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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=">AAADAnicjVHLahRBFD3pqInxNerSTeEgjCBDdxB0E0gUwWVEJwmkY6ju1MwU0y+qqoUwzC5/kp07cesPuNWF+Af6F54qO6AG0Wq6+9S595yqe2/WFNq6OP66FC1fuHhpZfXy2pWr167f6N28tWPr1uRqlNdFbfYyaVWhKzVy2hVqrzFKllmhdrPZUx/ffaOM1XX1yh036qCUk0qPdS4dqcPeRlqosRsI9Xqe6sqJrcUD8Sw1ejJ19wUZ8XI0WCdyulRWpKV007GRs7ltSS8Oe/14GIclzoOkA310a7vufUGKI9TI0aKEQgVHXEDC8tlHghgNuQPMyRkiHeIKC6xR2zJLMUOSnfE74W6/YyvuvacN6pynFHwNlQL3qKmZZ4j9aSLE2+Ds2b95z4Onv9sx/1nnVZJ1mJL9l+4s8391vhaHMR6HGjRragLjq8s7lzZ0xd9c/FKVo0NDzuMjxg1xHpRnfRZBY0PtvrcyxL+FTM/6fd7ltvjub8kBJ3+O8zzYWR8m8TB58bC/+aQb9Sru4C4GnOcjbOI5tjGi9yk+4hM+RyfR2+hd9P5narTUaW7jtxV9+AFW4aY9</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=">AAAC43icjVHLSsNAFD2Nr1pfVZeCBIvgQksigm6EogguK9gHtLUk6bSGpkmYTIRSunPnTtz6A271X8Q/0L/wzphCtYhOSHLuufecmTvXDj03EobxltKmpmdm59LzmYXFpeWV7OpaOQpi7rCSE3gBr9pWxDzXZyXhCo9VQ86snu2xit09lfnKDeORG/iXoh+yRs/q+G7bdSxBVDO7WTvbLV0N6i2r02F82NCP9b2xuJnNGXlDLX0SmAnIIVnFIPuKOloI4CBGDww+BGEPFiJ6ajBhICSugQFxnJCr8gxDZEgbUxWjCovYLn07FNUS1qdYekZK7dAuHr2clDq2SRNQHScsd9NVPlbOkv3Ne6A85dn69LcTrx6xAtfE/qUbVf5XJ3sRaONI9eBST6FiZHdO4hKrW5En18e6EuQQEidxi/KcsKOUo3vWlSZSvcu7tVT+XVVKVsZOUhvjQ56SBmz+HOckKO/nTSNvXhzkCifJqNPYwBZ2aJ6HKOAcRZTI+xZPeMaLxrQ77V57+CrVUolmHd+W9vgJMaqbDQ==</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=">AAAC2HicjVHLSsNAFD2Nr1pf0S7dBIvgQkriRjdC0Y3LCqYttrUk6bSGpkmYTIRSCu7ErT/gVr9I/AP9C++MKahFdEKSM+fec2buvW4c+IkwzdecNje/sLiUXy6srK6tb+ibW7UkSrnHbC8KIt5wnYQFfshs4YuANWLOnKEbsLo7OJXx+g3jiR+FF2IUs/bQ6Yd+z/ccQVRHLzbtfftq3Oo6/T7jk7ZxbJgdvWSWTbWMWWBloIRsVSP9BS10EcFDiiEYQgjCARwk9DRhwURMXBtj4jghX8UZJiiQNqUsRhkOsQP69mnXzNiQ9tIzUWqPTgno5aQ0sEuaiPI4YXmaoeKpcpbsb95j5SnvNqK/m3kNiRW4JvYv3TTzvzpZi0APR6oGn2qKFSOr8zKXVHVF3tz4UpUgh5g4ibsU54Q9pZz22VCaRNUue+uo+JvKlKzce1luind5Sxqw9XOcs6B2ULbMsnV+UKqcZKPOYxs72KN5HqKCM1Rhk/cIj3jCs3ap3Wp32v1nqpbLNEV8W9rDB9VWlj8=</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=">AAAC1nicjVHLSsNAFD2Nr1pfrS7dBIvgQksigm6EohuXldoH1BKSdFoH0yQkE7UW3Ylbf8CtfpL4B/oX3hlTUIvohCRnzr3nzNx7ndDjsTCM14w2MTk1PZOdzc3NLywu5QvL9ThIIpfV3MALoqZjx8zjPqsJLjzWDCNm9x2PNZzzQxlvXLAo5oF/IgYha/ftns+73LUFUVa+0NqqWtebVWvQ3te5XrWurHzRKBlq6ePATEER6aoE+RecooMALhL0weBDEPZgI6anBRMGQuLaGBIXEeIqznCDHGkTymKUYRN7Tt8e7Vop69NeesZK7dIpHr0RKXWskyagvIiwPE1X8UQ5S/Y376HylHcb0N9JvfrECpwR+5dulPlfnaxFoIs9VQOnmkLFyOrc1CVRXZE3179UJcghJE7iDsUjwq5SjvqsK02sape9tVX8TWVKVu7dNDfBu7wlDdj8Oc5xUN8umUbJPN4plg/SUWexijVs0Dx3UcYRKqiR9yUe8YRnrandanfa/Weqlkk1K/i2tIcPZs+VTg==</latexit>
<latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit>
<latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">AAAC1nicjVHLSsNAFD2Nr1pfrS7dBIvgQksigm6EohuXldoH1BKSdFoH0yQkE7UW3Ylbf8CtfpL4B/oX3hlTUIvohCRnzr3nzNx7ndDjsTCM14w2MTk1PZOdzc3NLywu5QvL9ThIIpfV3MALoqZjx8zjPqsJLjzWDCNm9x2PNZzzQxlvXLAo5oF/IgYha/ftns+73LUFUVa+0NqqWtebVWvQ3te5XrWurHzRKBlq6ePATEER6aoE+RecooMALhL0weBDEPZgI6anBRMGQuLaGBIXEeIqznCDHGkTymKUYRN7Tt8e7Vop69NeesZK7dIpHr0RKXWskyagvIiwPE1X8UQ5S/Y376HylHcb0N9JvfrECpwR+5dulPlfnaxFoIs9VQOnmkLFyOrc1CVRXZE3179UJcghJE7iDsUjwq5SjvqsK02sape9tVX8TWVKVu7dNDfBu7wlDdj8Oc5xUN8umUbJPN4plg/SUWexijVs0Dx3UcYRKqiR9yUe8YRnrandanfa/Weqlkk1K/i2tIcPZs+VTg==</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=">AAAC13icjVHLSsNAFD2N7/qqunQTLIILLYkb3QiiG5dKra3UEpJx1ME0CZOJtBZxJ279Abf6R+If6F94Z0xBLaITkpw5954zc+8NklCkynFeC9bQ8Mjo2PhEcXJqema2NDd/lMaZZLzG4jCWjcBPeSgiXlNChbyRSO63g5DXg8tdHa9fcZmKODpU3YS32v55JM4E8xVRXmm+uVb1rlerXqe1Za8Ju+p1vVLZqThm2YPAzUEZ+dqPSy84wSliMGRogyOCIhzCR0pPEy4cJMS10CNOEhImznGDImkzyuKU4RN7Sd9z2jVzNqK99kyNmtEpIb2SlDaWSRNTniSsT7NNPDPOmv3Nu2c89d269A9yrzaxChfE/qXrZ/5Xp2tROMOmqUFQTYlhdHUsd8lMV/TN7S9VKXJIiNP4lOKSMDPKfp9to0lN7bq3vom/mUzN6j3LczO861vSgN2f4xwER+sV16m4B+vl7Z181ONYxBJWaJ4b2MYe9lEj7w4e8YRn69i6te6s+89Uq5BrFvBtWQ8f+6uVgw==</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 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=">AAAC+HicjVHLLgRBFD3aa7wHS5uKicRq0o2EjUTYWJIYJAapbmVU9HR3qqsnYcyH2NmJrR+w5QvEH/AXbpWSYCJUp7tPnXvPqbr3hlksc+37Lz1eb1//wGBpaHhkdGx8ojw5tZunhYpELUrjVO2HPBexTERNSx2L/UwJ3gxjsReeb5j4XkuoXKbJjr7IxGGTNxJ5KiOuiTouL9bYFau3uMrOJKsrnjRiwVaZOGpLx3a64sflil/17WLdIHCgAre20vIz6jhBiggFmhBIoAnH4MjpOUAAHxlxh2gTpwhJGxfoYJi0BWUJyuDEntO3QbsDxya0N565VUd0SkyvIiXDHGlSylOEzWnMxgvrbNjfvNvW09ztgv6h82oSq3FG7F+6z8z/6kwtGqdYsTVIqimzjKkuci6F7Yq5OftSlSaHjDiDTyiuCEdW+dlnZjW5rd30ltv4q800rNlHLrfAm7klDTj4Oc5usLtQDfxqsL1UWVt3oy5hBrOYp3kuYw2b2EKNvK/xgEc8eZfejXfr3X2kej1OM41vy7t/Bwc3pBQ=</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=">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</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=">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=">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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=">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</latexit>
