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

From Quantum Cellular Automata to Quantum Field Theory Alessandro Bisio Frontiers of Fundamental Physics Marseille, July 15-18th 2014

in collaboration with Giacomo Mauro D’Ariano o University of Pavia Paolo Perinotti QUIT group Alessandro Tosini University of Montreal Alexandre Bibeau-Delisle supported by

Outline 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

Quantum Theory Von Neumann, 1932 Each physical system is associated with a Hilbert space Unit vectors are associated with states of the system Physical observables are represented by self adjoint operators The Hilbert space of a composite system is the tensor product of the state spaces associated with the component systems The probabilities of the outcomes are given by the Born rule

Quantum Theory Operational Probabilistic Theory composition preparations σ parallel sequence transformations T T T T P i measurements T systems theory of information σ outcomes T P i probability ρ 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)

Quantum Theory Operational Probabilistic Theory composition preparations σ parallel sequence transformations T T T T P i measurements T systems theory of information σ outcomes T P i probability ρ Dynamics? Space? Time? Energy? 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)

Processing of Quantum Information Quantum Circuit Quantum Computer Simulating Physics with Computers (Feynman 1982) Can a Quantum Computer exactly simulate physical systems? Simulation as a guideline for discovery R. P. Feynman, Int. J. Theo. Phys. 21, 467 (1982)

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, ...

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, ...

What kind of computer? Quantum Circuit Quantum Cellular Automaton 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, ...

QCA for the Dirac field (1+1)-dimensional case Linearity ψ i ( t + 1) = U i,j ψ j ( t ) ψ ( x, t + 1) = U ψ ( x, t ) ✓ nS ◆ − im U = nS † − im S ψ ( x ) = ψ ( x + 1) ✓ ψ R ( x ) ◆ n 2 + m 2 = 1 , ψ ( x ) = ψ L ( x ) 0 6 m 6 1 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).

QCA for the Dirac field (1+1)-dimensional case Linearity ψ i ( t + 1) = U i,j ψ j ( t ) ψ ( x, t + 1) = U ψ ( x, t ) ✓ nS Z π ◆ − im Fourier U = d k U ( k ) ⌦ | k ih k | U = nS † − im - π ✓ ne ik ◆ − im S ψ ( x ) = ψ ( x + 1) U ( k ) = ✓ ψ R ( x ) ◆ ne − ik − im n 2 + m 2 = 1 , ψ ( x ) = ψ L ( x ) 0 6 m 6 1 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).

Dirac QCA vs Dirac evolution m, k → 0 U ( k ) = exp( − i H A ( k )) H D ( k ) + O ( m 2 k ) H A ( k ) ✓ − k ◆ m H D ( k ) = m k 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).

Dirac QCA vs Dirac evolution m, k → 0 U ( k ) = exp( − i H A ( k )) H D ( k ) + O ( m 2 k ) H A ( k ) ✓ − k ◆ m H D ( k ) = m k Dispersion relation 1 − m 2 k 2 − m 2 ✓ ◆ m, k → 0 cos 2 ( ω A ) = (1 − m 2 ) cos 2 ( k ) ω A ω D k 2 + m 2 6 D = k 2 + m 2 ω 2 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).

Dirac QCA vs Dirac evolution m, k → 0 U ( k ) = exp( − i H A ( k )) H D ( k ) + O ( m 2 k ) H A ( k ) ✓ − k ◆ m H D ( k ) = m k Dispersion relation 1 − m 2 k 2 − m 2 ✓ ◆ m, k → 0 cos 2 ( ω A ) = (1 − m 2 ) cos 2 ( k ) ω A ω D k 2 + m 2 6 D = k 2 + m 2 ω 2 Discrimination between black boxes U i U A exp( − iH A t ) Automaton = P ˆ ρ i i = A, D exp ( − iH D t ) U D Dirac = ¯ ρ ∈ S ¯ N particles less than k, ¯ N ¯ k momentum smaller than ✓ ◆ p err = 1 ≥ 1 1 − 1 ⇣ ⌘ 6 m 2 kNt p ( A | D ) + p ( D | A ) 2 2 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).

