Sunday, May 21, 2017

The algebraic structure of quantum and classical mechanics

Let's recap on what we derived so far. We started by considering time as a continous functor and we derived Leibniz identity from it. Then for a particular kind of time evolution which allows a representation as a product we were able to derive two products \(\alpha\) and \(\sigma\) for which we derived the fundamental bipartite relations.

Repeated applications of Leibniz identity resulted in proving \(\alpha\) as a Lie algebra, and \(\sigma\) as a Jordan algebra and an associator identity between them:

\([A,B,C]_{\sigma} + \frac{J^2 \hbar^2}{4}[A,B,C]_{\alpha} = 0\)

where \(J\) is a map between generators and observables encoding Noether's theorem.

Now we can combine the Jordan and Lie algebra as:

\(\star = \sigma\pm \frac{J \hbar}{2}\alpha\)

and it is not hard to show that this product is associative (pick \(\hbar = 2\) for convenience):

\([f,g,h]_{\star} = (f\sigma g \pm J f\alpha g)\star h - f\star(g\sigma h \pm J g\alpha h)=\)
\((f\sigma g)\sigma h \pm J(f\sigma g)\alpha h \pm J(f\alpha g)\sigma h + J^2 (f\alpha g)\alpha h \)
\(−f\sigma (g\sigma h) \mp J f\sigma (g\alpha h) \mp J f\alpha (g\sigma h) − J^2 f\alpha (g\alpha h) =\)
\([f, g, h]_{\sigma} + J^2 [f, g, h]_{\alpha} ±J\{(f\sigma g)\alpha h + (f\alpha g)\sigma h − f\sigma (g\alpha h) − f\alpha (g\sigma h)\} = 0\)

because the first part is zero by associator identity and the second part is zero by applying Leibniz identity. In Hilbert space representation the star product is nothing but the complex number multiplication in ordinary quantum mechanics

Now we can introduce the algebraic structure of quantum (and classical) mechanics:

A composability two-product algebra is a real vector space equipped with two bilinear maps \(\sigma \) and \(\alpha \) such that the following conditions apply:

- \(\alpha \) is a Lie algebra,
- \(\sigma\) is a Jordan algebra,
- \(\alpha\) is a derivation for \(\sigma\) and \(\alpha\),
- \([A, B, C]_{\sigma} + \frac{J^2 \hbar^2}{4} [A, B, C]_{\alpha} = 0\),
where \(J \rightarrow (−J)\) is an involution mapping generators and observables, \(1\alpha A = A\alpha 1 = 0\), \(1\sigma A = A\sigma 1 = A\)

For quantum mechanics \(J^2 = -1\). In the finite dimensional case the composability two-product algebra is enough to fully recover the full formalism of quantum mechanics by using the Artin-Wedderburn theorem.

The same structure applies to classical mechanics with only one change: \(J^2 = 0\).

In classical mechanics case, in phase space, the usual Poisson bracket representation for product \(\alpha\) can be constructively derived from above:
\(f\alpha g = \{f,g\} = f \overset{\leftrightarrow}{\nabla} g = \sum_{i=1}^{n} \frac{\partial f}{\partial q^i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q^i}\)

and the product \(\sigma\) is then the regular function multiplication.

In quantum mechanics case in the Hilbert space representation we have the commutator and the Jordan product:

\(A\alpha B = \frac{i}{\hbar}  (AB − BA)\)
\(A\sigma B = \frac{1}{2} (AB + BA)\)

or in the phase space representation the Moyal and cosine brackets:

\(\alpha = \frac{2}{\hbar}\sin (\frac{\hbar}{2} \overset{\leftrightarrow}{\nabla})\)
\(\sigma = \cos (\frac{\hbar}{2} \overset{\leftrightarrow}{\nabla})\)

where the associative product is the star product.

Monday, May 15, 2017

The Jordan algebra of observables

Last time, from concrete representations of the products \(\alpha\) and \(\sigma\) we derived this identity:

\([A,B,C]_{\sigma} + \frac{i^2 \hbar^2}{4}[A,B,C]_{\alpha} = 0\)

Let's use this in a particular case when \(C = A\sigma A\). What does the left hand side say?

