Equivalence of the Klein-Gordon random field and the complex Klein-Gordon quantum field

The difference between a Klein-Gordon random field and the complex Klein-Gordon quantum field is characterized, explicitly comparing the roles played by negative frequency modes of test functions in creation and annihilation operator presentations of the two theories. The random field and the complex quantum field can both be constructed from the same creation and annihilation operator algebra, making them equivalent in that sense. Introduction. – The “Klein-Gordon random field” is taken here to be a commutative quantum field for which [φ̂(x), φ̂(y)] = 0 for all x and y, satisfying microcausality trivially. The random field structure is of interest partly because it admits a Lie field deformation that preserves commutativity of the random field [1], whereas there is a no-go theorem proving that Lie field deformations of Wightman fields that preserve nontrivial microcausality are not possible [2]. A state over a random field can be presented in more directly probabilistic ways, but it is advantageous to give an algebraic and Hilbert space presentation of both when the aim is to show how closely a random field model may parallel a quantum field model. A discussion of the mathematics of random fields in the quantum field context may be found in [3]. A selection of approaches that are more-or-less in terms of random fields is listed in [4], to which may be added [5–7]. An argument that Bell inequalities are generally not satisfied by random fields may be found in [8]. A relatively abstract comparison of the Klein-Gordon random field with the complex Klein-Gordon quantum field shows that both fields involve negative-frequency modes of test functions. Negative-frequency modes have generally been understood as positive-frequency antimatter modes [9–11], however we here engage with the algebraic structure in a way that clarifies the parallel with random fields. The distinction between positiveand negative-frequency modes is somewhat problematic for quantum fields because it is not well-defined in curved space-times and for accelerating observers [9], whereas we will see that there is no need for a distinction between positiveand negative-frequency modes in the more natural mathematics of random fields. Part of the motivation for this Letter is that we might, in time, use Lie random fields to construct models for experiments, following the principles of Bell’s polemic [12], in contrast to accepting the focus of quantum theory on constructing models for measurement and preparation apparatuses that in principle are not perfectly separable in the context of a given experiment (see also [1, 8]). The idea that measurement apparatuses should be modeled explicitly as part of models of experiments is also expressed by Feynman & Hibbs, “The usual separation of observer and observed which is now needed in analyzing measurements in quantum mechanics should not really be necessary, or at least should be even more thoroughly analyzed. What seems to be needed is the statistical mechanics of amplifying apparatus.” [13, pp22-23]; such a model constructed in a quantum mechanical formalism may be found, for example, in [14]. A detailed thermodynamics of measurement apparatuses is also required if we take seriously the insistence of the Copenhagen interpretation, which has recently been given fresh life by [15], that we should give a classical description of an experimental apparatus that is sufficient for us to reproduce experimental results — in this context, a thermodynamic or kinetic theory model of the preparation and measurement apparatus and the raw measurement results is required to be classical. As we introduce classical models of increasing detail for an experiment, we effectively move the Heisenberg cut to smaller scales, in contrast to the more common approach that moves the Heisenberg cut to larger scales to include more