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Towards the loop quantum gravity
with compact phase space
Jakub Mielczarek
Jagiellonian University
Featuring:
• G. Czelusta, J. M. Quantum variational solving of the Wheeler-DeWitt equation,
arXiv: 2111.03038 (2021)
• D. Artigas, J. M. and C. Rovelli, A minisuperspace model of compact phase
space gravity, Phys. Rev. D 100, 043533 (2019)
Compact phase spaces arise in the semiclassical description of
quantum systems with finite dimensional Hilbert spaces.
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<latexit sha1_base64="WPdPUMlSpvGUjKpqLEsfjgmdehA=">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</latexit>
Compact phase space Finite Hilbert space
We are focused on the compact case rather on the phase space with boundaries.
V =
Z
!
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<latexit sha1_base64="VVaE4Zhqb2T2gdbIGK9eZ+MwARI=">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</latexit>
dim H ⇠
V
(2⇡})m
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<latexit sha1_base64="19319sJZKVy34wFFtEUMh88exUc=">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</latexit>
dim = 2m
<latexit sha1_base64="0C/rV5vg21JJjCKQVvfevV0kEZQ=">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</latexit>
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<latexit sha1_base64="0C/rV5vg21JJjCKQVvfevV0kEZQ=">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</latexit>
<latexit sha1_base64="oVAdWF4w/M7B3DtPDa8VFSPciJE=">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</latexit>
dim H < 1 , V < 1
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Nonlinear Field Space Theory (NFST)
J. M. and T. Trześniewski, The Nonlinear Field Space Theory, Phys. Lett. B 759, 424 (2016)
J. M., Spin-Field Correspondence, Universe 3, 29 (2017)
J. M. and T. Trześniewski, Nonlinear Field Space Cosmology, Phys. Rev. D 96, 043522 (2017)
J. Bilski, S. Brahma, A. Marcianò and J. M., Klein–Gordon field from the XXZ
Heisenberg model, Int. J. Mod. Phys. D 28, 1950020 (2018)
D. Artigas, J. M. and C. Rovelli, A minisuperspace model of compact phase space gravity,
Phys. Rev. D 100, 043533 (2019)
Grant Sonata Bis DEC-2017/26/E/ST2/00763 of the
National Science Centre, Compact phase space field theories,
2018-2022
D. Artigas, J. Bilski, S. Crowe, J. M. and T. Trześniewski, 
General-relativistic spin system, Phys. Rev. D 102, 125029 (2020)
D. Artigas, S. Crowe and J. M., Quantum fluctuations of the compact phase
space cosmology, Class. Quantum Grav. 38, 085007 (2021)
- compact phase space generalization of field theories
Why compact phase space for gravity?
Locally finite dimensional Hilbert space of quantum gravitational
field is supported at the theoretical ground.
Spikes (IR divergences)
Singularities (UV divergences)
Spikes in Ponzano-Regge amplitudes
are cured by q-deformations, which
introduce finite dimension of a Hilbert
space -> Turaev-Viro amplitude.
Compact phase spaces introduce
bounds on the the phase space
variables, which regularize curvature,
energy density, etc.
Model-independent arguments: N. Bao, S. M. Carroll and A. Singh, The Hilbert
Space of Quantum Gravity Is Locally Finite-Dimensional,
Int. J. Mod. Phys. D 26 (2017) no.12, 1743013
Finite horizon entropy/holography
SBH =
1
4
A
l2
Pl
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SdS ⇠
1
l2
Pl⇤
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S ⇠ ln dim H
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Loop Quantum Gravity
C. Rovelli and F. Vidotto, Compact phase space, cosmological constant, and discrete time,
Phys. Rev. D 91 (2015) no.8, 084037
⇣
e
R
A
, E
⌘
2 SU(2) ⇥ su(2)
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<latexit sha1_base64="CdkIUU6wRZtB7sjCMykF2u3+RLU=">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</latexit>
<latexit sha1_base64="IPYPn5ZESeSSS5EWKQUGOC7lfpg=">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</latexit>
The phase space of LQG per link is partially compact:
The compact phase space LQG proposal:
⇣
e
R
A
, e
R
E
⌘
2 SU(2) ⇥ SU(2)
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It has been justified that the compacatification can be associated with the
q-deformations of the SU(2) group, leading to the upper limit on j.
