Objects and arrows

With one day delay, let's continue the discussion about category theory. One way to look at category theory is as a generalization of the notion of equivalence: category theory = equivalence on steroids. 

It is informative to look at the original motivation for category theory and also to look at a problem around 1900. Suppose you go back in time without knowing any modern math except group theory and you are aware of Mobius strip, and Klein bottle. Your task is to try to figure out what else is possible? In other words, classify all two dimensional surfaces. Who can help you on this quest? Well, clothes are two dimensional surfaces made by tailors. How do they make them? By two operations: cutting and stitching. Knowing group theory, you realize cutting and stitching are opposite operations, and they do respect the axioms of a group. This is how homology theory actually came from: associating groups with topological spaces in order to classify them. Now fast forward to 1940s, several homology theories were known and the problem was why the groups involved in them are the same? How do we axiomatize homology theory and how do we know if two homologies are equivalent? The answer lies in the concept of natural transformation which requires the concept of functor, which in turn needs the idea of a category. 

So what is a category? A category consists of objects and morphisms (arrows) such that the morphisms can be composed. Here are some examples:

-examples from math:

• sets and functions
• groups and group homomorphisms
• Hilbert space and operators
• partial order sets and monotone functions
• manifolds and cobordisms

-examples from logic

• propositions and proofs

-example from physics

• physical systems and physical processes

-examples from computer science

• data types and programs

Now a functor maps a category to another category by mapping objects to objects and arrows to arrows in a way that preserves structure. This is how for example in algebraic topology we associate groups to topological spaces. 

A natural transformation is a arrow (morphism) between functors subject to some (natural) conditions.  

Apart of naturality, another key concept is universality which means a unique (up to an isomorphism) solution to problems of constructions. We will encounter that when we will express quantum mechanics in category formalism.

Category theory reveals surprising relationships:

• Cartesian  products of sets are like greater lower bounds of partial order sets
• Proofs in logic are like programs in functional programming

Back to quantum mechanics, unitary evolution preserves information and it should be no surprise that there quantum information can be represented in diagramatic fashion. However this is not the path I am going to take and I will make use of universality in deriving quantum mechanics from a simple principle - composition: a theory T describing two physical systems A and B must described the composite system A+B as well. This is very intuitive principle but in the formalism of category theory it has extremely powerful mathematical consequences, spelling out the complete internal details of the theory T. Quantum mechanics comes out of this in its full detail. 

Please stay tuned...

Monoids: the root of it all

Let's start talking about category theory. We will start from set theory and in the end try to get away from it. The first thing we need to discuss is magma. Basically you have a binary operation on a set and that's all: \(M \times M\rightarrow M\). One problem with magmas is that there is no associativity. Now not all mathematical operations lacking associativity are inherently primitive. Think of Lie algebras: the operation is not associative. However there you have something else: the Jacobi identity. But a pure magma without any additional structure is a rather inert object. The other problem with magmas is the lack of a unit. Add associativity and a unital element and category theory comes alive. 

To link the discussion to physics, nature obeys the structure of a (commutative) monoid: two physical systems can be composed into a larger physical system:

- composition is the binary operation 

- associativity guarantees our ability to reason about physical systems regardless of how we split a physical system into subsystems: quantum mechanics is valid for both an electron or an atom containing an electron

- the unital element is nothingness: composition with nothing leaves the original physical system intact.

In later posts I will show how quantum mechanics is a logical consequence of the commutative monoid above. In other words, quantum mechanics is inescapable and nature is quantum all the way.