\([A,B,C]_{\sigma} = (A\sigma B) \sigma (A\sigma A)) - A\sigma (B \sigma (A \sigma A))\) 

which if we drop \(\sigma\) for convenience sake reads:

\((AB)(AA) - A(B(AA))\)

If the right hand side is zero then we get the Jordan identity:

\((xy)(xx) = x(y(xx))\) where \(xy = yx\)

Now let's compute the right hand side and show it is indeed zero:

\([A,B,A\sigma A]_{\alpha} =  (A\alpha B) \alpha (A\sigma A)) - A\alpha (B \alpha (A \sigma A))\)

Using Leibniz identity in the second term we get:

\((A\alpha B) \alpha (A\sigma A)) - (A\alpha B) \alpha (A\sigma A) - B \alpha (A\alpha (A\sigma A))) = - B \alpha (A\alpha (A\sigma A))\)

But \(A\alpha (A\sigma A) = 0 \) because

\(A\alpha (A\sigma A) = (A\alpha A) \sigma A + A\sigma (A\alpha A) \)

and \(A\alpha A = -A\alpha A = 0\) by skew symmetry.

Therefore due to the associator identity, the product \(\sigma\) is a Jordan algebra. Now we need to arrive at the associator identity using only the ingredients derived so far. This is tedious but it can be done using only Jacobi and Leibniz identity. Grgin and Petersen derived it in 1976 and you can see the proof here

The associator identity is better written as:

\([A,B,C]_{\sigma} + \frac{J^2 \hbar^2}{4}[A,B,C]_{\alpha} = 0\)

where \(J\) is a map from the the product \(\alpha\) to the product \(\sigma\). The existence of this map is equivalent with Noether's theorem. It just happens that in quantum mechanics case \(J^2 = -1\) and the imaginary unit maps anti-Hermitean generators to Hermitean observables. 

In classical physics case, \(J^2 = 0\) and this means that the product \(\sigma\) is associative (in fact it is the ordinary function multiplication) and the product \(\alpha\) can be proven to be the Poisson bracket, but that is a topic for another day as we will continue to derive the mathematical structure of quantum mechanics. Please stay tuned.  

Sunday, May 7, 2017

Lie, Jordan algebras and the associator identity

Before I continue the quantum mechanics algebraic series, I want to first state my happiness for the defeat of the far (alt)-right candidate in France despite Putin's financial and hacking support. Europe has much better antibodies against the scums like Trump than US. In US the diseases caused by the inoculation of hate perpetuated over many years by Fox News has to run its course before things will get better.

Back to physics, first I will show that the product \(\alpha\) is indeed a Lie algebra. This is utterly trivial because we need to show antisymmetry and the Jacobi identity:

\(a\alpha b = -b\alpha a\)
\(a\alpha (b\alpha c) + c\alpha (a\alpha b) + b\alpha (c\alpha a) = 0\)

We already know that  the product \(\alpha\) is antisymmetric and we know that the it obeys Leibniz identity:

\(a\alpha (b\circ c) =  (a\alpha b) \circ c + b\circ (a\alpha c) \)

where \(\circ\) can stand for either \(\alpha\) or \(\sigma\). When \(\circ = \alpha\) we get:

\(a\alpha (b\alpha c) =  (a\alpha b) \alpha c + b\alpha (a\alpha c) \)

which by antisymmetry becomes

\(a\alpha (b\alpha c) = - c \alpha (a\alpha b) - b\alpha (c\alpha a) \)

In other words, the Jacobi identity.

Therefore the product \(\alpha\) is in fact a Lie algebra. Now we want to prove that the product \(\sigma\) is a Jordan algebra.

This is not as simple as proving the Lie algebra, and we will do it with the help of a new concept: the associator. Let us first define it. The associator of an arbitrary product \(\circ\) is defined as follows:

\([a,b,c]_{\circ} = (a\circ b)\circ c - a\circ (b\circ c)\)

as such it measures the lack of associativity. 