In the 2+1 case the deformations introduce the IR scale, which is the
cosmological constant: q = ei⇤l2
Pl
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M. Artymowski and J.M, Quantum Hubble horizon, Eur. Phys. J. C 79 (2019) no.7, 632
The deformations lead to possible solution to the initial homogeneity problem:
(holonomies and fluxes)
Loop Quantum Cosmology
In the homogeneous and isotropic case, the holonomy-flux algebra reduces
to the „cylindrical algebra”:
[E, U] = U
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<latexit sha1_base64="UnOmbfCvKbf+ke14qKoH2lMnUYU=">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</latexit>
<latexit sha1_base64="pNRfvwiVPWNQtqw00nzjRV7WgJQ=">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</latexit>
[E, U†
] = U†
<latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit>
<latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit>
<latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit>
<latexit sha1_base64="e09Ffe0xrCrEK1hrx/3U1hKrT94=">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</latexit>
[U, U†
] = 0
<latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">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</latexit>
<latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">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</latexit>
<latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">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</latexit>
<latexit sha1_base64="cnHTk9HrnF7nVajidzm+Fvfga8w=">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</latexit>
where (U, E) 2 U(1) ⇥ R
<latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">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</latexit>
<latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">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</latexit>
<latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">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</latexit>
<latexit sha1_base64="sbXSnnPGHmF4tIja13kFciurF8g=">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</latexit>
Employing the fact that , one can equivalently write:
What we consider in what follows is the SU(2) generalization of the
above algebra, so that
[Sx, Sy] = iSz
<latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit>
<latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit>
<latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit>
<latexit sha1_base64="PqwCuxVi4bc0TST3FEaDyljTv/0=">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</latexit>
[ Sz, Sy] = iSx
<latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit>
<latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit>
<latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit>
<latexit sha1_base64="MFRVY5XfOc9ROQOKHua2kMEDA9E=">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</latexit>
[ Sz, Sx] = iSy
<latexit sha1_base64="xcQnj3IKJcMU7jmF5FhQjM2t0yE=">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</latexit>
<latexit sha1_base64="xcQnj3IKJcMU7jmF5FhQjM2t0yE=">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</latexit>
<latexit sha1_base64="xcQnj3IKJcMU7jmF5FhQjM2t0yE=">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</latexit>
<latexit sha1_base64="TD7yzAuB2WBWvh7scwExLF4v7wc=">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</latexit>
The associated phase space is .
S2
<latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit>
<latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit>
<latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit>
<latexit sha1_base64="FQdV073B++8mJBySUjDjnu6Lhjs=">AAACz3icjVHLTsJAFD3UF+ILdemmkZi4Ii0bXRLduIQojwTQTMsAjX2lnWoIwbj1B9zqXxn/QP/CO+OQqMToNG3PnHvPmbn3OrHvpcKyXnPGwuLS8kp+tbC2vrG5VdzeaaZRlri84UZ+lLQdlnLfC3lDeMLn7TjhLHB83nKuT2W8dcOT1IvCCzGOeS9gw9AbeC4TRHW7ARMjx5mcTy8rV8WSVbbUMueBrUEJetWi4gu66COCiwwBOEIIwj4YUno6sGEhJq6HCXEJIU/FOaYokDajLE4ZjNhr+g5p19FsSHvpmSq1S6f49CakNHFAmojyEsLyNFPFM+Us2d+8J8pT3m1Mf0d7BcQKjIj9SzfL/K9O1iIwwLGqwaOaYsXI6lztkqmuyJubX6oS5BATJ3Gf4glhVylnfTaVJlW1y94yFX9TmZKVe1fnZniXt6QB2z/HOQ+albJtle16pVQ90aPOYw/7OKR5HqGKM9TQIO8Yj3jCs1E3bo074/4z1chpzS6+LePhAxBIlBY=</latexit>
U|'i = ei'
|'i
<latexit sha1_base64="feG20PHt+/O2ArMBR3S++OcJEYg=">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</latexit>
<latexit sha1_base64="feG20PHt+/O2ArMBR3S++OcJEYg=">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</latexit>
<latexit sha1_base64="feG20PHt+/O2ArMBR3S++OcJEYg=">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</latexit>
<latexit sha1_base64="nUzdepdA7p2pYh5nTBPWGaR0Ha8=">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</latexit>
[E, cos '] = i sin '
<latexit sha1_base64="ArEWWdXLtcY0nKzD5di8Ro+QadM=">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</latexit>
<latexit sha1_base64="ArEWWdXLtcY0nKzD5di8Ro+QadM=">AAAC63icjVHLLgRBFD3a+z1Y2lRMiAWTbpGwkQiRWJIYJDMT6S6Fin6lqnoSEX9gZye2fsCW/xB/wF+4VZp4RKhOd586955Tde+N8lhq4/tPHV5nV3dPb1//wODQ8MhoZWx8V2eF4qLOszhT+1GoRSxTUTfSxGI/VyJMoljsRafrNr7XFkrLLN0xZ7loJeFxKo8kDw1RB5WZxsYca/JMs2Y7VPmJZC22wuaZZE0t0w/yoFL1a75b7CcISlBFubayyiOaOEQGjgIJBFIYwjFCaHoaCOAjJ66Fc+IUIeniAhcYIG1BWYIyQmJP6XtMu0bJprS3ntqpOZ0S06tIyTBNmozyFGF7GnPxwjlb9jfvc+dp73ZG/6j0Sog1OCH2L9175n91thaDIyy7GiTVlDvGVsdLl8J1xd6cfarKkENOnMWHFFeEuVO+95k5jXa1296GLv7sMi1r97zMLfBib0kDDr6P8yfYXagFfi3YXqyurpWj7sMkpjBL81zCKjaxhTp5X+IO93jwEu/Ku/Zu3lK9jlIzgS/Lu30FK5Sc1w==</latexit>
<latexit sha1_base64="ArEWWdXLtcY0nKzD5di8Ro+QadM=">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</latexit>
<latexit sha1_base64="6c7IDnr1TFztEgyO2wMw0p+gCrI=">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</latexit>
[E, sin '] = i cos '
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[cos ', sin '] = 0
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Spherical phase space
2-sphere is a phase space of spin - SU(2)
variable. The 2-sphere is not a cotangent
bundle, but is equipped with symplectic
form and is a well defined symplectic
manifold.
