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

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

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

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

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

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)

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2 28Spin sS l d X

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

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

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

“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

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

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

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!

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

Thank you very much!

• Special thank to Keisuke Izumi for his kind suggestion!

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