It is helpful now to look at the concrete realizations of the products \(\alpha\) and \(\sigma\) in quantum mechanics to know where we want to arrive. In quantum mechanics the product alpha is the commutator, and the product sigma is the anticommutator:

\(A \alpha B = \frac{i}{\hbar}[A,B] = \frac{i}{\hbar}(AB - BA)\)
\(A\sigma B = \frac{1}{2}\{A, B\} = \frac{1}{2}(AB+BA)\)

Let's compute alpha and sigma associators:

\([A,B,C]_{\alpha} = \frac{-1}{\hbar^2}([AB-BA, C] - [A, BC-CB]) = \)
\(=\frac{-1}{\hbar^2}(ABC-BAC-CAB+CBA - ABC+ACB+BCA-CBA)\)
\(= \frac{-1}{\hbar^2}(-BAC-CAB +ACB+BCA)\)

\([A,B,C]_{\sigma} = \frac{1}{4}(\{AB+BA, C\} - \{A, BC+CB\}) = \)
\(=\frac{1}{4}(ABC+BAC+CAB+CBA - ABC-ACB-BCA-CBA) = \)
\(=\frac{1}{4}(BAC+CAB -ACB-BCA)  \)

and so we have the remarkable relationship:

\([A,B,C]_{\sigma} + \frac{i^2 \hbar^2}{4}[A,B,C]_{\alpha} = 0\)

What is remarkable about this is that the Jordan and Lie algebras lack associativity in precisely the same way and because of this they can be later combined into a single operation. The identity above also holds the key for proving the Jordan identity.

Next time I'll show how to derive the identity above using only the ingredients we proved so far and then I'll show how Jordan identity arises out of it. Please stay tuned.

Sunday, April 30, 2017

The origin of the symmetries of the quantum products

Quantum mechanics has three quantum products: 
  • the Jordan product of observables
  • the commutator product used for time evolution
  • the complex number multiplication of operators 
The last product is a composite construction of the first two and it is enough to study the Jordan product and the commutator. In the prior posts notation, the Jordan product is called \(\sigma\), and the commutator is called \(\alpha\). We will derive their full properties using category theory arguments and the Leibniz identity. Bur before doing this, I want to review a bit the two products. The commutator is well known and I will not spend time on it. Instead I will give the motovation for the Jordan product. 

In quantum mechanics the observables are represented as self-adjoint operators: \(O = O^{\dagger}\) If we want to create another self-adjoint operator out of two self-adjoint operators A and B, the simple multiplication won't work because \((AB)^{\dagger} = B^{\dagger} A^{\dagger} = BA \ne AB\). The solution is to have a symmetrized product: \(A\sigma B = (AB+BA)/2\). A lot of the quantum mechanics formalism transfers to the Jordan algebra of observables, but this is a relatively forgotten approach because it is rather cumbersome (the Jordan product is not associative but power associative) and (as it is expected) it does not produce any different predictions than the standard formalism based on complex numbers.

Now back to obtaining the symmetry properties of the Jordan product \(\sigma\) and commutator \(\alpha\), at first we cannot say anything about the symmetry of the product \(\sigma\). However we do know that the product \(\alpha\) obeys the Leibniz identity. We have already use it to derive the fundamental composition relationships, so what else can we do? We can apply it to a bipartite system:

\(f_{12}\alpha_{12}(g_{12}\alpha_{12}h_{12}) = g_{12}\alpha_{12}(f_{12}\alpha_{12}h_{12}) + (f_{12}\alpha_{12}g_{12})\alpha_{12} h_{12}\)


\(\alpha_{12} = \alpha\otimes \sigma + \sigma\otimes\alpha\)

Now the key observation is that in the right hand side, \(f\) and \(g\) appear in reverse order. Remember that the functions involved in the relationship above are free of constraints, by judicious picks of their value can lead to great simplifications because \(1 \alpha f = f\alpha 1 = 0\). The computation is tedious and I will skip it, but what you get in the end is this:

\(f_1\alpha h_1 \otimes [f_2 \alpha g_2 + g_2 \alpha f_2 ] = 0\)

which means that the product alpha is anti-symmetric \(f\alpha g = -g\alpha f\)

If we use this property in the fundamental bypartite relationship we obtain in turn that the product sigma is symmetric: \(f\sigma g = g\sigma f\)

Next time we will prove that \(\alpha\) is a Lie algebra and that \(\sigma\) is a Jordan algebra. Please stay tuned.