<latexit sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">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</latexit>
<latexit sha1_base64="+cKEhSbu8x1k6ugIVW++eJt129I=">AAAC8HicjVHLSsNAFD2Nr/quunQTLIIglKQbXRbduKyPqmCrTOLYDubFZKKU0o9w507c+gNu9SvEP9C/8M40BR+ITkhy5tx7zsy910sCkSrHeS1YI6Nj4xPFyanpmdm5+dLC4mEaZ9LnDT8OYnnssZQHIuINJVTAjxPJWegF/Mi73NbxoysuUxFHB6qb8FbI2pG4ED5TRJ2V1pshUx3P6+31T3vVsG83pWh3FJMyvraHsf1B7KxUdiqOWfZP4OagjHzV49ILmjhHDB8ZQnBEUIQDMKT0nMCFg4S4FnrESULCxDn6mCJtRlmcMhixl/Rt0+4kZyPaa8/UqH06JaBXktLGKmliypOE9Wm2iWfGWbO/efeMp75bl/5e7hUSq9Ah9i/dMPO/Ol2LwgU2TQ2CakoMo6vzc5fMdEXf3P5UlSKHhDiNzykuCftGOeyzbTSpqV33lpn4m8nUrN77eW6Gd31LGrD7fZw/wWG14joVd7darm3loy5iGStYo3luoIYd1NEg7xs84gnPlrRurTvrfpBqFXLNEr4s6+EDzpih0A==</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=">AAAC+XicjVHLSsQwFD3W93vUpZviILgaWhF0I4huXI7gjIKjQxqjhumLJhWl9EfcuRO3/oBb/QHxD/QvvIkVfCCa0vbk3HtOcu8N0lAq7XnPfU7/wODQ8Mjo2PjE5NR0bWa2rZI846LFkzDJ9gOmRChj0dJSh2I/zQSLglDsBb0tE987F5mSSbyrL1NxGLHTWJ5IzjRR3dpKJ2L6jLOw2C67RUeLC130ZFyW7rr7OaSOik6iZSSUG5XdWt1reHa5P4FfgTqq1UxqT+jgGAk4ckQQiKEJh2BQ9BzAh4eUuEMUxGWEpI0LlBgjbU5ZgjIYsT36ntLuoGJj2htPZdWcTgnpzUjpYpE0CeVlhM1pro3n1tmwv3kX1tPc7ZL+QeUVEatxRuxfuo/M/+pMLRonWLM1SKoptYypjlcuue2Kubn7qSpNDilxBh9TPCPMrfKjz67VKFu76S2z8RebaViz51VujldzSxqw/32cP0F7ueF7DX9npb6xWY16BPNYwBLNcxUb2EYTLfK+wj0e8OgUzrVz49y+pzp9lWYOX5Zz9wbMVKXx</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=">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</latexit>
<latexit sha1_base64="FtnA9M+wDi07g5/jW3UWXXbNWeE=">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</latexit>
Compactified flat de Sitter cosmology
Kinematics (symplectic form)
Dynamics (scalar constraint)
Friedmann
equation:
where
The procedure is
ambiguous! We made
the simplest choice.
Affine
2-sphere
:=
4
3
⇤
R2
12
2 [0, 1]
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<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>
C = q
✓
3
4
p2
+
⇤

◆
⇡ 0
<latexit sha1_base64="Drg0ok2XHaUlzso2eqLHAYLIprw=">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</latexit>
<latexit sha1_base64="Drg0ok2XHaUlzso2eqLHAYLIprw=">AAADEHicjVFNT9VAFD0UVMCvhy7ZTHwxwRhfWiSBhSYkbEh0gYkPSCiS6bx5j8nra8fplEia/gn+iTt3hq1/gLjTHfwL7wxDghKj07Q999x7zsydm+lcVTaOv09F0zO3bt+ZnZu/e+/+g4edhUfbVVkbIfuizEuzm/FK5qqQfatsLne1kXyS5XInG2+4/M6RNJUqi/f2WMv9CR8VaqgEt0QddN5ssNfsY5rLoV16wdKh4aJ52TYrbTrmWnOmPyw/D3T6lmwHvG0uU21q1OjQPmMpRab8xOKDTjfuxX6xmyAJoIuwtsrOGVIMUEKgxgQSBSzhHBwVPXtIEEMTt4+GOENI+bxEi3nS1lQlqYITO6bviKK9wBYUO8/KqwXtktNrSMnwlDQl1RnCbjfm87V3duzfvBvv6c52TP8seE2ItTgk9l+6q8r/1bleLIZY8z0o6kl7xnUngkvtb8WdnF3rypKDJs7hAeUNYeGVV/fMvKbyvbu75T5/7isd62IRamtcuFPSgJM/x3kTbC/3kriXvFvprr8Ko57FIp5giea5inVsYgt98v6MM/zAz+gk+hJ9jU4vS6OpoHmM31b07RdAqqzb</latexit>
<latexit sha1_base64="Drg0ok2XHaUlzso2eqLHAYLIprw=">AAADEHicjVFNT9VAFD0UVMCvhy7ZTHwxwRhfWiSBhSYkbEh0gYkPSCiS6bx5j8nra8fplEia/gn+iTt3hq1/gLjTHfwL7wxDghKj07Q999x7zsydm+lcVTaOv09F0zO3bt+ZnZu/e+/+g4edhUfbVVkbIfuizEuzm/FK5qqQfatsLne1kXyS5XInG2+4/M6RNJUqi/f2WMv9CR8VaqgEt0QddN5ssNfsY5rLoV16wdKh4aJ52TYrbTrmWnOmPyw/D3T6lmwHvG0uU21q1OjQPmMpRab8xOKDTjfuxX6xmyAJoIuwtsrOGVIMUEKgxgQSBSzhHBwVPXtIEEMTt4+GOENI+bxEi3nS1lQlqYITO6bviKK9wBYUO8/KqwXtktNrSMnwlDQl1RnCbjfm87V3duzfvBvv6c52TP8seE2ItTgk9l+6q8r/1bleLIZY8z0o6kl7xnUngkvtb8WdnF3rypKDJs7hAeUNYeGVV/fMvKbyvbu75T5/7isd62IRamtcuFPSgJM/x3kTbC/3kriXvFvprr8Ko57FIp5giea5inVsYgt98v6MM/zAz+gk+hJ9jU4vS6OpoHmM31b07RdAqqzb</latexit>
<latexit sha1_base64="grBQZepvEz5kuSQWIOKUXopGRZU=">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</latexit>
C =
S3
R1

3
4

S2
2
R2
2
⇤

⇡ 0
<latexit sha1_base64="zb/oE4jEq2O3VrO4QzItGJ03nkA=">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</latexit>
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Cylindrical limit (LQC)
This is the case of loop quantum cosmology.
R2 ! 1
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<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 scalar constraint
reduces to:
where the
polymerization
scale is: :=
1
R1
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<latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit>
<latexit sha1_base64="IpcgmU5scOhVCdLM71hbptMDtxU=">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</latexit>
The Friedmann
equation is:
where
C = q
✓
3
4
sin2
( p)
2
+
⇤

◆
⇡ 0
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<latexit sha1_base64="PPEVbLHbpaukz+XJa6s2xu2oZfE=">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</latexit>
<latexit sha1_base64="PPEVbLHbpaukz+XJa6s2xu2oZfE=">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</latexit>
<latexit sha1_base64="ouKKeo79jgC2YBg9Te2mhh8QX/U=">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</latexit>
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=">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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.
Action of the operator is defined on the kinematical Hilbert space .