Back on monoids, let's fall back on the usual example: composable functions: the image of a function is the domain of the next function. The link with programming is obvious: the output of one computation is plugged in as the input of another computation. As a side note, because of this functional programming is best explained in the language of category theory. When we talk function composition we usually write: \(f \circ g\)  which means \(f(g(x))\). To jump in abstraction and eliminate the nature of the elements considerations, there is an elementary trick to help navigate complex composition chains: call \(\circ\): AFTER like this: \(f~composed~with~g = f\circ g = f~ AFTER~ g\)

Now let's review the usual properties of injectivity and surjectivity:

Injectivity: for any elements \(x, x^{'}\), a function is injective if \(f(x) = f(x^{'}) ~implies~ x=x^{'}\)

Surjectivity: for any \(y\) in the range, there is an \(x\) in the domain such that \(f(x)=y\)

So how can we abstract this away and eliminate the talk about the elements? The corresponding category theory concepts are monic and epic:

Monic: a morphism is monic if for any \(g, h\) \(f\circ g = f\circ h ~implies~g=h\)

Epic: a morphism is epic if for any \(g, h\) \(g\circ f = h\circ f ~implies~g=h\)

Can you prove that if \(f:X\rightarrow Y\) then \(f\) is injective if and only if it is monic and it surjective if and only if it is epic? The proof can be found in many places but it is instructive to try to prove it yourself without looking it up first as this will help you better understand category theory. 

The last point I want to make today is that in category theory we move away from functions into abstract morphisms. The key point of morphisms is that they preserve mathematical structures. As such they can be used to jump between categories of very different nature. This is how category theory is a unifying structure of mathematics where the same patterns of reasoning can be replicated from logic to computer science, to algebraic topology, to quantum mechanics.

To be continued...

A new way to look at mathematics

I want to start today  a series of posts about category theory. This is a vast area of mathematics which unifies logic, computer programming, combinatorics, cohomology, etc, and quantum mechanics into a cohesive paradigm. It also settles the problem of interpretation for quantum mechanics. By its very construction category theory has no need for any realism baggage. The entire mathematics can be expressed not in the language of sets (which are abstractions based on our classical intuition) but in the language of categories free of any considerations about the nature of elements. Regarding physics, the paradigm of category theory is best expressed by a famous Bohr quote:

"It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we say about Nature."

Let me start slow. The usual usage of math is on the practical side to solve problems. How many times did we hear the lazy student complaint: why should we learn this? Math is not about memorization and math is very easy once we absorb its content. Learning math is a journey in mastering abstractions and general ways of reasoning. For example when you learn about Lie groups you can extract a lot of key result by elementary methods simply by studying matrices. However you hit a wall with octonions because they are not associative and do not admit a matrix representation for this very reason. In turn this precludes the proper understanding of exceptional Lie groups.

Or consider a simpler example, topology. A lot of functional analysis can be done using the concept of distance and metric spaces. For example a space in \(R^n\) is compact iff it is closed and bounded. Then the metric spaces are generalized by the concept of topological spaces which are based on the idea of neighborhoods, unions, and intersections. In this case compactness is defined much more abstractly: a space is compact iff any open covering has a finite subcover. 

A similar thing happens in category theory. Patterns of reasoning in various mathematical domains are abstracted away in a formalism which does not care about the nature of the elements. On one end this is harder and to help navigate this in the beginning you hold on particular examples; the typical examples are functions. However at some point you let go of the examples just like in topology you let go the notion of distance. At that point you learn to reason properly in category theory and a lot can be achieved in this way. Then we can make the journey backwards from abstract to concrete. There is a big bonus in this: we have the flexibility to pick the concrete examples we want. And in our case we will pick quantum mechanics. Quantum mechanics is best and most naturally expressed in the language of category theory. Goodbye sets, goodbye classical realism, let the category journey begin. Please stay tuned. 

Surreal Trajectories: the main argument against Bohmian ontology

I was extremely busy for the past two weeks and I simply did not have any time to write the weekly post. But without any more delays, as promised, here is the argument against Bohmian interpretation. The argument comes from a famous paper by Englert, Scully, Süssmann, and Walther: Surrealistic Bohm Trajectories.

For clarity, here are the original paper, the rebuttal, and the response.

