Editing Wave function (section)

===Wave functions and wave equations in modern theories===
All these wave equations are of enduring importance. The Schrödinger equation and the Pauli equation are under many circumstances excellent approximations of the relativistic variants. They are considerably easier to solve in practical problems than the relativistic counterparts.

The [[Klein–Gordon equation]] and the [[Dirac equation]], while being relativistic, do not represent full reconciliation of quantum mechanics and special relativity. The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called [[relativistic quantum mechanics]], while very successful, has its limitations (see e.g. [[Lamb shift]]) and conceptual problems (see e.g. [[Dirac sea]]).

Relativity makes it inevitable that the number of particles in a system is not constant. For full reconciliation, [[quantum field theory]] is needed.<ref>{{harvtxt|Weinberg|2002}} takes the standpoint that quantum field theory appears the way it does because it is the ''only'' way to reconcile quantum mechanics with special relativity.</ref>
In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called ''field operators'' (or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that the original relativistic wave equations and their solutions are still needed to build the Hilbert space. Moreover, the ''free fields operators'', i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases.

Thus the Klein–Gordon equation (spin {{math|0}}) and the Dirac equation (spin {{math|{{frac|1|2}}}}) in this guise remain in the theory. Higher spin analogues include the [[Proca equation]] (spin {{math|1}}), [[Rarita–Schwinger equation]] (spin {{math|{{frac|3|2}}}}), and, more generally, the [[Bargmann–Wigner equations]]. For ''massless'' free fields two examples are the free field [[Maxwell equation]] (spin {{math|1}}) and the free field [[Einstein equation]] (spin {{math|2}}) for the field operators.<ref>{{harvtxt|Weinberg|2002}} See especially chapter 5, where some of these results are derived.</ref>
All of them are essentially a direct consequence of the requirement of [[Lorentz invariance]]. Their solutions must transform under [[Lorentz transformation]] in a prescribed way, i.e. under a particular [[Representation theory of the Lorentz group#Common representations|representation of the Lorentz group]] and that together with few other reasonable demands, e.g. the [[cluster decomposition|cluster decomposition property]],<ref>{{harvnb|Weinberg|2002}} Chapter 4.</ref>
with implications for [[causality]] is enough to fix the equations.

This applies to free field equations; interactions are not included. If a Lagrangian density (including interactions) is available, then the Lagrangian formalism will yield an equation of motion at the classical level. This equation may be very complex and not amenable to solution. Any solution would refer to a ''fixed'' number of particles and would not account for the term "interaction" as referred to in these theories, which involves the creation and annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory.

In [[string theory]], the situation remains analogous. For instance, a wave function in momentum space has the role of Fourier expansion coefficient in a general state of a particle (string) with momentum that is not sharply defined.{{sfn|Zwiebach|2009}}