Questioning Quantum Mechanics? Kurt Barry SASS Talk January 25 th, 2012

Transcription

1 Questioning Quantum Mechanics? Kurt Barry SASS Talk January 25 th, 2012

2 2 Model of the Universe Fundamental Theory Low-Energy Limit Effective Field Theory Quantum Mechanics Quantum Mechanics is presently the framework for physical law. What if this isn t the case?

3 What if: K.Barry - SASS - Questioning QM? 3 Fundamental Theory Some Limit (almost) Effective Field Theory NOT Quantum Mechanics (almost) Quantum Mechanics What deviations from quantum mechanics are possible? What are their observable consequences?

4 4 What Is Quantum Mechanics? A state ψ is a ray in a Hilbert space Probabilities: φ ψ 2 Measured quantities are eigenvalues of Hermitian operators Time evolution described by Schrödinger equation: i d dt ψ = H ψ Linear

5 5 Possible generalization: linearity Nonlinearity is common in everyday phenomena; why not in quantum systems as well? To what extent can we test for non-linearity in a modelindependent manner? Can we preserve logical consistency in a non-linear quantum theory?

6 6 Outline Quantum Mechanics from non-linear dynamics? Concrete example: Weinberg s framework. Observable consequences of a fundamentally new character? Yes; state-dependent phases, nonlocality. parallel universe communication Can a nonlinear theory be logically consistent? Remove nonlocality? (Polchinski, Kibble) Fundamental challenges

7 7 Steven Weinberg, Annals Phys. 194 (1989) 336 A Nonlinear Extension: Weinberg s Framework

8 8 Making QM Nonlinear i dψ k dt = H klψ l

9 9 Making QM Nonlinear i dψ k dt = ψ k ψ j H jl ψ l real, bilinear function

10 10 Making QM Nonlinear i dψ k dt = ψ k h(ψ j, ψ l ) real, NON-bilinear function

11 One Further Requirement: Homogeneity K.Barry - SASS - Questioning QM? 11 Require that any nonlinear observable a satisfy: a Z ψ, Zψ = Z Za ψ, ψ Why? It makes it easy to keep wavefunction s normalization arbitrary Now what? Build, build, build.

12 12 Achievements Constant in time: Wavefunction s normalization, Hamiltonian Can find many nonlinear Hamiltonians that respect Galilean invariance Requires taking the standard linear generators for momentum, angular momentum, etc. Solutions with harmonic time dependence (stationary states)

13 13 What are the observable consequences of Weinberg nonlinearities?

14 N-atom coherent system, neglecting all but two closely split states 14 K.Barry - SASS - Questioning QM? 1 0 Rabi flop 1 a a Free evolution 1 a ae iδ If the free evolution is non-linear a la Weinberg, δ depends on a, a feature that is qualitatively distinct from any linear quantum mechanical evolution.

15 15 Experiment by Bollinger et al* Used a hyperfine nuclear transition of 9 Be + Model non-linearity: 2εa 2 Set a limit: ε 8 ± ev *Physical Review Letters 63, 1031 (1989)

16 Nonlinearity and Nonlocality 16 ψ I II φ a II (φ) I II a II ψ φ =???

17 17 Non-locality How can we combine separated systems? Weinberg proposes a form that permits separable solutions However, this form is basis-dependent Polchinski has proven that it permits instantaneous communication

18 18 Polchinski and Kibble Can we get logical consistency?

19 19 Remove nonlocality: Polchinski s proposal* K.Barry - SASS - Questioning QM? Basis-independent solution: reduced density matrices Eliminates nonlocality, but causes communication between wavefunction branches Binary messages of arbitrary length can be exchanged between the same observer in parallel worlds(!) In practice, wavefunction-branch coupling might be so complex that effects would be unobservable *Physical Review Letters 66, 397 (1991)

20 Wavefunction Branch Communication Observer + two-state system K.Barry - SASS - Questioning QM? 20 Nonlinear time evolution Observe spin +1/2-1/2 Choose action A or B Observe spin again-- outcome depends on choice of A or B! Many Worlds interpretation seems natural But is this consistent?

21 Kibble s Nonlinear QFT* Start in Schrödinger picture, introduce nonlinearity by allowing the Hamiltonian to depend on the wavefunction: i d dt ψ = H ψ ψ Preserve locality: write Hamiltonian as an integral over a density: H ψ = 1 2 π φ 2 + m 2 φ2 + g 4! φ4 +α + β : φ 2 : ψ + λ 2 : φ2 : ψ φ 2 *Commun. Math. Phys. 64, (1978) d 3 x 21

22 22 Physical Consequences (g=0) ω k 2 = k 2 + m 2 + λ : φ 2 (x): ψ Effects will be strongest for localized wavefunctions Wavepackets will show anomalous dispersion Scattering can occur even in the absence of interactions No free theory, really

23 23 Generic Difficulties for NLQM Perturbation theory difficult Can t get to a Heisenberg or interaction picture easily Consistent probabilistic interpretation challenging Scattering not a well-posed question Basic logical consistency is uncertain

24 24 Conclusions

25 25 Questioning Quantum Mechanics? Can Quantum Mechanics from non-linear dynamics? Concrete example: Weinberg s framework. Are there observable consequences of a fundamentally new character? Yes; state-dependent phases, nonlocality, parallel universe communication Can a nonlinear theory be logically consistent? Remove nonlocality? (Polchinski, Kibble) Fundamental challenges

26 The End! Thanks for listening! 26 Another consequence of non-linear quantum mechanics?

27 Details of New Phase K.Barry - SASS - Questioning QM? n ψ ψ 2 2 a ψ 2 2 (definitions) n 27 Model Hamiltonian function: h ψ, ψ = n 1 a E 1 + ae 2 + 2εa 2 Phase difference between the two components after a time t: δ = t E 1 E 2 4εa

28 Nonlinearity and Nonlocality 28 K.Barry - SASS - Questioning QM? I II I Ψ kl II ψ k φ l a II = φ a II = Ψ k1 4 k=1 Ψ = ψ 1 φ 1 a II = 1 Ψ = 1 ε 2 ψ 1 + ε ψ 2 φ 1 a II = 1 2ε 2 + 2ε 4