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Quantum cosmology
with compact phase space
Jakub Mielczarek
Jagiellonian University, Cracow
Based on:
D. Artigas, J. M. and C. Rovelli,
Phys. Rev. D 100, 043533 (2019)
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PoToR 2019
Szczecin, 25.09.19
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)
T. Trześniewski, Nonlinear Field Space Theory and Quantum Gravity,
Acta Phys. Pol. B Proc. Suppl. 10, 329 (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,
2018-2021
Scalar field
Mechanical
systems
Gauge fields
Fermionic field (in progress)
Minisuperspace
models
Cosmology
Midisuperspace
models
(in progress)
General
Relativity
Scalar field on cosmological background
Where we are?
Scalar multiplet (in progress)
Compact phase spaces arise in the semiclassical description of
quantum systems with finite dimensional Hilbert spaces.
dimH ⇠
V
(2⇡})n
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V =
Z
!
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Compact phase space Finite dim Hilbert space
We are focused on the compact case rather on the phase space with boundaries.
dim = 2n<latexit sha1_base64="di84GEhON1ThgBVYa5K+Prggwhk=">AAAC2nicjVHLSsQwFD3W93tUXLkpDoKroRVBN8KgC10qOI5gB0k7cQz2RZqKw+DGnbj1B9zqB4l/oH/hTcyAD0RT2p6ce89J7r1hHotCed7LgDM4NDwyOjY+MTk1PTNbmZs/KrJSRrwRZXEmj0NW8FikvKGEivlxLjlLwpg3w4sdHW9eclmILD1U3Zy3EtZJxZmImCLqtLIYKH6lem2RXAe7LEmYu+WuEV/1ap5Z7k/gW1CFXftZ5RkB2sgQoUQCjhSKcAyGgp4T+PCQE9dCjzhJSJg4xzUmSFtSFqcMRuwFfTu0O7FsSnvtWRh1RKfE9EpSulghTUZ5krA+zTXx0jhr9jfvnvHUd+vSP7ReCbEK58T+petn/lena1E4w6apQVBNuWF0dZF1KU1X9M3dT1UpcsiJ07hNcUk4Msp+n12jKUzturfMxF9Npmb1PrK5Jd70LWnA/vdx/gRHazXfq/kH69X6th31GJawjFWa5wbq2MM+GuTdwwMe8eQEzo1z69x9pDoDVrOAL8u5fwdrV5et</latexit><latexit sha1_base64="di84GEhON1ThgBVYa5K+Prggwhk=">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</latexit><latexit sha1_base64="di84GEhON1ThgBVYa5K+Prggwhk=">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</latexit><latexit sha1_base64="OltJ7dcckV/ivblgW+2UZZZLVxI=">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</latexit>
Why compact phase space for gravity?
Locally finite dimensional Hilbert space of quantum gravitational
field is supported at the theoretical ground.
Finite horizon entropy
Spikes (IR divergences)Singularities (UV divergences)
(But the result may depend on how microstates are counted, e.g. LQG)
S ⇠ A ln dimH<latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">AAAC6XicjVHLTtwwFD2kDyjQkpZlN1YHJFajpEKC5UA3sxxUBkYiCDkeD1jjPJQ4CDSaH+iuu6pbfoBt+yOIPyh/wbUxUgtCxVGS43PvOfa9Ny21qk0UXc8EL16+ej0792Z+YfHtu6Xw/Ye9umgqIfui0EU1SHkttcpl3yij5aCsJM9SLffT8Rcb3z+VVa2KfNecl/Iw48e5GinBDVFH4cpXltQqY1ss0TlLjDwzk6HKpizJuDkRXE+6U3YUtqJ25BZ7DGIPWvCrV4RXSDBEAYEGGSRyGMIaHDU9B4gRoSTuEBPiKkLKxSWmmCdtQ1mSMjixY/oe0+7AszntrWft1IJO0fRWpGRYJU1BeRVhexpz8cY5W/Yp74nztHc7p3/qvTJiDU6I/Z/uPvO5OluLwQibrgZFNZWOsdUJ79K4rtibs7+qMuRQEmfxkOIVYeGU931mTlO72m1vuYv/cZmWtXvhcxvc2FvSgOOH43wM9j6346gd76y3Ott+1HP4iE9Yo3luoIMueuiT9zdc4hd+B+Pge/Aj+HmXGsx4zTL+WcHFLbnBnYw=</latexit><latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">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</latexit><latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">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</latexit><latexit sha1_base64="ATa3zLCqMDarwQmSOjvO3CJ0w4w=">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</latexit>
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 argument: 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
SBH =
1
4
A
l2
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SdS ⇠
1
l2
