From Quantum Cellular Automata to Quantum Field Theory

Transcription

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

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

3 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

4 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

5 Quantum Theory T P i Operational Probabilistic Theory preparations transformations measurements composition parallel sequence T T T T systems T P i outcomes probability theory of information G. Ludwig, Foundations of Quantum Mechanics (Springer, New York, 1985). L. Hardy, e-print arxiv:quant-ph/ G. Chiribella, G. M. D Ariano, P. Perinotti, Phys. Rev. A 84, (2011)

6 Quantum Theory T P i Operational Probabilistic Theory preparations transformations measurements composition parallel sequence T T T T systems T P i outcomes probability theory of information Space? Dynamics? Energy? Time? G. Ludwig, Foundations of Quantum Mechanics (Springer, New York, 1985). L. Hardy, e-print arxiv:quant-ph/ G. Chiribella, G. M. D Ariano, P. Perinotti, Phys. Rev. A 84, (2011)

7 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)

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,...

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,...

10 What kind of computer? Quantum Circuit Quantum Cellular Automaton B. Schumacher, R.F. Werner e-print arxiv: 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,...

11 QCA for the Dirac field (1+1)-dimensional case Linearity i(t + 1) = U i,j j (t) (x, t + 1) = U (x, t) ns im U = im ns S (x) = (x + 1) n 2 + m 2 =1, 0 6 m 6 1 (x) = R (x) L (x) AB, G. M. D Ariano, A. Tosini, e-print arxiv: AB, G. M. D Ariano, A. Tosini, Phys. Rev. A 88, (2013).

12 QCA for the Dirac field (1+1)-dimensional case Linearity i(t + 1) = U i,j j (t) (x, t + 1) = U (x, t) ns im U = im ns S (x) = (x + 1) n 2 + m 2 =1, 0 6 m 6 1 (x) = R (x) L (x) Fourier U = Z - U(k) = dk U(k) kihk ne ik im im ne ik AB, G. M. D Ariano, A. Tosini, e-print arxiv: AB, G. M. D Ariano, A. Tosini, Phys. Rev. A 88, (2013).

13 Dirac QCA vs Dirac evolution U(k) =exp( ih A (k)) m, k! 0 H A (k) H D (k)+o(m 2 k) H D (k) = k m m k AB, G. M. D Ariano, A. Tosini, e-print arxiv: AB, G. M. D Ariano, A. Tosini, Phys. Rev. A 88, (2013).

14 Dirac QCA vs Dirac evolution U(k) =exp( ih A (k)) m, k! 0 H A (k) H D (k)+o(m 2 k) Dispersion relation! A m, k! 0 cos 2 (! A )=(1 m 2 ) cos 2 (k)! D 1 m 2 6 k 2 m 2 k 2 + m 2 H D (k) = k m m k! 2 D = k 2 + m 2 AB, G. M. D Ariano, A. Tosini, e-print arxiv: AB, G. M. D Ariano, A. Tosini, Phys. Rev. A 88, (2013).

15 Dirac QCA vs Dirac evolution U(k) =exp( ih A (k)) m, k! 0 H A (k) H D (k)+o(m 2 k) Dispersion relation! A m, k! 0 cos 2 (! A )=(1 m 2 ) cos 2 (k)! D 1 m 2 6 k 2 m 2 k 2 + m 2 H D (k) = k m m k! 2 D = k 2 + m 2 Discrimination between black boxes U A U D p err = 1 2 = = exp( ih A t) exp ( ih D t) Automaton Dirac p(a D)+p(D A) U i Pî 2 S k, N i = A, D N less than particles momentum smaller than 1 6 m2 knt AB, G. M. D Ariano, A. Tosini, e-print arxiv: AB, G. M. D Ariano, A. Tosini, Phys. Rev. A 88, (2013). k

