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February 20, 2018

CATEGORY:
What's Ultimately Possible in Physics? Essay Contest (2009)
[back]

TOPIC: Possibilistic Physics by Tobias Fritz [refresh]

TOPIC: Possibilistic Physics by Tobias Fritz [refresh]

I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is found that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The essay ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one.

Tobias Fritz is a graduate student at the Max Planck Institute for Mathematics in Bonn, Germany. He works mainly on the foundations of quantum mechanics.

I like this essay. However, I am not at all convinced why giving up probabilities in favor of possibilities will do any good. After all, the very basis of quantitative science (including physics) comprises the natural numbers. Natural numbers are intrinsically and fundamentally probabilistic, which is due to the random nature of the primes. What would be the fundamental counterpart of possibilities? Why giving up something fundamental and specific (probabilities) for something that lacks both a fundamental basis and specificity (your possibilities)?

Anyway--nice job.

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Anyway--nice job.

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Andreas Martin,

thank you for your feedback. You've made a very interesting point.

I agree that most theories of physics are based on the natural numbers. There are a few exceptions, though, for example the causal set approach to quantum gravity. I do not agree, though, that the natural numbers are instrinsically and fundamentally probabilistic. Let me explain why.

First of all, certainly the primes are a very interesting subset of the natural numbers &ndash mathematically. But how are they relevant for fundamental physics?

Then, yes, it seems that the probabilistic Cramér model accurately captures many aspects of the primes. But, as Terry Tao mentions in his structure and randomness in the primes, there are some problems with this model like "most primes are not divisible by 5". If the primes were totally pseudo-randomly probabilistic, exactly 1/5 of them should be divisible by 5. This aspect can indeed better be captured in a possibilistic model. Consider the following possibilities:

So a natural number is considered to be a possible prime if and only if it is not divisible by 2, 3 and 5. This yields the sieve of Eratosthenes.

These are the reasons why I don't believe that the natural numbers are enough justification for the use of probabilities. This, together with the arguments given in the essay, is everything that I can say right now in the favor of possibility. Personally, I don't believe that the concept of possibility is necessarily superior. But I do believe that the concept of probability shouldn't be taken for granted.

thank you for your feedback. You've made a very interesting point.

I agree that most theories of physics are based on the natural numbers. There are a few exceptions, though, for example the causal set approach to quantum gravity. I do not agree, though, that the natural numbers are instrinsically and fundamentally probabilistic. Let me explain why.

First of all, certainly the primes are a very interesting subset of the natural numbers &ndash mathematically. But how are they relevant for fundamental physics?

Then, yes, it seems that the probabilistic Cramér model accurately captures many aspects of the primes. But, as Terry Tao mentions in his structure and randomness in the primes, there are some problems with this model like "most primes are not divisible by 5". If the primes were totally pseudo-randomly probabilistic, exactly 1/5 of them should be divisible by 5. This aspect can indeed better be captured in a possibilistic model. Consider the following possibilities:

So a natural number is considered to be a possible prime if and only if it is not divisible by 2, 3 and 5. This yields the sieve of Eratosthenes.

These are the reasons why I don't believe that the natural numbers are enough justification for the use of probabilities. This, together with the arguments given in the essay, is everything that I can say right now in the favor of possibility. Personally, I don't believe that the concept of possibility is necessarily superior. But I do believe that the concept of probability shouldn't be taken for granted.

Hello dear Tobias Fritz,

Very interesting essay and discussions about numbers .Congratulations.

They are fascinatings these numbers indeed.

I think what these numbers have a correlation ,a pure correlation,with the physicality and their specific fractals .

The serie and its oscillations si specific .

The pairs are relevant .

I think rerally what the primes are finites like a specific number .

For me the same what our quantum entanglement of spheres and our universal spheres with its cosmological spheres .

The number of spheres is specific ,finite .

If our fundamental mathematics are physicals ,the synchronization is possible .

I have always thought that the infinty ,the zero ,the imaginaries are just a human extrapolation ,the physicality ,rational is finite ,

thus in our serie too we must insert the limits .

Of course the naturals ,reals, there ,are a multiplication or addition of primes .

Thus a finite system with primes and inside an infinite systems of naturals ,products ....for the complexification .The closed system is evidently a spherical system with the center ,the volume ....we can insert the thermodynamic ,the evolution too the mass .....

Best Regards

Steve

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Very interesting essay and discussions about numbers .Congratulations.

They are fascinatings these numbers indeed.

I think what these numbers have a correlation ,a pure correlation,with the physicality and their specific fractals .

The serie and its oscillations si specific .

The pairs are relevant .

I think rerally what the primes are finites like a specific number .