Change of variables
R1R2 = S
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Dimensional parameters
Symplectic form on 2-sphere:
Volume of phase space:
V =
Z
S2
! = 4⇡S
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Symplectic form
is, therefore, the angular momentum (spin) vector with the norm S.
Employing the above, the su(2) algebra is obtained:
with the following components:
Poisson bracket:
~
S = (Sx, Sy, Sz)
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<latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit>
<latexit sha1_base64="TJZ5tBzh19LPZSWaEzYMfbqYv0I=">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</latexit>
~
S
<latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit>
<latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit>
<latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit>
<latexit sha1_base64="TBxPMM/yPmf0T6MMvEYGyXsdj0c=">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</latexit>
One can introduce the vector
{Si, Sj} = ✏ijkSk
<latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit>
<latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit>
<latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit>
<latexit sha1_base64="I7UOLjFAAj//XAWx8T1brYw7Vag=">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</latexit>
! =
m
X
i=1
!i =
m
X
i=1
dpi ^ dqi
<latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">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</latexit>
<latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">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</latexit>
<latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">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</latexit>
<latexit sha1_base64="oqL9Gk9p0S3MMtDwUZms6ml1w+Y=">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</latexit>
The symplectic for the system of m DOFs is:
By performing compactification for every canonical pair the system’s
phase space changes as follows:
R2m
! S2m
<latexit sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">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</latexit>
<latexit sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">AAAC8HicjVHLSsNAFD2N73fVpZtgEQShJCLosujGZX30AVZlEsd2MC8mE0VKP8KdO3HrD7jVrxD/QP/CO9MUfCA6IcmZc+85M/deLwlEqhzntWANDY+Mjo1PTE5Nz8zOFecX6mmcSZ/X/DiIZdNjKQ9ExGtKqIA3E8lZ6AW84V3s6HjjkstUxNGhuk74ccjakTgXPlNEnRbXWiFTHc/r7vdOuuthz25J0e4oJmV8ZQ9iB/3YabHklB2z7J/AzUEJ+arGxRe0cIYYPjKE4IigCAdgSOk5ggsHCXHH6BInCQkT5+hhkrQZZXHKYMRe0LdNu6OcjWivPVOj9umUgF5JShsrpIkpTxLWp9kmnhlnzf7m3TWe+m7X9Pdyr5BYhQ6xf+kGmf/V6VoUzrFlahBUU2IYXZ2fu2SmK/rm9qeqFDkkxGl8RnFJ2DfKQZ9to0lN7bq3zMTfTKZm9d7PczO861vSgN3v4/wJ6utl1ym7exulynY+6nEsYRmrNM9NVLCLKmrkfYNHPOHZktatdWfd91OtQq5ZxJdlPXwAzzih0g==</latexit>
<latexit sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">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</latexit>
<latexit sha1_base64="+cKEhSbu8x1k6ugIVW++eJt129I=">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</latexit>
In consequence, the regularized kinematical Hilbert space for the system is:
Hkin = H⌦m
s
<latexit sha1_base64="krvugiKSYYl+DTn0qt0IhTvyJOY=">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</latexit>
<latexit sha1_base64="krvugiKSYYl+DTn0qt0IhTvyJOY=">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</latexit>
<latexit sha1_base64="krvugiKSYYl+DTn0qt0IhTvyJOY=">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</latexit>
<latexit sha1_base64="Iag4kIhX7mzUoKui3dcbTnqNRPE=">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</latexit>
where , so that:
Hs = span{s, si, ..., |s, si}
<latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit>
<latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit>
<latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit>
<latexit sha1_base64="iz+nNgiphiM7VeAF/vg5DllYvtY=">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</latexit>
Ŝ2
|s, szi = s(s + 1)|s, szi
<latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit>
<latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit>
<latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit>
<latexit sha1_base64="R3s46+qWvDocRDolfsve4fnR5BY=">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</latexit>
Ŝz|s, szi = sz|s, szi