Following the Dirac quantization of constrained systems, the physical Hilbert
space is constructed by solving the Wheeler-deWitt (WdW) equation:
H[N, ~
N] = S[N] + V [ ~
N] =
Z
⌃
d3
xNC +
Z
⌃
d3
x ~
N · ~
C
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The Hamiltonian of General Relativity (GR) is a sum of constraints:
In the quantum case, the constraints are promoted to self-adjoint operators.
In homogeneous quantum cosmology only the scalar constraint remains.
Ĉ
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Hkin
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<latexit sha1_base64="Na0/oxoSKj00IU7ESisf6WwBq+o=">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</latexit>
Hphys
<latexit sha1_base64="uembQmfG4JOjJgwpETn6L/ZOZjc=">AAAC3nicjVHLSsNAFD3GV31HXYmbYBFclUQEXRbddFnBPsBKnUynNpgXyUQsobhzJ279Abf6OeIf6F94Z4zgA9EJSc6ce8+Zufe6se+l0rafx4zxicmp6dLM7Nz8wuKSubzSTKMs4aLBIz9K2i5Lhe+FoiE96Yt2nAgWuL5ouecHKt66EEnqReGRHMbiJGBnodf3OJNEdc21TsDkgDM/r426eUeKS5nHg2E6GnXNsl2x9bJ+AqcAZRSrHplP6KCHCBwZAgiEkIR9MKT0HMOBjZi4E+TEJYQ8HRcYYZa0GWUJymDEntP3jHbHBRvSXnmmWs3pFJ/ehJQWNkkTUV5CWJ1m6XimnRX7m3euPdXdhvR3C6+AWIkBsX/pPjL/q1O1SPSxp2vwqKZYM6o6Xrhkuivq5tanqiQ5xMQp3KN4Qphr5UefLa1Jde2qt0zHX3SmYtWeF7kZXtUtacDO93H+BM3timNXnMOdcnW/GHUJ69jAFs1zF1XUUEeDvK9wjwc8GqfGtXFj3L6nGmOFZhVflnH3Bj7MmtA=</latexit>
<latexit sha1_base64="uembQmfG4JOjJgwpETn6L/ZOZjc=">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</latexit>
<latexit sha1_base64="uembQmfG4JOjJgwpETn6L/ZOZjc=">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</latexit>
<latexit sha1_base64="modhOdsYBh9Pr4zzgyDvRRw/xKw=">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</latexit>
so that .
ker Ĉ = Hphys ✓ Hkin
<latexit sha1_base64="9nUrmvOk+TrJZACl0Ar2JLXc438=">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</latexit>
<latexit sha1_base64="9nUrmvOk+TrJZACl0Ar2JLXc438=">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</latexit>
<latexit sha1_base64="9nUrmvOk+TrJZACl0Ar2JLXc438=">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</latexit>
<latexit sha1_base64="yYHBzdcJNncpUuIFwbAIup3vm3A=">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</latexit>
The WdW is difficult to solve in general case. Solutions are known for
certain minisuperspace models.
Complexity of the problem grows exponentially with the #DOF.
Ĉ| i = 0
<latexit sha1_base64="BKuUD4iZtyu6Fefg7Le24HSm5II=">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</latexit>
<latexit sha1_base64="BKuUD4iZtyu6Fefg7Le24HSm5II=">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</latexit>
<latexit sha1_base64="BKuUD4iZtyu6Fefg7Le24HSm5II=">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</latexit>
<latexit sha1_base64="rTBbVb5aEd/ohM8h/5/sgT34p+0=">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</latexit>
Our quantum constraint (m=1)
The symmetrized quantum scalar constraint:
We have shown that the WdW has always solutions for any for the bosonic
representations. For the fermionic representations the solutions do not exist,
except some particular values of .
Rescaled classical constraint:
<latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit>
<latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit>
<latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit>
<latexit sha1_base64="3cSg74jivEBcXT93HXVaPjnpdNI=">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</latexit>
<latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit>
<latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit>
<latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit>
<latexit sha1_base64="3cSg74jivEBcXT93HXVaPjnpdNI=">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</latexit>
C !
4S2
3R1
C = S3S2
2 S2
S3.
<latexit sha1_base64="KveFKTKWFFNpcVJujvoGLEj0QpU=">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</latexit>
<latexit sha1_base64="KveFKTKWFFNpcVJujvoGLEj0QpU=">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</latexit>
<latexit sha1_base64="KveFKTKWFFNpcVJujvoGLEj0QpU=">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</latexit>
<latexit sha1_base64="Cd6FykPgCQgyN+ScgP63phTvClA=">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</latexit>
Ĉ =
1
3
⇣
ŜzŜyŜy + ŜyŜzŜy + ŜyŜyŜz
⌘
Ŝ2
Ŝz ⇡ 0
<latexit sha1_base64="CNOTuOXZRc2+5AjBNaMPGUeXVnY=">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</latexit>
<latexit sha1_base64="CNOTuOXZRc2+5AjBNaMPGUeXVnY=">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</latexit>
<latexit sha1_base64="CNOTuOXZRc2+5AjBNaMPGUeXVnY=">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</latexit>
<latexit sha1_base64="U+Oc5M+yQzvGxjCdLFQ3RvafHD0=">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</latexit>
The simplest non-trivial case is s=1, for which:
Ĉ = 2
✓
1
6
◆
0
@
1 0 0
0 0 0
0 0 1
1
A = 2
✓
1
6
◆
Ŝz
<latexit sha1_base64="KdDizENsWZavvS6Az/o4+WX0wrU=">AAADYnicjVFBT9swFH5pGGNhgzKO4xBRDcGByikT22GTkLjsCGIFJIwqx3Vbq24SOQ5SF+Wn7HdNu++w/Ys9G1eCIQSOknz+3vs+v+eXFkqWhpBfQStcerH8cuVVtPr6zdp6e+PteZlXmos+z1WuL1NWCiUz0TfSKHFZaMFmqRIX6fTYxi9uhC5lnn0z80Jcz9g4kyPJmUFq0P5BJ8zUx00cf+nFVImR2aUjzXidNPVhs0+HQhlGtRxPzN4inoqxzGqmNZs3Nee8iZIdskMojcid/36CgIps6DMXJs85x9Z01gy+D9od0iVuxQ9B4kEH/DrJ2z+BwhBy4FDBDARkYBArYFDicwUJECiQu4YaOY1IuriABiLUVpglMIMhO8XvGHdXns1wbz1Lp+Z4isJXozKG96jJMU8jtqfFLl45Z8s+5l07T1vbHP+p95oha2CC7FO6ReZzdbYXAyP45HqQ2FPhGNsd9y6VuxVbeXynK4MOBXIWDzGuEXOnXNxz7DSl693eLXPxPy7TsnbPfW4Ff22VOODk/3E+BOe9bkK6yemHztFnP+oVeAfbsIvz/AhH8BVOoA88CIO9oBcctH6HUbgRbt6mtgKv2YR7K9z6B3Xhwy8=</latexit>
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The physical state, which satisfy the constraint, is:
| i =
0
@
0
1
0
1
A = |s = 1, sz = 0i =
1
p
2
(|01i + |10i)
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Transition amplitude for s=2
The transition amplitude is peaked for ⇤ ! 0+
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- SU(2) coherent states
Outlook of our method of solving the WdW equation
for arbitrary #DOF (m) on a quantum computer:
1. Regularize theory to make the Hilbert space finite.
a) Replace the flat (affine) phase space for every classical degree of
freedom with a sphere. The spherical phase space is a phase space of
angular momentum (spin). The flat phase space case is recovered in
the large spin limit.
b) Construct regularized quantum kinematics for the system under
consideration.
c) Express the Hamiltonian constraint in terms of the spin variables.
d) Quantize the regularized Hamiltonian constraint.
e) Represent the spin operators in terms of qubits.
2. Apply Variational Quantum Eigensolver (VQE) to
find the states minimizing „Hamiltonian” :
Ĉ| 0i = 0 () h 0|Ĉ2
| 0i = 0
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3. Study the large spin limit to recover the affine case.
Ĉ2
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Our constraint squared in general case:
The action of the spin operators on n-qubit (spin-1/2) quantum register
is given by:
where n = 2s.
