Abstract
Now that we have reached the subject of non-local field theory, it is important that we discuss some peculiar features that distinguish it from local field theory. This chapter is based on, and extends, Jaccard et al. (Phys. Rev. D 88:4, 044033, 2013 [1]), Dirian and Mitsou, (JCAP 10:065, 2014 [2]), Foffa et al. (Phys. Lett. B 733:76, 2014 [3]).
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- 1.
More precisely, in perturbative QFT the propagator \({\sim } {\bigg (} k^2 + m^2 {\bigg )}^{-1}\) corresponds to a non-local operator \({\bigg (} \square - m^2 {\bigg )}^{-1}\) in real space, so the loop corrections will in general be non-local. For scales \(k^2 \ll m^2\) however one can expand
$$\begin{aligned} \frac{1}{k^2 + m^2} = \frac{1}{m^2} {\bigg (} 1 - \frac{k^2}{m^2} + \mathcal{O}(k^4) {\bigg )}, \end{aligned}$$(3.1.5)in which case the corresponding real-space corrections are a series of local, but higher-derivative operators. In the presence of massless particles however, such as in the case of gravity for example, the propagator becomes non-analytic in \(k^2\) around \(k^2 = 0\), so these corrections are non-local at all scales.
- 2.
Or the latter is not even known.
- 3.
This is why \(\Gamma _{\mathrm{in}{-}\mathrm{out}}\) can be used for computing the lowest order quantum corrections to a potential \(V(\varphi )\) on flat space-time, because then \(| 0_\mathrm{out} \rangle \sim | 0_\mathrm{in} \rangle \) and one can restrict to the cases \(\phi = \mathrm{const}\) where the time-non-locality is irrelevant [11].
- 4.
As explained in [12], even in the case of scattering amplitudes what is physically observable is not the amplitude, but the corresponding probability
$$\begin{aligned} |\langle \Psi _\mathrm{out} | \Psi _\mathrm{in} \rangle |^2 = \langle \Psi _\mathrm{in} | {\big (} |\Psi _\mathrm{out} \rangle \langle \Psi _\mathrm{out} | {\big )}| \Psi _\mathrm{in} \rangle , \end{aligned}$$(3.1.8)which also takes the form of an expectation value of some operator.
- 5.
In general one finds arbitrary powers of different Green’s functions, but always such that the corresponding integration kernel is zero when its second argument is outside the past light-cone of its first argument.
- 6.
Of course, for quadratic fields, this has precisely the effect of replacing the fields by the solution to their equation of motion, although with the Feynman prescription if some \(\square - m^2\) has been inverted in the process.
- 7.
What one should not do in this case however, is consider the times \(t < t_i\) because for them the Green’s function will be advanced.
- 8.
More precisely, since by construction \(\psi \rightarrow 0\) if \(\phi \rightarrow 0\), we would have that \(\psi ^\mathrm{hom}\) is a linear functional of \(\phi \) and the new \(\square ^{-1}\) can thus still be written as the convolution with a Green’s function.
- 9.
As we saw, this is nothing but the information of the “retardedness” of the Green’s function, which was also appended to our formal action.
- 10.
The only exception are precisely the non-local formulations of local theories since then the localizing fields are the Stückelbergs that are pure-gauge.
- 11.
Indeed, in the quantum theory, a ghost gives rise to a negative-energy state, and therefore the vacuum can decay into ghosts plus ordinary (positive-energy) particles, as long as the total energy remains zero. The corresponding decay rate is infinite because the kinematic integral is unbounded, so this instability is fatal. More precisely, putting a cut-off on momenta we get, by dimensional analysis, that the decay probability per unit time and unit volume is \(\Gamma \sim \Lambda _\mathrm{c}^4\). This actually holds for ghosts with tachyonic mass, so that the corresponding field oscillates and there is a notion of particle, although with negative-definite energy \(E = - \sqrt{\mathbf {p}^2 + m^2}\). For ghosts with non-tachyonic mass part of the modes are diverging instead of oscillating so in that case one cannot even define particles.
- 12.
The notions of “small” and “close enough” are of course subjective since they depend on the choice of a distance in field space and can be taken from either an absolute or a relative point of view.
- 13.
This can be expected whenever the hidden dynamical field has a corresponding pole in the saturated propagator of the non-local theory.
- 14.
That is, since the non-localities take the form \(\int _{t_i}^t \dots \), they all vanish at \(t = t_i\).
- 15.
In the localized formulation this would have been deduced by simply noting that the initial conditions of the auxiliary scalars vanish at \(t_i = 0\). We will see later on a concrete example of this using a similar model.
- 16.
According to the definitions of Sect. 2.1.2.
- 17.
Note that this holds also on non-trivial space-times.
- 18.
Indeed, the trends \({\sim } e^{\omega t_\mathrm{E}}\) at \(t_\mathrm{E} \rightarrow -\infty \) and \({\sim } e^{-\omega t_\mathrm{E}}\) at \(t_\mathrm{E} \rightarrow +\infty \), with \(\omega > 0\), for the boundary conditions in Euclidean time turn into \({\sim } e^{i \omega t}\) at \(t \rightarrow -\infty \) and \({\sim } e^{-i\omega t}\) at \(t \rightarrow +\infty \) in Lorentzian time, i.e. no ingoing positive-frequency waves and no outgoing negative-frequency waves.
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Mitsou, E. (2016). Subtleties of Non-local Field Theory. In: Infrared Non-local Modifications of General Relativity . Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-31729-8_3
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