Pl⇤<latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">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</latexit><latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">AAAC+nicjVHLThRBFD20qCMojrJ0U3FC4mrSTSC6JLBx4WLIOEBCw6S6pgYqVD9SXU0gbX+JO3fGLT/AVj+A8Af6F94qexKUEKxOd586955Tde9NCq1KG4bXc8GD+YePHneeLCw+fbb0vPvi5U6ZV0bIkch1bvYSXkqtMjmyymq5VxjJ00TL3eRky8V3T6UpVZ59tOeFPEj5UaamSnBL1Li7PhzXsZVntp4Mm4bFpUoZi6eGizpqan24OgsPdNPEH8h4whs27vbCfugXuw2iFvTQrkHevUKMCXIIVEghkcES1uAo6dlHhBAFcQeoiTOElI9LNFggbUVZkjI4sSf0PaLdfstmtHeepVcLOkXTa0jJsEKanPIMYXca8/HKOzv2Lu/ae7q7ndM/ab1SYi2Oib1PN8v8X52rxWKKd74GRTUVnnHVidal8l1xN2c3qrLkUBDn8ITihrDwylmfmdeUvnbXW+7jP32mY91etLkVfrlb0oCjf8d5G+ys9qOwH22v9TY221F38Aqv8Ybm+RYbeI8BRuT9GZf4jh/Bp+BL8DX49ic1mGs1y/hrBRe/AVzipbs=</latexit><latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">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</latexit><latexit sha1_base64="y+20wniumFcasvp6MEg/CagtrNE=">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</latexit>
Simple counting of microstates:
De Sitter model
related to scale factorrelated to Hubble factor
Symplectic form:
Hamiltonian:
Poisson bracket:
Hubble factor:
Friedmann equation:
(|q| := V0a3
)<latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="OZow8DheX0vGhJLueanSFsU/vd4=">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</latexit>
Equation of motion: ˙q = {q, HGR} =
3
2
Nqp
<latexit sha1_base64="dQGZ6rQVprP6fnWSrS8ey2qQovc=">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</latexit><latexit sha1_base64="dQGZ6rQVprP6fnWSrS8ey2qQovc=">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</latexit><latexit sha1_base64="dQGZ6rQVprP6fnWSrS8ey2qQovc=">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</latexit><latexit sha1_base64="7E0r767JOJVZ8w5yFk9y7KjNN+8=">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</latexit>
H :=
1
3
˙q
q
=
1
2
p
<latexit sha1_base64="R3S10L1lSoNu/NXPP2tCsBep7TU=">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</latexit><latexit sha1_base64="R3S10L1lSoNu/NXPP2tCsBep7TU=">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</latexit><latexit sha1_base64="R3S10L1lSoNu/NXPP2tCsBep7TU=">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</latexit><latexit sha1_base64="zBJRKpQ03XuZBmfnNyPv3JIi0Ao=">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</latexit>
N = 1<latexit sha1_base64="89gxBv5EJwAAmy2v6N5F1YpMsUk=">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</latexit><latexit sha1_base64="89gxBv5EJwAAmy2v6N5F1YpMsUk=">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</latexit><latexit sha1_base64="89gxBv5EJwAAmy2v6N5F1YpMsUk=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdCMU3XQlFW0r1CLJdFqH5sVkopQi+ANu9dPEP9C/8M6YglpEJyQ5c+49Z+be6yeBSJXjvBasmdm5+YXiYmlpeWV1rby+0UrjTDLeZHEQy0vfS3kgIt5UQgX8MpHcC/2At/3hiY63b7lMRRxdqFHCu6E3iERfME8RdX565F6XK07VMcueBm4OKshXIy6/4Ao9xGDIEIIjgiIcwENKTwcuHCTEdTEmThISJs5xjxJpM8rilOERO6TvgHadnI1orz1To2Z0SkCvJKWNHdLElCcJ69NsE8+Ms2Z/8x4bT323Ef393CskVuGG2L90k8z/6nQtCn0cmhoE1ZQYRlfHcpfMdEXf3P5SlSKHhDiNexSXhJlRTvpsG01qate99Uz8zWRqVu9ZnpvhXd+SBuz+HOc0aO1VXafqnu1Xasf5qIvYwjZ2aZ4HqKGOBprkPcAjnvBs1a3Iyqy7z1SrkGs28W1ZDx9ZBo/V</latexit><latexit sha1_base64="MRDBXiwF92ZrKOXPN4+xcBsvANw=">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</latexit>
for
0 =
@HGR
@N
= q
✓
3
4
p2
+
⇤

◆
<latexit sha1_base64="DdmPK75dnWwXln/T+j8aPIYFxuE=">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</latexit><latexit sha1_base64="DdmPK75dnWwXln/T+j8aPIYFxuE=">AAADMXicjVFNT9wwEJ2kLR/bQpf2yMXqqhIIsUooElyQUHsoB4QoYgGJwMrxehdrvYlxHFQU5VfxT7hU3BAnUP9Ajx0bo36gqjhK8vxm3vOMJ1VSFCaKroLw2fMXY+MTk42Xr6amXzdn3uwVeakZ77Bc5vogpQWXIuMdI4zkB0pzOkol30+Hn2x8/4zrQuTZrjlX/GhEB5noC0YNUt2mjNaSvqasShTVRlBJNrpVYvhXU33eqev6F79VkzVymkjeN3OL5F70oa6W62RIlaJEHS8teDrZxAJ6FMUuVCdaDE7MfKPbbEXtyC3yGMQetMCv7bz5DRLoQQ4MShgBhwwMYgkUCnwOIYYIFHJHUCGnEQkX51BDA7UlZnHMoMgO8TvA3aFnM9xbz8KpGZ4i8dWoJPAeNTnmacT2NOLipXO27L+8K+dpazvHf+q9RsgaOEH2f7qHzKfqbC8G+rDqehDYk3KM7Y55l9Ldiq2c/NaVQQeFnMU9jGvEzCkf7pk4TeF6t3dLXfzWZVrW7pnPLeHOVokDjv8e52Owt9SOo3b8Zbm1/tGPegJm4R3M4TxXYB02YBs66H0JP4IgCMOL8Cq8Dm/uU8PAa97CHyv8/hPParlU</latexit><latexit sha1_base64="DdmPK75dnWwXln/T+j8aPIYFxuE=">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</latexit><latexit sha1_base64="2KwQlw+G9cQV1+ysZgu2TekWafQ=">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</latexit>
Scalar constraint:
Spherical phase space
The spherical space is consider mostly because of
its relation to spin. It is also a non-trivial case
since the sphere is not a cotangent bundle, but
equipped with symplectic form is a well defined
symplectic manifold.