Ŝi =
1
2
n
X
j=1
I1
⌦ ...Ij 1
⌦ ˆj
i ⌦ Ij+1
⌦ ...In
,
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For convenience, we introduced:
…
Here, we need n=2s qubits to represent spin s.
But, we know how to do it better, i.e. 2s + 1 = 2n
<latexit sha1_base64="bp3es4QWHSkndV6WQaNqUItVCls=">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</latexit>
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Because is not a unitary operator, the expectation values cannot be
evaluated directly with the use of quantum computing. For this purpose
the operator has to be expanded into unitarities (here, the Pauli matrices):
Ĉ =
X
j
cj
O
i
ˆk
ij
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<latexit sha1_base64="0Cs+rOwkKD1gf2j17EesVnemQqU=">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</latexit>
hĈi =
X
j
cjh
O
i
ˆk
iji
<latexit sha1_base64="HVnRU6ES3gjp3pLeWHWRLMbQBNo=">AAADFnicjVFNTxUxFD0MCIgKD1y6mfBi4uplxpjAhoTAhiUkPiBhcNIpZSiv85G2Q0Je3v/gn7BjZ9y6My7cgv4Lb0tJVGK0k5m599xzTnt7i1ZJY5Pk61Q0PfNkdm7+6cKz5y8Wl3rLK/um6TQXQ96oRh8WzAglazG00ipx2GrBqkKJg2K07eoHF0Ib2dTv7WUrjitW1vJUcmYJynt7mWJ1qUScnTE73p7Emfb5Rma6Kj+POb2BkhWybKyshMnlPT0zsqzYJB/L88mHUVDmvX4ySPyKHwdpCPoIa7fpfUGGEzTg6FBBoIalWIHB0HOEFAlawo4xJkxTJH1dYIIF0nbEEsRghI7oW1J2FNCacudpvJrTLopeTcoYr0nTEE9T7HaLfb3zzg79m/fYe7qzXdK/CF4VoRZnhP5L98D8X53rxeIU674HST21HnHd8eDS+VtxJ49/6cqSQ0uYi0+orinmXvlwz7HXGN+7u1vm698906Eu54Hb4Yc7JQ04/XOcj4P9t4M0GaR77/qbW2HU83iFVbyhea5hEzvYxZC8r/ENt7iLrqKb6GP06Z4aTQXNS/y2os8/AVsvsc0=</latexit>
<latexit sha1_base64="HVnRU6ES3gjp3pLeWHWRLMbQBNo=">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</latexit>
<latexit sha1_base64="HVnRU6ES3gjp3pLeWHWRLMbQBNo=">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</latexit>
<latexit sha1_base64="iN0LQ0JLsBPZAs9GiH4vnmaKENo=">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</latexit>
so that
Every contributing expectation value has to be evaluated individually. The
so-called Hadamard test can be used for this purpose:
= hÛi
<latexit sha1_base64="T8WhDTjASSw9waZqq1rFQYka75I=">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</latexit>
<latexit sha1_base64="T8WhDTjASSw9waZqq1rFQYka75I=">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</latexit>
<latexit sha1_base64="T8WhDTjASSw9waZqq1rFQYka75I=">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</latexit>
<latexit sha1_base64="nGORFQg+QanZ6vf2amof1VvF7yU=">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</latexit>
Ĉ2
<latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit>
<latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit>
<latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit>
<latexit sha1_base64="/fRjTsS43q5BgcFlTBcHi9YBB/I=">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</latexit>
Similarly for .
Ĉ2
<latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit>
<latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit>
<latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit>
<latexit sha1_base64="/fRjTsS43q5BgcFlTBcHi9YBB/I=">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</latexit>
The variational procedure
c (↵) =
a
max | i|
2 h (↵) |Ĉ†
Ĉ| (↵)i
+ b 1
h (↵) |
ˆ
~
S2
| (↵)i
s (s + 1)
!
<latexit sha1_base64="g8bBO9OStOL5U6TgNc9QlX/+0hY=">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</latexit>
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a, b 2 (0, 1)
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<latexit sha1_base64="HJFbPwxv2wadA9FbrOYgd89Yl9Q=">AAAC0HicjVHLSsNAFD2Nr1pfVZdugkWoUErSjS6LblxWsQ9oiyTTaR2aJjGZiKWIuPUH3OpXiX+gf+GdMQW1iE5Icubce87MvdcNPRFLy3rNGHPzC4tL2eXcyura+kZ+c6sRB0nEeJ0FXhC1XCfmnvB5XQrp8VYYcWfkerzpDo9VvHnNo1gE/rkch7w7cga+6AvmSKK6Tsk1O8I3i1bJ3r/IF6yypZc5C+wUFJCuWpB/QQc9BGBIMAKHD0nYg4OYnjZsWAiJ62JCXERI6DjHLXKkTSiLU4ZD7JC+A9q1U9anvfKMtZrRKR69ESlN7JEmoLyIsDrN1PFEOyv2N++J9lR3G9PfTb1GxEpcEvuXbpr5X52qRaKPQ12DoJpCzajqWOqS6K6om5tfqpLkEBKncI/iEWGmldM+m1oT69pVbx0df9OZilV7luYmeFe3pAHbP8c5CxqVsm2V7dNKoXqUjjqLHeyiSPM8QBUnqKFO3ld4xBOejTPjxrgz7j9TjUyq2ca3ZTx8AOGIkrs=</latexit>
a + b = 1
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<latexit sha1_base64="wu1wK1Qeba63pdtAX6R1YKmLnIw=">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</latexit>
<latexit sha1_base64="MOWKC3rmtbGshqR7KNb2jatStmU=">AAACyXicjVHLSsNAFD2Nr1pfVZdugkUQhJJ0oxuh6EZwU8E+oBZJptMamyZxMhFrceUPuNUfE/9A/8I74whqEZ2Q5My595yZe6+fhEEqHeclZ01Nz8zO5ecLC4tLyyvF1bVGGmeC8TqLw1i0fC/lYRDxugxkyFuJ4N7QD3nTHxyqePOaizSIo1M5Snhn6PWjoBcwTxLV8Hb8fdc+L5acsqOXPQlcA0owqxYXn3GGLmIwZBiCI4IkHMJDSk8bLhwkxHUwJk4QCnSc4w4F0maUxSnDI3ZA3z7t2oaNaK88U61mdEpIryCljS3SxJQnCKvTbB3PtLNif/Mea091txH9feM1JFbigti/dJ+Z/9WpWiR62NM1BFRTohlVHTMume6Kurn9pSpJDglxCncpLggzrfzss601qa5d9dbT8VedqVi1ZyY3w5u6JQ3Y/TnOSdColF2n7J5UStUDM+o8NrCJbZrnLqo4Qg118r7EAx7xZB1bV9aNdfuRauWMZh3flnX/DpVfkLE=</latexit>
We iteratively search for the minimum of the cost function:
where:
The second contribution fixes the spin-s
subspace of a quantum register.
↵min := argmin↵c(↵)
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<latexit sha1_base64="0QsrMeuIAUsyGZNQlONtESwHyHY=">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</latexit>
| 0i = | (↵min)i
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Acting iteratively we find:
so that
Quantum chip
Here, the IBM 5-qubit Yorktown
superconducting quantum chip
has been used. The connectivity
of the quantum processor is the following:
The expectation values are evaluated
on a quantum processor.
The s=1 case
hĈ2
i = 2
✓
1
6
◆2
(1 + h z ⌦ zi)
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Here:
The Hadamard test becomes:
The ansatz for the state
| (↵)i = | (✓1, ✓2)i = V̂ |00i
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is
Cost function during minimalization
for 10 runs, with randomly initialized
parameters:
The cost function landscape:
The quantum fidelity of the found state is:
| i =
1
p
2
(|01i + |10i)
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Averaged (over 10 runs) amplitudes of the final state:
The theoretically predicted
solution to the WdW equation:
In the simulations, 1024 shots for each circuit have been made.
0.997 ± 0.003
<latexit sha1_base64="GPvoqq66YDXQTCovjlG/s9cKdk4=">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</latexit>
<latexit sha1_base64="GPvoqq66YDXQTCovjlG/s9cKdk4=">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</latexit>
<latexit sha1_base64="GPvoqq66YDXQTCovjlG/s9cKdk4=">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</latexit>
<latexit sha1_base64="hy34//W4Dx9dUBLYHASk6gRqsfM=">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</latexit>
Summary
• Theoretical considerations support the compact phase space of gravity
- locally finite dimensional Hilbert space of gravity.
