Gravitational Interactions and Fine-Structure Constant Ulrich D. Jentschura Missouri University of Science and Technology Rolla, Missouri (Fellow/APS) Bled Workshop: Beyond Standard Model 22-JUL-2014 (Research Supported by NSF, NIST, Missouri Research Board, 2009-2014)
In exploring nature, a possible starting point is the beauty of mathematics! French physicist: He/she will start from mathematics, then explore the beauty of mathematics, and if, in the end, there is an application to be found somewhere, it is purely accidental but welcome.
Recap How it all Started Dirac Theory for Electromagnetic Coupling
Theory of Bound Systems: Hydrogen with QED Schrödinger theory: Nonrelativistic quantum theory. Explains up to (Zα) 2. Dirac theory: Relativistic quantum theory; includes zitterbewegung, spin, and spin-orbit coupling. Explains up to (Zα) 4. QED includes the self-interaction of the electron and tiny corrections to the Coulomb force law [α (Zα) 4 and beyond].
Schrödinger Theory (Nonrelativistic): Dirac-Coulomb Theory (Relativistic): Relativistic Correction Terms QED (Quantized Fields, Relativistic, Recoil): Beyond the Dirac formalism. Self-energy effects, corrections to the Coulomb force law, So called recoil corrections, Feynman diagrams... f"
Nonrelativistic Limit of Dirac Theory Relativistic Correction Terms: Dirac and Foldy-Wouthuysen Obtained after Foldy-Wouthuysen Transformation (Unitary Transformation of the Dirac-Coulomb Hamiltonian). Expansion in the Momentum Operators p ~ Zα. One-Particle Theory in the Coulomb Potential:
Dirac-Coulomb Hamiltonian, Foldy-Wouthuysen Transformation and Correction Terms Dirac-Coulomb Hamiltonian: Coulomb Potential: Result of Foldy-Wouthuysen Transformation of the Dirac-Coulomb Hamiltonian (2 x 2 Block Structure): Particle-Antiparticle Symmetry (β Matrix): There is no particle-antiparticle symmetry (no universal prefactor β). Electrons and attracted, whereas positrons are repulsed by the Coulomb field.
Once More Particle-Antiparticle Symmetry (β Matrix): Expectation Value in a Positive-Energy Schrödinger Eigenstate: Relativistic Correction Terms: Foldy-Wouthuysen Transformation There is no particle-antiparticle symmetry (no universal prefactor β). Electrons are attracted, whereas positrons are repulsed by the Coulomb field.
Schrödinger Theory with and without Relativistic Corrections Without Relativistic Corrections (Coulomb Field): With Relativistic Corrections (Spin-Orbit/Thomas Precession): With Relativistic Corrections for Particles and Antiparticles:
Dirac Theory for Curved Space-Times
Dirac Representation (Tilde Here for Flat Space)
Classical Geodesic in Curved Space Dirac Equation and Dirac Action in Flat Space:
Vierbein and Affine Connection Matrix Covariant Derivative of Vector: Curved-Space Dirac Algebra: Covariant Derivative of Gamma Matrix: Well-Known Solution ( Spin Connection ):
Factor... The Dirac equation constitutes one of the most versatile instruments of physics...
Quantum Mechanical Dirac Particle in Curved Space Curved-Space Dirac Algebra (Overlined Matrices): Lorentz Invariance of Gamma Matrices: Curved-Space Dirac Lagrangian:
Schwarzschild Metric and Eddington Coordinates Result for the Affine Scalar Product:
In the Schwarzschild Metric: Fully Relativistic Symmetry Properties for Particles and Antiparticles Ansatz for the Bispinor Wave Function: Fully Relativistic Radial Equations in Curved Space: Radial Equations in Flat Space: Symmetry of the Spectum:
Spectrum of Particles and Antiparticles: E and E After reinterpretation, the same physical energies. Reinterpretation principle: An antiparticle falls upward in the gravitational field, but backward in time and with the same kinetic energy as the corresponding particle.
