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Probability Current in the Relativistic Hamiltonian Quantum Mechanics

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Probability Current in the Relativistic Hamiltonian Quantum Mechanics Probability current in the relativistic Hamiltonian quantum mechanics Jakub Rembieli´nski University ofL´od´z Max Born Symposium, University of Wroc law, 28{30 June, 2011 Formulation of relativistic quantum mechanics: Salpeter equation K. Kowalski and J. Rembieli´nski,Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics (accepted for publication in Phys. Rev. A (2011)); continuation of: K. Kowalski and J. Rembieli´nski, The relativistic massless harmonic oscillator, Phys. Rev. A 81, 012118 (2010) p @ H = c2p2 + m2c4 + V (x); H ! i ; x ! x; p ! −i r ~@t ~ The spinless Salpeter equation: @ p i φ(x; t) = [ m2c4 − 2c2∆ + V (x)]φ(x; t) ~@t ~ L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102, 568 (1956) Probability current 2/27 Formulation of relativistic quantum mechanics: Salpeter equation K. Kowalski and J. Rembieli´nski,Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics (accepted for publication in Phys. Rev. A (2011)); continuation of: K. Kowalski and J. Rembieli´nski, The relativistic massless harmonic oscillator, Phys. Rev. A 81, 012118 (2010) p @ H = c2p2 + m2c4 + V (x); H ! i ; x ! x; p ! −i r ~@t ~ The spinless Salpeter equation: @ p i φ(x; t) = [ m2c4 − 2c2∆ + V (x)]φ(x; t) ~@t ~ L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102, 568 (1956) Probability current 2/27 Formulation of relativistic quantum mechanics: Salpeter equation K. Kowalski and J. Rembieli´nski,Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics (accepted for publication in Phys. Rev. A (2011)); continuation of: K. Kowalski and J. Rembieli´nski, The relativistic massless harmonic oscillator, Phys. Rev. A 81, 012118 (2010) p @ H = c2p2 + m2c4 + V (x); H ! i ; x ! x; p ! −i r ~@t ~ The spinless Salpeter equation: @ p i φ(x; t) = [ m2c4 − 2c2∆ + V (x)]φ(x; t) ~@t ~ L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102, 568 (1956) Probability current 2/27 Formulation of relativistic quantum mechanics: Salpeter equation K. Kowalski and J. Rembieli´nski,Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics (accepted for publication in Phys. Rev. A (2011)); continuation of: K. Kowalski and J. Rembieli´nski, The relativistic massless harmonic oscillator, Phys. Rev. A 81, 012118 (2010) p @ H = c2p2 + m2c4 + V (x); H ! i ; x ! x; p ! −i r ~@t ~ The spinless Salpeter equation: @ p i φ(x; t) = [ m2c4 − 2c2∆ + V (x)]φ(x; t) ~@t ~ L.L. Foldy, Synthesis of covariant particle equations, Phys. Rev. 102, 568 (1956) Probability current 2/27 Leslie L. Foldy (1919{2001) Probability current 3/27 Edwin E. Salpeter (1924{2008) Probability current 4/27 The Salpeter equation can be written in the form of the integro-differential equation @φ(x; t) Z i = d 3y K(x − y)φ(y; t) + V (x)φ(x; t) ~ @t where the kernel is given by 2 3 mc 2m c K2( jx − yj) K(x − y) = − ~ (2π)2~ jx − yj2 The Salpeter equation presumes the Newton-Wigner localization scheme implying the standard quantization rule x^ ! x, p^ ! −i~r. Consequently, the Hilbert space of solutions to the Salpeter is L2(R3; d 3x): Z hφj i = d 3x φ∗(x) (x) Therefore, we should identify jφ(x; t)j2 with the probability density ρ(x; t) satisfying the normalization condition: Z d 3x ρ(x; t) = 1 Probability current 5/27 The Salpeter equation can be written in the form of the integro-differential equation @φ(x; t) Z i = d 3y K(x − y)φ(y; t) + V (x)φ(x; t) ~ @t where the kernel is given by 2 3 mc 2m c K2( jx − yj) K(x − y) = − ~ (2π)2~ jx − yj2 The Salpeter equation presumes the Newton-Wigner localization scheme implying the standard quantization rule x^ ! x, p^ ! −i~r. Consequently, the Hilbert space of solutions to the Salpeter is L2(R3; d 3x): Z hφj i = d 3x φ∗(x) (x) Therefore, we should identify jφ(x; t)j2 with the probability density ρ(x; t) satisfying the normalization condition: Z d 3x ρ(x; t) = 1 Probability current 5/27 The Salpeter equation can be written in the form of the integro-differential equation @φ(x; t) Z i = d 3y K(x − y)φ(y; t) + V (x)φ(x; t) ~ @t where the