• LQG provides a framework to introduce this idea.
• A compact phase space extension of LQC has been constructed.
• Spherical phase space introduces a cut-off on the Hilbert, which allows
to solve the WdW on a quantum computer.
• The method generates physical states on a quantum register. The
states can be further used to e.g. evaluate transition amplitudes.
• Degeneration of the kernel can be extracted employing the Gram-
Schmidt procedure. In has been tested for the s=2 case.
• The method is not effective (compared to the classical methods) for
small number of degrees of freedom (small m). However, the approach
becomes theoretically advantageous over classical methods
(exponential speedup) for large #DOF (large m).
• Inhomogeneous configurations from m>>1 interacting copies of the
ultralocal SU(2) patches (in progress)
Thank you!

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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=">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</latexit> <latexit sha1_base64="xTL/p8YT5xoSa0tUZfT8cwlJUaE=">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</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=">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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. 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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=">AAADAnicjVHLahRBFD3pqInxNerSTeEgjCBDdxB0E0gUwWVEJwmkY6ju1MwU0y+qqoUwzC5/kp07cesPuNWF+Af6F54qO6AG0Wq6+9S595yqe2/WFNq6OP66FC1fuHhpZfXy2pWr167f6N28tWPr1uRqlNdFbfYyaVWhKzVy2hVqrzFKllmhdrPZUx/ffaOM1XX1yh036qCUk0qPdS4dqcPeRlqosRsI9Xqe6sqJrcUD8Sw1ejJ19wUZ8XI0WCdyulRWpKV007GRs7ltSS8Oe/14GIclzoOkA310a7vufUGKI9TI0aKEQgVHXEDC8tlHghgNuQPMyRkiHeIKC6xR2zJLMUOSnfE74W6/YyvuvacN6pynFHwNlQL3qKmZZ4j9aSLE2+Ds2b95z4Onv9sx/1nnVZJ1mJL9l+4s8391vhaHMR6HGjRragLjq8s7lzZ0xd9c/FKVo0NDzuMjxg1xHpRnfRZBY0PtvrcyxL+FTM/6fd7ltvjub8kBJ3+O8zzYWR8m8TB58bC/+aQb9Sru4C4GnOcjbOI5tjGi9yk+4hM+RyfR2+hd9P5narTUaW7jtxV9+AFW4aY9</latexit> <latexit 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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) <latexit 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sha1_base64="6953ZNA4lMASasbAVideoDL3Vp8=">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</latexit> <latexit sha1_base64="+fdNu4hgoRepBJkMxKA2uZcpx1w=">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</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=">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</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 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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 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S2 <latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit> <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=">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="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 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 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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>
  • 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=">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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=">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</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 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sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit> <latexit sha1_base64="iz+nNgiphiM7VeAF/vg5DllYvtY=">AAADA3icjVHLTtwwFD0E2gJ9DbDsxmJUqYtplLChmyIEG5YgdQCJoJFjzBDhPGQ7VVGYJX/Cjh1i2x/oFiTEH9C/6LXJSLSoah0lOT73nmPfe9NKZcZG0d1EMDn17PmL6ZnZl69ev3nbmZvfNmWtheyLUpV6N+VGqqyQfZtZJXcrLXmeKrmTHq+7+M5XqU1WFl/sSSX3cz4sssNMcEvUoLOS5NweCa6ajdHAsM8ssfKbbUzFixFLGtP7aBLNi6GSPRaGYY+xU9MbU8lo0OlGYeQXewriFnTRrs2yc4sEByghUCOHRAFLWIHD0LOHGBEq4vbREKcJZT4uMcIsaWvKkpTBiT2m75B2ey1b0N55Gq8WdIqiV5OS4T1pSsrThN1pzMdr7+zYv3k33tPd7YT+aeuVE2txROy/dOPM/9W5WiwO8cnXkFFNlWdcdaJ1qX1X3M3Zo6osOVTEOXxAcU1YeOW4z8xrjK/d9Zb7+L3PdKzbiza3xk93Sxpw/Oc4n4LtpTCOwnhrqbu61o56Gu+wiA80z2WsYgOb6JP3OX7gGjfBWXARXAZXD6nBRKtZwG8r+P4LJgmnrw==</latexit> Ŝ2 |s, szi = s(s + 1)|s, szi <latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit> <latexit 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sha1_base64="R3s46+qWvDocRDolfsve4fnR5BY=">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</latexit> Ŝz|s, szi = sz|s, szi <latexit 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sha1_base64="5yIU9JdBaNxiS272TQL1NZ5ulQA=">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</latexit> <latexit sha1_base64="5yIU9JdBaNxiS272TQL1NZ5ulQA=">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</latexit> <latexit 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  • 10. Compactified flat de Sitter cosmology Kinematics (symplectic form) Dynamics (scalar constraint) Friedmann equation: where The procedure is ambiguous! We made 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> C = q ✓ 3 4 p2 + ⇤  ◆ ⇡ 0 <latexit sha1_base64="Drg0ok2XHaUlzso2eqLHAYLIprw=">AAADEHicjVFNT9VAFD0UVMCvhy7ZTHwxwRhfWiSBhSYkbEh0gYkPSCiS6bx5j8nra8fplEia/gn+iTt3hq1/gLjTHfwL7wxDghKj07Q999x7zsydm+lcVTaOv09F0zO3bt+ZnZu/e+/+g4edhUfbVVkbIfuizEuzm/FK5qqQfatsLne1kXyS5XInG2+4/M6RNJUqi/f2WMv9CR8VaqgEt0QddN5ssNfsY5rLoV16wdKh4aJ52TYrbTrmWnOmPyw/D3T6lmwHvG0uU21q1OjQPmMpRab8xOKDTjfuxX6xmyAJoIuwtsrOGVIMUEKgxgQSBSzhHBwVPXtIEEMTt4+GOENI+bxEi3nS1lQlqYITO6bviKK9wBYUO8/KqwXtktNrSMnwlDQl1RnCbjfm87V3duzfvBvv6c52TP8seE2ItTgk9l+6q8r/1bleLIZY8z0o6kl7xnUngkvtb8WdnF3rypKDJs7hAeUNYeGVV/fMvKbyvbu75T5/7isd62IRamtcuFPSgJM/x3kTbC/3kriXvFvprr8Ko57FIp5giea5inVsYgt98v6MM/zAz+gk+hJ9jU4vS6OpoHmM31b07RdAqqzb</latexit> <latexit sha1_base64="Drg0ok2XHaUlzso2eqLHAYLIprw=">AAADEHicjVFNT9VAFD0UVMCvhy7ZTHwxwRhfWiSBhSYkbEh0gYkPSCiS6bx5j8nra8fplEia/gn+iTt3hq1/gLjTHfwL7wxDghKj07Q999x7zsydm+lcVTaOv09F0zO3bt+ZnZu/e+/+g4edhUfbVVkbIfuizEuzm/FK5qqQfatsLne1kXyS5XInG2+4/M6RNJUqi/f2WMv9CR8VaqgEt0QddN5ssNfsY5rLoV16wdKh4aJ52TYrbTrmWnOmPyw/D3T6lmwHvG0uU21q1OjQPmMpRab8xOKDTjfuxX6xmyAJoIuwtsrOGVIMUEKgxgQSBSzhHBwVPXtIEEMTt4+GOENI+bxEi3nS1lQlqYITO6bviKK9wBYUO8/KqwXtktNrSMnwlDQl1RnCbjfm87V3duzfvBvv6c52TP8seE2ItTgk9l+6q8r/1bleLIZY8z0o6kl7xnUngkvtb8WdnF3rypKDJs7hAeUNYeGVV/fMvKbyvbu75T5/7isd62IRamtcuFPSgJM/x3kTbC/3kriXvFvprr8Ko57FIp5giea5inVsYgt98v6MM/zAz+gk+hJ9jU4vS6OpoHmM31b07RdAqqzb</latexit> <latexit sha1_base64="Drg0ok2XHaUlzso2eqLHAYLIprw=">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</latexit> <latexit sha1_base64="grBQZepvEz5kuSQWIOKUXopGRZU=">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</latexit> C = S3 R1  3 4  S2 2 R2 2 ⇤  ⇡ 0 <latexit sha1_base64="zb/oE4jEq2O3VrO4QzItGJ03nkA=">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</latexit> <latexit sha1_base64="zb/oE4jEq2O3VrO4QzItGJ03nkA=">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</latexit> <latexit sha1_base64="zb/oE4jEq2O3VrO4QzItGJ03nkA=">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</latexit> <latexit sha1_base64="ESTp+F2FVnF/Yj7MjD3zCnNdhu0=">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</latexit>
