Measure, Probability, Quantum - Stellenbosch...

Measure, Probability,Quantum

Kevin H. Knuth1 and John Skilling2

1. University at Albany (SUNY), USA2. Maximum Entropy Data Consultants Ltd, Kenmare, Ireland

1

A Question Posed

“Why is it that when I take two pencils and add one pencil, I always get three pencils?

And when I take two pennies and add one penny, I always get three pennies,

and so on with rocks and sticks and candy and monkeys and planets and stars.

Is this true by definition as in 2+1 defines 3? Or is it an experimental fact so that at some point in the distant past this observation needed to be verified again and again?”

- Kevin Knuth

2

A Question Posed

“I have tried, with little success, to get some of my friends to understand my amazement that the abstraction of integers for counting is both possible and useful.

Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, l find it both strange and unexplainable.”

- Hamming, 1980

3

A Question Posed

4

2 Pens 4 Pens

A Question Posed

5

2 Pens 4 Pens

2 Pens with 4 Pens

A Question Posed

6

2 Pens 4 Pens

2 Pens with 4 Pens2 ⊕ 4 Pens

How does one determine theoperator ⊕ ?

7

Knuth K.H. 2015. FQXi 2015 Essay Contest Entry (Third Place Prize)http://fqxi.org/community/forum/topic/2382

arXiv:1504.06686 [math.HO]

A Question Posed

http://fqxi.org/community/forum/topic/2382

https://arxiv.org/abs/1504.06686

8

“Measure that which is measurable and make measureable that which is not” --- Galileo

Our job is to make models of the world around us.

These models are often quantified by consistently assigning numbers so that the quantification captures relevant relationships and symmetries.

9

“Quantum mechanics will cease to look puzzling only when we will be able to derive the formalism of the theory from a set of simple assertions about the world." - Carlo Rovelli

Combination: -with-

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LimitlessEquivalent, but disjoint, objects can be combined without restrictionso that A is different from A-with-A is different from A-with-A-with-A, etc.

ClosureGiven stuff A and stuff B, then combining stuff A-with-B is still stuff.

CommutativityOrder in which stuff is combined doesn’t matter. A-with-B is the same as B-with-A

AssociativityCombination can be done in different but equivalent ways(A-with-B)-with-C is the same as A-with-(B-with-C)

N-Tuples

11

Quantify stuff with a set ofnumbers (N-tuple).

Stuff A is represented by 𝐚Stuff B is represented by 𝐛

How to represent A-with-B?

N-Tuples

12

Closure

Commutativity

Limitless

Associativity

Does the operator -with- satisfy: ?




N-Tuples

13

N-Tuple Sum Rule

A-with-Bis represented by

(up to isomorphism) a + b

component-wise summation

Closure

Commutativity

Limitless

Associativity

If the operator -with-satisfies the above properties




theorem

Does the operator -with- satisfy: ?

Measures

14

CommensurabilityIf stuff has only one relevant property, then quantification requires only onedimension and our vector representation reduces to a scalar.

Measures

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If -with- satisfies the following properties:

Closure

Commutativity

Limitless

Associativity

Commensurability

Measures

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A-with-Bis represented by

(up to isomorphism) a + b

scalar summation

This is *WHY* we add things when the five boxes above can be checked.


If -with- satisfies the following properties:

Measure Theory

Closure

Commutativity

Limitless

Associativity

Commensurability

theorem

Combination and Partition

17

Partition is the inverse operation to combination.Partition decomposes a composite object into components, which may goall the way down to a notional substrate of a-priori equivalent microstates.

Combination and Partition

18

Partition is the inverse operation to combination.Partition decomposes a composite object into components, which may goall the way down to a notional substrate of a-priori equivalent microstates.

Quantification of the divisions obeys the symmetries (Closure, Commutativity,Associativity, Limitless, and Commensurability) of a measure, so that we havea scalar sum rule:

𝑞 𝑋 ⊕ 𝑌 = 𝑞 𝑋 + 𝑞(𝑌)

Steps

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We are interested in source-to-destination steps: 𝒖 = 𝑑𝑒𝑠𝑡 ; 𝑠𝑜𝑢𝑟𝑐𝑒

ClosureSteps, 𝒖 and 𝒗 can be chained (combined) into a composite step 𝒖 ∘ 𝒗 with a from operator ∘when 𝒖 starts where 𝒗 ends.

Changing the how the steps are combined does not change the result.

𝒖⊕ 𝒗 ∘ 𝒘 = 𝒖 ∘ 𝒘 ⊕ (𝒗 ∘ 𝒘)Right-Distributivity

Steps

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We are interested in source-to-destination steps: 𝒖 = 𝑑𝑒𝑠𝑡 ; 𝑠𝑜𝑢𝑟𝑐𝑒

ClosureSteps, 𝒖 and 𝒗 can be chained (combined) into a composite step 𝒖 ∘ 𝒗 with a from operator ∘when 𝒖 starts where 𝒗 ends.

