Ahshct. Pure states of a free particle in non-relativistic quantum mechanics are demibed, in which the probability of finding the particle to have a negative x-coordinate increases over an arbivarily long, but finite. time interval, even though the x-component of the particle's velocity is certainly positive throughout that time internal It is shown that, for any s w of this type, the greatest amount of probability which can Bow back f" positive to negative x-values in this counter-intuitive way, over any given time interval, is equal b the largest eigenvalue of a certain Hermitian operator, and it is estimated numerically to have a value near 0.04. This value is not only independent of the length of the time interval and the mass of the particle, but is also independent of the value of Planck's constant. It r e f l a the smcture of Schriidinger's equation, rather than the values of the paramete~s appearing there. Backtlow of positive probability is related to the non-positivity of Wigner's density function, and can be rewded as arising from a ROW of negative probability in the same direction as the velocity. Generalizations are indicated, to the relativistic free electron, and to non-relativistic cases in which probability backflow occurs even in opposition to an arbitrarily strong constant force.

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Abstract:A new lattice model is presented for correlated electrons on the unrestricted 4 L -dimensional electronic Hilbert space ⊗ L n=1 C 4 (where L is the lattice length). It is a supersymmetric generalization of the Hubbard model, but differs from the extended Hubbard model proposed by Essler, Korepin and Schoutens. The supersymmetry algebra of the new model is superalgebra gl(2|1).The model contains one symmetry-preserving free real parameter which is the Hubbard interaction parameter U , and has its origin here in the one-parameter family of inequivalent typical 4-dimensional irreps of gl(2|1). On a one-dimensional lattice, the model is exactly solvable by the Bethe ansatz.

A method is developed for systematically constructing trigonometric and rational solutions of the Yang-Baxter equation using the representation theory of quantum supergroups. New quantum R-matrices are obtained by applying the method to the vector representations of quantum osp (1/2) and gl (m/n).

We present a systematic technique to construct solutions to the Yang-Baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form.We exploit the fact that quantum non-compact algebras such as U q (su(1, 1)) and type-I quantum superalgebras such as U q (gl(1|1)) and U q (gl(2|1)) are known to admit non-trivial one-parameter families of infinite-dimensional and finite dimensional irreps, respectively, even for generic q.We develop a technique for constructing the corresponding spectral-dependent R-matrices. As examples we work out the the R-matrices for the three quantum algebras mentioned above in certain representations.