A formalism based on real octonions is developed in order to construct an octonionic Hilbert space for the description of colored quark states. The various possible forms of scalar products and related scalars are discussed. The choice of a direction in the space of octonion units leads naturally to a representation of the Poincaré group in terms of complex scalar products and complex scalars. The remaining octonion directions span the color degrees of freedom for quarks and anti‐quarks. In such a Hilbert space, product states associated with color singlets are shown to form a physical quantum mechanical Hilbert space for the description of hadrons. Color triplets, on the other hand, correspond to unobservable parafermion states of order three.
REFERENCES
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G. t’Hooft, unpublished.
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H. Fritzsch and M. Gell‐Mann, in Proceedings of the XVI International Conference on High Energy Physics, Chicago‐Batavia, Ill. 1972, edited by J. D. Jackson and A. Roberts (NAL, Batavia, Ill., 1973), Vol. 2, p. 135;
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17.
This conjugation corresponds to an automorphism of the quaternion subalgebras generated by the imaginary units for and it corresponds to an antiautomorphism of the other possible quaternion subalgebras.
18.
By bilinearity here, we mean linearity of both of the arguments of (X, Y) with respect to multiplication by real numbers.
19.
However, this bilinear form has a property unique to octonions namely and (X, Y) are in triality with each other. For definition of triality and details see Refs. 8 and 16 and M. Günaydin, Scuola Normale Superiore preprint (1975), to be published in Nuovo Cimento.
20.
This also guarantees the associativity of scalar multiplication which is part of the definition of a Hilbert space.
21.
An equivalent construction over the split octonionic Hilbert space was given in Ref. 9.
22.
See E. P. Wigner, “Unitary Representations of the Inhomogeneous Lorentz Group including Reflections,” in Group Theoretical Concepts and Methods in Elementary Particle Physics, edited by F. Gürsey (Gordon and Breach, New York, 1964).
23.
Here we should note that the necessity of fixing an imaginary unit, in our case is not only dictated by the invariance of the complex scalar product under the automorphism group of the Hilbert space but also by the requirement that the translations be unitarily implementable. In fact the representation matrices of any group action on our octonionic Hilbert space can involve at most one imaginary unit since the group action must be associative.
24.
If one uses the split octonions as the underlying composition algebra, such inconsistent product states vanish automatically; see Refs. 6, 9, and 16.
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J. M. Jauch, Foundations of Quantum Mechanics (Addison‐Wesley, Reading, Mass., 1968).
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