In this case, kinematics and dynamics can be
associated with a certain spin system. This is
important due to both practical reason (convenient
to perform calculations) and fundamental reason
(spin degrees of freedom for QG, relation to
condensed-matter systems - analog models).
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 a sphere:
Volume of phase space:
V =
Z
⌦
! = 4⇡S
<latexit sha1_base64="4nzSd+70WiXvprD30R23AtL0GwU=">AAAC6XicjVHLSsNAFD2Nr1pfVZduBqvgqiRS0I1QdOPOirYVGilJHOvQvEgmQin9AXfuxK0/4FZ/RPwD/QvvjBF8IDohM2fOvefM3Llu7ItUmuZzwRgbn5icKk6XZmbn5hfKi0utNMoSjze9yI+SE9dJuS9C3pRC+vwkTrgTuD5vu/09FW9f8iQVUXgsBzE/DZxeKM6F50iiuuW1Ftthtghld2gfBLznjJgdqZXoGrNjwY5YqVuumFVTD/YTWDmoIB+NqPwEG2eI4CFDAI4QkrAPByl9HVgwERN3iiFxCSGh4xwjlEibURanDIfYPs092nVyNqS98ky12qNTfPoTUjKskyaivISwOo3peKadFfub91B7qrsNaHVzr4BYiQti/9J9ZP5Xp2qROMe2rkFQTbFmVHVe7pLpV1E3Z5+qkuQQE6fwGcUTwp5Wfrwz05pU167e1tHxF52pWLX38twMr+qW1GDrezt/gtZm1TKr1mGtUt/NW13EClaxQf3cQh37aKBJ3le4xwMejb5xbdwYt++pRiHXLOPLMO7eAJirm9I=</latexit><latexit sha1_base64="4nzSd+70WiXvprD30R23AtL0GwU=">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</latexit><latexit sha1_base64="4nzSd+70WiXvprD30R23AtL0GwU=">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</latexit><latexit sha1_base64="5Sw4G9hfpyiVYD+BUvEC2Jei7IY=">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</latexit>
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 sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">AAACy3icjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl040aoaB9QiyTTaR2aJiGZFGrt0h9wq/8l/oH+hXfGKahFdEKSM+eec2fuvX4ciFQ6zmvOmptfWFzKLxdWVtfWN4qbW/U0yhLGaywKoqTpeykPRMhrUsiAN+OEewM/4A2/f6rijSFPUhGFV3IU8/bA64WiK5gniWpeDzkbX04KN8WSU3b0smeBa0AJZlWj4guu0UEEhgwDcISQhAN4SOlpwYWDmLg2xsQlhISOc0xQIG9GKk4Kj9g+fXu0axk2pL3KmWo3o1MCehNy2tgjT0S6hLA6zdbxTGdW7G+5xzqnutuI/r7JNSBW4pbYv3xT5X99qhaJLo51DYJqijWjqmMmS6a7om5uf6lKUoaYOIU7FE8IM+2c9tnWnlTXrnrr6fibVipW7ZnRZnhXt6QBuz/HOQvqB2XXKbsXh6XKiRl1HjvYxT7N8wgVnKGKmp7jI57wbJ1bqXVn3X9KrZzxbOPbsh4+AFnwkjo=</latexit><latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit><latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit><latexit sha1_base64="TBxPMM/yPmf0T6MMvEYGyXsdj0c=">AAACy3icjVHLTsJAFD3UF+ILdemmkZi4Ii0bXRLduDHBKI8EiWmHASeUtmmnJIgs/QG3+l/GP9C/8M44JCoxOk3bM+eec2fuvX4ciFQ6zmvOWlhcWl7JrxbW1jc2t4rbO400yhLG6ywKoqTleykPRMjrUsiAt+KEe0M/4E1/cKrizRFPUhGFV3Ic887Q64eiJ5gniWpdjzibXE4LN8WSU3b0sueBa0AJZtWi4guu0UUEhgxDcISQhAN4SOlpw4WDmLgOJsQlhISOc0xRIG9GKk4Kj9gBffu0axs2pL3KmWo3o1MCehNy2jggT0S6hLA6zdbxTGdW7G+5JzqnutuY/r7JNSRW4pbYv3wz5X99qhaJHo51DYJqijWjqmMmS6a7om5uf6lKUoaYOIW7FE8IM+2c9dnWnlTXrnrr6fibVipW7ZnRZnhXt6QBuz/HOQ8albLrlN2LSql6Ykadxx72cUjzPEIVZ6ihruf4iCc8W+dWat1Z959SK2c8u/i2rIcPWVCSOA==</latexit>
One can introduce a vector
{Si, Sj} = ✏ijkSk<latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit><latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">AAAC43icjVHLSsNAFD3Gd31FXQoSLIILKYkIuhGKblwqtbZgS0jiWKfNi2QilNCdO3fi1h9wq/8i/oH+hXfGFHwgOiHJmXPvOTP3Xjf2eSpM82VEGx0bn5icmi7NzM7NL+iLS6dplCUeq3uRHyVN10mZz0NWF1z4rBknzAlcnzXc3oGMN65YkvIoPBH9mLUDpxPyC+45gihbX23lNZtv1uxua7DXYnHKfaJz3u0NanavZOtls2KqZfwEVgHKKNZRpD+jhXNE8JAhAEMIQdiHg5SeM1gwERPXRk5cQoirOMMAJdJmlMUowyG2R98O7c4KNqS99EyV2qNTfHoTUhpYJ01EeQlheZqh4plyluxv3rnylHfr098tvAJiBS6J/Us3zPyvTtYicIFdVQOnmmLFyOq8wiVTXZE3Nz5VJcghJk7ic4onhD2lHPbZUJpU1S5766j4q8qUrNx7RW6GN3lLGrD1fZw/welWxTIr1vF2ubpfjHoKK1jDBs1zB1Uc4gh18r7GAx7xpDHtRrvV7j5StZFCs4wvS7t/B9vwm7o=</latexit><latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit><latexit sha1_base64="I7UOLjFAAj//XAWx8T1brYw7Vag=">AAAC43icjVHLSsNAFD3GV62vqktBgkVwISXpRjdC0Y3LSq0KTQlJOq1j0yQkE6GE7ty5E7f+gFv9F/EP9C+8M0ZQi+iEJGfOvefM3HvdyOeJMIyXCW1yanpmtjBXnF9YXFourayeJmEae6zphX4Yn7tOwnwesKbgwmfnUcycgeuzM7d/KONnVyxOeBiciGHE2gOnF/Au9xxBlF3asLKGzXca9qU12rdYlHCf6Ixf9kcNu1+0S2WjYqiljwMzB2Xkqx6WnmGhgxAeUgzAEEAQ9uEgoacFEwYi4trIiIsJcRVnGKFI2pSyGGU4xPbp26NdK2cD2kvPRKk9OsWnNyalji3ShJQXE5an6SqeKmfJ/uadKU95tyH93dxrQKzABbF/6T4z/6uTtQh0sadq4FRTpBhZnZe7pKor8ub6l6oEOUTESdyheEzYU8rPPutKk6jaZW8dFX9VmZKVey/PTfEmb0kDNn+OcxycViumUTGPq+XaQT7qAtaxiW2a5y5qOEIdTfK+xgMe8aQx7Ua71e4+UrWJXLOGb0u7fwfbUJu4</latexit>
