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 -