Sunday, April 23, 2017

Politics and a bit on the symmetry properties of the commutator and the Jordan products

This week I am thorn between a physics and politics. On one hand I have the scheduled physics topics to talk about, and on the other hand there are very juicy political topics. So let me start with some political commentary which I will attempt to keep at the minimum.

If you do not live in US, it is hard to understand the amount of political pressure on science which comes from the right. GOP is at war with science because of three factors:  the political elites are corrupt and depend on lobby money from corporations who make more money when they wreck the environment, the religious right is at war with evolution, and lastly the inbred rednecks swallow hook, line, and sinker the toxic sludge of propaganda of Fox News in the name of "freedom". 

So it was a breath of fresh air the recent march for science in which people sick and tired of the GOP war on science took a stand for the facts that 2+2 is still 4, humans cause global warming, and Earth is older than 5000 years. And then I opened my email and I see an alert about a new post by Lubos Motl defending Bill O'Reilly. I normally delete those notifications and I don't really know how I am subscribed to them because I only get about one a week - it is strangely inconsistent. So I said to myself: how fitting. The (naked) emperor of physics who wanted once to reclassify an archive paper to the general physics section after it was published in PRL, the climate change denier and the open apologist of the murderer Putin, con-man Trump, and white trash Sarah Palin throws his support behind the another toxic sorry propagandist like himself. Would have been too much to expect him to defend science instead? Out of curiosity I followed the link to see the pro O'Reilly rant, and I was not disappointed: it was choke full of imbecilic nonsense as I expected. But then I saw the icing on the cake: I see in the history list that Lubos did write a rant against the march for science too calling it misguided and unethical. Wow! Now in France (like in US or UK) there is no shortage of stupidity which just propelled Marine LePen into the final for presidency. The global village idiots will flock to her side and I have no doubt Lubos will support her too.

OK, the political topics took too much and I want to continue with the series topic on quantum mechanics reconstruction. Let me just say what the products \(\alpha\) and \(\sigma\) will turn out to be. In the classical mechanics case it can be constructively proven that \(\alpha\) is the Poisson bracket while in the quantum case, \(\alpha\) is the commutator. The other product \(\sigma\) is the regular function multiplication in classical mechanics and the Jordan product (the anti-commutator) in quantum mechanics. 

Now the Poisson bracket and the commutators are anti-symmetric: \(f\alpha g = - g\alpha f\), and the regular function multiplication and the Jordan product are symmetric products: \(f\sigma g = g\sigma f\). The symmetry properties are preserved under system composition as we can see from the fundamental relationships:

\(\Delta (\alpha) = \alpha \otimes \sigma + \sigma \otimes \alpha \)
\(\Delta (\sigma) = \sigma \otimes \sigma - \alpha \otimes \alpha\)

because S*S = S, S*A = A, A*A = S

Incidentally, this observation opens up another way into quantum mechanics reconstruction (from the operational point of view) but I will not talk about it in this series. Instead next time I will show how to prove the fact that the product \(\alpha\) is anti-symmetric. Again Leibniz identity will come to the rescue. Then using the fundamental relationship we must have that the product \(\sigma\) is symmetric.  Eventually all their mathematical properties will be obtained. Please stay tuned. 