The is therefore a vector of angular momentum (spin) with the norm S.
Compact phase space FRW
Kinematics (symplectic form)
Dynamics (Hamiltonian)
Friedmann
equation:
where
The procedure is
ambiguous! We make
the simplest choice.
Affine 2-sphere
:=
4
3
⇤
R2
12
2 [0, 1]
<latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="89Xa6GSnIb2JPqG3sm0wmExRhx0=">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</latexit>
The affine case is recovered in the limit.R1, R2 ! 1<latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="MaR+F5j7siTjGesrWgPJZwhvMBE=">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</latexit>
Polymer limit
This is the case of loop quantum cosmology.
R2 ! 1<latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">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</latexit><latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">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</latexit><latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">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</latexit><latexit sha1_base64="i0BtGXKYQ1woH+zsYAye3Punzfc=">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</latexit>
In the limit the so-called
polymerization of momentum is obtained.
The su(2) algebra reduces to the cylindrical case:
The Hamiltonian reduces to
where the polymerization
scale is:
:=
1
R1<latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit><latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit><latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit><latexit sha1_base64="IpcgmU5scOhVCdLM71hbptMDtxU=">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</latexit>
The Friedmann equation is where
Cosmological evolution
The leading correction behaves as a
phantom matter with the equation of state:
~ de Sitter expansion ~ de Sitter contraction
Cosmological recollapse (IR effect)
(UV effect)
(UV effect)
⇤ ! ⇤e↵ = ⇤(1 )<latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">AAADAnicjVHNTtwwGJwNtCzbvy09crFYVaKHrhJUqb0goXLh0MMisYBE0MpJvItFNokchxZFe+ub9MYNceUFuMIB8QbwFnw2XokWVa2jJOP5Zsb+7KhIZal9/6bhzcw+ez7XnG+9ePnq9Zv224XtMq9ULPpxnuZqN+KlSGUm+lrqVOwWSvBxlIqd6HDd1HeOhCplnm3p40Lsj/kok0MZc03UoL0afiNxwlmo5OhAc6Xy78xxgzrU4oeuxXA4mbDVKc2Wg49hIlLNPwzaHb/r28GegsCBDtzo5e1rhEiQI0aFMQQyaMIpOEp69hDAR0HcPmriFCFp6wITtMhbkUqQghN7SN8RzfYcm9HcZJbWHdMqKb2KnAzvyZOTThE2qzFbr2yyYf+WXdtMs7dj+kcua0ysxgGx//JNlf/rM71oDPHF9iCpp8IyprvYpVT2VMzO2aOuNCUUxBmcUF0Rjq1zes7Mekrbuzlbbuu3VmlYM4+dtsKd2SVdcPDndT4F2yvdwO8Gm586a1/dVTexiCUs031+xho20EOfsn/hApe48n56J96pd/Yg9RrO8w6/De/8HuE5p50=</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">AAADAnicjVHNTtwwGJwNtCzbvy09crFYVaKHrhJUqb0goXLh0MMisYBE0MpJvItFNokchxZFe+ub9MYNceUFuMIB8QbwFnw2XokWVa2jJOP5Zsb+7KhIZal9/6bhzcw+ez7XnG+9ePnq9Zv224XtMq9ULPpxnuZqN+KlSGUm+lrqVOwWSvBxlIqd6HDd1HeOhCplnm3p40Lsj/kok0MZc03UoL0afiNxwlmo5OhAc6Xy78xxgzrU4oeuxXA4mbDVKc2Wg49hIlLNPwzaHb/r28GegsCBDtzo5e1rhEiQI0aFMQQyaMIpOEp69hDAR0HcPmriFCFp6wITtMhbkUqQghN7SN8RzfYcm9HcZJbWHdMqKb2KnAzvyZOTThE2qzFbr2yyYf+WXdtMs7dj+kcua0ysxgGx//JNlf/rM71oDPHF9iCpp8IyprvYpVT2VMzO2aOuNCUUxBmcUF0Rjq1zes7Mekrbuzlbbuu3VmlYM4+dtsKd2SVdcPDndT4F2yvdwO8Gm586a1/dVTexiCUs031+xho20EOfsn/hApe48n56J96pd/Yg9RrO8w6/De/8HuE5p50=</latexit><latexit sha1_base64="tMpofGH0WR6riQZeZTjXuvBAuG0=">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</latexit>
⇤ ! ⇤e↵ = ⇤(1 )<latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="tMpofGH0WR6riQZeZTjXuvBAuG0=">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</latexit>
de Sitter expansion may terminate…
=
⇢⇤
⇢c
2 [0, 1]
<latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">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</latexit><latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">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</latexit><latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">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</latexit><latexit sha1_base64="j4/HWccULijMCRJsJrBIHjusaik=">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</latexit>
and the effective phantom can be a sign of it.
