Topos of Quantum Gravity

[Philip Thrift]

Some Remarks on the Logic of Quantum Gravity

Andreas Doering

https://www.academia.edu/5456772/Some_Remarks_on_the_Logic_of_Quantum_Gravity

We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism. We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.

...

Finally, when we commit ourselves to describing the whole universe using structures in a topos, and if we use the internal logic of the topos to assign truth values to propositions etc., we do not have to do all our proofs and mathematical arguments internally in the topos, i.e., constructively. Doing physics necessarily means to separate oneself from the system to be described, even if this system is the whole universe. Since we have to ‘step out’ of the system, we have to argue using the (typically Boolean) metalogic in which we define the mathematical structures, e.g. topoi and state objects, that we use in the mathematical description of the system at hand. It is this Boolean metalogic in which we do physics.

@philipthrift

[Bruno Marchal]

On 21 Sep 2020, at 13:44, Philip Thrift <cloud...@gmail.com> wrote:

Some Remarks on the Logic of Quantum Gravity

Andreas Doering

https://www.academia.edu/5456772/Some_Remarks_on_the_Logic_of_Quantum_Gravity

We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism.

There are realist interpretation of the physical reality in the Hilbert formalism, or at least with his “von Neumann Algebra”. I am not sure the Hilbert formalism is really a problem for realism, but of course we know the price (the multiplication of histories in physics). The apparent non-realism is phenomenological, and is the result of abstracting ourselves from the global picture, which is a web of histories/computations.

We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.

...

Finally, when we commit ourselves to describing the whole universe using structures in a topos, and if we use the internal logic of the topos to assign truth values to propositions etc., we do not have to do all our proofs and mathematical arguments internally in the topos, i.e., constructively. Doing physics necessarily means to separate oneself from the system to be described, even if this system is the whole universe. Since we have to ‘step out’ of the system, we have to argue using the (typically Boolean) metalogic in which we define the mathematical structures, e.g. topoi and state objects, that we use in the mathematical description of the system at hand. It is this Boolean metalogic in which we do physics.

That seems quite reasonable, although it is hard to see if the boolean structure described a first person plural or not. Mechanism assume an arithmetical booleanism (a machine stop or does not stop), and the topos is or should be canonically derived by the intuitionist logic related to the knower ([]p & p).

If this could help to drive gravity from the numbers, why not, but that is premature today. The goal of mechanism was more for the urgent question: to get qualia which makes sense with the quanta derived from arithmetic.

I will remember the link when I have more time. Goldblatt, who is responsible for the relation between the B modal logic and quantum logic, used in the derivation of quantum logic from the mechanist mind-body problem, has written a book on the toposes, also with the plural Topoï (and “Topoï” is the title of Goldblatt’s book). The book is readable for logiciens (!), and it has a good introduction to Category Theory. It is not needed to grasp my contribution though, despite I do use it a little bit in my French more detailed exposition (available in my URL).

Category adds another layer of abstraction and mathematical sophistication which I try to avoid. I work top->bottom, starting from the computationalist theory of consciousness to solve the computationalist “matter” problem. There, all use of any mathematical structure must be shown necessary in Arithmetic (+ Mechanism).

Bruno

@philipthrift

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[Brent Meeker]

On 9/21/2020 4:44 AM, Philip Thrift wrote:

Some Remarks on the Logic of Quantum Gravity

Andreas Doering

https://www.academia.edu/5456772/Some_Remarks_on_the_Logic_of_Quantum_Gravity

We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism. We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.

...

Finally, when we commit ourselves to describing the whole universe using structures in a topos, and if we use the internal logic of the topos to assign truth values to propositions etc., we do not have to do all our proofs and mathematical arguments internally in the topos, i.e., constructively. Doing physics necessarily means to separate oneself from the system to be described, even if this system is the whole universe.

Where does that "necessarily" come from?  Is it a theorem?...from what axioms?

Since we have to ‘step out’ of the system, we have to argue using the (typically Boolean) metalogic in which we define the mathematical structures, e.g. topoi and state objects, that we use in the mathematical description of the system at hand. It is this Boolean metalogic in which we do physics.