  • 11. Cylindrical limit (LQC) 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=">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=">AAAC2XicjVHLTsJAFD3UF+ILHzs3jcTEFWnZ6JLoxiUaeSRASFsGmFDaZjrVIHHhzrj1B9zqDxn/QP/CO2NJVGJ0mrZnzr3nzNx73cjnsbSs14wxN7+wuJRdzq2srq1v5De3anGYCI9VvdAPRcN1YubzgFUllz5rRII5I9dndXd4ouL1SyZiHgYXchyx9sjpB7zHPUcS1cnvnHdKLcH7A+kIEV61eNCT406+YBUtvcxZYKeggHRVwvwLWugihIcEIzAEkIR9OIjpacKGhYi4NibECUJcxxlukCNtQlmMMhxih/Tt066ZsgHtlWes1R6d4tMrSGlinzQh5QnC6jRTxxPtrNjfvCfaU91tTH839RoRKzEg9i/dNPO/OlWLRA9HugZONUWaUdV5qUuiu6Jubn6pSpJDRJzCXYoLwp5WTvtsak2sa1e9dXT8TWcqVu29NDfBu7olDdj+Oc5ZUCsVbaton5UK5eN01FnsYg8HNM9DlHGKCqrkfY1HPOHZaBq3xp1x/5lqZFLNNr4t4+EDGhKYAA==</latexit> In the limit the so-called polymerization of momentum is obtained. The su(2) algebra reduces to the cylindrical case: The scalar constraint reduces to: where the polymerization scale is: := 1 R1 <latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit> <latexit 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sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit> <latexit sha1_base64="IpcgmU5scOhVCdLM71hbptMDtxU=">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</latexit> The Friedmann equation is: where C = q ✓ 3 4 sin2 ( p) 2 + ⇤  ◆ ⇡ 0 <latexit sha1_base64="PPEVbLHbpaukz+XJa6s2xu2oZfE=">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</latexit> <latexit sha1_base64="PPEVbLHbpaukz+XJa6s2xu2oZfE=">AAADNHicjVHPSxwxFH471VbtD7d69BJcCiuly8wqtBdB9OKhBwuuCo5KJptdw2YnMZMRZZh/q/+JiOBNCu3Ff8BLX2IW2kppM8zM9773vi95eZmWorBxfNOInk1NP38xMzv38tXrN/PNtwt7hSoN4z2mpDIHGS24FDnvWWElP9CG03Em+X422nL5/XNuCqHyXXup+dGYDnMxEIxapE6aaouskzOSSj6wbfKBpANDWbWajqjWtK7W6sCkhciPu+1UonWfEr1SVwEfd2vyflL12VOYe9ST1IjhqV0hKYZGXZD4pNmKO7Ff5ClIAmhBWDuqeQ0p9EEBgxLGwCEHi1gChQKfQ0ggBo3cEVTIGUTC5znUMIfaEqs4VlBkR/gdYnQY2Bxj51l4NcNdJL4GlQTeoUZhnUHsdiM+X3pnx/7Nu/Ke7myX+M+C1xhZC6fI/ks3qfxfnevFwgA++R4E9qQ947pjwaX0t+JOTn7pyqKDRs7hPuYNYuaVk3smXlP43t3dUp//7isd62IWakv44U6JA07+HOdTsNftJHEn+bLW2tgMo56BJViGNs7zI2zANuxAD72v4KEx1ZiOvka30V307bE0agTNIvy2ovufnSK41A==</latexit> <latexit sha1_base64="PPEVbLHbpaukz+XJa6s2xu2oZfE=">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</latexit> <latexit 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  • 12. 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 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  • 13. Action of the operator is defined on the kinematical Hilbert space . Following the Dirac quantization of constrained systems, the physical Hilbert space is constructed by solving the Wheeler-deWitt (WdW) equation: H[N, ~ N] = S[N] + V [ ~ N] = Z ⌃ d3 xNC + Z ⌃ d3 x ~ N · ~ C <latexit sha1_base64="3iTbiNcaX1O3JtpPpKOc7S4nzoI=">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</latexit> <latexit 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In homogeneous quantum cosmology only the scalar constraint remains. 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sha1_base64="uembQmfG4JOjJgwpETn6L/ZOZjc=">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</latexit> <latexit sha1_base64="uembQmfG4JOjJgwpETn6L/ZOZjc=">AAAC3nicjVHLSsNAFD3GV31HXYmbYBFclUQEXRbddFnBPsBKnUynNpgXyUQsobhzJ279Abf6OeIf6F94Z4zgA9EJSc6ce8+Zufe6se+l0rafx4zxicmp6dLM7Nz8wuKSubzSTKMs4aLBIz9K2i5Lhe+FoiE96Yt2nAgWuL5ouecHKt66EEnqReGRHMbiJGBnodf3OJNEdc21TsDkgDM/r426eUeKS5nHg2E6GnXNsl2x9bJ+AqcAZRSrHplP6KCHCBwZAgiEkIR9MKT0HMOBjZi4E+TEJYQ8HRcYYZa0GWUJymDEntP3jHbHBRvSXnmmWs3pFJ/ehJQWNkkTUV5CWJ1m6XimnRX7m3euPdXdhvR3C6+AWIkBsX/pPjL/q1O1SPSxp2vwqKZYM6o6Xrhkuivq5tanqiQ5xMQp3KN4Qphr5UefLa1Jde2qt0zHX3SmYtWeF7kZXtUtacDO93H+BM3timNXnMOdcnW/GHUJ69jAFs1zF1XUUEeDvK9wjwc8GqfGtXFj3L6nGmOFZhVflnH3Bj7MmtA=</latexit> <latexit sha1_base64="modhOdsYBh9Pr4zzgyDvRRw/xKw=">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</latexit> so that . ker Ĉ = Hphys ✓ Hkin <latexit 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sha1_base64="9nUrmvOk+TrJZACl0Ar2JLXc438=">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</latexit> <latexit sha1_base64="yYHBzdcJNncpUuIFwbAIup3vm3A=">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</latexit> The WdW is difficult to solve in general case. Solutions are known for certain minisuperspace models. Complexity of the problem grows exponentially with the #DOF. Ĉ| i = 0 <latexit sha1_base64="BKuUD4iZtyu6Fefg7Le24HSm5II=">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</latexit> <latexit sha1_base64="BKuUD4iZtyu6Fefg7Le24HSm5II=">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</latexit> <latexit sha1_base64="BKuUD4iZtyu6Fefg7Le24HSm5II=">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</latexit> <latexit sha1_base64="rTBbVb5aEd/ohM8h/5/sgT34p+0=">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</latexit>
  • 14. Our quantum constraint (m=1) The symmetrized quantum scalar constraint: We have shown that the WdW has always solutions for any for the bosonic representations. For the fermionic representations the solutions do not exist, except some particular values of . Rescaled classical constraint: <latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit> <latexit 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sha1_base64="3cSg74jivEBcXT93HXVaPjnpdNI=">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</latexit> <latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit> <latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit> <latexit sha1_base64="44McbfHylNlnhDKVfg2BdxbNze4=">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</latexit> <latexit sha1_base64="3cSg74jivEBcXT93HXVaPjnpdNI=">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</latexit> C ! 