From Quantum Cellular Automata to Quantum Field Theory Alessandro - - PowerPoint PPT Presentation

▶
from quantum cellular automata to quantum field theory
SMART_READER_LITE
LIVE PREVIEW

From Quantum Cellular Automata to Quantum Field Theory Alessandro - - PowerPoint PPT Presentation

Dec 06, 2023 488 likes •913 views

From Quantum Cellular Automata to Quantum Field Theory Alessandro Bisio Frontiers of Fundamental Physics Marseille, July 15-18th 2014 in collaboration with Giacomo Mauro DAriano o University of Pavia Paolo Perinotti QUIT group

slide-1
SLIDE 1

From Quantum Cellular Automata to Quantum Field Theory

Alessandro Bisio

Marseille, July 15-18th 2014

Frontiers of Fundamental Physics

slide-2
SLIDE 2

in collaboration with

Giacomo Mauro D’Ariano Paolo Perinotti Alessandro Tosini

supported by

Alexandre Bibeau-Delisle

  • University of Pavia

QUIT group University of Montreal

slide-3
SLIDE 3

Motivation QCA for the 1D Dirac free evolution The fate of Lorentz covariance: from QCA to deformed relativity models Final remarks and (many) open problems

Outline

slide-4
SLIDE 4

Quantum Theory

Unit vectors are associated with states of the system The Hilbert space of a composite system is the tensor product

  • f the state spaces associated with the component systems

Physical observables are represented by self adjoint operators Each physical system is associated with a Hilbert space

Von Neumann, 1932

The probabilities of the outcomes are given by the Born rule

slide-5
SLIDE 5

systems

Quantum Theory

  • G. Ludwig, Foundations of Quantum Mechanics (Springer, New York, 1985).
  • L. Hardy, e-print arXiv:quant-ph/0101012.
  • G. Chiribella, G. M. D’Ariano, P. Perinotti, Phys. Rev. A 84, 012311 (2011)

Operational Probabilistic Theory

preparations measurements transformations

Pi

σ

T

probability

T

ρ

Pi

σ

  • utcomes

composition

parallel

T T

sequence

T T

theory of information

slide-6
SLIDE 6

systems

Quantum Theory

Dynamics? Energy? Space? Time?

  • G. Ludwig, Foundations of Quantum Mechanics (Springer, New York, 1985).
  • L. Hardy, e-print arXiv:quant-ph/0101012.
  • G. Chiribella, G. M. D’Ariano, P. Perinotti, Phys. Rev. A 84, 012311 (2011)

Operational Probabilistic Theory

preparations measurements transformations

Pi

σ

T

probability

T

ρ

Pi

σ

  • utcomes

composition

parallel

T T

sequence

T T

theory of information

slide-7
SLIDE 7

Processing of Quantum Information

Quantum Circuit Quantum Computer

  • R. P. Feynman, Int. J. Theo. Phys. 21, 467 (1982)

Can a Quantum Computer exactly simulate physical systems? Simulation as a guideline for discovery

Simulating Physics with Computers (Feynman 1982)

slide-8
SLIDE 8

What kind of computer?

Quantum Circuit

Rules of the game

“[...] everything that happens in a finite volume of space and time would have to be exactly analyzable with a finite numbers of logical operations” R. Feynman

Each system interacts with a finite number of neighbors: locality

Reversible Quantum Computation: unitary evolution

isotropy, homogeneity, ...

slide-9
SLIDE 9

What kind of computer?

Quantum Circuit

Rules of the game

“[...] everything that happens in a finite volume of space and time would have to be exactly analyzable with a finite numbers of logical operations” R. Feynman

Each system interacts with a finite number of neighbors: locality

Reversible Quantum Computation: unitary evolution

isotropy, homogeneity, ...

slide-10
SLIDE 10

What kind of computer?

Quantum Cellular Automaton

Quantum Circuit

  • B. Schumacher, R.F. Werner

e-print arXiv:0405174.

Rules of the game

“[...] everything that happens in a finite volume of space and time would have to be exactly analyzable with a finite numbers of logical operations” R. Feynman

Each system interacts with a finite number of neighbors: locality

Reversible Quantum Computation: unitary evolution

isotropy, homogeneity, ...

slide-11
SLIDE 11

QCA Dirac for field the

case (1+1)-dimensional

AB, G. M. D’Ariano, A. Tosini, e-print arXiv:1212.2839.