Formalism of the Foldy-Wouthuysen Method
Calculation by Foldy-Wouthuysen Method! How does the Foldy-Wouthuysen Transformation Work? Example: Free Dirac Hamiltonian: Differential Operators and Spin Odd Even
Free Dirac Hamiltonian: Foldy-Wouthuysen Transformation is Unitary
Converting the Gravitationally Coupled Dirac Equation to Hamiltonian Form [see also U.D.J., Phys. Rev. A 87, 032101 (2013)] Dirac-Schwarzschild Hamiltonian [r s = 2 G M = Schwarzschild Radius]
Quantum Particle in a Gravitational Field General relativity yields the following result for the Dirac-Schwarzschild central-field problem: [Note: Cannot simply insert the gravitational potential on the basis of the correspondence principle] Now do the Foldy-Wouthuysen transformation.
Now for Gravitational Coupling First Transformation
Now for Gravitational Coupling Second Transformation Relativistic Kinetic Correction Leading Gravitational Term Gravitational Breit Term Gravitational Spin-Orbit Coupling Gravitational Zitterbewegung (Darwin) Term [U.D.J. and J. H. Noble, Phys. Rev. A 88, 022121 (2013)]
Foldy-Wouthuysen Transformed Dirac-Schwarzschild Hamiltonian: Another Result from the Literature with a Spurious Spin-Gravity Coupling: [This term breaks parity. Why? Well, spin is pseudo-vector but g vector.] [Y. N. Obukhov, PRL (2001)] Rather Subtle Mistake: Obukhov uses a parity-breaking Foldy-Wouthuysen transformation, which is mathematically valid (still unitary) but changes the physical interpretation of the spin operator.
Discussion of the Foldy-Wouthuysen Transformed Gravitationally Coupled Dirac-Schwarzschild Hamiltonian (with reference to the Dirac-Coulomb Hamiltonian)
Quantum Particle in a Gravitational Field Nonrelativistic Theory (r s is the Schwarzschild Radius): Result of Foldy-Wouthuysen Transformation: Perfect Particle-Antiparticle Symmetry (Overall Prefactor β):
Quantum Particle in a Gravitational Field Relativistic Kinetic Correction Leading Gravitational Term Gravitational Breit Term Gravitational Spin-Orbit Coupling [in agreement with the Classical Geodesic Precession derived in 1920 by A.D.Fokker] Gravitational Zitterbewegung (Darwin) Term [U.D.J. and J. H. Noble, Phys.Rev.A 88, 022101 (2013) ]
Quantum Result [Phys.Rev.A 88, 022101 (2013)]: Classical Result (Fokker, de Sitter, Schouten):
Spectrum of Gravitational Bound States
Spectrum of Gravitational Bound States but larger for other mass configurations
Spectrum of Gravitational Bound States
Gravitational Quantum Bound States [e-print 1403.2955 (2014), to appear in the Annalen der Physik (Berlin)]:
Spectrum of Gravitational Bound States Physical"Rei" [Phys. Rev. A (2014), in press]
Gravitational Correction to the Current
Photon Emission Vertex in Flat Spacetime Photon Emission Vertex in Curved Spacetime
Gravitational Corrections to the Transition Current Gravitational Correction to the Dipole Coupling The terms without r s have been known before and are used in Lamb shift calculations Gravitational Correction to the Quadrupole Coupling Gravitational Correction to the Magnetic Coupling
Global Dilaton
Global Dilaton
Might conjecture that the current Universe corresponds to a particular point in the family of models related by a global dilation transformation: The latter proportionality finds some motivation in string theory. Invariance Properties under the Global Dilation:
Global Dilaton and Variational Principle: Might conjecture that the current value of λ is determined by a variational principle:
Conclusions
Conclusions Dirac particles in a Coulomb Field: Understood since the 1920s and 1930s, with zitterbewegung and Thomas precession Dirac particles in a central gravitational field: Understood since very recently, with the spectrum lifting the (n,j) degeneracy Global dilaton transformation might relate the fine-structure constant to the gravitational interaction and suggest a variational principle
Thanks for Your Attention!
Kaluza-Klein Theories
Take four space-time dimension, and an extra one which is compactified.
Take four space-time dimension, and an extra one which is compactified. κ =(16"π"G) 1/2 " The fine-structure constant is predicted to be proportional to G!