Evolution on the phase space
Exact solutions:
= 0.1<latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="zs0Q4bWdZsUJj9bry7G2E35WgXA=">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</latexit>
= 0.5<latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="n7rCGcVm8TztJDnGePDDtHfx+AI=">AAACznicjVHLSsNAFD2Nr1pfVZdugkVwFZKC6EYounFZwT6gLTKZTmtoXiSTQinFrT/gVj9L/AP9C++MKahFdEKSM+eec2fuvW7se6m07deCsbS8srpWXC9tbG5t75R395pplCVcNHjkR0nbZanwvVA0pCd90Y4TwQLXFy13dKnirbFIUi8Kb+QkFr2ADUNv4HEmiep0+8KX7Ny2Tkq35Ypt2XqZi8DJQQX5qkflF3TRRwSODAEEQkjCPhhSejpwYCMmrocpcQkhT8cFZiiRNyOVIAUjdkTfIe06ORvSXuVMtZvTKT69CTlNHJEnIl1CWJ1m6nimMyv2t9xTnVPdbUJ/N88VECtxR+xfvrnyvz5Vi8QAZ7oGj2qKNaOq43mWTHdF3dz8UpWkDDFxCvcpnhDm2jnvs6k9qa5d9Zbp+JtWKlbtea7N8K5uSQN2fo5zETSrlmNbznW1UrvIR13EAQ5xTPM8RQ1XqKOhO/6IJzwbdWNszIz7T6lRyD37+LaMhw+TuJKn</latexit>
= 0.9<latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="/8b2BwzT4ViCy1omV12vrBaQYhg=">AAACznicjVHLSsNAFD2Nr1pfVZdugkVwFZJu1IVQdOOygn1AW2QyndbQvEgmhVKKW3/ArX6W+Af6F94ZU1CL6IQkZ849587ce93Y91Jp268FY2l5ZXWtuF7a2Nza3inv7jXTKEu4aPDIj5K2y1Lhe6FoSE/6oh0nggWuL1ru6FLFW2ORpF4U3shJLHoBG4bewONMEtXp9oUv2bltnZVuyxXbsvUyF4GTgwryVY/KL+iijwgcGQIIhJCEfTCk9HTgwEZMXA9T4hJCno4LzFAib0YqQQpG7Ii+Q9p1cjakvcqZajenU3x6E3KaOCJPRLqEsDrN1PFMZ1bsb7mnOqe624T+bp4rIFbijti/fHPlf32qFokBTnUNHtUUa0ZVx/Msme6Kurn5pSpJGWLiFO5TPCHMtXPeZ1N7Ul276i3T8TetVKza81yb4V3dkgbs/BznImhWLce2nOtqpXaRj7qIAxzimOZ5ghquUEdDd/wRT3g26sbYmBn3n1KjkHv28W0ZDx+dPJKr</latexit>
There are four symmetric solutions
for the four quadrants of a sphere.