U = ✓ nS −im −im nS† ◆

0 6 m 6 1

n2 + m2 = 1,

Sψ(x) = ψ(x + 1)

ψ(x, t + 1) = Uψ(x, t)

ψ(x) = ✓ψR(x) ψL(x) ◆

Linearity

ψi(t + 1) = Ui,jψj(t)

AB, G. M. D’Ariano, A. Tosini, Phys. Rev. A 88, 032301 (2013).

slide-12
SLIDE 12

QCA Dirac for field the

case (1+1)-dimensional

AB, G. M. D’Ariano, A. Tosini, e-print arXiv:1212.2839.

U = ✓ nS −im −im nS† ◆

0 6 m 6 1

n2 + m2 = 1,

Sψ(x) = ψ(x + 1)

ψ(x, t + 1) = Uψ(x, t)

ψ(x) = ✓ψR(x) ψL(x) ◆

Linearity

ψi(t + 1) = Ui,jψj(t)

U(k) = ✓neik −im −im ne−ik ◆

U = Z π

  • π

dk U(k) ⌦ |kihk|

Fourier

AB, G. M. D’Ariano, A. Tosini, Phys. Rev. A 88, 032301 (2013).

slide-13
SLIDE 13

U(k) = exp(−iHA(k))

m, k → 0

HA(k) HD(k) + O(m2k)

HD(k) = ✓−k m m k ◆

Dirac QCA vs Dirac evolution

AB, G. M. D’Ariano, A. Tosini, e-print arXiv:1212.2839. AB, G. M. D’Ariano, A. Tosini, Phys. Rev. A 88, 032301 (2013).

slide-14
SLIDE 14

U(k) = exp(−iHA(k))

m, k → 0

HA(k) HD(k) + O(m2k)

HD(k) = ✓−k m m k ◆

Dirac QCA vs Dirac evolution

Dispersion relation

ω2

D = k2 + m2

cos2(ωA) = (1 − m2) cos2(k) ωD ✓ 1 − m2 6 k2 − m2 k2 + m2 ◆ ωA

m, k → 0

AB, G. M. D’Ariano, A. Tosini, e-print arXiv:1212.2839. AB, G. M. D’Ariano, A. Tosini, Phys. Rev. A 88, 032301 (2013).

slide-15
SLIDE 15

U(k) = exp(−iHA(k))

m, k → 0

HA(k) HD(k) + O(m2k)

HD(k) = ✓−k m m k ◆

Dirac QCA vs Dirac evolution

Dispersion relation

ω2

D = k2 + m2

cos2(ωA) = (1 − m2) cos2(k) ωD ✓ 1 − m2 6 k2 − m2 k2 + m2 ◆ ωA

m, k → 0

black boxes Discrimination between

exp (−iHDt) exp(−iHAt)

UA UD

Automaton Dirac

= =

ρ ∈ S¯

k, ¯ N

¯ k

¯ N

less than particles momentum smaller than

ρ

Ui

Pˆ

i

i = A, D

perr = 1 2 ⇣ p(A|D) + p(D|A) ⌘ ≥ 1 2 ✓ 1 − 1 6m2kNt ◆

AB, G. M. D’Ariano, A. Tosini, e-print arXiv:1212.2839. AB, G. M. D’Ariano, A. Tosini, Phys. Rev. A 88, 032301 (2013).

slide-16
SLIDE 16

summary Mid-term

Quantum Theory Quantum Cellular Automata Free fields linearity

m, k → 0

Dirac automaton usual theory

slide-17
SLIDE 17

summary Mid-term

Quantum Theory Quantum Cellular Automata Free fields linearity

m, k → 0

Dirac automaton usual theory

lattice discrete coordinates

Lorentz invariant equations Relativity

slide-18
SLIDE 18

summary Mid-term

Quantum Theory Quantum Cellular Automata Free fields linearity

m, k → 0

Dirac automaton usual theory

coordinates change? boost?