Sunday, April 16, 2017

The fundamental bipartite relations

Continuing from where we left off last time, we introduced the most general composite products for a bipartite system:

\(\alpha_{12} = a_{11}\alpha \otimes \alpha + a_{12} \alpha\otimes\sigma + a_{21} \sigma\otimes \alpha + a_{22} \sigma\otimes\sigma\)
\(\sigma_{12} = b_{11}\alpha \otimes \alpha + b_{12} \alpha\otimes\sigma + b_{21} \sigma\otimes \alpha + b_{22} \sigma\otimes\sigma\)

The question now becomes: are the \(a\)'s and \(b\)'s parameters free, or can we say something abut them? To start let's normalize the products \(\sigma\) like this:

\(f\sigma I = I\sigma f = f\)

which can always be done. Now in:

\((f_1 \otimes f_2)\alpha_{12}(g_1\otimes g_2) = \)
\(=a_{11}(f_1 \alpha g_1)\otimes  (f_2 \alpha g_2) + a_{12}(f_1 \alpha g_1) \otimes (f_2 \sigma g_2 ) +\)
\(+a_{21}(f_1 \sigma g_1)\otimes  (f_2 \alpha g_2) + a_{22}(f_1 \sigma g_1) \otimes (f_2 \sigma g_2 )\)

if we pick \(f_1 = g_1 = I\) :

\((I \otimes f_2)\alpha_{12}(I\otimes g_2) = \)
\(=a_{11}(I \alpha I)\otimes  (f_2 \alpha g_2) + a_{12}(I \alpha I) \otimes (f_2 \sigma g_2 ) +\)
\(+a_{21}(I \sigma I)\otimes  (f_2 \alpha g_2) + a_{22}(I \sigma I) \otimes (f_2 \sigma g_2 )\)

and recalling from last time that \(I\alpha I = 0\) from Leibniz identity we get:

\(f_2 \alpha g_2 = a_{21} (f_2 \alpha g_2 ) + a_{22} (f_2 \sigma g_2)\)

which demands \(a_{21} = 1\) and \(a_{22} = 0\).

If we make the same substitution into:

 \((f_1 \otimes f_2)\sigma_{12}(g_1\otimes g_2) = \)
\(=b_{11}(f_1 \alpha g_1)\otimes  (f_2 \alpha g_2) + b_{12}(f_1 \alpha g_1) \otimes (f_2 \sigma g_2 ) +\)
\(+b_{21}(f_1 \sigma g_1)\otimes  (f_2 \alpha g_2) + b_{22}(f_1 \sigma g_1) \otimes (f_2 \sigma g_2 )\)

we get:

\(f_2 \sigma g_2 = b_{21} (f_2 \alpha g_2 ) + b_{22} (f_2 \sigma g_2)\)

which demands \(b_{21} = 0\) and \(b_{22} = 1\)

We can play the same game with \(f_2 = g_2 = I\) and (skipping the trivial details) we get two additional conditions: \(a_{12} = 1\) and \(b_{12} = 0\).

In coproduct notation what we get so far is:

\(\Delta (\alpha) = \alpha \otimes \sigma + \sigma \otimes \alpha + a_{11} \alpha \otimes \alpha\)
\(\Delta (\sigma) = \sigma \otimes \sigma + b_{11} \alpha \otimes \alpha\)

By applying Leibniz identity on a bipartite system, one can show after some tedious computations that \(a_{11} = 0\). The only remaining free parameters is \(b_{11}\) which can be normalized to be ether -1, 0, or 1 (or elliptic, parabolic, and hyperbolic). Each choice corresponds to a potential theory of nature. For example 0 corresponds to classical mechanics, and -1 to quantum mechanics.

Elliptic composability is quantum mechanics! The bipartite products obey:

\(\Delta (\alpha) = \alpha \otimes \sigma + \sigma \otimes \alpha \)
\(\Delta (\sigma) = \sigma \otimes \sigma - \alpha \otimes \alpha\)

Please notice the similarity with complex number multiplication. This is why complex numbers play a central role in quantum mechanics.

Now at the moment the two products do not respect any other properties. But we can continue this line of argument and prove their symmetry/anti-symmetry. And from there we can derive their complete properties arriving constructively at the standard formulation of quantum mechanics. Please stay tuned.