Changing the how the steps are combined does not change the result.

𝒖⊕ 𝒗 ∘ 𝒘 = 𝒖 ∘ 𝒘 ⊕ (𝒗 ∘ 𝒘)Right-Distributivity

Right-Distributivity implies that the quantification of a step is linear in the destination. Thus a step can be quantified by

𝑝 𝑋 ; 𝑍 = 𝑞 𝑋 𝑓(𝑍)

where the only remaining freedom is a scale factor that may depend on the source.

Combining Steps

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Steps must be quantified by

and, for other arguments:

𝑝 𝑋 ; 𝑌 = 𝑞 𝑋 𝑓 𝑌 𝑝 𝑌 ; 𝑍 = 𝑞 𝑌 𝑓(𝑍)

𝑝 𝑋 ; 𝑍 = 𝑞 𝑋 𝑓(𝑍)

The quantification of 𝑝 𝒖 ∘ 𝒗 depends on the quantification of 𝑝 𝒖 and 𝑝 𝒗so that

𝑝 𝒖 ∘ 𝒗 = 𝜙(𝑝 𝒖 , 𝑝 𝒗 )

and

𝑝(𝑋; 𝑍) = 𝜙(𝑝(𝑋 ; 𝑌), 𝑝(𝑌 ; 𝑍))

𝑞 𝑋 𝑓 𝑍 = 𝜙(𝑞 𝑋 𝑓(𝑌), 𝑞 𝑌 𝑓(𝑍))

Combining Steps

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The left-hand side is linear in 𝑞(𝑋) and linear in 𝑓(𝑍), which implies that the right-hand side must also be linear in those terms.

Thus 𝜙 𝑥, 𝑦 = 𝛾𝑥𝑦 for some coefficient 𝛾, which amounts to a choice of units.

Choosing 𝛾 = 1, we have

𝑝 𝒖 ∘ 𝒗 = 𝑝 𝒖 𝑝(𝒗)

𝑝(𝑋; 𝑍) = 𝜙(𝑝(𝑋 ; 𝑌), 𝑝(𝑌 ; 𝑍))

which is the product rule.

Combining Steps

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Letting 𝑋 = 𝑌 and 𝑍 = 𝑌, we have that

𝑝(𝑋; 𝑍) = 𝜙(𝑝(𝑋 ; 𝑌), 𝑝(𝑌 ; 𝑍))

𝑞(𝑌)𝑓 𝑌 = 𝜙(𝑞 𝑌 𝑓(𝑌), 𝑞 𝑌 𝑓(𝑌))

= 𝛾 [𝑞 𝑌 𝑓(𝑌)]2

So that for 𝛾 = 1 we have that 𝑓 𝑌 = 𝑞−1(𝑌) and

𝑝 𝑑𝑒𝑠𝑡 ; 𝑠𝑜𝑢𝑟𝑐𝑒 =𝑞(𝑑𝑒𝑠𝑡)

𝑞(𝑠𝑜𝑢𝑟𝑐𝑒)

24

“Probability is expectation founded upon partial knowledge”- George Boole

Probability

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Inferences combine using -or- (probspeak for -with-)

Inferences chain using -from- (probspeak for reversed -then-)

p(A | C) = p(A | B) p(B | C)

p(A-or-B) = p(A) + p(B) Sum Rule

Product Rule

This is *WHY* we use probability here (and everywhere) when making inferences. Bayes rules!

Closure Commutativity

Limitless

Associativity

Commensurability

Closure Distributivity Associativity

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“This theoretical failure to find a plausible alternative to quantum mechanics, even more than the precise experimental verification of linearity, suggests to me that quantum mechanics is the way it is because any small change in quantum mechanics would lead to logical absurdities.“

- Steven Weinberg

Quantum Theory

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Our job is to make sense of our observations of objects.

We do not observe objects directly. We observe the interaction between the object and a probe.

As a result, this objective of this theory is to quantify interactions.

Pair Postulate

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We adopt the pair postulate, where interactions are quantified by two numbers.

Connection with scalar observation is mediated probabilistically through scalar quantities 𝑞(𝜓) arising from the underlying pairs 𝜓.

Detail is lost, which we may try to recover through inference.

Pair Postulate

29

We adopt the pair postulate, where interactions are quantified by two numbers.

Connection with scalar observation is mediated probabilistically through scalar quantities 𝑞(𝜓) arising from the underlying pairs 𝜓.

Detail is lost, which we may try to recover through inference.

Given a production of objects 𝑋, we partition the objects and quantify the extraction of objects 𝑌 by the pair 𝜓 𝑌 𝑋).