lattice discrete coordinates

Lorentz invariant equations Relativity

slide-19
SLIDE 19

and QCA Lorentz transformation

Consider the 1D Dirac automaton

the dispersion relation is clearly non Lorentz invariant

cos2(ω) = (1 − m2) cos2(k)

Lorentz invariance is violated ultra-relativistic scales

U(k) = ✓neik −im −im ne−ik ◆

✓ω0 k0 ◆ = γ ✓ 1 −β −β 1 ◆ ✓ω k ◆

γ := 1 p 1 − β2

Lorentz transformation

classical mechanics emergent from the automaton

at

slide-20
SLIDE 20

and QCA Lorentz transformation

Consider the 1D Dirac automaton

the dispersion relation is clearly non Lorentz invariant

cos2(ω) = (1 − m2) cos2(k)

Lorentz invariance is violated ultra-relativistic scales

U(k) = ✓neik −im −im ne−ik ◆

✓ω0 k0 ◆ = γ ✓ 1 −β −β 1 ◆ ✓ω k ◆

γ := 1 p 1 − β2

Lorentz transformation

classical mechanics emergent from the automaton

at

  • r

different transformation privileged reference frame

slide-21
SLIDE 21

A simple speculation from Quantum Gravity

`P = r ~G c3 EP = r ~c5 G

Threshold for quantum spacetime

slide-22
SLIDE 22

A simple speculation from Quantum Gravity

`P = r ~G c3 EP = r ~c5 G

Threshold for quantum spacetime

BUT length energy are not Lorentz invariant and

slide-23
SLIDE 23

A simple speculation from Quantum Gravity

`P = r ~G c3 EP = r ~c5 G

Threshold for quantum spacetime

BUT length energy are not Lorentz invariant and relativity principle? In whose reference frame

EP

threshold a for new phenomena? is Give up

slide-24
SLIDE 24

Deformed relativity

Preserve relativity principle Lorentz group invariant energy scale

  • J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).
  • G. Amelino-Camelia, Physics Letters B 510, 255 (2001).

AND

slide-25
SLIDE 25

Deformed relativity

Preserve relativity principle Lorentz group invariant energy scale

  • J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).
  • G. Amelino-Camelia, Physics Letters B 510, 255 (2001).

AND Modify action Lorentz group the

  • f the
slide-26
SLIDE 26

Deformed relativity

Preserve relativity principle Lorentz group invariant energy scale

  • J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).
  • G. Amelino-Camelia, Physics Letters B 510, 255 (2001).

AND Modify action Lorentz group the

  • f the

non-linear action in momentum space

LD

β := D−1 Lβ D, Lβ = γ ✓ 1 −β −β 1 ◆

momentum space fundamental is more

D is a non-linear map JD(0, 0) = I

invertible singular invariant point energy

slide-27
SLIDE 27

Deformed relativity

Preserve relativity principle Lorentz group invariant energy scale

  • J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).
  • G. Amelino-Camelia, Physics Letters B 510, 255 (2001).

AND Modify action Lorentz group the

  • f the

non-linear action in momentum space

LD

β := D−1 Lβ D, Lβ = γ ✓ 1 −β −β 1 ◆

momentum space fundamental is more

D is a non-linear map JD(0, 0) = I

invertible singular invariant point energy

which D ?

slide-28
SLIDE 28

Deformed relativity and QCA

Automaton dispersion relation

  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760

cos2(ω) = (1 − m2) cos2(k) sin2(ω) cos2(k) − tan2(k) = m2

˜ ω2 − ˜ k2 = m2

slide-29
SLIDE 29

Deformed relativity and QCA

Automaton dispersion relation

  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760

cos2(ω) = (1 − m2) cos2(k) sin2(ω) cos2(k) − tan2(k) = m2

˜ ω2 − ˜ k2 = m2

D ✓ω k ◆ =

sin(ω) cos(k)

tan(k) !

