a quantization of spacetime based on spin(3,1) symmetry · •simple promotion from so(3,1) to...
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A Quantization of Spacetime based on Spin(3,1) symmetry
Hsu-Wen Chiang based on works in collaboration with
Yao-Chieh Hu and Pisin Chen
Leung Center for Cosmology and Particle Astrophysics (LeCosPA) National Taiwan University (NTU)
Phys. Rev. D93,084043 (2016), arxiv:1512.03157 1
NTU-Fudan Mini-Workshop 2016
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Mr. Riemann Walks to H-Bar
• Many theories suggest that there exists a minimum distance, e.g. LQG and string theory
• In Hořava-Lifshitz gravity and superfluid vacuum theory Lorentz symm. is broken as well
Image from “Hyperspace” by Michio Kaku 2
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Mr. Riemann Walks to H-Bar
• Many theories suggest that there exists a minimum distance, e.g. LQG and string theory
• In Hořava-Lifshitz gravity and superfluid vacuum theory Lorentz symm. is broken as well
Image from “Hyperspace” by Michio Kaku
Perhaps the spacetime structure itself should be modified
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Non-commutativity
1) Curvature = Non-commutativity for translation operator Curvature on momentum space non-commutative position operators
Snyder[1], Amelino-Camelia[2], etc
2) Directly modifying the group structure
A. Connes[3], Majid[4], etc
But how to avoid catastrophic Lorentz invariance violation?
=
[1] H.S. Snyder, “Quantized Space-Time” (1947) [2] Kowalski-Glikman, hep-th/0405273 [3] A. Connes, hep-th/9603053 [4] E. Batista and S. Majid, hep-th/0205128 4
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Introducing Adler’s Approach[1]
• Distance function is quadratic
• Following Dirac’s approach one introduces “linear line element”
• is a matrix reinterpret as QM object
• Definition of expectation value matters!
• Adler uses which is not SO(3,1) invariant
5
Lds dx
As x x
†
1
2ds g dx dx
[1] Ronald J. Adler, “A quantum theory of distance along a curve”, arxiv:1402.5921
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Deficiencies of the Original Attempt
1) Normalization is not SO(3,1) invariant
2) Uncertainty of the curve breaks Lorentz inv.
3) Proper distance of null eigenstates is zero
– The choice of normalization sets it to zero
4) Proper distance is not rep. indep.
6
2
2 1A A AVar s s x s
†
1
The theory can be cured by a L.I. normalization and reinterpreted as a Spin(3,1) spacetime theory
x
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Spacetime w/ Spinorial Worldline
• Spinor plays the key role in null structure of GR
• Simple promotion from SO(3,1) to Spin(3,1) modifies nothing classically.
• Global structure affects quantum properties.
• From bi-fermionic structure of the Adler’s theory fermionization is the natural tool[2].
• Consider K.K. 0th state of cylindrical Spin(3,1) Polyakov action
7 [2] S. Coleman, “Quantum sine-Gordon equation as the massive Thirring model” (1975)
21
2 28Spin sS l d X
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Extension of Worldsheet Action
Much more combinations!
String
SO(n,1) Spin(n,1)
“SpinString”
IR IR
IR
classically
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Spacetime w/ Spinorial Worldline
• Spinor plays the key role in null structure of GR
• Simple promotion from SO(3,1) to Spin(3,1) modifies nothing classically.
• Global structure affects quantum properties.
• From bi-fermionic structure of the Adler’s theory fermionization is the natural tool[2].
• Consider K.K. 0th state of cylindrical Spin(3,1) Polyakov action
• resembles linear line element 9
21
2 28Spin sS l d X
X
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Re-Fermionization
• Fix the group by tetrad: where and
• KK0 state is trivial Vanilla rules are enough
• Since for KK0 state & , from bosonization rules we have
• spacetime is quantized with characteristic length
10
2icDc DX 2c c D X
44 iX
L Lc c e
I J
J ID i
1
2 28 sl d D XD X
I
IX e X
TpM indices
0D X v 1 0D X 0
0c v 44 iX
L Lc c e
2sl
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A Spinorial Spacetime
• The worldline action becomes
• Arrive at a spacetime interval operator
and
• Lorentz invariant normalization condition is set to be (time-like, null, space-like)
• is similar to spin operator: They both have 3 quantum number!
• E.g. , , helicity
• states within lightcone holographic 11
ˆ I IX 2 2ˆ ˆ 4I
Ids X X
1/ 0 / 1
0S i D d
ˆ I IX
0 2X̂ S Z ZX̂ S ZV̂
3t
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“Lattice” Spacetime and BH Entropy
12
t
2SR
22
2 2 2 2
2 2ln ln ln ln
4 4 4
S SS S S
S P
A AS l R l
l l
0
r rˆˆ ˆˆ A VX
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Spinorial Spacetime & Smearing Effect
• Uncertainties are covariant!
,
1) For 100 KeV GRB photon @ z~10 smearing is of the order Not detectable!
– Since Lorentz inv. is exact, no dispersive effect
2) For EeV neutrino @ z~10 size of possible source
– Smearing of the arrival time is ~ 1 picosec 13
2var 3ds ds 22var I
Ir X r r n n r n
1310 mL
210L m
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Fourier Analysis and U(su(2)) theory
• May link Fourier k of the ext. deriv. to particle p
• Fourier analysis of spin(3,1) group Dim’l reduction to SU(2) or projection to SL(2,C)
• Has been done for U(su(2)) type theory[3]
• Starting from commutator
they obtain exterior derivative &
• Commutation relations are and
14 [3] E. Batista and S. Majid, J. Math. Phys. 44, 107 (2003), hep-th/0205128
2
iidx
2
2
Id
, 2i j ij k
kx x i x
i j j i ijk k ijx dx dx x i dx d
i i ix d d x dx
Supposedly time
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Generalized Uncertainty Relation
• Now let us apply exterior derivative to Fourier
modes
and define momentum accordingly as
with
• GUP at low E limit
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1,
2 4i j ij j ijk k j ij j
ix p i x x p t p
2sin 8
sin2
ik x ik xk d k
de i k dx ek
1sini i i i i iip ip dx i dx d dx dxk k k
22i j ij ij i jx p p p
Max p!
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Future Work
• Next Step: Fourier modes and QFT on spinorial spacetime
• So far the model is still kinematic, dynamics needs to be added through either
– Incorporating LQG into our picture
–Considering excited states of worldsheet
• Excited states cannot be fermionized As expected…
• Virasoro algebra? 16
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Thank you very much!
• Special thank to Keisuke Izumi for his kind suggestion!
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