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Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space
Towards the loop quantum gravity with compact phase space

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Towards the loop quantum gravity with compact phase space

  • 1. Towards the loop quantum gravity with compact phase space Jakub Mielczarek Jagiellonian University Featuring: • G. Czelusta, J. M. Quantum variational solving of the Wheeler-DeWitt equation, arXiv: 2111.03038 (2021) • D. Artigas, J. M. and C. Rovelli, A minisuperspace model of compact phase space gravity, Phys. Rev. D 100, 043533 (2019)
  • 2. Compact phase spaces arise in the semiclassical description of quantum systems with finite dimensional Hilbert spaces. <latexit sha1_base64="vYDMe4ccJRre9xMa8Jwkjr6rBaI=">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</latexit> <latexit sha1_base64="vYDMe4ccJRre9xMa8Jwkjr6rBaI=">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</latexit> <latexit sha1_base64="vYDMe4ccJRre9xMa8Jwkjr6rBaI=">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</latexit> <latexit sha1_base64="WPdPUMlSpvGUjKpqLEsfjgmdehA=">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</latexit> Compact phase space Finite Hilbert space We are focused on the compact case rather on the phase space with boundaries. V = Z ! <latexit sha1_base64="xTL/p8YT5xoSa0tUZfT8cwlJUaE=">AAAC53icjVHLSsNAFD2Nr/quunQTLIKrkoigG6HoQpcVbCs0UibTsQ7mRTIRSunenTtx6w+41T8R/0D/wjtjBB+ITkhy5tx7zsy9108CmSnHeS5ZY+MTk1Pl6ZnZufmFxcrSciuL85SLJo+DOD3xWSYCGYmmkioQJ0kqWOgHou1f7Ot4+1KkmYyjYzVIxGnI+pE8k5wporqVNS9k6pyzYNga2bu2JyPVHXoHLAzZyPbiUPRZt1J1ao5Z9k/gFqCKYjXiyhM89BCDI0cIgQiKcACGjJ4OXDhIiDvFkLiUkDRxgRFmSJtTlqAMRuwFffu06xRsRHvtmRk1p1MCelNS2lgnTUx5KWF9mm3iuXHW7G/eQ+Op7zagv194hcQqnBP7l+4j8786XYvCGXZMDZJqSgyjq+OFS266om9uf6pKkUNCnMY9iqeEuVF+9Nk2mszUrnvLTPzFZGpW73mRm+NV35IG7H4f50/Q2qy5Ts092qrW94pRl7GKNWzQPLdRxyEaaJL3Fe7xgEdLWtfWjXX7nmqVCs0Kvizr7g3ohJ1D</latexit> <latexit sha1_base64="xTL/p8YT5xoSa0tUZfT8cwlJUaE=">AAAC53icjVHLSsNAFD2Nr/quunQTLIKrkoigG6HoQpcVbCs0UibTsQ7mRTIRSunenTtx6w+41T8R/0D/wjtjBB+ITkhy5tx7zsy9108CmSnHeS5ZY+MTk1Pl6ZnZufmFxcrSciuL85SLJo+DOD3xWSYCGYmmkioQJ0kqWOgHou1f7Ot4+1KkmYyjYzVIxGnI+pE8k5wporqVNS9k6pyzYNga2bu2JyPVHXoHLAzZyPbiUPRZt1J1ao5Z9k/gFqCKYjXiyhM89BCDI0cIgQiKcACGjJ4OXDhIiDvFkLiUkDRxgRFmSJtTlqAMRuwFffu06xRsRHvtmRk1p1MCelNS2lgnTUx5KWF9mm3iuXHW7G/eQ+Op7zagv194hcQqnBP7l+4j8786XYvCGXZMDZJqSgyjq+OFS266om9uf6pKkUNCnMY9iqeEuVF+9Nk2mszUrnvLTPzFZGpW73mRm+NV35IG7H4f50/Q2qy5Ts092qrW94pRl7GKNWzQPLdRxyEaaJL3Fe7xgEdLWtfWjXX7nmqVCs0Kvizr7g3ohJ1D</latexit> <latexit sha1_base64="xTL/p8YT5xoSa0tUZfT8cwlJUaE=">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</latexit> <latexit sha1_base64="VVaE4Zhqb2T2gdbIGK9eZ+MwARI=">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</latexit> dim H ⇠ V (2⇡})m <latexit sha1_base64="00hKEIMIixq/X7jSgJtaZCbNJE4=">AAADAHicjVHLbtQwFD0NLR1KgQGWbKKOkNrNKKkqFYlNBZtZTiVmOlJTKsfj6Vh1HrIdpCrKpn/Cjh1iyw+wpSvEH7R/wbVJ1ZcqcJTk+Nx7jn3vTUsljY2i3wvBg8Wlh8udRyuPV588fdZ9/mJsikpzMeKFKvQkZUYomYuRlVaJSakFy1Il9tLj9y6+90loI4v8gz0pxUHGjnI5k5xZog67b5OpzMIkY3bOmaoHTZgYR8w04/UVP26aen0zKWWYzI1iZr7xMWsOu72oH/kV3gVxC3po17Do/kKCKQpwVMggkMMSVmAw9OwjRoSSuAPUxGlC0scFGqyQtqIsQRmM2GP6HtFuv2Vz2jtP49WcTlH0alKGeE2agvI0YXda6OOVd3bsfd6193R3O6F/2nplxFrMif2X7jLzf3WuFosZ3vgaJNVUesZVx1uXynfF3Ty8VpUlh5I4h6cU14S5V172OfQa42t3vWU+fu4zHev2vM2tcOFuSQOOb4/zLhhv9uOoH+9u9XbetaPu4BXWsE7z3MYOBhhiRN6f8QM/cRacBl+Cr8G3v6nBQqt5iRsr+P4HjMSnmQ==</latexit> <latexit sha1_base64="00hKEIMIixq/X7jSgJtaZCbNJE4=">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</latexit> <latexit sha1_base64="00hKEIMIixq/X7jSgJtaZCbNJE4=">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</latexit> <latexit sha1_base64="19319sJZKVy34wFFtEUMh88exUc=">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</latexit> dim = 2m <latexit sha1_base64="0C/rV5vg21JJjCKQVvfevV0kEZQ=">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</latexit> <latexit sha1_base64="0C/rV5vg21JJjCKQVvfevV0kEZQ=">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</latexit> <latexit sha1_base64="0C/rV5vg21JJjCKQVvfevV0kEZQ=">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</latexit> <latexit 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  • 3. Nonlinear Field