1

I want to give a brief overview as to where I am coming from with this question.

When we go from classical mechanics to QM, one the crucial things we consider is that $\vec x$ and $\vec p$ are replaced with operators $\hat{\vec x}$ and $\hat{\vec p}$ and that the poisson brackets transform in the following way:

$$\{x_i,p_j\}=i\delta_{ij}\rightarrow [\hat x_i,\hat p_j]=i\hbar\delta_{ij}. (Eq.1)$$

When we go from QM to QFT the canonical commutation relations are:

$$[\phi(x),\pi(y)]_{x_0=y_0}=i\delta^{3}(\vec x - \vec y) (Eq.2)$$

$$[\phi(x),\phi(y)]_{x_0=y_0}=0 (Eq.3)$$

$$[\pi(x),\pi(y)]_{x_0=y_0}=0 (Eq.4)$$

From (2) one can say $\phi(x)$ plays the role of $x_i$ (or $\vec x$) and $\pi(x)$ that of $p_j$ (or $\vec p$).

But then we can consider the Heisenberg equations in QFT, such as:

$$\partial_\mu \phi(x)=[iP^\mu,\phi(x)] (Eq.5)$$

$$k^\mu a^\dagger(\vec k)=[P^\mu,a^\dagger(\vec k)] (Eq.6)$$

$$-k^\mu a^\dagger(\vec k)=[P^\mu,a(\vec k)] (Eq.7)$$

I have two questions:

  1. Is there any relation between the two sets of commutation relations? And can we make any claim about the framework/picture we are in, i.e Schroedinger picture or Heisenberg one?

  2. If $P^\mu$ the four momentum operator is present in QFT, why don't we have a 4 position operator as well $X^\mu$ ?

Qmechanic
  • 220,844
imbAF
  • 1,912
  • 13
  • 30

1 Answers1

3

Indeed, one can define a triple of selfadjoint operators, in the one-particle space, that satisfies the usual CCR of momentum and position.

A. Wightman proved that it is uniquely defined due to its covariance properties under the Euclidean group of the rest space of Minkowski reference frames. It is the well known Newton-Wigner position observable.

The point is whether or not it mathematically describes the physical notion of position associated to localized detectors.

There is an extensive literature accumulated over the years against this possibility (Hegerfeldt-like and Malament-like theorems). The nature of the obstructions concerns causality: the possibility of macroscopic superluminal propagation of information.

This long standing issue is still open and, in my view, it did not receive the necessary attention up to now as it would deserve a similar fundamental issue.

In any cases, if a notion of position can be defined, it is not possible in terms of selfadjoint operators (projector valued measures labeled by spatial sets), but more sophisticated notions are necessary (POVMs).

Valter Moretti
  • 80,330