Sunday, April 9, 2017

Time evolution for a composite system

Continuing where we left off last time, let me first point out one thing which I glossed over too fast: the representation of \(D\) as a product \(\alpha\): \(Dg = f\alpha g\). This is highly nontrivial and not all time evolutions respect it. In fact, the statement above is nothing but a reformulation of Noether's theorem in the Hamiltonian formalism. I did not build up the proper mathematical machinery to easily show this, so take my word on it for now. I might revisit this at a later time.

Now what I want to do is explore what happens to the product \(\alpha\) when we consider two physical systems 1 and 2. First, let's introduce the unit element of our category, and let's call it "I":

\(f\otimes I = I\otimes f = f\)

for all \(f \in C\)

Then we have \((f_1\otimes I) \alpha_{12} (g_1\otimes I) = f \alpha g\)

On the other hand suppose in nature there exists only the product \(\alpha\). Then the only way we can construct a composite product \(\alpha_{12}\) out of \(\alpha_1\) and \(\alpha_2\) is:

\((f_1\otimes f_2) \alpha_{12} (g_1 \otimes g_2) = a(f_1 \alpha_1 g_1)\otimes (f_2\alpha_2 g_2)\)

where \(a\) is a constant. 

Now if we pick \(f_2 = g_2 = I\) we get:

\((f_1\otimes I) \alpha_{12} (g_1 \otimes I) = a(f_1 \alpha_1 g_1)\otimes (I \alpha_2 I)  \)
which is the same as \(f \alpha g\) by above. 

But what is \(I\alpha I\)? Here we use the Leibniz identity and prove it is equal with zero:

\(I \alpha (I\alpha A) = (I \alpha I) \alpha A + I \alpha (I \alpha A)\)

for all \(A\) and hence \(I\alpha I = 0\)

But this means that a single product alpha by itself is not enough! Therefore we need a second product \(\sigma\)! Alpha will turn out to be the commutator, and sigma the Jordan product of observables, but we will derive this in a constructive fashion.

Now that we have two products in our theory of nature, let's see how can we build the composite products out of individual systems. Basically we try all possible combinations:

\(\alpha_{12} = a_{11}\alpha \otimes \alpha + a_{12} \alpha\otimes\sigma + a_{21} \sigma\otimes \alpha + a_{22} \sigma\otimes\sigma\)
\(\sigma_{12} = b_{11}\alpha \otimes \alpha + b_{12} \alpha\otimes\sigma + b_{21} \sigma\otimes \alpha + b_{22} \sigma\otimes\sigma\)

which is shorthand for (I am spelling out only the first case):

\((f_1 \otimes f_2)\alpha_{12}(g_1\otimes g_2) = \)
\(=a_{11}(f_1 \alpha g_1)\otimes  (f_2 \alpha g_2) + a_{12}(f_1 \alpha g_1) \otimes (f_2 \sigma g_2 ) +\)
\(+a_{21}(f_1 \sigma g_1)\otimes  (f_2 \alpha g_2) + a_{22}(f_1 \sigma g_1) \otimes (f_2 \sigma g_2 )\)

For the mathematically inclined reader we have constructed what it is called a coalgebra where the operation is called a coproduct: \(\Delta : C \rightarrow C\otimes C\). In category theory a coproduct is obtained from a product by reversing the arrows.

Now the task is to see if we can say something about the coproduct parameters: \(a_{11},..., b_{22}\). In general nothing can constrain their values, but in our case we do have an additional relation: Leibniz identity which arises out the functoriality of time evolution. This will be enough to fully determine the products \(\alpha\) and \(\sigma\), and from them the formalism of quantum mechanics. Please stay tuned.

Sunday, March 26, 2017

Time as a continous functor

To recall from prior posts, a functor maps objects to objects and arrows to arrows between two categories. In other words, it is structure preserving. In the case of a monoidal category, suppose there is an arrow * from \(C\times C \rightarrow C\). Then a functor T makes the diagram below commute:

This is all fancy abstract math which has a simple physical interpretation when T corresponds to time evolution: the laws of physics do not change in time. Moreover it can be shown with a bit of effort and knowledge of C* algebras that Time as a functor = unitarity.