Smells like Platonism.

Brent

[Philip Thrift]

On Monday, September 21, 2020 at 12:49:22 PM UTC-5 Brent wrote:

On 9/21/2020 4:44 AM, Philip Thrift wrote:

Some Remarks on the Logic of Quantum Gravity

Andreas Doering

https://www.academia.edu/5456772/Some_Remarks_on_the_Logic_of_Quantum_Gravity

We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism. We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.

...

Finally, when we commit ourselves to describing the whole universe using structures in a topos, and if we use the internal logic of the topos to assign truth values to propositions etc., we do not have to do all our proofs and mathematical arguments internally in the topos, i.e., constructively. Doing physics necessarily means to separate oneself from the system to be described, even if this system is the whole universe.

Where does that "necessarily" come from?  Is it a theorem?...from what axioms?

Since we have to ‘step out’ of the system, we have to argue using the (typically Boolean) metalogic in which we define the mathematical structures, e.g. topoi and state objects, that we use in the mathematical description of the system at hand. It is this Boolean metalogic in which we do physics.

Smells like Platonism.

Brent

Topos (Category) language is an alternative language for physics.

 

There is no God that handed down the language physics must adopt.

But it is  a language closer to programming:

     https://github.com/AlgebraicJulia/Catlab.jl

Review of the Topos Approach to Quantum Theory

https://www.researchgate.net/publication/51913885_Review_of_the_Topos_Approach_to_Quantum_Theory

Topos theory has been suggested as an alternative mathematical structure with which to formulate physical theories. In particular, the topos approach suggests a radical new way of thinking about what a theory of physics is and what its conceptual framework looks like. The motivation of using topos theory to express quantum theory lies in the desire to overcome certain interpretational problems inherent in the standard formulation of the theory. In particular, the topos reformulation of quantum theory overcomes the instrumentalist/Copenhagen interpretation thereby rendering the theory more realist. In the process one ends up with a multivalued/intuitionistic logic rather than a Boolean logic. In this article we shall review some of these developments. 

@philipthift

[Bruno Marchal]

On 21 Sep 2020, at 22:48, Philip Thrift <cloud...@gmail.com> wrote:

On Monday, September 21, 2020 at 12:49:22 PM UTC-5 Brent wrote:

On 9/21/2020 4:44 AM, Philip Thrift wrote:

Some Remarks on the Logic of Quantum Gravity

Andreas Doering

https://www.academia.edu/5456772/Some_Remarks_on_the_Logic_of_Quantum_Gravity

We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism. We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.

...

Finally, when we commit ourselves to describing the whole universe using structures in a topos, and if we use the internal logic of the topos to assign truth values to propositions etc., we do not have to do all our proofs and mathematical arguments internally in the topos, i.e., constructively. Doing physics necessarily means to separate oneself from the system to be described, even if this system is the whole universe.

Where does that "necessarily" come from?  Is it a theorem?...from what axioms?

Since we have to ‘step out’ of the system, we have to argue using the (typically Boolean) metalogic in which we define the mathematical structures, e.g. topoi and state objects, that we use in the mathematical description of the system at hand. It is this Boolean metalogic in which we do physics.

Smells like Platonism.

Brent

Topos (Category) language is an alternative language for physics.

Topos is typically not a language. Indeed it is a mathematical structure which associates to itself its own language. Language is derivative in topos or other rich categories. A language name expression becomes itself an arrow, a “natural object” of some kind, in a category.

Bruno

 

There is no God that handed down the language physics must adopt.

But it is  a language closer to programming:

     https://github.com/AlgebraicJulia/Catlab.jl

Review of the Topos Approach to Quantum Theory

https://www.researchgate.net/publication/51913885_Review_of_the_Topos_Approach_to_Quantum_Theory

Topos theory has been suggested as an alternative mathematical structure with which to formulate physical theories. In particular, the topos approach suggests a radical new way of thinking about what a theory of physics is and what its conceptual framework looks like. The motivation of using topos theory to express quantum theory lies in the desire to overcome certain interpretational problems inherent in the standard formulation of the theory. In particular, the topos reformulation of quantum theory overcomes the instrumentalist/Copenhagen interpretation thereby rendering the theory more realist. In the process one ends up with a multivalued/intuitionistic logic rather than a Boolean logic. In this article we shall review some of these developments. 

