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Feynman's Integral is About Mutually Unbiased Bases

George Svetlichny

The Feynman integral can be seen as an attempt to relate, under certain circumstances, the quantum-information-theoretic separateness of mutually unbiased bases to causal proximity of the measuring processes.

arXiv:0708.3079Aug 22, 20077 citations

Spin Path Integrals and Generations

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. 1Two 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

Jan 1, 20090 citations

Why Lagrangians?

George Svetlichny, Piotr Kielanowski, Anatol Odzijewicz +2 more

We argue that Feynman's Integral imposes the condition of being mutually unbiased on pairs of bases that are causally proximal. This sheds light on the nature of Lagrangian theories from the emergent space‐time perspective.

AIP conference proceedingsJan 1, 20071 citation

MUTUALLY UNBIASED BASES 1

Ingemar Bengtsson

This is a review of the problem of Mutually Unbiased Bases in finite dimensional Hilbert spaces, real and complex. Also a geometric measure of “mubness” is introduced, and applied to some explicit calculations in six di

Jan 1, 20060 citations

On approximations of Feynman path integrals

Tapani Hyttinen

We study approximations of Feynman path integrals in finite dimensional spaces and how the approximations determine the propagator.

arXiv:2208.06206Aug 12, 20220 citations

Position, Momentum, and the Standard Model Fermions

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

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.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.

AIP conference proceedingsJan 1, 20100 citations

Feynman's path integral and mutually unbiased bases

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Feynman's path integral and mutually unbiased bases

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