But what can we derive from the commutative diagram above? With the additional help of two more very simple and natural ingredients we will be able to reconstruct the complete formalism of quantum mechanics!!! Today I will introduce the first one: time is a continuous parameter. Just like in group theory adding continuity results in the theory of Lie groups we will consider continous functors and we will investigate what happens in the neighborhood of the identity element.

In the limit of time evolution going to zero T becomes the identity. For infinitesimal time evolution we can then write:

\(T = I + \epsilon D\)

We plug this back into the diagram commutativity condition \(T(A)*T(B) = T(A*B)\) and we obtain in first order the chain rule of differentiation:

\(D(A*B) = D(A)*B + A*D(B)\)

There is not a single kind of time evolution and \(D\) is not unique (think of various hamiltonians). There is a natural transformation between different time evolution functors  and we can express D as an operation like this: \(D_A = A\alpha\) where \((\cdot \alpha \cdot)\) is a product.

\(\alpha : C\times C \rightarrow C\)

Then we obtain the Leibniz identity:

\(A\alpha (B * C) = (A\alpha B) * C + B * (A \alpha C)\)

This is extremely powerful, as it is unitarity in disguise.  Next time we'll use the tensor product and the second ingredient to obtain many more mathematical consequences. Please stay tuned.

Sunday, March 19, 2017

Monoidal categories and the tensor product

Last time we discussed the category theory product which forms another category from two categories. Suppose now that we start with one category \(C\) and form the product with itself \(C\times C\). It is natural to see if there is a functor from \(C\times C\) to \(C\). If such a functor exists and moreover it respects associativity and unit elements, then the category \(C\) is called a monoidal category. By abuse of notation, the functor above is called the tensor product, but this is not the usual tensor product of vector space. The tensor product of vector space is only one concrete example of a monoidal product. To get to the ordinary tensor product we need to inject physics into the problem. 

The category \(C\) we are interested in is that of physical systems where the objects are physical systems, and arrows are compositions of physical systems. The key physical concepts needed are that of time and dynamical degree of freedom inside Hamiltonian formalism.

Time plays an distinguished role in quantum mechanics both in terms of formalism (remember that there is no time operator) and in how quantum mechanics can be reconstructed. 

The space in Hamiltonian formalism is a Poisson manifold which is not necessarily a vector space but because the Hilbert space \(L^2 (R^3\times R^3)\) is isomorphic to \(L^2 (R^3 ) \otimes L^2 (R^3 )\) let's discuss monoidal categories for vector spaces obeying an equivalence relationship. Hilbert spaces form a category of their own and there is a functor mapping physical systems into Hilbert spaces. This is usually presented as the first quantum mechanics postulate: each physical system is associated with a complex Hilbert space H.

For complete generality of the definition of the tensor product we consider two distinct vector space V and W for which we first consider the category theory product (in this case the Cartesian product) but for which we make the following identifications:
  • \((v_1, w)+(v_2, w) = (v_1 + v_2, w)\)
  • \((v, w_1)+(v, w_2) = (v, w_1 + w_2)\)
  • \(c(v,w) = (cv, w) = (v, cw)\)
For physical justification think of V and W as one dimensional vector spaces corresponding to distinct dynamical degrees of freedom. Linearity is a property of vector spaces and we expect this property to be preserved if vector spaces are to describe nature. Bilinearity in the equivalence relationship above arises because the degrees of freedom are independent.

Now a Cartesian product of vector spaces respecting the above relationships is a new mathematical object: a tensor product.

The tensor product is unique up to isomorphism and respects the following universal property:

There is a bilinear map \(\phi : V\times W \rightarrow V\otimes W\) such that given any other vector space Z and a bilinear map \(h: V\times W \rightarrow Z\) there is a unique linear map \(h^{'}: V\otimes W \rightarrow Z\) such that the diagram below commutes.

