Intrinsic randomness - Carl Hoefer

Objective chance and quantum
randomness
Carl Hoefer
ICREA - U. Barcelona
Sept. 27, 2016
1

Philosophers of science:
1. Try to understand the notion of law of nature or physical
law.
2. Try to understand the notions of objective or intrinsic
randomness, and objective chance.
Physicists are often curious about these topics too!
2

Goals of [the long version of ] this talk:
1. Explore the notion of an (irreducible, fundamental)
probabilistic law of nature, and argue that we don’t have a
good grasp of what it means to postulate such a thing.
Valerio Scarani: “What does it mean, a physical law that
has statistical character?”
A. Einstein: “Der Gott würfelt nicht.”
3

Goals of this talk:
1. Explore the notion of an (irreducible, fundamental)
probabilistic law of nature, and argue that we don’t have a
good grasp of what it would mean to postulate such a
thing.
2. Show that, by contrast, a probabilistic law that is grounded
on underlying determinism can be grasped easily - and we
have many good examples of such laws.
3. Offer some comments on how the randomness found in
quantum mechanics could be of this variety.
4

Objective probabilities or ‘chances’
• Pr(Heads) = 0.5, flipping a ‘fair’ coin
• Pr(00) in throw of 38-slot roulette wheel = 1/38
• Pr(Pu241 decay in 1 year) = 0.05
• Pr(spin-z up | spin-x up earlier) = 0.5
– ... what kind of facts are these?
– Can we make sense of them as objec*ve and ground-level
truths?
5

De Finetti on ‘probability’:
“My thesis, paradoxically, and a little provocatively, but
nonetheless genuinely, is simply this:
PROBABILITY DOES NOT EXIST
The abandonment of superstitious beliefs about the
existence of the Phlogiston, the Cosmic Ether, Absolute
Space and Time, . . . or Fairies and Witches was an
essential step along the road to scientific thinking.
Probability, too, if regarded as something endowed with
some kind of objective existence, is no less a misleading
misconception, an illusory attempt to exteriorize or
materialize our true probabilistic beliefs.”
• (But we must leave this topic to one side for now.)
6

Distinctions and notations
• indeterminism = ¬ determinism
• indeterminism vs random behavior
• indeterminism vs probabilistic (or ‘stochastic’) laws
7

Non-random indeterminisms
• Classical Mechanics (CM):
–Space Invaders (5-particle system in t-reverse)
–Norton’s Dome (& other symmetry-breaking situations)
8

Non-random indeterminisms
• General Relativity (GR):
–Naked singularities (e.g. “white holes”)
–Other types of ‘hole’
–models with no Cauchy surface
• What all these CM and GR cases have in common:
• No involvement of probability; indeterminism = simple breakdown of
determinism.
9

• indeterminism vs probabilistic laws
• randomness in general vs probabilistic-law-randomness
–examples: cancer incidence, vs radioacCve decay rate
10

• product randomness vs process randomness
–[apparent randomness vs intrinsic randomness]
11

• product randomness vs process randomness
–[apparent randomness vs intrinsic randomness]
12

Outline:
I. The dialectics of primitive chance laws.
II. Chances from underlying determinism
III. QM & intrinsic randomness
13

Outline:
III. QM & intrinsic randomness
14

Chance from Determinism
15
Diaconis’ coin-flip analysis
VerCcal: rate of rotaCon Horizontal: iniCal upward velocity

16

17

18

19

Galton board & IC distributions
21
Any of these IC distribuCons leads to the same sta*s*cal outcome paNern.

Galton board & IC distributions
22
Some possible IC distribuCons however do not give ‘right’ results
⇍

Outline:
III. QM& intrinsic randomness: the heterodox view
23

Bohmian Mechanics
• De Broglie 1927;
• David Bohm 1952
24

Bohmian Mechanics
• QM represents physical
systems with a
‘wavefunction’, Ψ.
• Basic idea of BM: In addition
to Ψ, there are in fact point-
like particles with well-
defined positions at all
times, moving on continuous
paths. Ψ acts on the
particles like a ‘pilot wave’,
determining their velocities
at every moment.
25

Standard two-slit experiment of QM
allegedly shows wave-like behavior
of things like photons, electrons:
26

Bohmian particles passing through
double slit
27

A theory with some appeal …
“Is it not clear from the smallness of the scintillation on the
screen that we have to do with a particle? And is it not clear,
from the diffraction and interference patterns, that the motion
of the particle is directed by a wave? De Broglie showed in
detail how the motion of a particle, passing through just one
of two holes in screen, could be inﬂuenced by waves
propagating through both holes. And so inﬂuenced that the
particle does not go where the waves cancel out, but is
attracted to where they cooperate. This idea seems to me so
natural and simple, to resolve the wave-particle dilemma in
such a clear and ordinary way, that it is a great mystery to me
that it was so generally ignored.” (J. Bell 1986)
28

