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AIP conference proceedings

Why Lagrangians?

George Svetlichny, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmeier, Theodore Voronov

Published Jan 1, 2007• 1 citation
Publisher

Abstract

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.
DOI
10.1063/1.2820959

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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.
I offer some historical comments about the origins of Feynman's path integral approach, as an alternative approach to standard quantum mechanics. Looking at the interaction between Einstein and Feynman, which was mediated by Feynman's thesis supervisor John Wheeler, it is argued that, contrary to what one might expect, the significance of the interaction between Einstein and Feynman pertained to a critique of classical field theory, rather than to a direct critique of quantum mechanics itself. Nevertheless, the critical perspective on classical field theory became a motivation and point of departure for Feynman's space-time approach to non-relativistic quantum mechanics.
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.
The Feynman path integral is an essential part of our mathematical description of fundamental nature at small scales. However, what it seems to say about the world is very much at odds with our classical intu-itions, and exactly why nature requires us to describe her in this way is currently unknown. We will describe here a possibility according to which the path integral may be the spacetime manifestation of objects existing in a lower-dimensional analog of spacetime until they give rise to the emergence of spacetime objects under a process that is currently labeled a ‘Quantum Measure-ment’. This idea is based on a mathematical distinction which at present does not appear to be widely appreci-ated.
The existence problem for mutually unbiased bases is an unsolved problem in quantum information theory. A related question is whether every pair of bases admits vectors that are unbiased to both. Mathematically this translates to the question whether two Lagrangian Clifford tori intersect, and a body of results exists concerning it. These (deep!) results are however rather weak when viewed from the point of view of the first problem. We make a detailed study of how the intersections behave in the simplest non-trivial case, that of complex projective 2-space (the qutrit), for which the set of Clifford tori can be usefully parametrized by the unistochastic subset of Birkhoff's polytope. An interesting picture emerges. A foray into higher dimensions is included.