4S2 3R1 C = S3S2 2 S2 S3. <latexit sha1_base64="KveFKTKWFFNpcVJujvoGLEj0QpU=">AAADCHicjVHNTtwwGBwCLX/9WeDIxWJVqZeukgWpHFoJaS8cgWUBiUDkeL2LtdkkcpxWaLUvwJtw44a49gV6BN4A3oLPJki0qCqOkoznmxn7s+M8UYXx/dsJb3Lqzdvpmdm5+XfvP3ysLSzuFVmpheyILMn0QcwLmahUdowyiTzIteTDOJH78aBl6/s/pC5Ulu6a01weDXk/VT0luCEqqrVaLNSqf2K41tlPFvY0F6O19nFzPFpl4YDnOWc7UTBmre/taJW1o+Zxk30JuzIxnJGMyEZUq/sN3w32EgQVqKMaW1ntGiG6yCBQYgiJFIZwAo6CnkME8JETd4QRcZqQcnWJMebIW5JKkoITO6Bvn2aHFZvS3GYWzi1olYReTU6GT+TJSKcJ29WYq5cu2bL/yh65TLu3U/rHVdaQWIMTYv/ne1K+1md7Mehh3fWgqKfcMbY7UaWU7lTsztmzrgwl5MRZ3KW6Jiyc8+mcmfMUrnd7ttzV75zSsnYuKm2Je7tLuuDg7+t8CfaajcBvBNtr9Y1v1VXPYBkr+Ez3+RUb2MQWOpR9jt+4wa135l14l97Vo9SbqDxL+GN4vx4AZdiniw==</latexit> <latexit sha1_base64="KveFKTKWFFNpcVJujvoGLEj0QpU=">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</latexit> <latexit sha1_base64="KveFKTKWFFNpcVJujvoGLEj0QpU=">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</latexit> <latexit sha1_base64="Cd6FykPgCQgyN+ScgP63phTvClA=">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</latexit> Ĉ = 1 3 ⇣ ŜzŜyŜy + ŜyŜzŜy + ŜyŜyŜz ⌘ Ŝ2 Ŝz ⇡ 0 <latexit sha1_base64="CNOTuOXZRc2+5AjBNaMPGUeXVnY=">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</latexit> <latexit sha1_base64="CNOTuOXZRc2+5AjBNaMPGUeXVnY=">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</latexit> <latexit sha1_base64="CNOTuOXZRc2+5AjBNaMPGUeXVnY=">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</latexit> <latexit sha1_base64="U+Oc5M+yQzvGxjCdLFQ3RvafHD0=">AAADYXicjVFNT9tAEB3HBUL4cumRy6oRUhAistNDORQpEpceqUoACUO03mySFRvbWq8RwfI/4X+hnnuBf9HZrQO0EYK1bL99897szkyUSpFp3//l1NwPC4tL9eXGyura+ob3cfMkS3LFeI8lMlFnEc24FDHvaaElP0sVp5NI8tPo6tDET6+5ykQSH+tpyi8mdBSLoWBUI9X37sIx1cVhSQ5IOFSUFUFZfClDyYe6RWzsZ9m/nYHpE9ido55Fjfngs4qESozGemcvHHCp6eyMy84LBU1TldwQv+81/bZvF5kHQQWaUK2jxLuHEAaQAIMcJsAhBo1YAoUMn3MIwIcUuQsokFOIhI1zKKGB3hxVHBUU2Sv8jnB3XrEx7k3OzLoZniLxVegksI2eBHUKsTmN2HhuMxv2tdyFzWnuNsV/VOWaIKthjOxbvpnyvT5Ti4Yh7NsaBNaUWsZUx6osue2KuTl5UZXGDClyBg8wrhAz65z1mVhPZms3vaU2/mCVhjV7VmlzeDS3xAEH/49zHpx02oHfDn50mt1v1ajrsAWfoYXz/Apd+A5H0APm1JyWEzid2m932fXczb/SmlN5PsE/y936A7Y6yTw=</latexit> The simplest non-trivial case is s=1, for which: Ĉ = 2 ✓ 1 6 ◆ 0 @ 1 0 0 0 0 0 0 0 1 1 A = 2 ✓ 1 6 ◆ Ŝz <latexit sha1_base64="KdDizENsWZavvS6Az/o4+WX0wrU=">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</latexit> <latexit sha1_base64="KdDizENsWZavvS6Az/o4+WX0wrU=">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</latexit> <latexit sha1_base64="KdDizENsWZavvS6Az/o4+WX0wrU=">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</latexit> <latexit sha1_base64="e68BTZ2igHhNB76olDUReisrSFQ=">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</latexit> The physical state, which satisfy the constraint, is: | i = 0 @ 0 1 0 1 A = |s = 1, sz = 0i = 1 p 2 (|01i + |10i) <latexit 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  • 15. Transition amplitude for s=2 The transition amplitude is peaked for ⇤ ! 0+ <latexit sha1_base64="IG+SRGoYutfRBkkZn5bqZs3ycOY=">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</latexit> <latexit sha1_base64="IG+SRGoYutfRBkkZn5bqZs3ycOY=">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</latexit> <latexit sha1_base64="IG+SRGoYutfRBkkZn5bqZs3ycOY=">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</latexit> <latexit sha1_base64="nEyCG6VLsi9AtxzQP/j5DMotKeY=">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</latexit> - SU(2) coherent states
  • 16. Outlook of our method of solving the WdW equation for arbitrary #DOF (m) on a quantum computer: 1. Regularize theory to make the Hilbert space finite. a) Replace the flat (affine) phase space for every classical degree of freedom with a sphere. The spherical phase space is a phase space of angular momentum (spin). The flat phase space case is recovered in the large spin limit. b) Construct regularized quantum kinematics for the system under consideration. c) Express the Hamiltonian constraint in terms of the spin variables. d) Quantize the regularized Hamiltonian constraint. e) Represent the spin operators in terms of qubits. 2. Apply Variational Quantum Eigensolver (VQE) to find the states minimizing „Hamiltonian” : Ĉ| 0i = 0 () h 0|Ĉ2 | 0i = 0 <latexit sha1_base64="4kWZ63d0DhNJ5FiduOctCqkKz6w=">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</latexit> <latexit sha1_base64="4kWZ63d0DhNJ5FiduOctCqkKz6w=">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</latexit> <latexit sha1_base64="4kWZ63d0DhNJ5FiduOctCqkKz6w=">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</latexit> <latexit sha1_base64="uxDquaTlsI24XG6q2rQzHbzsRl8=">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</latexit> 3. Study the large spin limit to recover the affine case. Ĉ2 <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit> <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">AAACzHicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6LLYjSupYB/SVknSaTs0L5KJUEq3/oBb/S7xD/QvvDNOQS2iE5KcOfeeM3PvdWOfp8KyXnPG0vLK6lp+vbCxubW9U9zda6ZRlnis4UV+lLRdJ2U+D1lDcOGzdpwwJ3B91nLHNRlv3bMk5VF4LSYx6wXOMOQD7jmCqJvuyBHT2uy2clcsWWVLLXMR2BqUoFc9Kr6giz4ieMgQgCGEIOzDQUpPBzYsxMT1MCUuIcRVnGGGAmkzymKU4RA7pu+Qdh3NhrSXnqlSe3SKT29CShNHpIkoLyEsTzNVPFPOkv3Ne6o85d0m9He1V0CswIjYv3TzzP/qZC0CA5ypGjjVFCtGVudpl0x1Rd7c/FKVIIeYOIn7FE8Ie0o577OpNKmqXfbWUfE3lSlZufd0boZ3eUsasP1znIugWSnbVtm+OilVz/Wo8zjAIY5pnqeo4gJ1NMg7wCOe8GxcGsKYGrPPVCOnNfv4toyHD5ibkrk=</latexit> <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit> <latexit sha1_base64="/fRjTsS43q5BgcFlTBcHi9YBB/I=">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</latexit>
  • 17. Our constraint squared in general case:
  • 18. The action of the spin operators on n-qubit (spin-1/2) quantum register is given by: where n = 2s. Ŝi = 1 2 n X j=1 I1 ⌦ ...Ij 1 ⌦ ˆj i ⌦ Ij+1 ⌦ ...In , <latexit sha1_base64="k1JHeQGS59w78zyXLYHOyPLHJNw=">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</latexit> <latexit sha1_base64="k1JHeQGS59w78zyXLYHOyPLHJNw=">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</latexit> <latexit sha1_base64="k1JHeQGS59w78zyXLYHOyPLHJNw=">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</latexit> <latexit sha1_base64="pjon7Llk2Pf9p2iFAQnLKK0tyyE=">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</latexit> For convenience, we introduced: … Here, we need n=2s qubits to represent spin s. But, we know how to do it better, i.e. 2s + 1 = 2n <latexit sha1_base64="bp3es4QWHSkndV6WQaNqUItVCls=">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</latexit> <latexit sha1_base64="bp3es4QWHSkndV6WQaNqUItVCls=">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</latexit> <latexit sha1_base64="bp3es4QWHSkndV6WQaNqUItVCls=">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</latexit> <latexit sha1_base64="wE42MHttPNie6mpk7dB2zH0Vc5M=">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</latexit>
