Approximating the infinite - a model theoretic look at quantum physics

Transcription

1 Approximating the infinite - a model theoretic look at quantum physics Åsa Hirvonen (Joint work with Tapani Hyttinen) Department of Mathematics and Statistics University of Helsinki, Finland Crossing Worlds, Helsinki, June 4, 2016 Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

2 Quantum physics Odd phenomena when looking at the small scale light as particles (and wawes) electrons as wawes (and particles) determining one property of the system changes another Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

3 Models in physics Physicists want to explain the phenomena by a model Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

4 Models in physics Physicists want to explain the phenomena by a model Explain: present a model that is able to predict measurements Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

5 Models in physics Physicists want to explain the phenomena by a model Explain: present a model that is able to predict measurements Current consensus: present the state of the system as a vector in a complex Hilbert space (wavefunction, solution to Schrödinger equation) State: everything there is to know about the system. Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

6 Hilbert space complete inner product space over given field of scalars Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

7 Hilbert space complete inner product space over given field of scalars vectors with lengths and angles e.g. R n is a n-dimensional real Hilbert space elements can be written as (finite or countable) linear combinations over a basis a = a n e n n=0 where the vectors e 0, e 1, e 2,... form a basis in a complex Hilbert space the coefficients a n are complex numbers Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

8 Operators Operators are linear functions on a Hilbert space H T (ax + by) = at (x) + bt (y) for a, b C (or R in a real Hilbert space) and x, y H In quantum physics properties are modelled by operators (e.g. position operator Q, momentum operator P) Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

9 Eigenvalues and eigenvectors A vector x H is an eigenvector for the operator T if T (x) = ax for some scalar a. Then a is an eigenvlue of T. In quantum physics eigenvalues correspond to measured outcomes. Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

10 Eigenbasis If there is a basis of a Hilbert space H consisting of eigenvectors of an operator T, this basis is an eigenbasis. In finite-dimensional complex Hilbert spaces, for most operators one can find eigenbases. In quantum physics measures are thought of as projections to a basic vector in the eigenbasis for the operator corresponding to the measured property. If x = n=0 a ne n, the probability, that the outcome is the eigenvalue of e n is a n 2. Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

11 The idea states are length one vectors; a state is a complete description of the system the system evolves in time; this is modelled by a unitary operator we get information from the model, by performing measurements measuring a property projects the state onto an eigenvector corresponding to the measured property Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

12 The standard model The standard model is built on the Hilbert space L 2 (R) = {[f ] f : R C and f (x) 2 dx < } where the equivalence class corresponds to identify functions that differ on a measure zero set Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

13 The problem The standard space does not have eigenvalues and eigenvectors for the operators one is interested in. Physicists use the eigenvalue approach only as an intuitive idea, but actually calculate by other methods Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

14 The propagator of the free particle Consider a particle in (one-dimensional) space. The time evolution operator of this system is K t = e itp2 /2m If the state of the system at time 0 is ψ(t 0 ), then the state at time t is ψ(t)(x) = K(x, y, t)ψ(y, 0)dy R where K(x, y, t) is the propagator, which gives the probability amplitude for the particle to have travelled from y to x in time t. Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

15 Calculating the propagator various approaches: path integrals: calculate the probability amplitude for each path from x to y and sum these (integrate) calculate by using approximations of the time evolution operator our approach is to build a model where there actually are eigenvectors Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

16 Ultrafilters Let I be a set. D P(I ) is an ultrafilter if 1 if X, Y D then X Y D, 2 if X D and X Y I, then Y D, 3 for every X D, either C D or I \X D. Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

17 Ultraproducts If I is a set and for each i I, M i is an L-model, we get a new model by taking as its elements where (b i ) i I [(a i ) i I ] iff {[(a i ) i I ] a i M i {i I a i = b i } D and defining the structure accordingly. Use: what holds in almost all small models, holds in the ultraproduct. Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

18 Metric ultraproducts of Hilbert spaces let the small models be hilbert spaces H i take the ultraproduct of these cut out the infinite elements (those of infinite length) mod out the infinitesimals Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

19 Our method build finite Hilbert spaces H i with two bases that are Fourier transforms of each other take the ultraproduct of these spaces and find something that looks like a Hilbert space structure in this also look at the metric utraproduct of the spaces H i show that you can embed the standard model into the metric ultraproduct (often; with suitable scaling) Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

20 Results calculating the propagator in H i, we get N 1/2 times the value given by physicists this gives the same probability amplitudes in our model as the traditional result gives the standard model (when comparing them via the embedding) Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18

21 Happy birthday, Juliette! Å. Hirvonen (University of Helsinki) QM and model theory June 4, / 18