Approximating the infinite - a model theoretic look at ... · If there is a basis of a Hilbert space H consisting of eigenvectors of an operator T, this basis is an eigenbasis. In

Approximating the infinite - a modeltheoretic look at quantum physics

Asa Hirvonen(Joint work with Tapani Hyttinen)

Department of Mathematics and StatisticsUniversity of Helsinki, Finland

Crossing Worlds, Helsinki, June 4, 2016

A. Hirvonen (University of Helsinki) QM and model theory June 4, 2016 1 / 18

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 changesanother


Models in physics

Physicists want to explain the phenomena by a model

Explain: present a model that is able to predictmeasurementsCurrent consensus: present the state of the system as avector in a complex Hilbert space (wavefunction, solutionto Schrodinger equation)State: everything there is to know about the system.


Models in physics


Explain: present a model that is able to predictmeasurements

Current consensus: present the state of the system as avector in a complex Hilbert space (wavefunction, solutionto Schrodinger equation)State: everything there is to know about the system.


Models in physics


Explain: present a model that is able to predictmeasurementsCurrent consensus: present the state of the system as avector in a complex Hilbert space (wavefunction, solutionto Schrodinger equation)State: everything there is to know about the system.


Hilbert spacecomplete inner product space over given field ofscalars

vectors with lengths and angles

e.g. Rn is a n-dimensional real Hilbert space

elements can be written as (finite or countable)linear combinations over a basis

a =∞∑n=0

anen

where the vectors e0, e1, e2, . . . form a basis

in a complex Hilbert space the coefficients an arecomplex numbers


Hilbert spacecomplete inner product space over given field ofscalars

vectors with lengths and angles

e.g. Rn is a n-dimensional real Hilbert space

elements can be written as (finite or countable)linear combinations over a basis

a =∞∑n=0

anen

where the vectors e0, e1, e2, . . . form a basis

in a complex Hilbert space the coefficients an arecomplex numbers


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)


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 measuredoutcomes.


Eigenbasis

If there is a basis of a Hilbert space H consisting ofeigenvectors of an operator T , this basis is an eigenbasis.In finite-dimensional complex Hilbert spaces, for mostoperators one can find eigenbases.

In quantum physics measures are thought of asprojections to a basic vector in the eigenbasis for theoperator corresponding to the measured property.If x =

∑∞n=0 anen, the probability, that the outcome is

the eigenvalue of en is |an|2.


The idea

states are length one vectors; a state is a completedescription of the system

the system evolves in time; this is modelled by aunitary operator

we get information from the model, by performingmeasurements

measuring a property projects the state onto aneigenvector corresponding to the measured property


The standard model

The standard model is built on the Hilbert space

L2(R) = {[f ] | f : R→ C and

∫ ∞

−∞|f (x)|2dx <∞}

where the equivalence class corresponds to identifyfunctions that differ on a measure zero set


The problem

The standard space does not have eigenvalues andeigenvectors for the operators one is interested in.Physicists use the eigenvalue approach only as anintuitive idea, but actually calculate by other methods


The propagator of the free particleConsider 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(t0), then thestate at time t is

ψ(t)(x) =

∫R

K (x , y , t)ψ(y , 0)dy

where K (x , y , t) is the propagator, which gives theprobability amplitude for the particle to have travelledfrom y to x in time t.


Calculating the propagator

various approaches:

path integrals: calculate the probability amplitudefor each path from x to y and sum these (integrate)

calculate by using approximations of the timeevolution operator

our approach is to build a model where thereactually are eigenvectors


Ultrafilters

Let I be a set. D ⊂ P(I ) is an ultrafilter if1 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.


Ultraproducts

If I is a set and for each i ∈ I , Mi is an L-model, we geta new model by taking as its elements

{[(ai)i∈I ] | ai ∈Mi

where (bi)i∈I ∈ [(ai)i∈I ] iff

{i ∈ I | ai = bi} ∈ D

and defining the structure accordingly.

Use: what holds in almost all small models, holds in theultraproduct.


Metric ultraproducts of Hilbert spaces

let the “small” models be hilbert spaces Hi

take the ultraproduct of these

cut out the infinite elements (those of infinitelength)

mod out the infinitesimals


Our method

build finite Hilbert spaces Hi with two bases thatare Fourier transforms of each other

take the ultraproduct of these spaces and findsomething that looks like a Hilbert space structurein this

also look at the metric utraproduct of the spaces Hi

show that you can embed the standard model intothe metric ultraproduct (often; with suitable scaling)


Results

calculating the propagator in Hi , we get N−1/2 timesthe value given by physicists

this gives the same probability amplitudes in ourmodel as the traditional result gives the standardmodel (when comparing them via the embedding)


Happy birthday, Juliette!


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Approximating the infinite - a model theoretic look at ... · If there is a basis of a Hilbert space H consisting of eigenvectors of an operator T, this basis is an eigenbasis. In