Quantum cosmology with compact phase space
Quantum cosmology with compact phase space
Quantum cosmology with compact phase space
Quantum cosmology with compact phase space

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  • 2. 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) T. Trześniewski, Nonlinear Field Space Theory and Quantum Gravity, Acta Phys. Pol. B Proc. Suppl. 10, 329 (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, 2018-2021
  • 3. Scalar field Mechanical systems Gauge fields Fermionic field (in progress) Minisuperspace models Cosmology Midisuperspace models (in progress) General Relativity Scalar field on cosmological background Where we are? Scalar multiplet (in progress)
  • 4. Compact phase spaces arise in the semiclassical description of quantum systems with finite dimensional Hilbert spaces. dimH ⇠ V (2⇡})n <latexit sha1_base64="4MYsiWL7NHMvlXnC+0d3Wg04Qhw=">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</latexit><latexit 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  • 5. Why compact phase space for gravity? Locally finite dimensional Hilbert space of quantum gravitational field is supported at the theoretical ground. Finite horizon entropy Spikes (IR divergences)Singularities (UV divergences) (But the result may depend on how microstates are counted, e.g. LQG) S ⇠ A ln dimH<latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">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</latexit><latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">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</latexit><latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">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</latexit><latexit sha1_base64="ATa3zLCqMDarwQmSOjvO3CJ0w4w=">AAAC6XicjVHLLgRBFD3a+z1Y2lQMidWk24alx8aSMEi0SHVNoTLVj3RXC5nMD9jZia0fsOVHxB/wF26VlnhEqE53nzr3nlN1740yrQrj+889Xm9f/8Dg0PDI6Nj4xGRtanqvSMtcyKZIdZofRLyQWiWyaZTR8iDLJY8jLfej9oaN75/LvFBpsmsuM3kU89NEnSjBDVHHtfkdFhYqZmss1AkLjbwwnZaKuyyMuTkTXHc2u+y4VvcbvlvsJwgqUEe1ttLaE0K0kEKgRAyJBIawBkdBzyEC+MiIO0KHuJyQcnGJLkZIW1KWpAxObJu+p7Q7rNiE9tazcGpBp2h6c1IyLJAmpbycsD2NuXjpnC37m3fHedq7XdI/qrxiYg3OiP1L95H5X52txeAEK64GRTVljrHVicqldF2xN2efqjLkkBFncYviOWHhlB99Zk5TuNptb7mLv7hMy9q9qHJLvNpb0oCD7+P8CfaWGoHfCLaX6qvr1aiHMIs5LNI8l7GKTWyhSd5XuMcDHr22d+3deLfvqV5PpZnBl+XdvQG5IZ2K</latexit> 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 argument: 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 SBH = 1 4 A l2 Pl<latexit sha1_base64="hlSDegc/77QgWSJ7UTIAb6biyE4=">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</latexit><latexit 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sha1_base64="hlSDegc/77QgWSJ7UTIAb6biyE4=">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</latexit><latexit 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sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">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</latexit><latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">AAAC+nicjVHLThRBFD20qCMojrJ0U3FC4mrSTSC6JLBx4WLIOEBCw6S6pgYqVD9SXU0gbX+JO3fGLT/AVj+A8Af6F94qexKUEKxOd586955Tde9NCq1KG4bXc8GD+YePHneeLCw+fbb0vPvi5U6ZV0bIkch1bvYSXkqtMjmyymq5VxjJ00TL3eRky8V3T6UpVZ59tOeFPEj5UaamSnBL1Li7PhzXsZVntp4Mm4bFpUoZi6eGizpqan24OgsPdNPEH8h4whs27vbCfugXuw2iFvTQrkHevUKMCXIIVEghkcES1uAo6dlHhBAFcQeoiTOElI9LNFggbUVZkjI4sSf0PaLdfstmtHeepVcLOkXTa0jJsEKanPIMYXca8/HKOzv2Lu/ae7q7ndM/ab1SYi2Oib1PN8v8X52rxWKKd74GRTUVnnHVidal8l1xN2c3qrLkUBDn8ITihrDwylmfmdeUvnbXW+7jP32mY91etLkVfrlb0oCjf8d5G+ys9qOwH22v9TY221F38Aqv8Ybm+RYbeI8BRuT9GZf4jh/Bp+BL8DX49ic1mGs1y/hrBRe/AVzipbs=</latexit><latexit sha1_base64="y+20wniumFcasvp6MEg/CagtrNE=">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</latexit> Simple counting of microstates:
  • 6. De Sitter model related to scale factorrelated to Hubble factor Symplectic form: Hamiltonian: Poisson bracket: Hubble factor: Friedmann equation: (|q| := V0a3 )<latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="OZow8DheX0vGhJLueanSFsU/vd4=">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</latexit> Equation of motion: ˙q = {q, HGR} = 3 2 Nqp <latexit 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sha1_base64="dQGZ6rQVprP6fnWSrS8ey2qQovc=">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</latexit><latexit sha1_base64="dQGZ6rQVprP6fnWSrS8ey2qQovc=">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</latexit><latexit sha1_base64="7E0r767JOJVZ8w5yFk9y7KjNN+8=">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</latexit> H := 1 3 ˙q q = 1 2 p <latexit 