−π 2 6 k 6 π 2

  • 3
  • 2
  • 1

1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k wHkL

ωinv = π 2

slide-30
SLIDE 30

Deformed relativity and QCA

  • 3
  • 2
  • 1

1 2 3

  • 1.0
  • 0.5

0.0 0.5 1.0 k vHkL

  • 3
  • 2
  • 1

1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k wHkL

B A B A B B

A and B exhibits the same kinematics

  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760.
slide-31
SLIDE 31

Deformed relativity and QCA

  • 3
  • 2
  • 1

1 2 3

  • 1.0
  • 0.5

0.0 0.5 1.0 k vHkL

  • 3
  • 2
  • 1

1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k wHkL

B A B A B B

A and B exhibits the same kinematics

|ψi = Z dµk ˆ g(k)|ki

LD

β

  • !

Z dµk ˆ g(k)|k0i = Z dµk0 ˆ g(k(k0))|k0i

State transformation

  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760.
slide-32
SLIDE 32

Deformed relativity in position space

  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760.

The model is defined in the momentum space Transformations in the position space?

?

slide-33
SLIDE 33

Deformed relativity in position space

  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760.

The model is defined in the momentum space Transformations in the position space?

?

Operational toy model of spacetime

point in spacetime coincidence of wavepackets

slide-34
SLIDE 34
  • R. Schutzhold, W. G. Unruh, JETP Lett. 78, 431 (2003).
  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760.
  • G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. D 84, 084010 (2011).
  • ne-particle evolution

Lorentz transformation |ki ! |k0i

Deformed relativity in position space

+

transformation of the coincidence points

transformation of wavepackets

slide-35
SLIDE 35
  • R. Schutzhold, W. G. Unruh, JETP Lett. 78, 431 (2003).
  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760.
  • G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. D 84, 084010 (2011).
  • ne-particle evolution

Lorentz transformation |ki ! |k0i

Deformed relativity in position space

+

transformation of the coincidence points

transformation of wavepackets

✓t0 x0 ◆ ≈ ✓−∂ω0k ∂k0k ∂ω0ω −∂k0ω ◆

k0=k0

✓ t x ◆

slide-36
SLIDE 36
  • R. Schutzhold, W. G. Unruh, JETP Lett. 78, 431 (2003).
  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760.
  • G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. D 84, 084010 (2011).
  • ne-particle evolution

Lorentz transformation |ki ! |k0i

Deformed relativity in position space

+

transformation of the coincidence points

transformation of wavepackets

✓t0 x0 ◆ ≈ ✓−∂ω0k ∂k0k ∂ω0ω −∂k0ω ◆

k0=k0

✓ t x ◆

Relative Locality

momentum-dependent spacetime

slide-37
SLIDE 37

k1 ≈ 0 (x, t) (x0

2, t0 2)

(x0

1, t0 1)

k0

2 ≈ −0.6

k2 ≈ π/5 k0

1 ≈ −1.2

β = . 9 9

5000 5000 5000 5000 20000 0 20000 5000 5000

Loss of coincidence

Relative locality Observer-dependent spacetime

  • A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760.
  • R. Schutzhold, W. G. Unruh, JETP Lett. 78, 431 (2003).

Deformed relativity in position space

slide-38
SLIDE 38

3-dimensional case and spinor field

AB, G. M. D’Ariano, P. Perinotti, in preparation

astrophysical objects Modified Dispersion relation

New phenomenology

  • C. J. Hogan Phys. Rev. D 85, 064007 (2012)

Open problems

non linear action of the rotation group

Real space formulation

k-Poincare algebra and k-Minkowski

  • perational characterization of boosts

light from distant Non-commutative spacetime holographic noise

  • G. Amelino-Camelia and L. Smolin
  • Phys. Rev. D 80, 084017 (2009)
slide-39
SLIDE 39

Quantum theory

“Quantum theory” computational field

Quantum “ab initio” theory of dynamics Main idea

A final overlook

slide-40
SLIDE 40

Quantum theory

“Quantum theory” computational field

Quantum “ab initio” theory of dynamics Main idea

Free QED Free Dirac Field Interactions

Energy? Momentum?

QCA model

A final overlook

slide-41
SLIDE 41

Quantum theory

“Quantum theory” computational field

Quantum “ab initio” theory of dynamics Main idea

Free QED Free Dirac Field Interactions

Energy? Momentum?

QCA model

A final overlook

Deformed relativity

momentum space DSR

  • perational toy-model
  • f spacetime

3D emergent spacetime

Boost?

slide-42
SLIDE 42

Thank you!