Space Theory (NFST) J. M. and T. Trześniewski, The Nonlinear Field Space Theory, Phys. Lett. B 759, 424 (2016) J. M., Spin-Field Correspondence, Universe 3, 29 (2017) J. M. and T. Trześniewski, Nonlinear Field Space Cosmology, Phys. Rev. D 96, 043522 (2017) J. Bilski, S. Brahma, A. Marcianò and J. M., Klein–Gordon field from the XXZ Heisenberg model, Int. J. Mod. Phys. D 28, 1950020 (2018) D. Artigas, J. M. and C. Rovelli, A minisuperspace model of compact phase space gravity, Phys. Rev. D 100, 043533 (2019) Grant Sonata Bis DEC-2017/26/E/ST2/00763 of the National Science Centre, Compact phase space field theories, 2018-2022 D. Artigas, J. Bilski, S. Crowe, J. M. and T. Trześniewski,  General-relativistic spin system, Phys. Rev. D 102, 125029 (2020) D. Artigas, S. Crowe and J. M., Quantum fluctuations of the compact phase space cosmology, Class. Quantum Grav. 38, 085007 (2021) - compact phase space generalization of field theories
  • 4. Why compact phase space for gravity? Locally finite dimensional Hilbert space of quantum gravitational field is supported at the theoretical ground. Spikes (IR divergences) Singularities (UV divergences) Spikes in Ponzano-Regge amplitudes are cured by q-deformations, which introduce finite dimension of a Hilbert space -> Turaev-Viro amplitude. Compact phase spaces introduce bounds on the the phase space variables, which regularize curvature, energy density, etc. Model-independent arguments: N. Bao, S. M. Carroll and A. Singh, The Hilbert Space of Quantum Gravity Is Locally Finite-Dimensional, Int. J. Mod. Phys. D 26 (2017) no.12, 1743013 Finite horizon entropy/holography SBH = 1 4 A l2 Pl <latexit sha1_base64="hlSDegc/77QgWSJ7UTIAb6biyE4=">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</latexit> <latexit 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  • 5. Loop Quantum Gravity C. Rovelli and F. Vidotto, Compact phase space, cosmological constant, and discrete time, Phys. Rev. D 91 (2015) no.8, 084037 ⇣ e R A , E ⌘ 2 SU(2) ⇥ su(2) <latexit sha1_base64="CdkIUU6wRZtB7sjCMykF2u3+RLU=">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</latexit> <latexit 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sha1_base64="6953ZNA4lMASasbAVideoDL3Vp8=">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</latexit> <latexit sha1_base64="6953ZNA4lMASasbAVideoDL3Vp8=">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</latexit> <latexit sha1_base64="+fdNu4hgoRepBJkMxKA2uZcpx1w=">AAADA3icjVHLSiNBFD224xgfM5PRpZvCIESQ0J3NuBJHEVxGnKiQaOjuVGKRflFdLUjI0j9xN7vBrT/gVkH8A/2LuVWWw4xBxmq6+9xz7zlVt26QRSJXrvsw4Ux+mPo4XZqZnZv/9PlL+evCQZ4WMuTNMI1SeRT4OY9EwptKqIgfZZL7cRDxw2CwrfOHZ1zmIk1+qPOMH8d+PxE9EfqKqE55ox3xnqoyfjJsi0Sx76O1P3hnxNpS9E/VKqOY7TerdUJKxDxnNuqUK27NNYuNA8+CCuxqpOV7tNFFihAFYnAkUIQj+MjpacGDi4y4YwyJk4SEyXOMMEvagqo4VfjEDujbp6hl2YRi7ZkbdUi7RPRKUjKskCalOklY78ZMvjDOmn3Le2g89dnO6R9Yr5hYhVNi/6d7qXyvTvei0MO66UFQT5lhdHehdSnMreiTs7+6UuSQEadxl/KScGiUL/fMjCY3veu79U3+0VRqVsehrS3wpE9JA/Zej3McHNRrnlvz9uqVzS076hKWsIwqzfMbNrGLBprkfYkb3OLOuXB+Or+cq+dSZ8JqFvHPcq5/Aza/pUU=</latexit> It has been justified that the compacatification can be associated with the q-deformations of the SU(2) group, leading to the upper limit on j. In the 2+1 case the deformations introduce the IR scale, which is the cosmological constant: q = ei⇤l2 Pl <latexit sha1_base64="2nyuevNyoTUk1OqwYqHtMYBkUQg=">AAAC43icjVHLSsNAFD2Nr/qOuhQkWARXJSmCboSCGxcuKtgqWFuSdKqD0yQmE7GE7ty5E7f+gFv9F/EP9C+8M6bgA9EJSc6ce8+Zufd6keCJtO2XgjEyOjY+UZycmp6ZnZs3FxYbSZjGPqv7oQjjI89NmOABq0suBTuKYub2PMEOvfMdFT+8ZHHCw+BA9iN20nNPA97lviuJapsrF9uslfHmHkk6riValXbWlOxKZjUxGAzaZsku23pZP4GTgxLyVQvNZzTRQQgfKXpgCCAJC7hI6DmGAxsRcSfIiIsJcR1nGGCKtCllMcpwiT2n7yntjnM2oL3yTLTap1MEvTEpLayRJqS8mLA6zdLxVDsr9jfvTHuqu/Xp7+VePWIlzoj9SzfM/K9O1SLRxZaugVNNkWZUdX7ukuquqJtbn6qS5BARp3CH4jFhXyuHfba0JtG1q966Ov6qMxWr9n6em+JN3ZIG7Hwf50/QqJQdu+zsb5Sq1XzURSxjFes0z01UsYsa6uR9jQc84slgxo1xa9x9pBqFXLOEL8u4fweimJwN</latexit> <latexit sha1_base64="2nyuevNyoTUk1OqwYqHtMYBkUQg=">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</latexit> <latexit sha1_base64="2nyuevNyoTUk1OqwYqHtMYBkUQg=">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</latexit> <latexit sha1_base64="xU3POQp8IkwtwL1Z3/qJLos/6Q8=">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</latexit> M. Artymowski and J.M, Quantum Hubble horizon, Eur. Phys. J. C 79 (2019) no.7, 632 The deformations lead to possible solution to the initial homogeneity problem: (holonomies and fluxes)
