Locality in Quantum Field Theory and Hadamard point splitting

Given the title of my blog, the first post should be on locality. In Quantum Field Theory (QFT) there are two different notions of locality. The first one, also often called causality, is the requirement that measurements of observables in causally disconnected (space-like) regions do not influence each other. This is expressed in the requirement that the corresponding operators commute, i.e., [\mathcal{O}(x), \mathcal{O}(y)] = 0 whenever x and y are space-like.

The second notion of locality, the one that I am interested in this post, is relevant in the context of QFT on curved space-times or in the presence of external (background) fields. It can be roughly stated as follows: For the definition of an observable \mathcal{O}(x) localized at x, only the local geometric data at x should be relevant. A precise definition in the context of QFT on curved space-times was provided by Hollands and Wald (2001) and Brunetti, Fredenhagen, and Verch (2003), and it can be straightforwardly generalized to gauge background fields.

Now even in the context of free fields, most relevant local observables are quadratic in the fields, i.e., non-linear. Examples are the stress-energy tensor,

\qquad T_{\mu \nu}(x) = \partial_\mu \phi(x) \partial_\nu \phi(x) - \frac{1}{2} g_{\mu \nu} \partial^\lambda \phi(x) \partial_\lambda \phi(x)

for a massless, minimally coupled scalar field, and the current,

\qquad j^\mu(x) = \bar \psi(x) \gamma^\mu \psi(x)

for a Dirac field. Both are relevant for the study of back-reaction effects, the former in the context of QFT on curved space-times, the latter for QFT in the presence of external electromagnetic fields.

In a QFT, the fields are singular, mathematically they have to be treated as distributions. In particular, one can not simply multiply them point-wise. This is easily seen from the fact that the two-point function \omega_\Phi(x, y) = \langle \Phi \rvert \phi(x) \phi(y) \lvert \Phi \rangle diverges in the limit x \to y of coinciding points. So how should one evaluate the above observables in the state \Phi?

The answer one finds in typical QFT textbooks is that, before taking the limit of coinciding points, one should subtract the vacuum two-point function \omega_\Omega(x, y) = \langle \Omega \rvert \phi(x) \phi(y) \lvert \Omega \rangle, i.e., for the definition of the expectation value of \phi^2(x) one should define

\qquad \langle \Phi \rvert \phi^2(x) \lvert \Phi \rangle = \lim_{y \to x} \left( \omega_{\Phi}(x,y) - \omega_\Omega(x,y) \right).

Analogously, one proceeds with expressions containing derivatives, such as the stress-energy tensor. This procedure is called point-splitting.

The prescription is equivalent to Wick normal ordering and perfectly sensible on Minkowski space (in the absence of any other background field). However, the difficulties begin already when one restricts to sub-regions of Minkowski space, as in the Casimir effect. There, one restricts the fields to space-times M_\Sigma = \mathbb{R} \times \Sigma with boundary \mathbb{R} \times \partial \Sigma on which one imposes, say, Dirichlet boundary conditions \phi|_{\partial \Sigma} = 0. On this space-time, there is again a vacuum state \Omega for the quantum field. However, this vacuum state should \emph{not} be used for point-splitting. This would lead to a vanishing of the energy density in the ground state and thus to the absence of the Casimir effect. The correct result is obtained by using the Minkowski space vacuum two-point function for the point-splitting.

Now this sounds paradoxical: In order to define a local observable in M_\Sigma, one has to use the two-point function of Minkowski space. How does the field in M_\Sigma know that M_\Sigma is embedded in Minkowski space? This seems to grossly violate the locality requirement that only the local data at x should enter the definition of the observable \phi^2(x).

The situation becomes even more confusing when one tries to define the stress-energy tensor or the current on generic curved space-times or in the presence of generic external potentials. In such a situation there is no preferred vacuum state, as there is no time-translation symmetry that can be used to single out positive energy modes. So how should one then define the above local observables?

In the context of QFT on curved space-times, it was realized in the 1970’s, that physically reasonable states \Phi have two-point functions of so-called Hadamard form, i.e., they can be locally, for x near y, written as \omega_\Phi(x,y) = H(x,y) + R_\Phi(x,y), where the Hadamard parametrix H contains all the singularities and is determined solely by the local geometric data, independently of the state \Phi. The idea is thus to point split with H, i.e., to define \langle \Phi \rvert \phi^2(x) \lvert \Phi \rangle = R_\Phi(x, x). Building on work of Hadamard (1923) on fundamental solutions to the wave equation and later work of DeWitt and Brehme (1960) and DeWitt (1975), this was worked out by Christensen (1976), Adler, Liebermann and Ng (1977) and Wald (1977). This procedure is called Hadamard point-splitting.

The above paradox concerning the Casimir effect is now easily resolved by noting that on flat space-time, the Hadamard parametrix (essentially) coincides with the Minkowski vacuum two-point function. A complete understanding of Hadamard point-splitting was achieved with the work of Hollands and Wald, which covers all local field polynomials (not just the stress-energy tensor) and also classifies the (renormalization) ambiguities inherent in the construction.

In the context of Quantum Electrodynamics (QED) in external potentials, the definition of the current j^\mu is usually achieved by Schwinger’s prescription: By exploiting the (anti-) symmetry of the two-point function under exchange of x and y, one can remove some of the singularities. However, some divergences remain if the current \partial_\nu F^{\mu \nu} of the external field does not vanish, and even if it does, the result in general depends on the direction from which the limit y \to x is taken. Even worse, in textbooks on the subject, a crucial parallel transport in Schwinger’s expression is omitted, leading to further divergences and gauge dependences.

Motivated by this unsatisfactory state of affairs, Jan Schlemmer and myself reconsidered QED in external potentials in the light of the lessons learned from QFT on curved space-times. Not surprisingly, the current can indeed be satisfactorily defined via Hadamard point-splitting, and it can even be used to perform concrete calculations. However, there was one surprising aspect. When trying to figure out the historical development, we found an article of Dirac (1934), in which he states:

A distribution [two-point function] R such as occurs in nature according to the above assumption can be divided naturally into two parts R = R_a + R_b,
where R_a contains all the singularities and is also completely fixed for any given [background] field, so that any alteration one can make in the distribution of electrons and positrons will correspond to an alteration in R_b but none in R_a.

[…]

It therefore appears reasonable to make the assumption that the electric and current densities corresponding to R_b are those which are physically present, arising from the distribution of electrons and positrons. In this way we can remove the infinities […].

This is, I think, an admirably clear description of the Hadamard point-splitting procedure. This conceptual breakthrough seems to have been forgotten quite rapidly, probably due to the difficulty of doing actual calculations in position space. The method was still used by Uehling (1935), Serber (1935) and partly also by Heisenberg and Euler (1936) in their seminal works on vacuum polarization. Nowadays, one finds historical reviews on QED in external potentials that do not even cite Dirac’s paper. The re-discoverers of Hadamard point-splitting in the 1970’s also seem to have been unaware of Dirac’s work.

It appears that important insights even of eminent researchers can be forgotten quite quickly.

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