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AIP conference proceedings

Position, Momentum, and the Standard Model Fermions

Carl A. Brannen, John Gideon Hartnett, Paul C Abbott

Published Jan 1, 2010• 0 citations

Abstract

Measuring the position of a free electron gives it an unknown momentum. A second measurement will give a new position far away. On the other hand, consecutive measurements of spin give the same result. In this talk we attempt to unify these very different behaviors.Position and momentum are complementary observables. An eigenstate of position has no information about momentum and vice versa. For an N‐dimensional Hilbert space we say that two bases |aj〉, and |bk〉 are “mutually unbiased” if all their transition probabilities are equal: |〈aj|bk〉|2 = 1/N. Position and momentum are similar to mutually unbiased bases, but for an infinite dimensional Hilbert space.Svetlichny discovered that Feynman’s (position) path integral can be interpreted as a sum over products of mutually unbiased bases. This raises the question “what happens when we compute spin‐1/2 path integrals over mutually unbiased bases?” There are three pairwise mutually unbiased bases for spin‐1/2. We compute the long time propagators of spin path integrals and show that a natural description of the three generations of elementary particles results. We discuss the lepton masses and other material.

DOI: 10.1063/1.3460199

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Feynman's path integral and mutually unbiased bases

J Tolar, G Chadzitaskos

Our previous work on quantum mechanics in Hilbert spaces of finite dimensions N is applied to elucidate the deep meaning of Feynman's path integral pointed out by G. Svetlichny. He speculated that the secret of the Feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases. We confirm the corresponding property of the short-time propagator by using a specially devised N x N -approximation of quantum mechanics in L^2(R) applied to our finite-dimensional analogue of a free quantum particle.

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Measuring the position of a free electron gives it an unknown momentum. A second measurement will give a new position far away. On the other hand, consecutive measurements of spin give the same result. In this talk we attempt to unify these very different behaviors.Position and momentum are complementary observables. An eigenstate of position has no information about momentum and vice versa. For an N‐dimensional Hilbert space we say that two bases |aj〉, and |bk〉 are “mutually unbiased” if all their transition probabilities are equal: |〈aj|bk〉|2 = 1/N. Position and momentum are similar to mutually unbiased bases, but for an infinite dimensional Hilbert space.Svetlichny discovered that Feynman’s (position) path integral can be interpreted as a sum over products of mutually unbiased bases. This raises the question “what happens when we compute spin‐1/2 path integrals over mutually unbiased bases?” There are three pairwise mutually unbiased bases for spin‐1/2. We compute the long time propagators of spin path integrals and show that a natural description of the three generations of elementary particles results. We discuss the lepton masses and other material.

We begin by defining mutually unbiased (MU) observables on a finite dimensional Hilbert space. We also consider the more general concept of parts of MU observables. The relationships between MU observables, value-complementary observables and two other conditions involving sequential products of observables are discussed. We next present a special motivating case of MU observables called finite position and momentum observables. These are atomic observables related by a finite Fourier transform. Finite position and momentum observables are employed to give examples of parts of MU observables that are value-complementary and those that are not value-complementary. Various open problems involving these concepts are presented. These problems mainly involve extending this work from sharp observables to unsharp observables.

Two consecutive measurements of the position of a non relativistic free particle will give entirely unrelated results. Recent quantum information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have explained this property of position observables as a result of a path in the Feynman integral being mathematically defined as a product of incompatible states; that is, as a product of mutually unbiased bases (MUBs). On the other hand, two consecutive measurements of a particle’s spin give identical results. This raises the question “what happens when spin path integrals are computed over products of MUBs? ” We show that the usual Pauli spin is obtained in the long time limit along with three generations of particles. We propose applications to the masses and mixing matrices of the elementary fermions. 1

With this second part we shall start more formal developments that allow us to improve our understanding of quantum theory. In particular, we shall deal with the two most basic quantum mechanical observables, position and momentum, which are crucial for understanding the state and the dynamics of a quantum system. Diﬀerent from classical mechanics, quantum mechanics tells us that it is not possible to measure both of the position and momentum simultaneously. This is a consequence of the non-commutativity of these two observables. Moreover, in this chapter we shall learn the basic formalism of integration and diﬀerentiation, a fundamental mathematical tool if one is really interested in understanding quantummechanics.

In quantum information, complementarity of quantum mechanical observables plays a key role. The eigenstates of two complementary observables form a pair of mutually unbiased bases (MUBs). More generally, a set of MUBs consists of bases that are all pairwise unbiased. Except for specific dimensions of the Hilbert space, the maximal sets of MUBs are unknown in general. Even for a dimension as low as six, the identification of a maximal set of MUBs remains an open problem, although there is strong numerical evidence that no more than three simultaneous MUBs do exist. Here, by exploiting a newly developed holographic technique, we implement and test different sets of three MUBs for a single photon six-dimensional quantum state (a "qusix"), encoded exploiting polarization and orbital angular momentum of photons. A close agreement is observed between theory and experiments. Our results can find applications in state tomography, quantitative wave-particle duality, quantum key distribution.

A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum S=(√3 ℏ)⁄2; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.

Our previous work on quantum mechanics in Hilbert spaces of finite dimensions N is applied to elucidate the deep meaning of Feynman's path integral pointed out by G. Svetlichny. He speculated that the secret of the Feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases. We confirm the corresponding property of the short-time propagator by using a specially devised N x N -approximation of quantum mechanics in L^2(R) applied to our finite-dimensional analogue of a free quantum particle.