  • 6. Loop Quantum Cosmology In the homogeneous and isotropic case, the holonomy-flux algebra reduces to the „cylindrical algebra”: [E, U] = U <latexit sha1_base64="UnOmbfCvKbf+ke14qKoH2lMnUYU=">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</latexit> <latexit sha1_base64="UnOmbfCvKbf+ke14qKoH2lMnUYU=">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</latexit> <latexit sha1_base64="UnOmbfCvKbf+ke14qKoH2lMnUYU=">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</latexit> <latexit sha1_base64="pNRfvwiVPWNQtqw00nzjRV7WgJQ=">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</latexit> [E, U† ] = U† <latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit> <latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit> <latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit> <latexit sha1_base64="e09Ffe0xrCrEK1hrx/3U1hKrT94=">AAAC43icjVHLSsNAFD2Nr1pfVZeCBIvgQkvSjW6EogguK9gHtLUk6bSGpkmYTIRSunPnTtz6A271X8Q/0L/wzphCtYhOSHLuufecmTvXDj03EobxltJmZufmF9KLmaXlldW17PpGJQpi7rCyE3gBr9lWxDzXZ2XhCo/VQs6svu2xqt07lfnqDeORG/iXYhCyZt/q+m7HdSxBVCu7XT/bL18NG22r22V81NSP9YOJuJXNGXlDLX0amAnIIVmlIPuKBtoI4CBGHww+BGEPFiJ66jBhICSuiSFxnJCr8gwjZEgbUxWjCovYHn27FNUT1qdYekZK7dAuHr2clDp2SRNQHScsd9NVPlbOkv3Ne6g85dkG9LcTrz6xAtfE/qUbV/5XJ3sR6OBI9eBST6FiZHdO4hKrW5En1ye6EuQQEidxm/KcsKOU43vWlSZSvcu7tVT+XVVKVsZOUhvjQ56SBmz+HOc0qBTyppE3Lwq54kky6jS2sIM9muchijhHCWXyvsUTnvGiMe1Ou9cevkq1VKLZxLelPX4CMQqbCw==</latexit> [U, U† ] = 0 <latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">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</latexit> <latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">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</latexit> <latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">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</latexit> <latexit sha1_base64="cnHTk9HrnF7nVajidzm+Fvfga8w=">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</latexit> where (U, E) 2 U(1) ⇥ R <latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">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</latexit> <latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">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</latexit> <latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">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</latexit> <latexit sha1_base64="sbXSnnPGHmF4tIja13kFciurF8g=">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</latexit> Employing the fact that , one can equivalently write: What we consider in what follows is the SU(2) generalization of the above algebra, so that [Sx, Sy] = iSz <latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit> <latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit> <latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit> <latexit sha1_base64="PqwCuxVi4bc0TST3FEaDyljTv/0=">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</latexit> [ Sz, Sy] = iSx <latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit> <latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit> <latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit> <latexit sha1_base64="MFRVY5XfOc9ROQOKHua2kMEDA9E=">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</latexit> [ Sz, Sx] = iSy <latexit 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sha1_base64="xcQnj3IKJcMU7jmF5FhQjM2t0yE=">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</latexit> <latexit sha1_base64="TD7yzAuB2WBWvh7scwExLF4v7wc=">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</latexit> The associated phase space is . S2 <latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit> <latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit> <latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit> <latexit sha1_base64="FQdV073B++8mJBySUjDjnu6Lhjs=">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</latexit> U|'i = ei' |'i <latexit 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sha1_base64="feG20PHt+/O2ArMBR3S++OcJEYg=">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</latexit> <latexit