If I look at ordinary nonrelativistic quantum mechanics, then the usual way of dealing with it is either the schroedinger, or the Heisenberg picture. To point out similarities first:

Observables $O$ are represented by a linear Operator acting uppon some element $| \Psi \rangle$ of a Hilbert space, and calculating matrix elemenents (this incorporates the propability-Interpretation of QM). In QM, the dynamics (the notion that Observables, like for example the position of a particle) can change over time, and the way to incorporate this is either to make the Operatoes $\hat{o}(t)$ time dependend, or to make the states $| \Psi(t) \rangle$ time dependent (or some strange mixture), which would be either Heisenberg or Schroedinger picture.

Now in QFT, we want to have changing observables in every point in space and in every point in time, so the analogy for QM in the Heisenberg picture would be to make either the $\hat{o}(\vec{x}, t)$ also spatially dependent (which is indeed referred to as Heisenberg picture). However, an analogy to the Schrödinger picture would be to have time and space independent Operators $\hat{o}$, and timely and spatially dependent states $| \Psi(\vec{x},t) \rangle$.

To put my question to the end:Does a Formulation of Quantum Field Theories exist with the Operatores being timely and spatially independent, and the states being carrying the whole dependence on time and space?


1 Answer 1


I assume below that we are considering pretty standard QFT with usual locality properties.

Let's look first at how the transition from the Heisenberg picture to the Schrödinger one works. We have time dependent operators and the evolution operator that connects them at different times, \begin{equation} \hat{O}(t,\mathbf{x})=e^{i\hat{H}t}\hat{O}(0,\mathbf{x})e^{-i\hat{H}t} \end{equation} Then we introduce time-dependent state for which we then can use operators at $t=0$, \begin{equation} |\psi(t)\rangle\equiv e^{i\hat{H}t}|\psi\rangle,\quad \langle\psi(t)|\hat{O}(0,\mathbf{x})|\psi(t)\rangle=\langle\psi|\hat{O}(t,\mathbf{x})|\psi\rangle \end{equation} This transition is useful because the Hamiltonian can be written as the function of the simultaneous field $\phi(t,\mathbf{x})$ and canonical momenta $\pi(t,\mathbf{x})$ operators where $t$ is arbitrary and can be set to zero. That gives $|\psi(t)\rangle$ very nice interpretation of the state defined on the timeslice like e.g. some functional $\Psi_t[\phi(\mathbf{x})]$.

Now naively in the translationally invariant QFT you have very similar relation, \begin{equation} \hat{O}(t,\mathbf{x})=e^{i\hat{H}t-\hat{\mathbf{P}}\mathbf{x}}\hat{O}(0,\mathbf{0})e^{-i\hat{H}t+\hat{\mathbf{P}}\mathbf{x}} \end{equation} where $\hat{\mathbf{P}}$ is the operator of total momentum/spatial translations, $[\hat{\mathbf{P}},\hat{H}]=0$. You may think of introducing $|\psi(t,\mathbf{x})\rangle\equiv e^{i\hat{H}t-\hat{\mathbf{P}}\mathbf{x}}|\psi\rangle$.

The difference is that both $\hat{\mathbf{P}}$ and $\hat{H}$ are integrals of some functions of the canonical operators over the whole space! That ruins all attempts to give $|\psi(t,\mathbf{x})\rangle$ any interpretation that would involve only this point in spacetime. Both temporal evolution and shift in space would involve concepts from the whole timeslice not just single point.

The reason of course is that you can't determine the evolution of the field by just fixing it at some point. You need to fix it on the whole timeslice


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