gravitational interactions and fine-structure...

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)

Gravitational Interactions and Fine-Structure Constant

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].

Beyond the Dirac formalism. Self-energy effects, corrections to the Coulomb force law, So–called recoil corrections, Feynman diagrams... f"

Schrödinger Theory (Nonrelativistic):

Dirac-Coulomb Theory (Relativistic):

QED (Quantized Fields, Relativistic, Recoil):

Relativistic Correction Terms

Obtained after Foldy-Wouthuysen Transformation

(Unitary Transformation of the Dirac-Coulomb Hamiltonian).

Expansion in the Momentum Operators p ~ Zα.

Nonrelativistic Limit of Dirac Theory

Relativistic Correction Terms: Dirac and Foldy-Wouthuysen

One-Particle Theory in the Coulomb Potential:

Dirac-Coulomb Hamiltonian:

Dirac-Coulomb Hamiltonian, Foldy-Wouthuysen Transformation and Correction Terms

Coulomb Potential:

Result of Foldy-Wouthuysen Transformation of the Dirac-Coulomb Hamiltonian (2 x 2 Block Structure):

There is no particle-antiparticle symmetry (no universal prefactor β). Electrons and attracted, whereas positrons are repulsed by the Coulomb field.

Particle-Antiparticle Symmetry (β Matrix):

Once More…

Relativistic Correction Terms: Foldy-Wouthuysen Transformation

Expectation Value in a Positive-Energy Schrödinger Eigenstate:

Particle-Antiparticle Symmetry (β Matrix):

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)

Dirac Equation and Dirac Action in Flat Space:

Classical Geodesic in Curved Space

Covariant Derivative of Vector:

Curved-Space Dirac Algebra:

Covariant Derivative of Gamma Matrix:

Vierbein and Affine Connection Matrix

Well-Known Solution (“Spin Connection”):

The Dirac equation constitutes one of the most versatile instruments of physics...

Factor...

Quantum Mechanical Dirac Particle in Curved Space

Lorentz Invariance of Gamma Matrices:

Curved-Space Dirac Lagrangian:

Curved-Space Dirac Algebra (Overlined Matrices):

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 [rs = 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…

Relativistic Kinetic Correction Leading Gravitational Term

Gravitational Breit Term

Gravitational Zitterbewegung (Darwin) Term

Gravitational Spin-Orbit Coupling

Now for Gravitational Coupling… Second Transformation…

[U.D.J. and J. H. Noble, Phys. Rev. A 88, 022121 (2013)]

[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.

[This term breaks parity. Why? Well, spin is pseudo-vector but g vector.]

Foldy-Wouthuysen Transformed Dirac-Schwarzschild Hamiltonian:

Another Result from the Literature with a Spurious Spin-Gravity Coupling:

Discussion of the Foldy-Wouthuysen Transformed Gravitationally Coupled

Dirac-Schwarzschild Hamiltonian

(with reference to the Dirac-Coulomb Hamiltonian)


Nonrelativistic Theory (rs is the Schwarzschild Radius):

Result of Foldy-Wouthuysen Transformation:

Perfect Particle-Antiparticle Symmetry (Overall Prefactor β):


Relativistic Kinetic Correction Leading Gravitational Term

Gravitational Breit Term

Gravitational Zitterbewegung (Darwin) Term

Gravitational Spin-Orbit Coupling [in agreement with the Classical Geodesic Precession derived in 1920 by A.D.Fokker]

[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


…but larger for other mass configurations…

Gravitational Quantum Bound States [e-print 1403.2955 (2014), to appear in the Annalen der Physik (Berlin)]:

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

Gravitational Correction to the Quadrupole Coupling Gravitational Correction to the Magnetic Coupling

The terms without rs have been known before and are used in Lamb shift calculations

Global Dilaton

Invariance Properties under the Global Dilation:

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.

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.

κ =(16"π"G)1/2"

Take four space-time dimension, and an extra one which is compactified.

The fine-structure constant is predicted to be proportional to G!

gravitational interactions and fine-structure...

Documents