@philipthift

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[Lawrence Crowell]

I downloaded Doering’s paper. In scanning this I see a mention of Chris Isham, who started this idea of Topos as a category system of physics. I think in a way this might be a way of looking at dualities, where if they have the same category or categorical topology of sheaves then these are dualities. While this can be elegant mathematics, such as how Grothendieke formulated algebraic geometry as cohomologies as topoi, this may come after the fact. I think honestly that physical ideas are a better way to blaze this trail.

The complex coupling constant τ = θ/2π + 4πi/g^2 is a case where there is a duality between the generator of a group, generally thought of as a Lie algebra, and an observable which technically is in a Jordan algebra. This coupling constant with some element H defines g = exp(-τH), where θ is the vacuum angle and g the standard coupling constant. This angle defines the constant wave function along orbits of gauge transformations. For A → UAU^{-1} - (dU)U^{-1} and a wave function for a field φ.

ψ(A, φ) → ψ(UAU^{-1} - (dU)U^{-1}, Uφ) ≃ e^{iθ} ψ(A, φ).

The angle θ is a winding number for the gauge orbits π_3(G) for the group. These orbits are defined for small gauge transformations by U = e^{iα}

UAU^{-1} - (dU)U^{-1} ≃ A + i([α, A] + dα)

Uφ ≃ φ + iαφ

That defines ψ(A, φ) → (1 + idθ) ψ(A, φ), where dθ is an orbit map.  The angle is a winding number.

For this orbit space for the operator e^{iτH} we then have the associated real valued -4π/g^2. The winding number, say at a nexus of a Penrose diagram, is then associated with a dual form. This duality is equivalent to the Euclideanization of time t → it so -t/ħ = -1/kT. This is a duality (Lie algebraic generator) ↔ (Jordan observable). This then can in principle be formulated according to a topoi.

LC

[Bruno Marchal]

On 22 Sep 2020, at 19:49, Lawrence Crowell <goldenfield...@gmail.com> wrote:

I downloaded Doering’s paper. In scanning this I see a mention of Chris Isham, who started this idea of Topos as a category system of physics.

Which is nice. Isham is also quite open for the “many-world”.

I think in a way this might be a way of looking at dualities, where if they have the same category or categorical topology of sheaves then these are dualities. While this can be elegant mathematics, such as how Grothendieke formulated algebraic geometry as cohomologies as topoi, this may come after the fact. I think honestly that physical ideas are a better way to blaze this trail.

Or philosophy of mind, also.

The complex coupling constant τ = θ/2π + 4πi/g^2 is a case where there is a duality between the generator of a group, generally thought of as a Lie algebra, and an observable which technically is in a Jordan algebra. This coupling constant with some element H defines g = exp(-τH), where θ is the vacuum angle and g the standard coupling constant.

And tau is 2*pi, I guess.

This angle defines the constant wave function along orbits of gauge transformations. For A → UAU^{-1} - (dU)U^{-1} and a wave function for a field φ.

ψ(A, φ) → ψ(UAU^{-1} - (dU)U^{-1}, Uφ) ≃ e^{iθ} ψ(A, φ).

The angle θ is a winding number for the gauge orbits π_3(G) for the group. These orbits are defined for small gauge transformations by U = e^{iα}

UAU^{-1} - (dU)U^{-1} ≃ A + i([α, A] + dα)

Uφ ≃ φ + iαφ

That defines ψ(A, φ) → (1 + idθ) ψ(A, φ), where dθ is an orbit map.  The angle is a winding number.

For this orbit space for the operator e^{iτH} we then have the associated real valued -4π/g^2. The winding number, say at a nexus of a Penrose diagram, is then associated with a dual form. This duality is equivalent to the Euclideanization of time t → it so -t/ħ = -1/kT. This is a duality (Lie algebraic generator) ↔ (Jordan observable). This then can in principle be formulated according to a topoi.