Bohmian mechanics: spin-measurement
29
Norsen (2013): how Bohmian parCcles get distributed (unequal weight state)

Bohmian mechanics
1. Characterization of state:
(Ψ, X)
2. Schrödinger’s evolution:
3. The guidance equation (GE):
– If the wave funcCon is wriNen in polar form , the guidance equaCon
simply reads .
4. The statistical postulate:
30
≡ … ∈
Ψ … ∈
! ! !
"
! ! !
"
3
1 2 N
3
1 2
X (X , X , , X )
( , x , , ; )
N
N
Nx x t
t
∂Ψ
= Ψ
∂
ˆHih
∇ Ψ
= =
Ψ
!""" #
v Im
kk
k
k
dx
dt m
ρ = Ψ
2
0 0( , ) (x, )x t t
Ψ = !/iS
Re
= ∇
!""
m v Skk k

Bohmian mechanics
• The Bohmian particles follow trajectories in 3d-
space according to the so-called Guidance
equation:
(GE)
31
∇ Ψ
= =
Ψ
!""" #
v Im
kk
k
k
dx
dt m

Bohmian mechanics
• BM makes probabilistic predictions because of the
‘Quantum Equilibrium’ or Statistical Postulate:
(SP)
32
ρ = Ψ
2
0 0( , ) (x, )x t t

QM & intrinsic randomness. . .
What about Bohmian Mechanics??
• In order to say we have certified randomness from
quantum phenomena, we have to rule out BM and
similar Det hidden variable theories. . . How?
• Usual complaints made against BM:
• BM requires conspiratorial iniCal condiCons
• BM does not allow ‘free choice’ of A, B
• BM allows faster-than-light signalling
33

What about Bohmian Mechanics?
• But these complaints are misguided:
• BM involves no conspiratorial super-determinism
• BM allows free choice of measurement se[ngs in Bell
experiments
• BM is a no-signalling theory
– . . . so the standard reasons for excluding ‘hidden
variable’ theories, like BM, seem ineﬀecCve.
34

• BM allows free choice of measurement se[ngs in Bell
experiments
35

Causal structure (assumed) of EPR-
type experiments
36
• A, B: settings for
measurement device
• X, Y: spin/polarization
measurements (spacelike
separation between A, X
and B, Y).
• Z: pre-existing states able to
causally influence A, X, B
and Y.
A B
X Y
Z
E F

Causal structure of EPR experiments
(for Bohmians)
37
• Determination of A, B
partly or fully affected by
external factors E, F.
• Red arrows: non-local
causal influence, a
feature of Bohmian
mechanics.
A B
X Y
Z
E F

Causal structure of EPRB experiments
(for Bohmians)
38
• Determination of A, B
can be by human agents’
free choices;
• Or from photons from
opposite sides of the
galaxy if you like.
• No ‘conspiratorial
conditions’ needed for
deterministic story.
A B
X Y
Z
E F

• Conspiracy (in the sense commonly invoked in the Bell Test
field) not needed. Non-local action at a distance is all that is
required to explain violation of Bell inequality.
• Experimenter ‘free choice’ perfectly OK (unless you think
determinism rules out free choice tout court)
39

. . . and what about signalling?
40
• BM is a “parameter
dependent” theory.
• Parameter dependence ≠
effective signalling
• Statistical postulate
ensures no effective
signalling possible
• SP ≠ conspiratorial initial
conditions in the
“superdeterminism” sense.
A B
X Y
Z
E F

Important note: I am not endorsing
Bohm’s theory
• There are reasons to be skeptical that BM is on the right
track.
• But - violation of no-signalling (at the surface level,
where we have reason to trust N-S), or of free choice,
are not among those reasons
• Plus: where there’s one theory, there may be more out
there waiting to be discovered.
41

Contrast: properties of standard QM
• Non-local
• Contextual
• Parameter-dependence [Copenhagen]
• [arguable] causation at spacelike separation
42

Summing up
1. I argued that it’s difficult to spell out what we mean when we postulate
intrinsic randomness of the propensity or chance-law variety.
2. By contrast, I argued, we can understand objective probability claims
if they arise from determinism + nicely-distributed initial conditions.
– Quantum “randomness” in Bohmian Mechanics is of exactly this sort.
3. I noted that while Bohmian QM has no intrinsic randomness (being
deterministic), it satisfies no-signalling in a pragmatic or effective
sense. At the surface, it’s a counterexample to randomness-
certification arguments; but at the deep level one could say it is a
“signalling” theory. But in this deep-level sense, we have no way to
rule out that nature herself is signalling.
4. Distinction: certified [intrinsic] randomness vs certified [effective]
randomness. BM is counterexample to former, but not the latter.
43

Intrinsic randomness - Carl Hoefer

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