For me the same what our quantum entanglement of spheres and our universal spheres with its cosmological spheres .

The number of spheres is specific ,finite .

If our fundamental mathematics are physicals ,the synchronization is possible .

I have always thought that the infinty ,the zero ,the imaginaries are just a human extrapolation ,the physicality ,rational is finite ,

thus in our serie too we must insert the limits .

Of course the naturals ,reals, there ,are a multiplication or addition of primes .

Thus a finite system with primes and inside an infinite systems of naturals ,products ....for the complexification .The closed system is evidently a spherical system with the center ,the volume ....we can insert the thermodynamic ,the evolution too the mass .....

Best Regards

Steve

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The link with the uniqueness and the unity vector is relevant .

The serie in a kind of physical fractal beginning with the number 1 is relevant .if the serie goes....... thus the thermodynamic link about the volume of the sphere is important,the serie increases in the numbers of primes but the volume prortionally decreases in its specific fractal in the two senses,quant. and cosm. 1 2 3 is fascinating it's the ultim begining even correlated with the Big Bang which is for me a kind of multiplication .If I had the volume of the main central sphere and the volume of our Universal sphere and the numbers of cosmological spheres ,it d be better and easier to calculate .

Regards

Steve

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The serie in a kind of physical fractal beginning with the number 1 is relevant .if the serie goes....... thus the thermodynamic link about the volume of the sphere is important,the serie increases in the numbers of primes but the volume prortionally decreases in its specific fractal in the two senses,quant. and cosm. 1 2 3 is fascinating it's the ultim begining even correlated with the Big Bang which is for me a kind of multiplication .If I had the volume of the main central sphere and the volume of our Universal sphere and the numbers of cosmological spheres ,it d be better and easier to calculate .

Regards

Steve

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Here are some comments by Robert Spekkens on my essay, divided into two posts for length reasons:

Dear Tobias,

Thank you for directing me to your paper. I largely agree withyour perspective. I am also of the opinion that it is best *not* toconsider the toy theory as a probabilistic theory. I have often saidthat all that is required for this theory is modal logic, that is,...

view entire post

Dear Tobias,

Thank you for directing me to your paper. I largely agree withyour perspective. I am also of the opinion that it is best *not* toconsider the toy theory as a probabilistic theory. I have often saidthat all that is required for this theory is modal logic, that is,...

view entire post

Third, I don't interpret the probabilities in the toy theory as subjective degrees of belief nor as objective propensities but rather as relative frequencies. My epistemic states describe the relative frequencies of ontic states in an ensemble of similarly preparedsystems, or, equivalently, what an agent knows about the ontic state of a single system drawn from this ensemble. Unlike the subjective...

view entire post

view entire post

Interesting answer from Spekkens. However, I do not see why modal logic automatically implies the binary, possibilistic view. Any model (or actual representation as a Kripke structure) of modal logic is equipped with a truth assignment function over all possible worlds. This function naturally leads to probabilities or, at the least, to beliefs in the sense of Dempster-Shafer theory.

See, e.g., "Conceptual Foundations of Quantum Mechanics:. the Role of Evidence Theory, Quantum Sets, and Modal Logic"

Resconi, Germano; Klir, George J.; Pessa, Eliano

International Journal of Modern Physics C, Volume 10, Issue 01, pp. 29-62 (1999).

Also, my own essay, "The Ultimate Physics of Number Counting", which too starts from a modal logic perspective.

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See, e.g., "Conceptual Foundations of Quantum Mechanics:. the Role of Evidence Theory, Quantum Sets, and Modal Logic"

Resconi, Germano; Klir, George J.; Pessa, Eliano

International Journal of Modern Physics C, Volume 10, Issue 01, pp. 29-62 (1999).

Also, my own essay, "The Ultimate Physics of Number Counting", which too starts from a modal logic perspective.

report post as inappropriate

Dear Tobias,

Let me mention with a grin that John Baetz could probably if possibly have trouble with replacement of his negative probability by negative possibility.

Regards,

Eckard

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Let me mention with a grin that John Baetz could probably if possibly have trouble with replacement of his negative probability by negative possibility.

Regards,

Eckard

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Dear Eckard,

thank you for your comment. Although I'm aware of John Baez' work, I don't know what "his negative probability" is. Could you point me to a reference?

Obviously, probabilities in the sense of asymptotic frequencies are always non-negative. And asymptotic frequencies are what the Born predicts. No negative probabilities here.

best, Tobias

thank you for your comment. Although I'm aware of John Baez' work, I don't know what "his negative probability" is. Could you point me to a reference?