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sha1_base64="zBJRKpQ03XuZBmfnNyPv3JIi0Ao=">AAAC/3icjVHNTtwwGBxCS5efwrYcuVhdIXFhlSwHEBLSqr1wpBILSCxCjtdLo80mxnEqoSiHvklv3BDXvkCvcEN9A/oW/eyaih+h4ijJeL6ZsT87VmlSmDD8NRFMvno99aYxPTM793Z+ofnu/V6Rl1rInsjTXB/EvJBpksmeSUwqD5SWfByncj8efbL1/a9SF0me7ZozJY/G/CRLhonghqjj5uY229xi/aHmoorqaq32uD/ITXVa1/SyLbb6T9Cp+yOuFGeKHTdbYTt0gz0FkQct+LGTN2/QxwA5BEqMIZHBEE7BUdBziAghFHFHqIjThBJXl6gxQ96SVJIUnNgRfU9odujZjOY2s3BuQauk9GpyMiyTJyedJmxXY65eumTLPpdduUy7tzP6xz5rTKzBF2L/57tTvtRnezEYYsP1kFBPyjG2O+FTSncqdufsXleGEhRxFg+orgkL57w7Z+Y8hevdni139VuntKydC68t8dvuki44enydT8Fepx2F7ehzp9X96K+6gSV8wArd5zq62MYOepT9HT9xhevgW3AeXASXf6XBhPcs4sEIfvwBNEmmTA==</latexit> N = 1<latexit 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sha1_base64="MRDBXiwF92ZrKOXPN4+xcBsvANw=">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</latexit> for 0 = @HGR @N = q ✓ 3 4 p2 + ⇤  ◆ <latexit 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sha1_base64="2KwQlw+G9cQV1+ysZgu2TekWafQ=">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</latexit> Scalar constraint:
  • 7. Spherical phase space The spherical space is consider mostly because of its relation to spin. It is also a non-trivial case since the sphere is not a cotangent bundle, but equipped with symplectic form is a well defined symplectic manifold. In this case, kinematics and dynamics can be associated with a certain spin system. This is important due to both practical reason (convenient to perform calculations) and fundamental reason (spin degrees of freedom for QG, relation to condensed-matter systems - analog models). change of variables R1R2 = S<latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit><latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">AAACzXicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6EYounFnrfaBtZQkndahaRKSiVCqbv0Bt/pb4h/oX3hnnIJaRCckOXPuPWfm3utGPk+EZb1mjJnZufmF7GJuaXlldS2/vlFPwjT2WM0L/TBuuk7CfB6wmuDCZ80oZs7Q9VnDHRzLeOOGxQkPgwsxilh76PQD3uOeI4i6rHbsaqdkHprnnXzBKlpqmdPA1qAAvSph/gVX6CKEhxRDMAQQhH04SOhpwYaFiLg2xsTFhLiKM9whR9qUshhlOMQO6NunXUuzAe2lZ6LUHp3i0xuT0sQOaULKiwnL00wVT5WzZH/zHitPebcR/V3tNSRW4JrYv3STzP/qZC0CPRyoGjjVFClGVudpl1R1Rd7c/FKVIIeIOIm7FI8Je0o56bOpNImqXfbWUfE3lSlZufd0bop3eUsasP1znNOgXiraVtE+2yuUj/Sos9jCNnZpnvso4wQV1Mg7wCOe8GycGqlxa9x/phoZrdnEt2U8fADRGJH0</latexit><latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit><latexit sha1_base64="EQQGZxdPbWiCIMWfS4rffAw+7yo=">AAACzXicjVHLSsNAFD2Nr1pfVZdugkVwVZJudCMU3bizVvtALSVJp3VoXkwmQqm69Qfc6m+Jf6B/4Z0xBbWITkhy5tx7zsy91419nkjLes0ZM7Nz8wv5xcLS8srqWnF9o5lEqfBYw4v8SLRdJ2E+D1lDcumzdiyYE7g+a7nDIxVv3TCR8Cg8l6OYdQJnEPI+9xxJ1EW9a9e7FfPAPOsWS1bZ0sucBnYGSshWLSq+4Ao9RPCQIgBDCEnYh4OEnkvYsBAT18GYOEGI6zjDHQqkTSmLUYZD7JC+A9pdZmxIe+WZaLVHp/j0ClKa2CFNRHmCsDrN1PFUOyv2N++x9lR3G9HfzbwCYiWuif1LN8n8r07VItHHvq6BU02xZlR1XuaS6q6om5tfqpLkEBOncI/igrCnlZM+m1qT6NpVbx0df9OZilV7L8tN8a5uSQO2f45zGjQrZdsq26eVUvUwG3UeW9jGLs1zD1Uco4YGeYd4xBOejRMjNW6N+89UI5dpNvFtGQ8f0HiR8g==</latexit> Dimensional parameters Symplectic form on a sphere: Volume of phase space: V = Z ⌦ ! = 4⇡S <latexit sha1_base64="4nzSd+70WiXvprD30R23AtL0GwU=">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</latexit><latexit sha1_base64="4nzSd+70WiXvprD30R23AtL0GwU=">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</latexit><latexit sha1_base64="4nzSd+70WiXvprD30R23AtL0GwU=">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</latexit><latexit sha1_base64="5Sw4G9hfpyiVYD+BUvEC2Jei7IY=">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</latexit>
  • 8. 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 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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 a vector {Si, Sj} = ✏ijkSk<latexit 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sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit><latexit sha1_base64="I7UOLjFAAj//XAWx8T1brYw7Vag=">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</latexit> The is therefore a vector of angular momentum (spin) with the norm S.
  • 9. Compact phase space FRW Kinematics (symplectic form) Dynamics (Hamiltonian) Friedmann equation: where The procedure is ambiguous! We make the simplest choice. Affine 2-sphere := 4 3 ⇤ R2 12 2 [0, 1] <latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="89Xa6GSnIb2JPqG3sm0wmExRhx0=">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</latexit> The affine case is recovered in the limit.R1, R2 ! 1<latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="MaR+F5j7siTjGesrWgPJZwhvMBE=">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</latexit>
