We offer an interpretation of superquantum correlations in terms of a “doubly” quantum theory.
We argue that string theory, viewed as a quantum theory with two deformation parameters, the string tension

In this paper, we present an observation relating two fields of physics which are ostensibly quite remote, namely, the study of the foundations of quantum mechanics (QM) centered around the violation of the celebrated Bell inequalities [

Consider two classical variables

Let us briefly review the simplest routes to these bounds. Following [

Indeed, Popescu and Rohrlich have demonstrated that one can concoct super-quantum correlations which violate the Cirel’son bound, while still maintaining consistency with relativistic causality [

But what type of theory would predict such correlations? It has been speculated that a specific superquantum theory could essentially be derived from the two requirements of relativistic causality and the saturation of the

A related development has been the proof by van Dam that superquantum correlations which saturate the

One proposal for a superquantum theory discussed in the literature uses a formal mathematical redefinition of the norms of vectors from the usual

At this point, we make the very simple observation that it is the procedure of “quantization,” which takes us from classical mechanics to QM, that increases the bound from the Bell/CHSH value of 2 to the Cirel’son value of

In the following, we will clarify which “quantization” procedure we have in mind, and how it can be applied for a second time onto QM, leading to a “doubly quantized” theory. We then argue that a physical realization of such a theory may be offered by nonperturbative open string field theory (OSFT).

Before going into the “double quantization” procedure, let us first observe that from the point of view of general mathematical deformation theory [

Second, superquantum correlations point to a nonlocality, which is more nonlocal, so to speak, than the aforementioned “quantum nonlocality” of QM and QFT. However, QFT’s are actually local theories, and true nonlocality is expected only in theories of quantum gravity. That quantum gravity must be nonlocal stems from the requirement of diffeomorphism invariance, as has been known from the pioneering days of that field [

Third, the web of dualities discovered in ST [

What follows is a heuristic attempt to make these expectations physically concrete. Our essential observation is as follows: the “quantization” procedure responsible for turning the classical Bell bound of 2 into the quantum Cirel’son bound of

This defines our “double quantization” procedure, through which two deformation parameters,

At this point, we make the observation that a “doubly quantized” theory may already be available in the form of Witten’s open string field theory (OSFT) [

The doubly deformed nature of the theory is explicit in the Witten action for the “classical” open string field

In addition to its manifestly “doubly” quantized path integral, OSFT has as massless modes the ordinary photons, which are used in the experimental verification of the violation of Bell’s inequalities [

We close this section with a caveat and a speculation. In the above reasoning, the two quantizations were taken to be independent with two independent deformation parameters. In the case of OSFT, they were

Given that a superquantum theory is supposedly more “quantum” than QM, let us now consider the the extreme quantum limit of QM,

What would the

However, this observation is perhaps a bit naïve, since the proof of the Cirel’son bound itself may no longer be valid under the wash-out of all phases. Let us invoke here an optical-mechanical analogy: geometric optics is the zero wavelength limit of electromagnetism, which would correspond to the

But before we ask what rule should replace that of Born, let us confront the obvious problem that in the limit

Could the

Another issue here is that of interpretation: in the classical (

Finally we offer some comments on possible experimental observations of such superquantum violations of Bell’s inequalities. The usual setup involves entangled photons [

Superquantum correlations could also be observable in cosmology. The current understanding of the large-scale structure of the universe, that is, the distribution of galaxies and galaxy clusters, is that they are seeded by quantum fluctuations. In standard calculations, it is assumed that the quantum correlations of these fluctuations are Gaussian (non-Gaussian correlations have also been considered). If the correlations were, in fact, superquantum, however, their signature could appear as characteristic deviations from the predicted large-scale structure based on Gaussian correlations. Such superquantum correlations would presumably be generated in the quantum gravity phase, and thus should be enhanced by the expansion of the universe at the largest possible scales. It would be interesting to look for evidence of such large-scale superquantum correlations in the existing WMAP [

We conclude with a few words regarding a new experimental “knob” needed to test our doubly quantized approach to superquantum correlations. In the classic experimental tests of the violation of Bell’s inequalities [

In this paper, we have obviously only scratched the surface of a possible superquantum theory, and many probing questions remain to be answered and understood. We hope to address some of them in future works.

The authors thank V. Balasubramanian, J. de Boer, J. Heremans, S. Mathur, K. Park, J. Polchinski, R. Raghavan, D. Rohrlich, V. Scarola, J. Simon, and A. Staples for helpful comments, interesting discussions, and salient questions. D. Minic acknowledges the hospitality of the Mathematics Institute at Oxford University and Merton College, Oxford, and his respective hosts, Philip Candelas and Yang-Hui He. D. Minic also thanks the Galileo Galilei Institute for Theoretical Physics, Florence, for the hospitality and the INFN for partial support. Z. Lewis, D. Minic, and T. Takeuchi are supported in part by the U.S. Department of Energy Grant no. DE-FG05-92ER40677, task A.