The argument is simple: in a double slit experiment with a which way detector present before the slit (which incidentally kills the interference pattern), the Bohmian trajectories do not cross the axis of symmetry. 

However the wavefunction is:

\(\Psi = \psi_>|detect~up\rangle + \psi_<|detect~down\rangle\)

and \(\psi_>\) does not vanish in the bottom part and \(\psi_<\) does not vanish in the top part. As such, the particle can be found in the down section while the particle was detected earlier by the upper which way detector. But this is at odds with Bohmian trajectories which by symmetry considerations do not connect the up with the down.

The conclusion is that Bohmian trajectories do not always have a correspondence in reality. The issue is not whether Bohmian quantum mechanics does or does not make the same prediction as standard quantum mechanics as the rebuttal seems to imply, but the issue is the ontology of Bohmian trajectories. The claimed advantage of Bohmian mechanics is its clarity rooted in realism, but if Bohmian trajectories are at odds with experiments, what is the value of Bohmian interpretation? Remember that in Bohmian interpretation the only thing "real" is the particle trajectory. I could not find a valid answer from the Bohmian community to the surrealistic paper challenge and in my opinion this paper it is a decisive clear cut argument against Bohmian interpretation. 

Book Review: "Making Sense of Quantum Mechanics"

After much delay I had found the time to finish reading Jean Bricmont's "Making Sense of Quantum Mechanics" book.

The book is the best presentation of Bohmian interpretation I have ever read. It masterly combines the philosophical ideas with a bit of math, famous quotes, and some historical perspective. 

After preliminary topics in chapter one, chapter two discusses the first quantum "mystery": superposition, while chapter four discuses the second "mystery": nonlocality.  It was chapter three, a philosophical "intermetzzo" which took me a very loooong time to read and prevented me to write this review much sooner: one one end I could not write this post without reading it, and on the other end I was loosing interest very quickly in it after a couple of pages of historical review. Then Bricmont proceeds into presenting Bohniam mechanics - the heart of the book. 

Let's dig a bit deeper into it. Chapter two is a very well written introduction into why quantum mechanics is counter-intuitive. This is presented in the style of modern quantum foundation undergrad classes. Chapter four main idea is this: to many physicists Bell's result proved the impossibility of non-contextual hidden variables (or local realism) while Bell should be understood in conjunction with EPR: EPR+Bell = nonlocality. But what does nonlocality mean? Is it just higher than expected correlations? Here Bricmont makes a very bold and provocative claim: 

"the conclusion of his [Bell's] argument, combined with the EPR argument is rather that there are nonlocal physical effects (and not just correlations between events) in Nature."

To support this chapter 4.2 discusses "Einstein's boxes" [I had a series of posts discussing why in my opinion they do not represent an argument for nonlocality. What EPR+Bell shows is that the composition of two physical systems into a larger physical system does not respect the rules of classical physics - parabolic composability but new rules - elliptic composability. Nature is not "nonlocal" but "non-parabolic composable"]. 

Onto the main topic, the presentation of Bohmian mechanics is standard and what it is surprising is the degree on which the underdetermination issue is addressed: there are an infinite number of alternative theories (like Nelson's stochastic theory) which are in the same realistic vein and which make the same predictions as Bohmian mechanics. Chapter three discussion is invoked here but I feel the argument is very weak (not even a handwaving).

Then the book talks about alternative approaches to Bohmian mechanics courageously taking (some well deserved some not) shots at alternative interpretations (like GRW, MWI, CH, Qbism), and wraps up with historical topics and sociological arguments.

