the isolated electron: de broglie's „hidden“ thermodynamics ...hofmann/r_hofmann...de...
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29 September 2017
ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
The isolated electron: De Broglie's „hidden“ Thermodynamics,
SU(2) Quantum Yang-Mills theory, and a strongly perturbed BPS Monopole
International Conference on Numerical Analysis and Applied Mathematics 2017, Thessaloniki
R. Hofmann, Institut für Theoretische Physik, Universität Heidelberg
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
outline
Louis de Broglie's arguments on existence of „thermodynamics of the isolated electron“
![Page 3: The isolated electron: De Broglie's „hidden“ Thermodynamics ...hofmann/R_Hofmann...De Broglie's „hidden“ Thermodynamics, SU(2) Quantum Yang-Mills theory, and a strongly perturbed](https://reader034.vdocuments.mx/reader034/viewer/2022051808/6008b2c09744561e364b3693/html5/thumbnails/3.jpg)
ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
outline
Louis de Broglie's arguments on existence of „thermodynamics of the isolated electron“
phase diagram of SU(2) Quantum Yang-Mills thermodynamics - deconfining, preconfining, and confining phases - thermal ground state and its excitations in deconfining phase - center-vortex loops in confining phase
![Page 4: The isolated electron: De Broglie's „hidden“ Thermodynamics ...hofmann/R_Hofmann...De Broglie's „hidden“ Thermodynamics, SU(2) Quantum Yang-Mills theory, and a strongly perturbed](https://reader034.vdocuments.mx/reader034/viewer/2022051808/6008b2c09744561e364b3693/html5/thumbnails/4.jpg)
ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
outline
Louis de Broglie's arguments on existence of „thermodynamics of the isolated electron“
phase diagram of SU(2) Quantum Yang-Mills thermodynamics - deconfining, preconfining, and confining phases - thermal ground state and its excitations in deconfining phase - center-vortex loops in confining phase
electric-magnetically dual interpretation of U(1) charges in SU(2) YMTD
- value of coupling , caloron action - finestructure constant → dual interpretation of
![Page 5: The isolated electron: De Broglie's „hidden“ Thermodynamics ...hofmann/R_Hofmann...De Broglie's „hidden“ Thermodynamics, SU(2) Quantum Yang-Mills theory, and a strongly perturbed](https://reader034.vdocuments.mx/reader034/viewer/2022051808/6008b2c09744561e364b3693/html5/thumbnails/5.jpg)
ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
outline
Louis de Broglie's arguments on existence of „thermodynamics of the isolated electron“
phase diagram of SU(2) Quantum Yang-Mills thermodynamics - deconfining, preconfining, and confining phases - thermal ground state and its excitations in deconfining phase - center-vortex loops in confining phase
electric-magnetically dual interpretation of U(1) charges in SU(2) YMTD
- value of coupling , caloron action - finestructure constant → dual interpretation of
strongly perturbed BPS monopole
- expanding spherical shells - breathing mode of monopole core
![Page 6: The isolated electron: De Broglie's „hidden“ Thermodynamics ...hofmann/R_Hofmann...De Broglie's „hidden“ Thermodynamics, SU(2) Quantum Yang-Mills theory, and a strongly perturbed](https://reader034.vdocuments.mx/reader034/viewer/2022051808/6008b2c09744561e364b3693/html5/thumbnails/6.jpg)
ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
outline
Louis de Broglie's arguments on existence of „thermodynamics of the isolated electron“
phase diagram of SU(2) Quantum Yang-Mills thermodynamics - deconfining, preconfining, and confining phases - thermal ground state and its excitations in deconfining phase - center-vortex loops in confining phase
electric-magnetically dual interpretation of U(1) charges in SU(2) YMTD
- value of coupling , caloron action - finestructure constant → dual interpretation of
strongly perturbed BPS monopole
- expanding spherical shells - breathing mode of monopole core
electron as self-intersecting, spatial center-vortex loop: size estimates
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Louis de Broglie: reminder of derivation of wavelength , I
- consider particle at rest as a `clock´ with frequency which extents through a large portion of space, that is, amplitude of oscillation is the same everywhere:
- by the rules of QM: rest energy equals quantum of action times :
- consider Lorentz-trafo in direction:
spatially “flat” oscillation becomes propagating wave with
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Louis de Broglie: reminder of derivation of wavelength , II
- on the other hand:
or upon substitution of
-
existence of `clock´ frequency essential in derivation of
de Broglie wavelength !
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Thermodynamics of isolated electron: Louis de Broglie
- from demand that entropy
is a fundamental invariant under the Poincare group and from
(considering an adiabatic variation of rest mass due to input of heat)
- increase of rest energy under boost can be decomposed into a decrease of internal heat to
and work invested to boost the particle
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Thermodynamics of isolated electron: Louis de Broglie
Particle with internal heat
can not be point-like.
How does SU(2) Yang-Mills thermodynamics
come in?