kernel is given by 2 3 mc 2m c K2( jx − yj) K(x − y) = − ~ (2π)2~ jx − yj2 The Salpeter equation presumes the Newton-Wigner localization scheme implying the standard quantization rule x^ ! x, p^ ! −i~r. Consequently, the Hilbert space of solutions to the Salpeter is L2(R3; d 3x): Z hφj i = d 3x φ∗(x) (x) Therefore, we should identify jφ(x; t)j2 with the probability density ρ(x; t) satisfying the normalization condition: Z d 3x ρ(x; t) = 1 Probability current 5/27 The Salpeter equation was discarded because of: I its nonlocality I the lack of manifest Lorentz covariance However I the nonlocality of the Salpeter equation does not disturb the light cone structure 2 3 3 I the space L (R ; d x) of solutions to the Salpeter equation is invariant under the Lorentz group I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation I the Salpeter equation is widely used in the phenomenological description of the quark-antiquark-gluon system as a hadron model Probability current 6/27 The Salpeter equation was discarded because of: I its nonlocality I the lack of manifest Lorentz covariance However I the nonlocality of the Salpeter equation does not disturb the light cone structure 2 3 3 I the space L (R ; d x) of solutions to the Salpeter equation is invariant under the Lorentz group I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation I the Salpeter equation is widely used in the phenomenological description of the quark-antiquark-gluon system as a hadron model Probability current 6/27 The Salpeter equation was discarded because of: I its nonlocality I the lack of manifest Lorentz covariance However I the nonlocality of the Salpeter equation does not disturb the light cone structure 2 3 3 I the space L (R ; d x) of solutions to the Salpeter equation is invariant under the Lorentz group I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation I the Salpeter equation is widely used in the phenomenological description of the quark-antiquark-gluon system as a hadron model Probability current 6/27 The Salpeter equation was discarded because of: I its nonlocality I the lack of manifest Lorentz covariance However I the nonlocality of the Salpeter equation does not disturb the light cone structure 2 3 3 I the space L (R ; d x) of solutions to the Salpeter equation is invariant under the Lorentz group I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation I the Salpeter equation is widely used in the phenomenological description of the quark-antiquark-gluon system as a hadron model Probability current 6/27 The Salpeter equation was discarded because of: I its nonlocality I the lack of manifest Lorentz covariance However I the nonlocality of the Salpeter equation does not disturb the light cone structure 2 3 3 I the space L (R ; d x) of solutions to the Salpeter equation is invariant under the Lorentz group I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation I the Salpeter equation is widely used in the phenomenological description of the quark-antiquark-gluon system as a hadron model Probability current 6/27 The Salpeter equation was discarded because of: I its nonlocality I the lack of manifest Lorentz covariance However I the nonlocality of the Salpeter equation does not disturb the light cone structure 2 3 3 I the space L (R ; d x) of solutions to the Salpeter equation is invariant under the Lorentz group I the Salpeter equation possesses solutions of positive energies only and we have no problems with paradoxes occuring in the case of the Klein-Gordon equation I the agreement of predictions of the spinless Salpeter equation with the experimental spectrum of mesonic atoms is as good as for the Klein-Gordon equation I the Salpeter equation is widely used in the phenomenological description of the quark-antiquark-gluon system as a hadron model Probability current 6/27 The Salpeter equation was discarded because of: I its nonlocality I the lack of manifest Lorentz covariance However I the nonlocality of the Salpeter equation does not disturb the light cone structure 2 3 3 I the space L (R ; d x) of solutions to the Salpeter equation is invariant under the Lorentz group I the Salpeter equation
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