  • 19. Because is not a unitary operator, the expectation values cannot be evaluated directly with the use of quantum computing. For this purpose the operator has to be expanded into unitarities (here, the Pauli matrices): Ĉ = X j cj O i ˆk ij <latexit sha1_base64="cwVPnr3AuQJBA+wS2f22UxHjjqs=">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</latexit> <latexit sha1_base64="cwVPnr3AuQJBA+wS2f22UxHjjqs=">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</latexit> <latexit sha1_base64="cwVPnr3AuQJBA+wS2f22UxHjjqs=">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</latexit> <latexit sha1_base64="0Cs+rOwkKD1gf2j17EesVnemQqU=">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</latexit> hĈi = X j cjh O i ˆk iji <latexit 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sha1_base64="HVnRU6ES3gjp3pLeWHWRLMbQBNo=">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</latexit> <latexit sha1_base64="HVnRU6ES3gjp3pLeWHWRLMbQBNo=">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</latexit> <latexit sha1_base64="iN0LQ0JLsBPZAs9GiH4vnmaKENo=">AAADFnicjVHLTtwwFD2khVLKY9ouu4k6QupqlLCBDdKobLoEqQNIhEaOMcGM85DtIKHR/Ef/pLvuKrbsEItu+/iLXhsjtUUIHCW599xzjn19i1ZJY5PkeiZ68nR27tn884UXi0vLK72Xr3ZN02kuRrxRjd4vmBFK1mJkpVViv9WCVYUSe8V4y9X3zoQ2sqk/2vNWHFasrOWx5MwSlPd2MsXqUok4O2F2sjWNM+3zzcx0VX4ac3oDJStk2VhZCZPLG3pmZFmxaT6Rp9NP46DMe/1kkPgV3w3SEPQR1nbTu0KGIzTg6FBBoIalWIHB0HOAFAlawg4xIUxTJH1dYIoF0nbEEsRghI7pW1J2ENCacudpvJrTLopeTcoYq6RpiKcpdrvFvt55Z4fe5z3xnu5s5/QvgldFqMUJoQ/pbpmP1bleLI6x4XuQ1FPrEdcdDy6dvxV38vivriw5tIS5+IjqmmLulbf3HHuN8b27u2W+/sszHepyHrgdfrtT0oDT/8d5N9hdG6TJIN1Z6w/fh1HP4w3e4h3Ncx1DfMA2RuT9Bd/xAz+jz9HX6Ft0cUONZoLmNf5Z0eUfWo+xyw==</latexit> so that Every contributing expectation value has to be evaluated individually. The so-called Hadamard test can be used for this purpose: = hÛi <latexit sha1_base64="T8WhDTjASSw9waZqq1rFQYka75I=">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</latexit> <latexit sha1_base64="T8WhDTjASSw9waZqq1rFQYka75I=">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</latexit> <latexit sha1_base64="T8WhDTjASSw9waZqq1rFQYka75I=">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</latexit> <latexit sha1_base64="nGORFQg+QanZ6vf2amof1VvF7yU=">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</latexit> Ĉ2 <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit> <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit> <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit> <latexit sha1_base64="/fRjTsS43q5BgcFlTBcHi9YBB/I=">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</latexit> Similarly for . Ĉ2 <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit> <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit> <latexit sha1_base64="JT725uatH3DlRe5NN1BdZV5kPRE=">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</latexit> <latexit sha1_base64="/fRjTsS43q5BgcFlTBcHi9YBB/I=">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</latexit>
  • 20. The variational procedure c (↵) = a max | i| 2 h (↵) |Ĉ† Ĉ| (↵)i + b 1 h (↵) | ˆ ~ S2 | (↵)i s (s + 1) ! <latexit sha1_base64="g8bBO9OStOL5U6TgNc9QlX/+0hY=">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</latexit> <latexit sha1_base64="g8bBO9OStOL5U6TgNc9QlX/+0hY=">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</latexit> <latexit sha1_base64="g8bBO9OStOL5U6TgNc9QlX/+0hY=">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</latexit> <latexit sha1_base64="ol21CeyssDZ7hk2IiUEzPwdmAN8=">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</latexit> a, b 2 (0, 1) <latexit 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sha1_base64="wu1wK1Qeba63pdtAX6R1YKmLnIw=">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</latexit> <latexit sha1_base64="MOWKC3rmtbGshqR7KNb2jatStmU=">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</latexit> We iteratively search for the minimum of the cost function: where: The second contribution fixes the spin-s subspace of a quantum register. ↵min := argmin↵c(↵) <latexit 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sha1_base64="N2O0NihKKfaJQR0qZH+nCeqb+ek=">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</latexit> Acting iteratively we find: so that
  • 21. Quantum chip Here, the IBM 5-qubit Yorktown superconducting quantum chip has been used. The connectivity of the quantum processor is the following: The expectation values are evaluated on a quantum processor.
  • 22. The s=1 case hĈ2 i = 2 ✓ 1 6 ◆2 (1 + h z ⌦ zi) <latexit sha1_base64="JJAKr985hp69oHEBm4zZ8XYX0kE=">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</latexit> <latexit 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sha1_base64="JJAKr985hp69oHEBm4zZ8XYX0kE=">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</latexit> <latexit sha1_base64="96YDB/zqoIH2uM9Y3NAMgmtMUXU=">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</latexit> Here: The Hadamard test becomes: The ansatz for the state | (↵)i = | (✓1, ✓2)i = V̂ |00i <latexit sha1_base64="l4Bjif5dvRI2Y4e5wiLek8DFCWQ=">AAADMnicjVHLSsQwFL3T8Tm+Rl26KQ6CggztIOhGGHTjUsEZBSNDWjPTYPogTQWp81f+iQvRnbhR8AfceRMj+EA0pe3Jueec5CZBJniuPO+u4lRHRsfGJyZrU9Mzs3P1+YVunhYyZJ0wFak8DmjOBE9YR3El2HEmGY0DwY6C811dP7pgMudpcqguM3Ya00HC+zykCqlePb4iWc6JYH21WpKg7xIqsogOieSDSK0RSZOBYO62+0lHVMQU7fnrFrS+i2skoqrsDnulNg2vPM+WevWG1/TMcH8C34IG2LGf1m+BwBmkEEIBMTBIQCEWQCHH5wR88CBD7hRK5CQibuoMhlBDb4EqhgqK7Dl+Bzg7sWyCc52ZG3eIqwh8JTpdWEFPijqJWK/mmnphkjX7W3ZpMvXeLvEf2KwYWQURsn/5PpT/9eleFPRhy/TAsafMMLq70KYU5lT0zt1PXSlMyJDT+AzrEnFonB/n7BpPbnrXZ0tN/ckoNavnodUW8Kx3iRfsf7/On6Dbavpe0z/YaLR37FVPwBIswyre5ya0YQ/2oYPZN/BacSpV59q5dx6cx3epU7GeRfgynJc3dUC6UA==</latexit> <latexit 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sha1_base64="l4Bjif5dvRI2Y4e5wiLek8DFCWQ=">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</latexit> <latexit sha1_base64="B7dnYlNf5JQCn938JzaWzcUTwcc=">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</latexit> is Cost function during minimalization for 10 runs, with randomly initialized parameters: The cost function landscape:
  • 23. The quantum fidelity of the found state is: | i = 1 p 2 (|01i + |10i) <latexit sha1_base64="CGI+9YhBlB5+i7AoLYVeiEw3JPs=">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</latexit> <latexit 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  • 24. Summary • Theoretical considerations support the compact phase space of gravity - locally finite dimensional Hilbert space of gravity. • LQG provides a framework to introduce this idea. • A compact phase space extension of LQC has been constructed. • Spherical phase space introduces a cut-off on the Hilbert, which allows to solve the WdW on a quantum computer. • The method generates physical states on a quantum register. The states can be further used to e.g. evaluate transition amplitudes. • Degeneration of the kernel can be extracted employing the Gram- Schmidt procedure. In has been tested for the s=2 case. • The method is not effective (compared to the classical methods) for small number of degrees of freedom (small m). However, the approach becomes theoretically advantageous over classical methods (exponential speedup) for large #DOF (large m). • Inhomogeneous configurations from m>>1 interacting copies of the ultralocal SU(2) patches (in progress)