Quite interesting. Still very mysterious, and probably related to the material modes ([]p & <>t) in arithmetic. This should be related to knot theory, and the quantum invariant of knots and braids, for the arithmetical-quantum origin of space, but this led me to the self-distributive algebra until I get stuck in complex questions of set theory and the very large cardinals, like the cardinal of Woodin and Laver… To be continued … in the next millenium, or the one after … (we are so slow…).

Bruno

LC

On Monday, September 21, 2020 at 6:44:11 AM UTC-5 cloud...@gmail.com wrote:

Some Remarks on the Logic of Quantum Gravity

Andreas Doering

https://www.academia.edu/5456772/Some_Remarks_on_the_Logic_of_Quantum_Gravity

We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism. We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.

...

Finally, when we commit ourselves to describing the whole universe using structures in a topos, and if we use the internal logic of the topos to assign truth values to propositions etc., we do not have to do all our proofs and mathematical arguments internally in the topos, i.e., constructively. Doing physics necessarily means to separate oneself from the system to be described, even if this system is the whole universe. Since we have to ‘step out’ of the system, we have to argue using the (typically Boolean) metalogic in which we define the mathematical structures, e.g. topoi and state objects, that we use in the mathematical description of the system at hand. It is this Boolean metalogic in which we do physics.

@philipthrift

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[Philip Thrift]

It is interesting approach to fundamentally replace the underlaying language (logic) of physics.

Topos Quantum Theory

Christopher J. Isham

https://fqxi.org/grants/large/awardees/view/__details/2006/isham

One important feature of topos theory is that a proposition such as "the physical quantity A has a value in a certain range" need not be simply true or false: rather, there are more possibilities that are given by the intrinsic logic that is possessed by a topos.

Attempt to integrate this into a programming language:

Catlab.jl - https://github.com/AlgebraicJulia/Catlab.jl

@philipthrift

[Bruno Marchal]

On 24 Sep 2020, at 13:20, Philip Thrift <cloud...@gmail.com> wrote:

It is interesting approach to fundamentally replace the underlaying language (logic) of physics.

Topos Quantum Theory

Christopher J. Isham

https://fqxi.org/grants/large/awardees/view/__details/2006/isham

One important feature of topos theory is that a proposition such as "the physical quantity A has a value in a certain range" need not be simply true or false: rather, there are more possibilities that are given by the intrinsic logic that is possessed by a topos.

That is all nice as long as you don’t confuse the language with what it is supposed to talk about. Using a non boolean topos is risky in that regards, but does make sense for the universal machine’s theory of qualia, including the quanta. This is not a problem as we have independent reason to put the physical reality in a first person (plural) realm, and the fundamental reality remains an independent Boolean structure (be it arithmetic, or a boolean topos with a "natural number object”).

Bruno

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[Lawrence Crowell]

The τ = θ/2π + 4πi/g^2 contains elements that are both Lie algebraic and Jordan. The Jordan algebra connects with the E8 and the Jordan matrix algebras, in particular J^3(O). These have the property of a multiplication X°Y = ½(XY + YX) , which is how we think of classical variables. For this and reasons of Bott peridocity E8 is purely real valued, We only get complex valued exceptional algebras with SO(32) = E8×E8, where these are a pair with one real and the other with i = √-1 times the real valued elements. In this manner τ = θ/2π + 4πi/g^2 has elements that are generators, such as θ and classical-like or Jordan algebraic such as g. 

As an historical sideline, Pascual Jordan was a brilliant man and he proposed a lot of this with quantum mechanics with Wigner in 1935. Jordan though saw favor in the Nazi party and was involved with the rocket program as Peenamunde with von Baun. Jordan went down in ignominy with the end of the war, and his work has been largely sidelined. He did recant. Teichmuller was similar and fanatically Nazi. He joined the Wehrmacht in 1943 after Stalingrad and found his end in the battle of Kursk. His mathematics though is not forgotten, even if he was a nasty man.

LC

On Thursday, September 24, 2020 at 4:05:43 AM UTC-5 Bruno Marchal wrote: