values at all positions. But even when you put it on an infinite volume lattice, a typical wavefunctional in infinite volume will have a divergent energy, because you will have a certain energy density at each point when the wavefuntional is not the vacuum, a finite energy density. Infinite energy configurations of the field theory, those with a finite energy density, are very complicated, because they do not decompose into free particles at asymptotic times, but keep knocking around forever.


• The actual equations of motion for the wavefuntional are not particularly illuminating, and do not have the manifest Lorentz symmetry, because you chose a time-slice to define the wavefunction relative to.

These problems are overcome by working with the path integral. In the path integral, if you are adamant that you want the wavefuntion, you can get it by doing the path integral imposing a boundary condition on the fields at a certain time. But a path integral Monte-Carlo simulation, or even with just a little bit of Wick rotation, will make the wavefunction settle to be the vacuum, and insertions will generally only perturb to finite energy configurations, so you get the things you care about for scattering problems.

Still the wavefunction of fields is used in a few places for special purposes, although, with one very notable exception, the papers tend to be on the obscure side. There are 1980s papers which attempted to find the string formulation of gauge theories which tried to work with the field Hamiltonian in the Schordinger representation, and these were by famous authors, but the name escapes me (somebody will know, maybe Lubos knows immediately).

The best example of where this approach bears fruit is when the reduction in dimension gives a field theory which has a relationship with known solvable models. This is the example of the 2+1 gauge vacuum, which was analyzed in the Schrodinger representation by Nair and collaborators in the past decade.

A recent paper which reviews and extends the results is here: http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.6376v1.pdf . This is, by far, the most significant use of Schrodinger wavefunctions in field theory to date.