𝑋𝑌

𝑍

𝜓(𝑌|𝑋)

Feynman Rules: Sum Rule

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The Sum Rule for pairs is

𝒖 + 𝒗 =𝑢1 + 𝑣1𝑢2 + 𝑣2

representing

𝜓 𝑋⊕ 𝑌 𝑍) = 𝜓 𝑋 𝑍) + 𝜓 𝑌 𝑍)

𝒖 + 𝒗 𝒖 𝒗

Feynman Rules: Product Rule

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The Product Rule for pairs is

𝒖 ∘ 𝒗 =

𝑗𝑘

𝛾𝑖𝑗𝑘 𝒖𝑗𝒗𝑘

representing

𝜓 𝑋 𝑍) = 𝜓 𝑋 𝑌) ∘ 𝜓 𝑌 𝑍)

𝒖 ∘ 𝒗 𝒖 𝒗

where 𝛾𝑖𝑗𝑘 represents 8 constant coefficients to be determined.


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The Product Rule for pairs is

𝒖 ∘ 𝒗 =

𝑗𝑘

𝛾𝑖𝑗𝑘 𝒖𝑗𝒗𝑘

where 𝛾𝑖𝑗𝑘 represents 8 constant coefficients to be determined.

Their arbitrariness can be reduced by a linear shear (2 x 2 matrix) instead of the rescaling, or regraduation, that is employed for scalars.

The shear removes four degrees of freedom, leaving four remaining.

Chaining is associative, and associativity reduces the bilinear product rule to Three classes that can be sheared into a standard form (Goyal, Knuth, Skilling 2010)(based on the discriminant 𝜇 = −1, 0, +1)


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The three possible Product Rules for pairs are

𝒖 ∘ 𝒗 =𝑢1𝑣1 − 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝒖 ∘ 𝒗 =𝑢1𝑣1

𝑢1𝑣2 + 𝑢2𝑣1

𝒖 ∘ 𝒗 =𝑢1𝑣1 + 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

The other two potentially allowable classes are

𝒖 ∘ 𝒗 =𝑢1𝑣1𝑢2𝑣1



34


𝒖 ∘ 𝒗 =𝑢1𝑣1 − 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝒖 ∘ 𝒗 =𝑢1𝑣1

𝑢1𝑣2 + 𝑢2𝑣1

𝒖 ∘ 𝒗 =𝑢1𝑣1 + 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

The other two potentially allowable classes are



ruled out since in each case one of the pairs acts as a scalar.


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𝒖 ∘ 𝒗 =𝑢1𝑣1 − 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝒖 ∘ 𝒗 =𝑢1𝑣1

𝑢1𝑣2 + 𝑢2𝑣1

𝒖 ∘ 𝒗 =𝑢1𝑣1 + 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝜓 = 𝜓12 + 𝜓2

2 tan−1𝜓2

𝜓1

𝜓1

𝜓2

𝜓1

𝜓12 − 𝜓2

2 tanh−1𝜓2

𝜓1

modulus phase


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By the symmetries

the log-moduli 𝜓 and phases arg(𝜓) for each of the product rule candidatesmust be linearly additive:


log 𝜓 𝑋 𝑍) = log 𝜓 𝑋 𝑌) + log 𝜓 𝑌 𝑍)

arg 𝜓 𝑋 𝑍) = arg(𝜓 𝑋 𝑌)) + arg(𝜓 𝑌 𝑍))


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log 𝜓 𝑋 𝑍) = log 𝜓 𝑋 𝑌) + log 𝜓 𝑌 𝑍)

arg 𝜓 𝑋 𝑍) = arg(𝜓 𝑋 𝑌)) + arg(𝜓 𝑌 𝑍))

The only freedom between conforming linear scales is a constant factor,so there exists a common scale (based on the symmetries) on which we may place our scalar

log 𝑞 = 𝛼 log 𝜓 + 𝛽 arg(𝜓)for some constants 𝛼 and 𝛽.


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Our knowledge about the object through 𝑞 is invariant under phase change with compensating modulus change.

Any prior probabilities we assign to the phase arg(𝜓) must be similarly invariant,that is uniform.

However, for the second and third cases,

𝒖 ∘ 𝒗 =𝑢1𝑣1

𝑢1𝑣2 + 𝑢2𝑣1𝜓1

𝜓2

𝜓1

𝒖 ∘ 𝒗 =𝑢1𝑣1 + 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝜓12 − 𝜓2

2 tanh−1𝜓2

𝜓1


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Our knowledge about the object through 𝑞 is invariant under phase change with compensating modulus change.

Any prior probabilities we assign to the phase arg(𝜓) must be similarly invariant,that is uniform.

However, for the second and third cases,

the phase has infinite range making the priors improper and inference impossible.

