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Ulrich D. Jentschura Missouri University of Science and Technology
Rolla, Missouri and MTA-DE Particle Physics Research Group,
P.O.Box 51, H-4001 Debrecen, Hungary
University of Melbourne School of Physics
Colloquium 30-JUL-2013
(Research Supported by NSF, NIST, MRB in the years 2009-2013)
Fundamental Constants: Theory and Experiment
U.D.J.’s Areas of Interest:
Particle Physics and Field Theory [B-Bar Mixing (1998), Neutrinos (2012-today)]
QED Theory of Bound States
[Two-Loop Effects (2003-2005), Helium g Factor (2012)]
Atomic Processes [Two-Photon Decay Rate (2007-2009), Casimir Effects (2010-today)]
Theory and Experiment: Fundamental Constants
[Variation of Fundamental Constants (2003-2004), with Ted Hänsch]
Further Activity Areas:
Theoretical Relativistic Laser Physics
[Laser-Dressed Lamb Shift, Laser-Assisted Bremsstrahlung,
Laser-Assisted Vacuum Polarization, Triple Compton Scattering and Entanglement (2003-today)]
Relativistic Quantum Mechanics and General Relativity [Gravitationally Coupled Dirac Equation for Antimatter,
Gravitational Spin-Orbit Coupling (starting 2013)]
Spectra of Anharmonic Oscillators [with Jean Zinn-Justin, Head of CEA Saclay/DAPNIA
(2004-today)]
Numerical Algorithms [e.g., CNC Transformation (1999)]
(U.D.J. is a Member of the Editorial Board/Physical Review A)
Upcoming Book:
The QED Book
Classical Electrodynamics, Bound-State QED, Casimir E!ects
Ulrich D. Jentschura
! !!E
Just for reference, that’s me:
For reference, that’s also me, not someone with the same name:
[1] The broad, acquired skill set can be
used with ease in different areas of physics.
[2] One can sometimes find connections between different areas that would not be visible otherwise.
[3] A wider perspective on physics is sometimes
useful when leading a research group.
What is the Relation of Nonrelativistic to Relativistic Physics?
What are the So-Called Relativistic
“Correction Terms” About?
One always hears about the “Relativistic Corrections” In Quantum Physics, notably, Atomic Physics…
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
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:
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:
With Relativistic Corrections for Particles and Antiparticles:
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 and attracted, whereas positrons are repulsed by the Coulomb field.
Quantum Particle in a Gravitational Field
Nonrelativistic Theory (rs is the Schwarzschild Radius):
With Relativistic Corrections (Spin-Orbit and Zitterbewegung):
Perfect Particle-Antiparticle Symmetry (Overall Prefactor β):
Quantum Particle in a Gravitational Field [ Result Derived in … 2013 (Surprisingly!) ] 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. Jentschura and J. H. Noble, e-print 1306.0479, Phys. Rev. A, in press (2013)]
Quantum Result [e-print 1306.0479 (2013)]:
Classical Result (Fokker, de Sitter, Schouten):
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]
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. Jentschura and J. H. Noble, e-print 1306.0479, Phys. Rev. A, in press (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:
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
Dirac-Coulomb Hamiltonian:
Comparison to the Well-Known Dirac-Coulomb Result:
Coulomb Potential:
Foldy-Wouthuysen Transformed Dirac-Coulomb Hamiltonian:
There is no particle-antiparticle symmetry (no universal prefactor β). Electrons and attracted, whereas positrons are repulsed by the Coulomb field.
Relativistic Correction Terms: Foldy-Wouthuysen Transformation
QED of Bound States
Beyond the Dirac formalism. Self-energy effects, corrections to the Coulomb force law, So–called recoil corrections, Feynman diagrams... f
Theory of Bound Systems: Three Developments
Schrödinger Theory:
Dirac Theory:
QED:
Relativistic Correction Terms: Dirac and Foldy-Wouthuysen
Lamb-Shift Phenomenology
Lifts 2S-2P degeneracy:
Lamb-Shift Phenomenology
Shifts nS-n’S transition frequencies:
The Predictive Power of QED…
For this level of accuracy, one needs bound-state QED, i.e., the formalism of relativistic bound-state quantum field theory.
