TY - GEN

T1 - The marvelous consequences of hardy spaces in quantum physics

AU - Bohm, Arno

AU - Bui, Hai Viet

PY - 2013

Y1 - 2013

N2 - Dynamical differential equations, like the Schrödinger equation for the states, or the Heisenbergequa tion for the observables, need to be solved under boundary conditions. The original boundary condition of von Neumann, the Hilbert space axiom, required that the allowed wave functions are Lebesgue square integrable. This leads by a mathematical theorem of Stone-von Neumann to the unitary group evolution meaning the time t extends over -∞ < t < +<. Physicists do not use Lebesgue integrals but followed a different path usinga lmost exclusively the Dirac formalism and well-behaved (Schwartz) functions. This led the mathematicians to Schwartz-Rigged Hilbert spaces (Gelfand triplets), which are the mathematical core of Dirac's bra-ket formalism. This is insufficient for a theory that includes resonance and decay phenomena, which requires analytic continuation in energy E in order to accommodate exponentially decayingG amow kets, Breit-Wigner (Lorentzian) resonances, and Lippmann-Schwinger kets. This leads to a pair of Rigged Hilbert Spaces of smooth Hardy functions, one representing the prepared states of scatteringe xperiments (preparation apparatus) and the other representingd etected observables (registration apparatus). A mathematical consequence of the Hardy space axiom is that the time evolution is asymmetric given by the semi-group, i.e., t0 ≤ t < +<, with a finite t0. What would the meaningo f that t0 be?

AB - Dynamical differential equations, like the Schrödinger equation for the states, or the Heisenbergequa tion for the observables, need to be solved under boundary conditions. The original boundary condition of von Neumann, the Hilbert space axiom, required that the allowed wave functions are Lebesgue square integrable. This leads by a mathematical theorem of Stone-von Neumann to the unitary group evolution meaning the time t extends over -∞ < t < +<. Physicists do not use Lebesgue integrals but followed a different path usinga lmost exclusively the Dirac formalism and well-behaved (Schwartz) functions. This led the mathematicians to Schwartz-Rigged Hilbert spaces (Gelfand triplets), which are the mathematical core of Dirac's bra-ket formalism. This is insufficient for a theory that includes resonance and decay phenomena, which requires analytic continuation in energy E in order to accommodate exponentially decayingG amow kets, Breit-Wigner (Lorentzian) resonances, and Lippmann-Schwinger kets. This leads to a pair of Rigged Hilbert Spaces of smooth Hardy functions, one representing the prepared states of scatteringe xperiments (preparation apparatus) and the other representingd etected observables (registration apparatus). A mathematical consequence of the Hardy space axiom is that the time evolution is asymmetric given by the semi-group, i.e., t0 ≤ t < +<, with a finite t0. What would the meaningo f that t0 be?

KW - Hardy space

KW - Rigged Hilbert space

KW - Semigroup

KW - Time asymmetry

KW - Unitary group

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U2 - 10.1007/978-3-0348-0448-6_17

DO - 10.1007/978-3-0348-0448-6_17

M3 - Conference contribution

AN - SCOPUS:84959239815

SN - 9783034804479

T3 - Trends in Mathematics

SP - 211

EP - 228

BT - Geometric Methods in Physics

A2 - Kielanowski, Piotr

A2 - Ali, S. Twareque

A2 - Odzijewicz, Anatol

A2 - Schlichenmaier, Martin

A2 - Voronov, Theodore

PB - Springer International Publishing

T2 - 30th Workshop on Geometric Methods in Physics, 2011

Y2 - 26 June 2011 through 2 July 2011

ER -