Mid-term summary Quantum Theory Quantum Cellular Automata Free fields linearity m, k → 0 usual theory Dirac automaton

Mid-term summary Quantum Theory Quantum Cellular Automata Free fields linearity m, k → 0 usual theory Dirac automaton discrete coordinates lattice Lorentz invariant equations Relativity

Mid-term summary Quantum Theory Quantum Cellular Automata Free fields linearity m, k → 0 usual theory Dirac automaton discrete coordinates lattice Lorentz invariant equations Relativity coordinates change? boost?

QCA and Lorentz transformation ✓ ne ik ◆ − im Consider the 1D Dirac automaton U ( k ) = ne − ik − im the dispersion relation cos 2 ( ω ) = (1 − m 2 ) cos 2 ( k ) is clearly non Lorentz invariant classical mechanics emergent from the automaton Lorentz transformation ✓ 1 ✓ ω 0 ◆ ◆ ✓ ω ◆ − β = γ Lorentz invariance is violated k 0 − β k 1 at ultra-relativistic scales 1 γ := p 1 − β 2

QCA and Lorentz transformation ✓ ne ik ◆ − im Consider the 1D Dirac automaton U ( k ) = ne − ik − im the dispersion relation cos 2 ( ω ) = (1 − m 2 ) cos 2 ( k ) is clearly non Lorentz invariant classical mechanics emergent from the automaton Lorentz transformation ✓ 1 ✓ ω 0 ◆ ◆ ✓ ω ◆ − β = γ Lorentz invariance is violated k 0 − β k 1 at ultra-relativistic scales 1 γ := p 1 − β 2 di ff erent privileged or transformation reference frame

A simple speculation from Quantum Gravity r r ~ c 5 ~ G ` P = E P = c 3 G Threshold for quantum spacetime

A simple speculation from Quantum Gravity r r ~ c 5 ~ G ` P = E P = c 3 G Threshold for quantum spacetime BUT and energy are not Lorentz invariant length

A simple speculation from Quantum Gravity r r ~ c 5 ~ G ` P = E P = c 3 G Threshold for quantum spacetime BUT and energy are not Lorentz invariant length In whose reference frame is a threshold for new phenomena? E P Give up relativity principle?

Deformed relativity Preserve relativity principle AND invariant energy scale Lorentz group G. Amelino-Camelia, Physics Letters B 510, 255 (2001). J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).

Deformed relativity Preserve relativity principle AND invariant energy scale Lorentz group Modify the action of the Lorentz group G. Amelino-Camelia, Physics Letters B 510, 255 (2001). J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).

Deformed relativity Preserve relativity principle AND invariant energy scale Lorentz group Modify the action of the Lorentz group non-linear action in momentum space D is a non-linear map J D (0 , 0) = I L D β := D − 1 � L β � D , ✓ 1 singular invariant ◆ − β L β = γ point energy − β 1 invertible momentum space is more fundamental G. Amelino-Camelia, Physics Letters B 510, 255 (2001). J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).

Deformed relativity Preserve relativity principle AND invariant energy scale Lorentz group Modify the action of the Lorentz group non-linear action in momentum space D is a non-linear map J D (0 , 0) = I L D β := D − 1 � L β � D , ✓ 1 singular invariant ◆ − β L β = γ point energy − β 1 invertible momentum space which D ? is more fundamental G. Amelino-Camelia, Physics Letters B 510, 255 (2001). J. Magueijo, L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).

Deformed relativity and QCA Automaton dispersion relation sin 2 ( ω ) cos 2 ( ω ) = (1 − m 2 ) cos 2 ( k ) cos 2 ( k ) − tan 2 ( k ) = m 2 k 2 = m 2 ω 2 − ˜ ˜ A. Bibeau-Delisle, AB, G. M. D’Ariano, P. Perinotti, A. Tosini, eprint arXiv:1310.6760