Is there any theorem that suggests that QM+SR has to be an operator theory?

Question

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UPDATE

To make my question more precise, I'll define what I mean by an operator theory:

• An operator theory is a theory in which the dynamical objects are operators, i.e., the equations of motion are imposed on operators.

• A wave function theory, on the other hand, is a theory in which the dynamical objects are functions of space-time, i.e., the equations of motion are imposed on functions.

In wave-function theories we have differential operators (e.g., $-i\nabla$), but these aren't dynamical, so there is a clear distinction between an operator and a differential operator. To make the distinction more evident, I'll place hats $\hat{\square}$ over dynamical operators.

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At the first stages of the old (non-relativistic) quantum theory Heisenberg developed a theory in which the fundamental objects were operators, namely, the position and momentum operators: $$ \begin{align} i\frac{\mathrm d}{\mathrm dt} \hat X(t)&=[\hat X,\hat H] \\ i\frac{\mathrm d}{\mathrm dt} \hat P(t)&=[\hat P,\hat H] \end{align} \tag{QM.1} $$

More or less one year later Schrödinger published a new theory in which the fundamental object is just a function of space-time: $$ -i\partial_t\psi(x)=-\frac{1}{2m}\Delta\psi(x)+V(x)\psi(x) \tag{QM.2} $$

Later on, Schrödinger proved that the two formulations are actually equivalent.

Nowadays we have QFT, which is an operator theory, because the equations of motion are imposed on fields, i.e., operators. For example, a scalar field $\hat\phi$ evolves through $$ \begin{align} i\frac{\mathrm d}{\mathrm dt}\hat\phi(x)=[\hat\phi,\hat H]\\ i\frac{\mathrm d}{\mathrm dt}\hat\pi(x)=[\hat\pi,\hat H] \end{align} \tag{QFT.1} $$ where $\hat \pi$ is the field conjugate to $\hat\phi$.

To me, it seems more or less natural to ask about a possible $(\mathrm{QFT.2})$, i.e., a formulation of QFT as a wave-function theory: $$ \begin{array} {}&\text{non-relativistic} & \text{relativistic}\\ \text{operator theory} & (\mathrm{QM.1}) & (\mathrm{QFT.1}) \\ \text{wave-function theory} & (\mathrm{QM.2}) & \quad ?? \end{array} $$

IMHO functional methods (i.e., path integrals) belong to an operator point of view. I believe it is not possible to use path integrals to calculate, for example, scattering amplitudes without using operators sooner or later, which means that path integrals don't actually fit in the $??$ slot.

My question(s):

• Has someone done anything similar to Schrödinger, in the sense of reformulating QFT using solely functions of space-time and differential operators? Is there any theory of relativistic quantum mechanics in which the formalisms consists of PDE's?

• If the answer to the first question is that no wave-function theory exists as for today, then is there any reason to expect that there will never be? I mean: is there a no-go theorem or an argument that suggests that it is impossible to formulate QFT as a wave-function theory?

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My thoughts on this

Any relativistic theory of interactions must be able to describe creation/annihilation phenomena, which operators easily do (through $a,a^\dagger$). But a single wave-function must have a fixed number of space-time arguments, so the number of particles must be fixed. This means that one wave-function will not suffice.

A wave-function theory of interactions must, therefore, consist in an infinite number of wave-functions, each with a different number of space-time arguments: $$ \begin{align} &\psi(x_1)\\ &\psi(x_1,x_2)\\ &\psi(x_1,x_2,x_3)\\ &\cdots \end{align} $$ which means that we must have an infinite number of PDE's, one for each $\psi$. And the solutions must be consistent with the operator formulation of QFT, i.e., we must prove that the new formulation is equivalent to the old one.

I believe that the easier way to archive this is to use the correlation functions of QFT (or $n$-point functions) $$ \begin{align} \psi(x_1)&=\langle\hat\phi(x_1)\rangle\\ \psi(x_1,x_2)&=\langle\hat\phi(x_1)\hat\phi(x_2)\rangle\\ \psi(x_1,x_2,x_3)&=\langle\hat\phi(x_1)\hat\phi(x_2)\hat\phi(x_3)\rangle\\ &\cdots \end{align} $$

With this, it should be possible in principle to find the PDE's for the $\psi$'s in terms of the PDE's for $\hat\phi$. This means, we should use $(\partial^2+m^2)\hat\phi=\hat j$ to get $$ \begin{align} \mathcal O_1\ \psi(x_1)&=\mathcal J(x_1)\\ \mathcal O_2\ \psi(x_1,x_2)&=\mathcal J(x_1,x_2)\\ \mathcal O_3\ \psi(x_1,x_2,x_3)&=\mathcal J(x_1,x_2,x_3)\\ &\cdots \end{align} $$ for some differential operators $\mathcal O_i$ and some sources $\mathcal J$. Once we find the appropriate $\mathcal O_i,\mathcal J$, there will be no explicit reference to any operator, and we only use wave-functions (of course, these wave-functions don't have the same probabilistic interpretation of non-relativistic QM, but this is irrelevant: the point is to find an equivalent formulation of QFT using just functions of spacetime).

I don't know if this makes much sense, or whether there is a better approach. Any comment will be appreciated.

Answer 1

What you call an operator theory is usually called the Heisenberg picture of quantum mechanics. What you call a wave function theory is usually called the Schrödinger picture of quantum mechanics.

It is well-known that for every quantum mechanical model, the Heisenberg picture and the Schrödinger picture are fully equivalent through a dual description, as long as one doesn't consider time correlations.

This therefore also holds for quantum field theories, which are particular cases of quantum theories. In the Schrödinger picture of quantum field theory the ordinary Schrödinger equation takes the form of a functional Schrödinger equation. Here states of a QFT are treated as functionals of the field coordinates in the same way as states in QM are treated as functions of the position coordinates. A thorough discussion of the functional Schrödinger picture is in the article by Jackiw, Analysis on infinite dimensional manifolds: Schrodinger representation for quantized fields (p.78-143 of the linked document).

On the other hand, the Heisenberg picture is more general than the Schrödinger picture as it allows the discussion of time correlations of observable quantities. This is important in statistical mechanics, and essential for the study of quantum field theory at finite time, through the close time path (CTP) formalism. For the latter, see, e.g., Introduction to the nonequilibrium functional renormalization group by Berges.

As mentioned in the comments, the case of 4D relativistic QFT is somewhat peculiar since there are still unresolved foundational problems related to nonperturbative renormalization. As Dirac says, the ''operators'' in the Heisenberg picture don't act anymore on Fock space, so there is no Fock space Schrödinger picture. However, and Dirac does not say this, the Heisenberg picture gives (presumably, proved only in lower dimensions) a valid operator description in a different Hilbert space, and using the time generator of the corresponding unitary representation of the Poincare group, one gets a corresponding Schrödinger picture on this renormalized Hilbert space.

Answer 2

There is indeed a theory of differential operators applied on wavefunctions (or more accurately wavefunctionals), the Schrodinger functional formalism, though it is not used a lot in most applications. It is defined by a wavefunctional at a time t

\begin{equation} \Psi[\phi(\vec x), t] \end{equation}

Upon which act differential operators, defined by

\begin{eqnarray} \hat\phi \Psi[\phi(\vec x), t] &=& \phi \Psi[\phi(\vec x), t]\\ \hat\pi \Psi[\phi(\vec x), t] &=& -i\hbar\frac{\delta}{\delta\phi(x)}\Psi[\phi(\vec x), t] \end{eqnarray}