𝒖 ∘ 𝒗 =𝑢1𝑣1

𝑢1𝑣2 + 𝑢2𝑣1𝜓1

𝜓2

𝜓1

𝒖 ∘ 𝒗 =𝑢1𝑣1 + 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝜓12 − 𝜓2

2 tanh−1𝜓2

𝜓1


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We are then left with a single possible Product Rule

𝒖 ∘ 𝒗 =𝑢1𝑣1 − 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝒖 ∘ 𝒗 =𝑢1𝑣1

𝑢1𝑣2 + 𝑢2𝑣1

𝒖 ∘ 𝒗 =𝑢1𝑣1 + 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝜓 = 𝜓12 + 𝜓2

2 tan−1𝜓2

𝜓1

𝜓1

𝜓2

𝜓1

𝜓12 − 𝜓2

2 tanh−1𝜓2

𝜓1

modulus phase

infinite rangeimproper prior

infinite rangeimproper prior

Ranges from 0 to 2𝜋proper prior!

Quantum Mechanics: Scalar Measure

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Symmetries force our scalar measure to be linear in the log modulus and phase when chaining

Periodicity in phase implies that 𝛽 = 0, so that

𝑞 = 𝜓 𝛼

Quantum Mechanics: Scalar Measure

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𝑞 = 𝜓 𝛼

𝑋𝑌

𝑍

𝜓(𝑌|𝑋)

Scalar 𝑞 quantifies what we observe, which is the mean rate at which objects ofone type of partition are produced.

rate 𝑞

𝑄 = 𝑞1 + 𝑞2 + …+ 𝑞𝑁

We can consider an ensemble of 𝑁 such experiments

By the Sum Rule the pairs sum

Ψ = 𝜓1 + 𝜓2 + …+ 𝜓𝑁

Quantum Mechanics: Born Rule

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𝑞 = 𝜓 2

𝑄 = 𝑞1 + 𝑞2 + …+ 𝑞𝑁

Ensemble of 𝑁 experiments

The pairs sum

Ψ = 𝜓1 + 𝜓2 + …+ 𝜓𝑁

The phases are randomized, so the mean is zero and variance grows linearly with 𝑁.

Thus quantity 𝑄 and variance 𝑣𝑎𝑟(Ψ) scale together with 𝑁.As a result 𝛼 = 2 and we have the Born Rule

Probability Assignments

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Production of objects is independent, so probabilities are Poisson

Pr 𝑞 𝑑𝑞 = exp −𝑞

𝑟

𝑑𝑞

𝑟with rate 𝑟

The probability distribution for 𝜓 is then Gaussian

Pr 𝜓 𝑑2𝜓 =1

𝜋𝑒𝑥𝑝 −

𝜓 2

𝑟

𝑑2𝜓

𝑟

which is preserved through combination and partition, as expected.

Measure, Probability, Quantum

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𝑝 𝐮 = 𝐮 2𝐮 ∘ 𝐯 =

𝑢1𝑣1 − 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

𝐮 + 𝐯 =𝑢1 + 𝑣1𝑢2 + 𝑣2

Measure Theory

Probability Theory

Quantum Theory : Feynman Rules & Born Rule

A-with-B = 𝑎 + 𝑏

p(A-or-B) = p(A) + p(B) p(A-from-C) = p(A-from-B) p(B-from-C)

theorems

“Quantum Mechanics will cease to look puzzling only when we will be able to derive the formalism of the theory from a set of simple physical assertions about the world.”

- Carlo Rovelli

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“Quantum Mechanics will cease to look puzzling only when we will be able to derive the formalism of the theory from a set of simple physical assertions about the world.”

- Carlo Rovelli

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Closure Commutativity

Reproducibility

Associativity

Commensurability

𝐮 + 𝐯 =𝑢1 + 𝑣1𝑢2 + 𝑣2


𝑝 𝐮 = 𝐮 2𝐮 ∘ 𝐯 =

𝑢1𝑣1 − 𝑢2𝑣2𝑢1𝑣2 + 𝑢2𝑣1

Quantum Mechanics

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We use combination and partition to construct a corresponding pattern of partially-known complex amplitudes and associated probabilities, as dictated by simple symmetries.

We then use standard probabilistic inference to restrict those possibilities in accordance with observations, thereby gaining predictive power over what our detectors will record.

That is what Quantum Mechanics is.

𝑝 𝐮 = 𝐮 2

𝐮 = 𝐯1 + 𝐯2 + 𝐯3 + 𝐯4

𝐮

𝐯1𝐯2𝐯3𝐯4

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THANK YOU

Ariel Caticha, Seth Chaiken, Keith Earle, Anton Garrett, Steve Gull, Oleg Lunin, Andrei Khrennikov, Julio Stern, Federico Holik, FQXi, and Ed Jaynes!

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