Now onto what the book covers poorly: surreal trajectories, and quantum field theory in Bohmian mechanics. Surreal trajectories are mentioned in passing in a quote while they are the major objection to the interpretation. As I said before, the very name "surreal trajectory" was a masterful catchy clever title for a paper but it backfired in the long term because it was attaching a stigma to Bohmian mechanics which in turned allowed Bohmian supporters to summarily and unfairly dismiss the argument. I will revisit the argument in next post. The key point of surreal trajectories paper is that the particle is detected where Bohmian mechanics predicts it must not go, and since the only thing "real" in Bohmian mechanics is the position of the particle, it represents a fatal blow to the Bohmian ontology. Currently, to my knowledge, there is no consensus inside the Bohmian community on the proper answer the surreal trajectory paper: some deny it is a problem at all while others acknowledge the problem and propose (faulty) ideas on how to deal with it. This is similar with the situation inside the MWI camp where the big pink elephant in the room there is the notion of probabilities: some in MWI disagree it is an issue while others attempt to solve it (but fail). 

Quantum field theory in Bohmian mechanics is another sore point which is not properly discussed. My take on the topic is that a Bohmian quantum field theory is impossible to be constructed, and I want to be proven wrong by a consistent proposal: show me the money, show me the archive paper where the problem is comprehensibly solved.

Bad points aside, overall I liked the book, I find it stimulating, and I enjoyed very much reading it (except chapter 3 which invariably succeeded putting me to sleep). The book is a must read for any person seriously interested in the foundations of quantum mechanics.

(The nonsense of) Joy Christian Reloaded

I was preparing the first physics posts of the year when I got some comments and a question on Joy Christian on an old blog post. In the words of late Yogi Berra, this is "deja vu all over again". Probably the best description of Joy Christian is given by the Monty Python: The Dead Parrot sketch:

The question I got is the following:

"I would like to understand whether the equations (67) - (75) in Joy Christian’s paper “Local Causality in a Friedmann-Robertson-Walker Spacetime” make any sense at all. I don't understand how the mathematical limes operation are carried out."

The paper which got past the referees by trickery is on the archive: https://arxiv.org/pdf/1405.2355v7.pdf and there you see the full derivation of the main faulty claim. Minus some obfuscation techniques, Eqs. 67-75 are nothing but the one-pager Joy preprint: https://arxiv.org/pdf/1103.1879v1.pdf

The main hand-waving trick in the "derivation" is a conversion inside of a sum of \(\lambda^k\) from a variable into an index which amounts to adding apples to oranges and obtaining the incorrect result (see the bottom of page 8 on my preprint: https://arxiv.org/pdf/1109.0535v3.pdf). 

The mistake happens on the transition from Eq 73 to Eq 74 because the L's belong to two distinct kinds of algebras: let's call them apples and oranges. Ignoring the axb, the troubled sum term is something like this:

\(L(\lambda^1) + L(\lambda^2)+L(\lambda^3)+L(\lambda^4)+L(\lambda^5)+...=\)

apple_1 + apple_2 + orange_3 +apple_4 + orange_5+...

with \(\lambda^1 = +1, \lambda^2\ = +1, \lambda^3 = -1, \lambda^4 = +1, \lambda^5 = -1...\)

and with the transformation rule: "apple = - oranges" when we convert to objects of the same kind (let's pick apples) we get:

\(apple(\lambda^1) + apple(\lambda^2)-apple(\lambda^3)+apple(\lambda^4)-apple(\lambda^5)+...=\)

\(apple(+1) + apple(+1)-apple(-1)+apple(+1)-apple(-1)+...=\)

\(apple+ apple+apple+apple+apple+...=\)

which no longer vanishes.

The preparation for this trick is on Eq. 49 which encodes the two distinct algebras (of apples and oranges) into a common formula.  In my preprint you can double check this by trying out the matrix representations of the two algebras (eqs 53-56).

Hopefully my explanation is clear enough. I know all Joy's mathematical tricks in all of his papers or in his blog debates, but I ran out of energy debunking his nonsense. Kudos to Richard Gill for pursuing this further. I was aware of the "Causality in a Friedmann-Robertson-Walker Spacetime" paper and it was on my to do list to write a rebuttal to it, but the journal withdrew it before I could get to it.

Happy New Year!