[U.D.J., S. Kotochigova. E. Le Bigot, P. J. Mohr and B. N. Taylor, Phys. Rev. Lett. 95, 163003 (2005)]
QED does not only predict transition frequencies (which are crucial for the determination of the Rydberg constant) but also
plays a crucial role in the determination of other fundamental physical constants.
Example: g factors and masses of fundamental particles.
(Research heavily supported by DFG
in the years 2004-2008)
Bound-Electron g Factor (S states)
Combination of the relativistic expansion with the loop expansion:
Terms marked in black correspond to the loop expansion of the free-electron g factor.
All Two-Loop Diagrams for the Bound-Electron g Factor
The two-loop coefficients read:
Analytic Results for the Two-Loop Correction
[K. Pachucki, A. Czarnecki, U.D.J. and V. A. Yerokhin, Phys. Rev. A 72, 022108 (2005)]
Two-Loop Self-Energy Corrections to the Bound-Electron g Factor
The logarithmic coefficient is generated exclusively by the displayed set of diagrams. Its value is surprisingly large and equal to 3.1.
A careful analysis of all contributing physical effects leads to the following table, for selected ions of current experimental interest:
New Theoretical Predictions for the g-Factor
Determination of the Electron Mass
The experimental uncertainty due to the Larmor-to-cyclotron frequency dominates over the theoretical uncertainty due to the g-factor. Consequence: The electron mass could be determined more accurately by measuring the bound-electron g-factor in carbon and/or oxygen more precisely; the current theory is not a limiting factor any more.
The following values may be derived from the recent carbon and oxygen-ion measurements, using the latest theory:
Muonic Hydrogen and Lamb Shift
Up to 2010: QED and experiment were
essentially in agreement, but then…
CODATA: rp = 0.8768(69) fm
electronic H: rp = 0.8802(80) fm
Scattering (Mainz, 2010): rp = 0.879(8) fm
Scattering (Jefferson Lab, 2011): rp = 0.875(10) fm
(essentially 0.88 fm) BUT
muonic H: rp = 0.84184(67) fm
(essentially 0.84 fm)
Muonic Hydrogen Puzzle
[R. Pohl et al., Nature 466 (2010) 213] [A. Antognini et al., Science 339 (2013) 417]
You calculate the spectrum. [Nonrelativistic Theory.]
You calculate the spectrum more accurately.
[Relativistic Effects.]
You calculate the spectrum even more accurately. [QED effects.]
At some point the nuclear size becomes important.
[Distortion of Coulomb Potential.]
Someone else measures the spectrum. [And then you can tell what the nuclear size is.]
Why Can You Determine Nuclear Radii from Spectroscopy?
Finite-Size Hamiltonian
[Affects S States with a Nonvanishing Probability Density at the Origin]
(proportional to the Dirac-δ function, measures probability density of the
electronic wave function at the origin)
[R. Pohl et al., Nature 466 (2010) 213] [A. Antognini et al., Science 339 (2013) 417]
Now the Theory: Vacuum Polarization Diagram
µ µ
p p
e e
µ µ
p p
Vacuum Polarization Effects.
The Coulomb law is incorrect at small distances.
Muonic hydrogen is smaller than atomic hydrogen by a factor of 207 (mass ratio of muon to electron).
The vacuum polarization energy shift is 40,000 times larger in muonic hydrogen.
Reason:
Generation of virtual electron-positron pairs in the vicinity of the proton.
The quantum vacuum has structure!
µ µ
p p
e e
µ µ
p p
µ µ
p p
e e
µ µ
p p
Generation of so-called virtual electron-positron pairs leads to modifications of the Coulomb force law at distance scales comparable to the electron Compton wavelength. For long distances, the modification in exponentially suppressed. (The 2P state is energetically higher, for muonic hydrogen)
(Coulomb Law)
(Mass Ratio)
(Quantum Correction)
Conspiracy of Self-Energy and Vacuum Polarization
µ µ
e e
pp
µ
p
e e
µ
e e
p
-0.0025 meV for 2P-2S µH
Dominant Theoretical Uncertainty: Recoil Correction to Vacuum Polarization
[Vacuum-Polarization Insertion in Two-Photon Exchange]
Now, that calculation is difficult. Logarithmic Terms Calculated: 0.0005 meV for 2P-2S µH
Latest News...
Muonic Hydrogen Discrepancy: 0.420 meV.
Largest Conceivable Uncertainty within Standard Model: +/- 0.010 meV.
All relevant theorists agree.
The fact that no theoretical explanation for the observed discrepancy exists was pointed out by U.D.J.
in two articles, published in Annals of Physics (N.Y.) in 2011.
Conundrum remains unsolved!
New fundamental forces!?!
Status Regarding 2S-2P Lamb Shift in mH
[U.D.J., Ann.Phys.(N.Y) 326 (2011) 500] [U.D.J., Ann.Phys.(N.Y) 326 (2011) 516]
Recently Found Discrepancies Mount:
Possible Variation of Fundamental Constants...
...as an Independent Subject
Frequency Measurement = Frequency Comparison Hydrogen Frequency versus Cesium Frequency
Problem: Everything may Drift
Does the fine-structure constant drift?
Does the nuclear magnetic moment drift?
How do we infer model-independent bounds on the drifts of the fundamental constants based on laboratory measurements? [Systematics are under better control in the laboratory than they are in astrophysics.]
Variation of Fundamental Constants
MPQ Measurement 1999 (1S-2S atomic hydrogen):
MPQ Measurement 2003 (Confirmed 2011):
Measured Atomic Transitions
MPQ (Max-Planck-Institute for Quantum Optics):
NIST (National Institute for Standards and Technology):
Repeat the measurements over intervals of two- to three years and infer the time variations of the “constants”!
At MPQ and NIST, there exist experimental possibilities to measure two narrow atomic resonances to very high accuracy,
using phase coherent comparison to the cesium time standard which defines the SI unit of time.
Einstein's Equivalence Principle (EEP)
In an inertial system, the result of any experiment not influenced by gravitation has to be independent of space and time. A variation of fundamental constants with time would lead to a violation of Einstein’s equivalence principle. [P. A. M. Dirac, Nature 192, 235 (1937)]
Left side: time It takes light to travel a distance of the order of the Electron Compton wavelength, right side: relative strength of electromagnetic interaction of electron and proton versus gravitation. Physicist’s courageous hypothesis: equal quantities are proportional!
Time Variation of Frequencies
General formula for an atomic transition frequency:
Hg, numerically (Atomic Physics Calculation):
H, analytically (Dirac):
Time Variation of Clock Transition
Numerical analysis:
H and Hg frequencies are compared to the clock transition.
The clock transition is a hyperfine transition:
Correlation-Independent Evaluation of MPQ and NIST Measurements
Define:
Input from MPQ and NIST:
Solving for x and y:
[PRL 92, 230802 (2004)] Third transition: Have one more property of the nucleus (one more unknown) and one more equation: Can solve for dα/dt.
Drifting Constants: Observations Involving Long Timescales
Murphy, Webb, Flambaum: MNRAS 345, 609 (2003): [meanwhile, improved and refined]
Fujii et al., Nucl. Phys. B 573, 377 (2000):
Natural fission reactor in Gabon, which operated some 2 Gyr ago.
Drifting Constants: RG arguments (W. Marciano in Springer Lecture Notes in Physics 648)
The (possible) drift of constants provides a relation of coupling constants in different space-time points. The RG joins the behaviour of coupling constants on different length/momentum scales. These aspects can be used to make an educated guess about where to look for drifts:
So, it might be assumed that a drift of the constants would rather be visible in the QED coupling at the Z boson scale than at the electron mass scale, and that a drift of the strong coupling constant would rather be visible in the nuclear magnetic moment of a hadron (which is inversely proportional to the hadronic mass) than in a high-energy experiment.
Drifting Constants: Relations Between Drifts (H. Fritzsch)
Assuming a unification of all fundamental forces at a GUT scale, one may, under additional assumptions, derive relations between the drift rate of various „constants“:
A drift of the fundamental scale of the strong interaction shifts hadronic masses and magnetic moments. Astrophysical observation: fine-structure constant may grow, the scale of the strong interaction and hadronic masses decrease, and the nuclear moments (of cesium) should increase.
Drifting Constants: Kaluza-Klein Theories
(C. Kiefer and others)
Take four space-time dimension, and an extra one which is compactified. A coordinate transformation in the fifth dimension corresponds to a gauge transformation of the electromagnetic field.
If and only if the scalar field assumes the constant value -e-2 , then we obtain the usual coupling of the e.m. field to gravity. Kaluza-Klein and string theories are consistent with drifting constants.
The Story Does Not End There…
Proposal for an Electron-EDM Experiment using an Atomic Fountain
Proposal for an Electron-EDM Experiment using an Atomic Fountain
The Radiatively Corrected QED Interaction:
The Vertex Function:
The Electron-EDM Term (Single e-):
The Electron-EDM Term (Enhanced for an Atom):
The Quantum Dynamical Hamiltonian:
Notion A: to use a strong electric rather than magnetic field, in order to interrogate the EDM term directly Notion B: to design an observable which can be Read out from the quantum dynamics in the apparatus. Notion C: To use an alkali atom with an upper S1/2|F=I+1/2, M> and a lower S1/2|F=I-1/2, M> hyperfine manifold
Proposal for an Electron-EDM Experiment using an Atomic Fountain
Step 1: Prepare the atom in a coherent superposition
of states within the upper hyperfine manifold
Step 2: Atom goes up-down in the fountain and evolves according to the EDM term.
Step 3: After many S1/2->P1/2 transitions (absorption
and spontaneous emission, conceivably to the lower hyperfine manifold, the atom jumps into the dark
state of the upper hyperfine manifold with probability pM, leading to a geometry-dependent survival probability S(θ).
Otherwise, it ends up in the lower hyperfine manifold.
Step 4. Using an S1/2->P3/2 cycling transition, one obtains a fluorescence signal proportional to S(θ).
One normalizes the signal by pumping atoms rom the lower hyperfine manifold into the upper manifold and
fluorescing the S1/2->P3/2 cycling transition again.
Illustration of Step 3: After many S1/2->P1/2 transitions the atom jumps into the darkstate of the upper hyperfine manifold with probability pM, leading to a survival probability S(θ).
How Can We Apply Relativistic Electron Theory Beyond Atomic Physics and Gravitational Coupling?
Answer: Laser Physics
Essential Ingredient: Dirac-Volkov Propagator
Furry Picture of Bound-State QED
Replacement of the Propagator:
Free Self-Energy: Bound-State Self-Energy:
Furry Picture of Laser Physics
Replacement of the Propagator:
Without Laser: With Laser:
Electron Propagator in the Laser-Dressed Furry Picture
Laser-Dressed QED (Aµ is time-dependent!)
Laser-dressed electron propagator can be expanded into plane waves.
[S. Schnez, E. Lotstedt, U.D.J. and C. H. Keitel, Phys.Rev.A 75, 053412 (2007)]
S-Matrix time and space integrations can be performed after this expansion.
The evaluation of the coefficients in the
expansion is notoriously problematic, as they involve so-called generalized Bessel functions
[E. Lötstedt and U.D.J., Phys.Rev.E 79, 026707 (2009)]
General Dirac-Volkov Solution
Averaging:
Laser-Dressed Mass:
Free Bispinor Solution
Phase Factor:
General Dirac-Volkov Propagator
Adjoint:
Sum Rules:
General Formula (But Not Useful for Practical Calculations):
Dirac-Volkov Solution (Linear Polarization)
Linear polarization. The Dirac-Volkov solution is a superposition of admixtures with s laser photons…
The admixture amplitudes are given by Generalized Bessel Functions…
Understanding Dirac-Volkov States: Quantum-Classical Correspondence
A0(s,α,β) gives the amplitude ω = 1 eV
I = 1016 W/cm2
α = 3.3 x 103
β = -2.7 x 103
Left: Classical Trajectory Right: Admixture Amplitude Quantum Classical Correspondence: [ Classical Emax ] = [ (q0 + s ω) at the cutoff index scut ]
A0(s,α,β)
s
[E. Lötstedt and U. D. Jentschura, Phys. Rev. E 79 (2009) 026707]
Generalized Bessel Functions
Generalized Bessel functions are sums over Bessel functions:
Recursion Relations connect different L:
Five-term recursion relations connecting different s are useful for numerical calculations! [See E. Lötstedt and U. D. Jentschura, Phys. Rev. E 79 (2009) 026707] The recursive method is akin to Miller’s method for arrays of Bessel functions, adapted to Generalized Bessel Functions (but there are caveats)
Dirac-Volkov Propagator (Linear Polarization)
We recall the generalized Bessel functions in the expansion of the Dirac-Volkov solution:
The explicit form of the propagator for linear polarization:
[S. Schnez, E. Lotstedt, U. D. J. and C. H. Keitel, Phys.Rev.A 75, 053412 (2007)]
Dirac-Volkov Propagator (Linear Polarization)
We recall the generalized Bessel functions in the expansion of the Dirac-Volkov solution:
The explicit form of the propagator for linear polarization:
[Once you have the propagator, you calculate anything you wish…]
Laser-Assisted Bremsstrahlung
Laser-Assisted Bremsstrahlung
0 1 2 3 4 5 6 7 810
0
1010
1020
!b/!
d"
/(d#
b d!
b)[
barn
/MeV
]
I=5.2! 1020
W/cm2
I=1.7! 1020
W/cm2
I=4.3! 1019
W/cm2
I=0
30000 30005 30010 30015 30020 30025 30030
105
1010
1015
1020
!b/!
d"
/(d#
b d!
b)[
barn
/MeV
]
Compton forward-scattering geometry Compton backscattering geometry
Feynman diagrams
[E. Lötstedt, U.D.J. and C.H. Keitel, Phys. Rev. Lett. 98 (2007) 043002]
Laser Channeling of Bethe-Heitler Pairs
Laser Channeling of Bethe-Heitler Pairs
[E. Lötstedt, U.D.J. and C.H. Keitel, Phys. Rev. Lett. 101 (2007) 203001]
e+
e!
!+
!!gamma photon
laser field k!
k!
q
q
p̃!
q!q!
q+ q+
p̃+
!"# !"#$ !"% !"%$ !"& !"&$!
!"$
#
#"$
%
%"$
'(#!!)
!+[*+,]
d"/d
!+[-./
!2]
# = 6# = 10
# = 0 [!102]
!"#$ % %"&% %"&' %"&( %"&$
!&!!$
!&!!(
!&!!'
!&!!%
!&!!&
!!/m
")*)[+,-
!2]
# = 10# = 0
ξ = -e E/m c ω$
Double-Compton Scattering with a Laser:
Correlated Two-Photon Emission [Some people call this „Unruh Radiation“]
Correlated Two-Photon Emission
[E. Lötstedt and U.D.J., Phys. Rev. Lett. 103 (2009) 110404] [E. Lötstedt and U.D.J., Phys. Rev. A 81 (2009) 053419]
Perturbative Double Compton Scattering:
Laser-Dressed Nonperturbative Double Compton Scattering:
Laser-Assisted Double Compton Backscattering The scattered photon energy is highest for direct backscattering.
The absorption of laser photons leads to additional bright lines. The space in between the bright lines is filled by two-photon
transitions of Dirac-Volkov states
[E. Lötstedt and U.D.J., Phys. Rev. Lett. 103 (2009) 110404] [E. Lötstedt and U.D.J., Phys. Rev. A 81 (2009) 053419]
Laser-Assisted Double Compton Backscattering The scattered photon energy is highest for direct backscattering.
The absorption of laser photons leads to additional bright lines. The space in between the bright lines is filled by two-photon
transitions of Dirac-Volkov states
Singularities in the Differential Two-Photon Emission Probability: Cascade Emission of Two Photons (Just like in Atomic Physics)
Triple-Compton Scattering with a Laser:
Correlated Three-Photon Emission
Correlated Three-Photon Emission:
…+ 21 more diagrams (commutations)…
[E. Lötstedt and U.D.J., Phys.Rev.Lett. 108 (2012) 233201] [E. Lötstedt and U.D.J., Phys.Rev.A 87 (2013) 033401]
Conclusions
Relativistic Quantum Field Theory, Low-Energy Precision Physics and High-Field Laser Science
Highly Accurate QED Predictions for Atomic Transition Frequencies in Simple Atomic Systems [Loop Corrections, Radiative-Recoil]. Also: g factor, Hyperfine Splitting. Determination of Fundamental Physical Constants with Unprecedented Accuracy.
Nonrelativistic Limit and Correction Terms in Electromagnetic and Gravitational Fields
Fruitful Direct Collaboration on Experimental Setups and Systematic Effects
Dirac-Volkov Propagator For Intense Laser Fields
Beyond “Testing QED”
Recommended