Obviously, probabilities in the sense of asymptotic frequencies are always non-negative. And asymptotic frequencies are what the Born predicts. No negative probabilities here.

best, Tobias

Dear Tobias,

John Baez is a prolific writer. I am sorry. I did not consider it worth to note where I read claimed negative energy and even most nonsensical negative probability, maybe in sci.physics.research, maybe via his home. While I did not find physical items without a natural zero, experts like Baez are claiming that virtually anything can be positive as well as negative.

I understand that e.g. negative pressure is reasonable for instance:

- if we consider an instantaneous value of the alternating component

- if we measure sound pressure re 20 microPascal in dB.

In general, application of formal mathematics needs obedience of common sense if necessary in the aftermath. Joke: Just 3 people are sitting in a room, then 5 of them leave it. Consequently 2 have to come in as to make the room empty.

If you have humor, read the longish lesson in two parts just written to me by Anton Biermans at my 527. I feel sick.

Regards,

Eckard

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John Baez is a prolific writer. I am sorry. I did not consider it worth to note where I read claimed negative energy and even most nonsensical negative probability, maybe in sci.physics.research, maybe via his home. While I did not find physical items without a natural zero, experts like Baez are claiming that virtually anything can be positive as well as negative.

I understand that e.g. negative pressure is reasonable for instance:

- if we consider an instantaneous value of the alternating component

- if we measure sound pressure re 20 microPascal in dB.

In general, application of formal mathematics needs obedience of common sense if necessary in the aftermath. Joke: Just 3 people are sitting in a room, then 5 of them leave it. Consequently 2 have to come in as to make the room empty.

If you have humor, read the longish lesson in two parts just written to me by Anton Biermans at my 527. I feel sick.

Regards,

Eckard

report post as inappropriate

Dear Andreas Martin,

unfortunately I'm having trouble retrieving the paper you mentioned... no online access and not in our library.

But I am currently reading your essay, which seems to contain some intriguing observations, and I will need a little more time to fully grasp it.

So do you mean that probability emerges from a Kripke model of modal logic, just as it emerges in many-worlds interpretations of quantum mechanics? If yes, then that implies that -- in modal logic -- probability is not a fundamental concept, but merely a derived quantity. If this is what you mean when you write "this function naturally leads to probabilities", then we agree: probabilities then are an extremely useful, but non-fundamental, concept.

Also remember that I'm not claiming possibilistic physics to be realistic in any sense; I'm just trying to question established concepts and see what one can do without them.

(By the way, both the long delay in posting Spekkens' email and the missing spaces therein are completely my bad.)

unfortunately I'm having trouble retrieving the paper you mentioned... no online access and not in our library.

But I am currently reading your essay, which seems to contain some intriguing observations, and I will need a little more time to fully grasp it.

So do you mean that probability emerges from a Kripke model of modal logic, just as it emerges in many-worlds interpretations of quantum mechanics? If yes, then that implies that -- in modal logic -- probability is not a fundamental concept, but merely a derived quantity. If this is what you mean when you write "this function naturally leads to probabilities", then we agree: probabilities then are an extremely useful, but non-fundamental, concept.

Also remember that I'm not claiming possibilistic physics to be realistic in any sense; I'm just trying to question established concepts and see what one can do without them.

(By the way, both the long delay in posting Spekkens' email and the missing spaces therein are completely my bad.)

Although it's been a few years since this essay has been written, it seems that there is still some interest in it and it is being read. So I would like to mention a few additional things about the possibilistic Bell inequality of the essay.

It turns out that this possibilistic Bell inequality had been well-known before and is actually due to Lucien Hardy. Nowadays it is known as "Hardy's paradox", and many papers have been written about it.

On the general idea of possibilistic Bell inequalities, there are also notes of a talk "Horn Formulas as Linear Inequalities and Nonlocality Paradoxes" on my webpage,

http://perimeterinstitute.ca/personal/tfritz/2011/ho

rn_formulas.pdf

and a paper that I coauthored,

http://arxiv.org/abs/1105.1819

On a related note, Abramsky and Hardy have subsequently explored "logical" Bell inequalities,

http://arxiv.org/abs/1203.1352

It turns out that this possibilistic Bell inequality had been well-known before and is actually due to Lucien Hardy. Nowadays it is known as "Hardy's paradox", and many papers have been written about it.

On the general idea of possibilistic Bell inequalities, there are also notes of a talk "Horn Formulas as Linear Inequalities and Nonlocality Paradoxes" on my webpage,

http://perimeterinstitute.ca/personal/tfritz/2011/ho

rn_formulas.pdf

and a paper that I coauthored,

http://arxiv.org/abs/1105.1819

On a related note, Abramsky and Hardy have subsequently explored "logical" Bell inequalities,

http://arxiv.org/abs/1203.1352

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