  • 10. Polymer limit This is the case of loop quantum cosmology. R2 ! 1<latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">AAAC2XicjVHLSsNAFD2Nr1pf9bFzEyyCq5KIoMuiG5dV7APaUpJ02g5NkzCZKLW4cCdu/QG3+kPiH+hfeGdMQS2iE5KcOfeeM3PvdSOfx9KyXjPGzOzc/EJ2Mbe0vLK6ll/fqMZhIjxW8UI/FHXXiZnPA1aRXPqsHgnmDF2f1dzBiYrXLpmIeRhcyFHEWkOnF/Au9xxJVDu/dd7ebwre60tHiPCqyYOuHLXzBato6WVOAzsFBaSrHOZf0EQHITwkGIIhgCTsw0FMTwM2LETEtTAmThDiOs5wgxxpE8pilOEQO6Bvj3aNlA1orzxjrfboFJ9eQUoTu6QJKU8QVqeZOp5oZ8X+5j3WnupuI/q7qdeQWIk+sX/pJpn/1alaJLo40jVwqinSjKrOS10S3RV1c/NLVZIcIuIU7lBcEPa0ctJnU2tiXbvqraPjbzpTsWrvpbkJ3tUtacD2z3FOg+p+0baK9tlBoXScjjqLbexgj+Z5iBJOUUaFvK/xiCc8Gw3j1rgz7j9TjUyq2cS3ZTx8ABqymAI=</latexit><latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">AAAC2XicjVHLSsNAFD2Nr1pf9bFzEyyCq5KIoMuiG5dV7APaUpJ02g5NkzCZKLW4cCdu/QG3+kPiH+hfeGdMQS2iE5KcOfeeM3PvdSOfx9KyXjPGzOzc/EJ2Mbe0vLK6ll/fqMZhIjxW8UI/FHXXiZnPA1aRXPqsHgnmDF2f1dzBiYrXLpmIeRhcyFHEWkOnF/Au9xxJVDu/dd7ebwre60tHiPCqyYOuHLXzBato6WVOAzsFBaSrHOZf0EQHITwkGIIhgCTsw0FMTwM2LETEtTAmThDiOs5wgxxpE8pilOEQO6Bvj3aNlA1orzxjrfboFJ9eQUoTu6QJKU8QVqeZOp5oZ8X+5j3WnupuI/q7qdeQWIk+sX/pJpn/1alaJLo40jVwqinSjKrOS10S3RV1c/NLVZIcIuIU7lBcEPa0ctJnU2tiXbvqraPjbzpTsWrvpbkJ3tUtacD2z3FOg+p+0baK9tlBoXScjjqLbexgj+Z5iBJOUUaFvK/xiCc8Gw3j1rgz7j9TjUyq2cS3ZTx8ABqymAI=</latexit><latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">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</latexit><latexit sha1_base64="i0BtGXKYQ1woH+zsYAye3Punzfc=">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</latexit> In the limit the so-called polymerization of momentum is obtained. The su(2) algebra reduces to the cylindrical case: The Hamiltonian reduces to where the polymerization scale is: := 1 R1<latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">AAAC3XicjVHLSsNAFD3Gd31V3QhugkVwVRIRFEEounGpYq3QljCZTjWYJmEyEUqoO3fi1h9wq78j/oH+hXfGCD4QnZDkzLnnnJk74ydhkCrHeR6yhkdGx8YnJktT0zOzc+X5hZM0ziQXdR6HsTz1WSrCIBJ1FahQnCZSsJ4fioZ/safrjUsh0yCOjlU/Ee0eO4uCbsCZIsorL7VCEneYvb1jt7qS8dwd5EeeO/DKFafqmGH/BG4BKijGQVx+QgsdxODI0INABEU4BENKTxMuHCTEtZETJwkFpi4wQIm8GakEKRixF/Q9o1mzYCOa68zUuDmtEtIryWljlTwx6SRhvZpt6plJ1uxv2bnJ1Hvr098vsnrEKpwT+5fvQ/lfn+5FoYst00NAPSWG0d3xIiUzp6J3bn/qSlFCQpzGHapLwtw4P87ZNp7U9K7Plpn6i1FqVs95oc3wqndJF+x+v86f4GS96jpV93CjUtstrnoCy1jBGt3nJmrYxwHqlH2Fezzg0fKsa+vGun2XWkOFZxFfhnX3BrBLmIs=</latexit><latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit><latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit><latexit sha1_base64="IpcgmU5scOhVCdLM71hbptMDtxU=">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</latexit> The Friedmann equation is where
  • 11. Cosmological evolution The leading correction behaves as a phantom matter with the equation of state: ~ de Sitter expansion ~ de Sitter contraction Cosmological recollapse (IR effect) (UV effect) (UV effect) ⇤ ! ⇤e↵ = ⇤(1 )<latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="tMpofGH0WR6riQZeZTjXuvBAuG0=">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</latexit> ⇤ ! ⇤e↵ = ⇤(1 )<latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="tMpofGH0WR6riQZeZTjXuvBAuG0=">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</latexit> de Sitter expansion may terminate… = ⇢⇤ ⇢c 2 [0, 1] <latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">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</latexit><latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">AAAC9XicjVHBTttAEH0xtNBAS1qOXFZESBxQZFdI9IIUlUsPPYBEIFIcRevNJlnh2NZ63SqK8hu9cUNc+YFe4RcQfwB/0dnFSNCoKmvZfvNm3tudnSiLVW58/67iLSy+ebu0/K66svr+w1rt46eTPC20kC2RxqluRzyXsUpkyygTy3amJR9HsTyNzg5s/vSH1LlKk2MzyWR3zIeJGijBDVG9mh/2ZWw422fhQHMxDfUo7U3D7+TQ57PZYyxmLFQJ6/g7QbdXq/sN3y02D4IS1FGuw7R2ixB9pBAoMIZEAkM4BkdOTwcBfGTEdTElThNSLi8xQ5W0BVVJquDEntF3SFGnZBOKrWfu1IJ2ienVpGTYIk1KdZqw3Y25fOGcLfsv76nztGeb0D8qvcbEGoyI/Z/uqfK1OtuLwQBfXA+KesocY7sTpUvhbsWenD3rypBDRpzFfcprwsIpn+6ZOU3uerd3y13+3lVa1sairC3wYE9JAw7+Huc8OPncCPxGcLRbb34tR72MDWxim+a5hya+4RAt8v6F37jGjffTO/cuvMvHUq9SatbxYnlXfwAbdqKP</latexit><latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">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</latexit><latexit sha1_base64="j4/HWccULijMCRJsJrBIHjusaik=">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</latexit> and the effective phantom can be a sign of it.
  • 12. Evolution on the phase space Exact solutions: = 0.1<latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="zs0Q4bWdZsUJj9bry7G2E35WgXA=">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</latexit> = 0.5<latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="n7rCGcVm8TztJDnGePDDtHfx+AI=">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</latexit> = 0.9<latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="/8b2BwzT4ViCy1omV12vrBaQYhg=">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</latexit> There are four symmetric solutions for the four quadrants of a sphere.