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Phase diagram of SU(2) Quantum Yang-Mills thermodynamics
deconfining preconfining confining
Hagedorn 2nd order
coupling
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Phase diagram of SU(2) Quantum Yang-Mills thermodynamics:Confining phase
excitations: n-fold self-intersecting center-vortex loops
stable
unstable
multiplicity [Bender & Wu, 1969]
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Proposal for electron's anatomy
(i) electric-magnetically dual interpretation of U(1) charges in SU(2)
(ii) classical dynamics of strongly perturbed BPS monopole [Fodor & Racz, 2004]
(iii) structure of deconfining thermal ground state, small-holonomy (anti)calorons, dissociation of large-holonomy (anti)calorons [Herbst & RH, 2004; RH, TQYM,2016]
based on:
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
(i) electric-magnetically dual interpretation: deconfining-phase gauge coupling and caloron action
thermodynamical consistency
small-mass expansion, pressure[Dolan et Jackiw (1974)]
transition to preconfining phase
SU(3)
SU(2)
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
thermodynamical consistency
small-mass expansion, pressure[Dolan et Jackiw (1974)]
transition to preconfining phase
SU(3)
SU(2)
- restore
coarse-graining dominated by
[Brodsky et al. (2011); Kaviani & RH 2012;RH 2012,2013]
(i) electric-magnetically dual interpretation: deconfining-phase gauge coupling and caloron action
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
if SU(2) something to do with photons [RH (2005), Grandou & RH (2015), etc] then electric-magnetically dual interpretation required: in units fine-structure constant
(i) electric-magnetically dual interpretation: argument involving QED fine-structure constant
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
if SU(2) something to do with photons [RH (2005), Grandou & RH (2015), etc] then electric-magnetically dual interpretation required: in units fine-structure constant
(i) electric-magnetically dual interpretation: argument involving QED fine-structure constant
![Page 18: The isolated electron: De Broglie's „hidden“ Thermodynamics ...hofmann/R_Hofmann...De Broglie's „hidden“ Thermodynamics, SU(2) Quantum Yang-Mills theory, and a strongly perturbed](https://reader034.vdocuments.mx/reader034/viewer/2022051808/6008b2c09744561e364b3693/html5/thumbnails/18.jpg)
ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
if SU(2) something to do with photons [RH (2005), Grandou & RH (2015), etc] then electric-magnetically dual interpretation required: in units fine-structure constant
for to be unitless:
( )
(i) electric-magnetically dual interpretation: argument involving QED fine-structure constant
![Page 19: The isolated electron: De Broglie's „hidden“ Thermodynamics ...hofmann/R_Hofmann...De Broglie's „hidden“ Thermodynamics, SU(2) Quantum Yang-Mills theory, and a strongly perturbed](https://reader034.vdocuments.mx/reader034/viewer/2022051808/6008b2c09744561e364b3693/html5/thumbnails/19.jpg)
ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
if SU(2) something to do with photons [RH (2005), Grandou & RH (2015), etc] then electric-magnetically dual interpretation required: in units fine-structure constant
for to be unitless:
But: magnetic coupling in SU(2)
SU(2) to be interpreted in an electric-magnetically dual way. (e.g., magnetic monopole electric monopole, etc.)
( )
(i) electric-magnetically dual interpretation: argument involving QED fine-structure constant
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
(ii) classical dynamics of strongly perturbed BPS monopole
- hyperboloidal conformal transformation of the original field equations for the profiles
and
of the adjoint Higgs and off-Cartan gauge fields fields, respectively
- under a spherically symmetric initial pulse in energy density: - high-frequency oscillations in forming expanding shells
[Fodor & Racz, 2004]
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
(ii) classical dynamics of strongly perturbed BPS monopole
- a localized breathing state appears in association with the energy density within the monopole core whose frequency approaches the mass
of the two off-Cartan modes in a power-like way in time
[Fodor & Racz, 2004]
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
(ii) classical dynamics of strongly perturbed BPS monopole
There is a generic circular frequency
associated with core breathing of the (initially perturbed) BPS monopole, that is, a clock in de Broglie's sense.
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
(iii) structure of deconfining thermal ground state
- If monopole liberated by dissociation of a large-holonomy caloron in deconfining phase at temperature then
[Diakonov et al 2004]
- Only (anti)calorons of scale contribute to emergent a priori estimate of thermal ground state coupling relevant in describing caloron dissociation and ensuing monopole- antimonopole
[Herbst & RH 2004; RH 2016]
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
(iii) structure of deconfining thermal ground state
[Diakonov et al 2004]We have:
(A)
(B)
with:
We assume static BPS monopole of mass
to be immersed into blob of deconfining phase of radius at .
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Size estimate for self-intersection region:
[RH, 2017]
Thus, from (A) and (B) we have:
Comparing this to the Compton wave length m,
(~ 4.12 core size of monopole)
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
hierarchy between Yang-Mills scale (~ ) and electron mass and Coulomb self-energy:
from
eV
For comparison: average electron temperature at ITER envisaged to be
eV !!
Coulomb self-energy correction:
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ICNAAM 2017 R. Hofmann, ITP, Uni Heidelberg
Summary and outlook ● brief review of Louis de Broglie's argument on electron's inherent
thermodynamics
● phase structure of SU(2) YMTD, deconfining phase, confining phase
● electron as a one-fold self-intersecting center-vortex loop
● strongly perturbed BPS monopole
● structure of deconfining thermal ground state
● size estimates
● Coulomb self-energy and relation between Yang-Mills scale and crit. Temp.
● open questions: - spherically symmetric perturbation of BPS monopole generalizable to other perturbations - spatial extent of loops (depending on environmentally provided resolution?) - Schrödinger wave mechanics facilitated by small core sizes vs. - What happens under acceleration? (radiation, radiation reaction,...)