sha1_base64="feG20PHt+/O2ArMBR3S++OcJEYg=">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</latexit> <latexit sha1_base64="nUzdepdA7p2pYh5nTBPWGaR0Ha8=">AAAC+HicjVHLLgRBFD3a+z1Y2lRMJFaT7rFgIxE2liQGiWFS3cpMZXq6O9XVEoYPsbMTWz9gyxeIP+Av3ColwUSoTnefOveeU3XvDbNY5tr3X/q8/oHBoeGR0bHxicmp6dLM7F6eFioStSiNU3UQ8lzEMhE1LXUsDjIleCeMxX7Y3jTx/TOhcpkmu/o8E0cd3kzkqYy4JqpRWq6xS1Y/4yprSVZXPGnGgq0xcdyVjr3qiTdKZb/i28V6QeBAGW5tp6Vn1HGCFBEKdCCQQBOOwZHTc4gAPjLijtAlThGSNi5whTHSFpQlKIMT26Zvk3aHjk1obzxzq47olJheRUqGRdKklKcIm9OYjRfW2bC/eXetp7nbOf1D59UhVqNF7F+6z8z/6kwtGqdYtTVIqimzjKkuci6F7Yq5OftSlSaHjDiDTyiuCEdW+dlnZjW5rd30ltv4q800rNlHLrfAm7klDTj4Oc5esFetBH4l2KmW1zfcqEcwjwUs0TxXsI4tbKNG3td4wCOevAvvxrv17j5SvT6nmcO35d2/AwaXpBI=</latexit> [E, cos '] = i sin ' <latexit sha1_base64="ArEWWdXLtcY0nKzD5di8Ro+QadM=">AAAC63icjVHLLgRBFD3a+z1Y2lRMiAWTbpGwkQiRWJIYJDMT6S6Fin6lqnoSEX9gZye2fsCW/xB/wF+4VZp4RKhOd586955Tde+N8lhq4/tPHV5nV3dPb1//wODQ8MhoZWx8V2eF4qLOszhT+1GoRSxTUTfSxGI/VyJMoljsRafrNr7XFkrLLN0xZ7loJeFxKo8kDw1RB5WZxsYca/JMs2Y7VPmJZC22wuaZZE0t0w/yoFL1a75b7CcISlBFubayyiOaOEQGjgIJBFIYwjFCaHoaCOAjJ66Fc+IUIeniAhcYIG1BWYIyQmJP6XtMu0bJprS3ntqpOZ0S06tIyTBNmozyFGF7GnPxwjlb9jfvc+dp73ZG/6j0Sog1OCH2L9175n91thaDIyy7GiTVlDvGVsdLl8J1xd6cfarKkENOnMWHFFeEuVO+95k5jXa1296GLv7sMi1r97zMLfBib0kDDr6P8yfYXagFfi3YXqyurpWj7sMkpjBL81zCKjaxhTp5X+IO93jwEu/Ku/Zu3lK9jlIzgS/Lu30FK5Sc1w==</latexit> <latexit sha1_base64="ArEWWdXLtcY0nKzD5di8Ro+QadM=">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</latexit> <latexit 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sha1_base64="6c7IDnr1TFztEgyO2wMw0p+gCrI=">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</latexit> [E, sin '] = i cos ' <latexit 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  • 7. Spherical phase space 2-sphere is a phase space of spin - SU(2) variable. The 2-sphere is not a cotangent bundle, but is equipped with symplectic form and is a well defined symplectic manifold. Change of variables R1R2 = S <latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit> <latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit> <latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit> <latexit sha1_base64="EQQGZxdPbWiCIMWfS4rffAw+7yo=">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</latexit> Dimensional parameters Symplectic form on 2-sphere: Volume of phase space: V = Z S2 ! = 4⇡S <latexit sha1_base64="kdigxS2y8Wm5xqtlFNneuXSHQ2Q=">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</latexit> <latexit sha1_base64="kdigxS2y8Wm5xqtlFNneuXSHQ2Q=">AAAC9XicjVHLSsNAFD3Gd31VXboJFsFVSUTQjSC6caloq2BqmYxjHcyLZKJI6G+4cydu/QG3+gviH+hfeGdMQS2iE5KcOfeeM3Pv9ZNAZspxXgeswaHhkdGx8crE5NT0THV2rpnFecpFg8dBnB75LBOBjERDSRWIoyQVLPQDcehfbOv44aVIMxlHB+o6Ea2QdSJ5JjlTRLWrjhcydc5ZUDS79obtyUi1C8P5frHfPVnpenEoOoxiq14i7f12tebUHbPsfuCWoIZy7cbVF3g4RQyOHCEEIijCARgyeo7hwkFCXAsFcSkhaeICXVRIm1OWoAxG7AV9O7Q7LtmI9tozM2pOpwT0pqS0sUSamPJSwvo028Rz46zZ37wL46nvdk1/v/QKiVU4J/YvXS/zvzpdi8IZ1k0NkmpKDKOr46VLbrqib25/qUqRQ0KcxqcUTwlzo+z12TaazNSue8tM/M1kalbveZmb413fkgbs/hxnP2iu1F2n7u6t1ja3ylGPYQGLWKZ5rmETO9hFg7xv8IgnPFtX1q11Z91/ploDpWYe35b18AHQ/KJu</latexit> <latexit sha1_base64="kdigxS2y8Wm5xqtlFNneuXSHQ2Q=">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</latexit> <latexit sha1_base64="SIZSykBRgJFSNceHdzgcKWHXv1c=">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</latexit> Symplectic form
  • 8. is, therefore, the angular momentum (spin) vector with the norm S. Employing the above, the su(2) algebra is obtained: with the following components: Poisson bracket: ~ S = (Sx, Sy, Sz) <latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit> <latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit> <latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit> <latexit sha1_base64="TJZ5tBzh19LPZSWaEzYMfbqYv0I=">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</latexit> ~ S <latexit 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sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit> <latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit> <latexit sha1_base64="TBxPMM/yPmf0T6MMvEYGyXsdj0c=">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</latexit> One can introduce the vector {Si, Sj} = ✏ijkSk <latexit 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  • 9. ! = m X i=1 !i = m X i=1 