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

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

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

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

20 QCA and Lorentz transformation Consider the 1D Dirac automaton the dispersion relation cos 2 (!) =(1 is clearly non Lorentz invariant Lorentz transformation! 0 k 0 := = 1 p ! k U(k) = ne ik im m 2 ) cos 2 (k) im ne ik Lorentz invariance is violated at classical mechanics emergent from the automaton ultra-relativistic scales different transformation or privileged reference frame

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

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

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

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

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

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

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

28 Deformed relativity and QCA Automaton dispersion relation cos 2 (!) =(1 m 2 ) cos 2 (k) sin 2 (!) cos 2 (k) tan 2 (k) =m 2! 2 k2 = m 2 A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

29 Deformed relativity and QCA Automaton dispersion relation cos 2 (!) =(1 m 2 ) cos 2 (k) sin 2 (!) cos 2 (k) tan 2 (k) =m 2 D! k = 2 6 k 6 2! inv = 2! sin(!) cos(k) tan(k) whkl ! 2 k2 = m k A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

30 Deformed relativity and QCA B A B B A B whkl k vhkl A and B exhibits the same kinematics k A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

31 Deformed relativity and QCA B A B B A B whkl k vhkl A and B exhibits the same kinematics k State transformation Z Z i = dµ k ĝ(k) ki LD! dµ k ĝ(k) k 0 i = Z dµ k 0 ĝ(k(k 0 )) k 0 i A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

32 Deformed relativity in position space The model is defined in the momentum space? Transformations in the position space? A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

33 Deformed relativity in position space The model is defined in the momentum space? Transformations in the position space? Operational toy model of spacetime coincidence of wavepackets point in spacetime A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

34 Deformed relativity in position space one-particle evolution + Lorentz transformation ki! k 0 i transformation of wavepackets transformation of the coincidence points R. Schutzhold, W. G. Unruh, JETP Lett. 78, 431 (2003). G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. D 84, (2011). A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

35 Deformed relativity in position space one-particle evolution + Lorentz transformation ki! k 0 i transformation of wavepackets transformation of the coincidence points t 0 x k k 0! k 0 =k 0 0 t x R. Schutzhold, W. G. Unruh, JETP Lett. 78, 431 (2003). G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. D 84, (2011). A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

36 Deformed relativity in position space one-particle evolution + Lorentz transformation ki! k 0 i transformation of wavepackets transformation of the coincidence points t 0 x k k 0! k 0 =k 0 0 t x momentum-dependent spacetime Relative Locality R. Schutzhold, W. G. Unruh, JETP Lett. 78, 431 (2003). G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. D 84, (2011). A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

37 Deformed relativity in position space Relative locality Observer-dependent spacetime Loss of coincidence k k 1 0 k 2 /5 (x 0 1,t 0 1) k (x, t) =0.99 (x 0 2,t 0 2) R. Schutzhold, W. G. Unruh, JETP Lett. 78, 431 (2003). A. Bibeau-Delisle, AB, G. M. D Ariano, P. Perinotti, A. Tosini, eprint arxiv:

38 Open problems 3-dimensional case and spinor field non linear action of the rotation group AB, G. M. D Ariano, P. Perinotti, in preparation Real space formulation operational characterization of boosts k-poincare algebra and k-minkowski New phenomenology Modified Dispersion relation Non-commutative spacetime light from distant astrophysical objects G. Amelino-Camelia and L. Smolin Phys. Rev. D 80, (2009) holographic noise C. J. Hogan Phys. Rev. D 85, (2012)

39 A final overlook Main idea Quantum theory Quantum computational field theory Quantum ab initio theory of dynamics

40 A final overlook Main idea Quantum theory Quantum computational field theory Quantum ab initio theory of dynamics Free Dirac Field QCA model Free QED Interactions Energy? Momentum?

41 A final overlook Main idea Quantum theory Quantum computational field theory Quantum ab initio theory of dynamics Free Dirac Field QCA model Free QED Boost? Interactions Energy? Momentum? momentum space DSR Deformed relativity operational toy-model of spacetime 3D emergent spacetime

42 Thank you!