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arXiv:1604.04950 · Physical Review A

Peres experiment using photons: No test for hypercomplex (quaternionic) quantum theories

Stephen L. Adler

Published Jun 19, 2017• 13 citations
arXivHTMLPDF

Abstract

Assuming the standard axioms for quaternionic quantum theory and a spatially localized scattering interaction, the $S$ matrix in quaternionic quantum theory is complex valued, not quaternionic. Using the standard connections between the $S$ matrix, the forward scattering amplitude for electromagnetic wave scattering, and the index of refraction, we show that the index of refraction is necessarily complex, not quaternionic. This implies that the recent optical experiment of Procopio et al. [Nat. Commun. 8, 15044 (2017)] based on the Peres proposal does not test for hypercomplex or quaternionic quantum effects arising within the standard Hilbert space framework. Such a test requires looking at near zone fields, not radiation zone fields.

DOI
10.1103/physreva.95.060101

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Comment on “Peres experiment using photons: No test for hypercomplex (quaternionic) quantum theories”

Lorenzo M. Procopio, Lee A. Rozema, Borivoje Dakić +1 more

In his recent article [Phys. Rev. A 95, 060101(R) (2017)], Adler questions the usefulness of the bound found in our experimental search for genuine effects of hypercomplex quantum mechanics [Nat. Commun. 8, 15044 (2017)]. Our experiment was performed using a black-box (instrumentalist) approach to generalized probabilistic theories; therefore, it does not assume a priori any particular underlying mechanism. From that point of view our experimental results do indeed place meaningful bounds on the possible effects of ``postquantum theories,'' including quaternionic quantum mechanics. In his article, Adler compares our experiment to nonrelativistic and M\"oller formal scattering theories within quaternionic quantum mechanics. With a particular set of assumptions, he finds that quaternionic effects would likely not manifest themselves in general. Although these assumptions are justified in the nonrelativistic case, a proper calculation for relativistic particles is still missing. Here, we provide a concrete relativistic example of Klein-Gordon scattering wherein the quaternionic effects persist. We note that when the Klein-Gordon equation is formulated using a Hamiltonian formalism it displays a so-called ``indefinite metric,'' a characteristic feature of relativistic quantum wave equations. In Adler's example this is directly forbidden by his assumptions, and therefore our present example is not in contradiction to his work. In complex quantum mechanics this problem of an indefinite metric is solved in a second quantization. Unfortunately, there is no known algorithm for canonical field quantization in quaternionic quantum mechanics.

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Assuming the standard axioms for quaternionic quantum theory and a spatially localized scattering interaction, the SSS matrix in quaternionic quantum theory is complex valued, not quaternionic. Using the standard connections between the SSS matrix, the forward scattering amplitude for electromagnetic wave scattering, and the index of refraction, we show that the index of refraction is necessarily complex, not quaternionic. This implies that the recent optical experiment of Procopio et al. [Nat. Commun. 8, 15044 (2017)] based on the Peres proposal does not test for hypercomplex or quaternionic quantum effects arising within the standard Hilbert space framework. Such a test requires looking at near zone fields, not radiation zone fields.
In his recent article [Phys. Rev. A 95, 060101(R) (2017)], Adler questions the usefulness of the bound found in our experimental search for genuine effects of hypercomplex quantum mechanics [Nat. Commun. 8, 15044 (2017)]. Our experiment was performed using a black-box (instrumentalist) approach to generalized probabilistic theories; therefore, it does not assume a priori any particular underlying mechanism. From that point of view our experimental results do indeed place meaningful bounds on the possible effects of ``postquantum theories,'' including quaternionic quantum mechanics. In his article, Adler compares our experiment to nonrelativistic and M\"oller formal scattering theories within quaternionic quantum mechanics. With a particular set of assumptions, he finds that quaternionic effects would likely not manifest themselves in general. Although these assumptions are justified in the nonrelativistic case, a proper calculation for relativistic particles is still missing. Here, we provide a concrete relativistic example of Klein-Gordon scattering wherein the quaternionic effects persist. We note that when the Klein-Gordon equation is formulated using a Hamiltonian formalism it displays a so-called ``indefinite metric,'' a characteristic feature of relativistic quantum wave equations. In Adler's example this is directly forbidden by his assumptions, and therefore our present example is not in contradiction to his work. In complex quantum mechanics this problem of an indefinite metric is solved in a second quantization. Unfortunately, there is no known algorithm for canonical field quantization in quaternionic quantum mechanics.
The axioms of quantum mechanics provide limited information regarding the structure of the Hilbert space, such as the underlying number system. The latter is generally regarded as complex, but generalizations of complex numbers, so-called hypercomplex numbers, cannot be ruled out in theory. Therefore, specialized experiments to test for hypercomplex quantum mechanics are needed. To date, experimental tests are limited to single-particle interference exploiting a closed phase relation in a three-path interferometer called the Peres test. The latter distinguishes complex quantum mechanics from quaternionic quantum mechanics. Here, we propose a general matrix formalism putting the Peres test on a solid mathematical ground. On this basis, we introduce multipath and multiparticle interference tests, which provide a direct probe for any dimension of the number system of quantum mechanics.
While complex numbers form the foundational number system of quantum mechanics, the theoretical possibility of higher-dimensional hypercomplex alternatives persists. In this paper, we present a test protocol designed to distinguish between complex and quaternionic quantum mechanics. The protocol is founded upon the quantum superdense coding scheme and functions by detecting the presence of multiple distinct quantum phase gates. We also provide the corresponding quantum circuit required to perform this test.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.
Abstract The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of a more general type. In particular, detailed discussion is given of the simplest generalization of the complex Hilbert space, that of the quaternion Hilbert module.