dpi ^ dqi <latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">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</latexit> <latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">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</latexit> <latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">AAADA3icjVHLSsNAFD3GV62vqks3g0VwVRIRdFMpunGpYFWwGvIY62BeJhOllC79E3fuxK0/4FZB/AP9C+9MI/hAdEKSM+fec2buvW4SiEya5suAMTg0PDJaGiuPT0xOTVdmZveyOE893vTiIE4PXCfjgYh4UwoZ8IMk5U7oBnzfPdtU8f0LnmYijnZlJ+FHodOOxInwHEmUXVlvxSFvO6zOWlke2l1Rt3rHIeuztvjO+wlxrUvutznzz21hV6pmzdSL/QRWAaoo1nZceUYLPmJ4yBGCI4IkHMBBRs8hLJhIiDtCl7iUkNBxjh7KpM0pi1OGQ+wZfdu0OyzYiPbKM9Nqj04J6E1JybBImpjyUsLqNKbjuXZW7G/eXe2p7tahv1t4hcRKnBL7l+4j8786VYvECdZ0DYJqSjSjqvMKl1x3Rd2cfapKkkNCnMI+xVPCnlZ+9JlpTaZrV711dPxVZypW7b0iN8ebuiUN2Po+zp9gb7lmmTVrZ6Xa2ChGXcI8FrBE81xFA1vYRpO8r/GARzwZV8aNcWvc9VONgUIzhy/LuH8H1X6nmQ==</latexit> <latexit sha1_base64="oqL9Gk9p0S3MMtDwUZms6ml1w+Y=">AAADA3icjVHLSsNAFD3GV31XXboZLIKrknSjm0rRjcsKVgWrIY9pHZqXyUSR4tI/cedO3PoDbhXEP9C/8M4YQSuiE5KcOfeeM3PvdZNAZNI0X4aM4ZHRsfHSxOTU9MzsXHl+YS+L89TjLS8O4vTAdTIeiIi3pJABP0hS7oRuwPfd3paK75/xNBNxtCsvEn4UOt1IdITnSKLs8kY7DnnXYXXWzvLQ7ou6dXkcsg/WFoO8nxDXPud+lzP/1BZ2uWJWTb3YT2AVoIJiNePyM9rwEcNDjhAcESThAA4yeg5hwURC3BH6xKWEhI5zXGKStDllccpwiO3Rt0u7w4KNaK88M6326JSA3pSUDCukiSkvJaxOYzqea2fF/ubd157qbhf0dwuvkFiJE2L/0n1m/lenapHoYF3XIKimRDOqOq9wyXVX1M3Zl6okOSTEKexTPCXsaeVnn5nWZLp21VtHx191pmLV3ityc7ypW9KArcFx/gR7taplVq2dWqWxWYy6hCUsY5XmuYYGttFEi7yv8YBHPBlXxo1xa9x9pBpDhWYR35Zx/w7U3qeX</latexit> The symplectic for the system of m DOFs is: By performing compactification for every canonical pair the system’s phase space changes as follows: R2m ! S2m <latexit sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">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</latexit> <latexit 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sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">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</latexit> <latexit sha1_base64="+cKEhSbu8x1k6ugIVW++eJt129I=">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</latexit> In consequence, the regularized kinematical Hilbert space for the system is: Hkin = H⌦m s <latexit 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sha1_base64="krvugiKSYYl+DTn0qt0IhTvyJOY=">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</latexit> <latexit sha1_base64="krvugiKSYYl+DTn0qt0IhTvyJOY=">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</latexit> <latexit sha1_base64="Iag4kIhX7mzUoKui3dcbTnqNRPE=">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</latexit> where , so that: Hs = span{s, si, ..., |s, si} <latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit> <latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit> <latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit> <latexit sha1_base64="iz+nNgiphiM7VeAF/vg5DllYvtY=">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</latexit> Ŝ2 |s, szi = s(s + 1)|s, szi <latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit> <latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit> <latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit> <latexit sha1_base64="R3s46+qWvDocRDolfsve4fnR5BY=">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</latexit> Ŝz|s, szi = sz|s, szi <latexit sha1_base64="5yIU9JdBaNxiS272TQL1NZ5ulQA=">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</latexit> <latexit sha1_base64="5yIU9JdBaNxiS272TQL1NZ5ulQA=">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</latexit> <latexit sha1_base64="5yIU9JdBaNxiS272TQL1NZ5ulQA=">AAAC73icjVHLSsNAFD3G97vq0k2wCCJSEhF0I4huXCpaFWwJkzhtg3k5MxFs9R/cuRO3/oBb/QvxD/QvvDNGsIrohCTnnnvPmblz/SwKpXKclx6rt69/YHBoeGR0bHxisjQ1fSDTXAS8GqRRKo58JnkUJryqQhXxo0xwFvsRP/RPt3T+8JwLGabJvrrIeD1mzSRshAFTRHmlxVqLqc7elde2L+WS9No1wZJmxO11m4IuyiuVnYpjlv0TuAUoo1g7aekZNZwgRYAcMTgSKMIRGCQ9x3DhICOujg5xglBo8hxXGCFtTlWcKhixp/RtUnRcsAnF2lMadUC7RPQKUtqYJ01KdYKw3s02+dw4a/Y3747x1Ge7oL9feMXEKrSI/Uv3Wflfne5FoYE100NIPWWG0d0FhUtubkWf3P7SlSKHjDiNTygvCAdG+XnPttFI07u+W2byr6ZSszoOitocb/qUNGD3+zh/goPliutU3N2V8sZmMeohzGIOCzTPVWxgGzuokvc1HvCIJ+vMurFurbuPUqun0Myga1n374n8oO4=</latexit> <latexit sha1_base64="FtnA9M+wDi07g5/jW3UWXXbNWeE=">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</latexit>