This universal property is very strong and several mathematical facts follows from it: the tensor product is unique up to isomorphism (instead of Z consider another tensor product \(V\otimes^{'}W\) ), the tensor product is associative, and there is a natural isomorphism between  \(V\otimes W\) and \(W\otimes V\) making the tensor product an example of a symmetric monoidal category, just like the category of physical systems under composition.

This may look like an insignificant trivial observation, but it is extremely powerful and it is the starting point of quantum mechanics reconstruction. On one hand we have composition of physical systems and theories of nature describing physical systems. On the other hand we have dynamical degrees of freedom and the rules of quantum mechanics. The two things are actually identical and each one can be derived from the other. To do this we need one additional ingredient: time viewed as a functor. Please stay tuned.

Monday, March 13, 2017

Category Theory Product

Before we discuss this week's topic, I want to make two remarks from the prior posts content. First, why we need natural transformations in algebraic topology? Associating groups to topological spaces (which incidentally describe the hole structure of the space) is done by the use of functors. Different (co)homology theories are basically different functors, and their equivalence is the same as proving the existence of a natural transformation. Second, the logic used in category theory is intuitionistic logic where truth is proved constructively. Since this is mapped into computer science by the Curry-Howard isomorphism, the fact that some statements have no constructive proof is equivalent with a computation running forever. In computation theory one encounters the halting problem. If the halting problem were decidable then category theory would have been mapped to ordinary logic instead of intuitionistic logic.

Now back to the topic of the day. We are still in pure math domain and we are looking at mathematical objects from 10,000 feet disregarding their nature and observing only their existence and their relationships (objects and arrows). The first question one asks is how to construct new categories from existing ones? One way is to simply reverse the direction of all arrows and the resulting category is unsurprisingly called the opposite category (or the dual). Another way is to combine two category into a new one. Enter the concept of a product of two categories: \(\mathbf{C}\times \mathbf{D}\). In set theory this would correspond with to the Cartesian product of two sets. However we need to give a definition which is independent of the nature of the elements. Moreover we want to give it in a way which guarantees uniqueness up to isomorphism. 

The basic idea is that of a projection from the elements of \(\mathbf{C}\times \mathbf{D}\) back to the elements of \(\mathbf{C}\) and \(\mathbf{D}\). So how do we know that those projections and the product is unique up to isomorphism? Suppose that there is another category \(\mathbf{Y}\) with maps \(f_C\) and \(f_D\). Then there is a unique map \(f\) such that the diagram below commutes

This diagram has to commute for all categories \(\mathbf{Y}\) and their maps \(f_C\) and \(f_D\). From this definition, can you prove uniqueness of the product up to isomorphism? It is a simple matter of "diagram reasoning". Just pretend that Y is now the "true incarnation" of the product. You need to find a morphisms f from Y to CxD and a morphism g from CxD to Y such that \(g\circ f =1_{C\times D}\), \(g\circ f = 1_Y\). See? Category theory is really easy and not harder than linear algebra.

Now what happens if we flip all arrows in the diagram above? We obtain a coproduct category \(\mathbf{C}\oplus \mathbf{D}\) and the projections maps become injection maps. 

OK, time for concrete examples:

  • sets: product = Cartesian product, coproduct = disjoint union
  • partial order sets: product = greatest lower bounds (meets), coproduct = least upper bounds (joins)
So where are we now? The concept of the product is very simple, but we need it as a stepping stone to the concept of tensor product and (symmetric) monoidal category. Why? Because physical systems form a symmetric monoidal category. Using categorical arguments we can derive the complete mathematical properties of any theory of nature describing a symmetric monoidal category. And the answer will turn out to be: quantum mechanics. Please stay tuned.

Saturday, March 4, 2017

The Curry–Howard isomorphism

Category theory may seem vary abstract and intimidating, but in fact it is extremely easy to understand. In category theory we look at concrete objects from far away without any regard for the internal structure. This is similar with Bohr's position on physics: physics is about what we can say about nature, and not decide what nature is. Surprisingly, a lot of information about the objects in category theory is derivable from the behavior of the objects and this is where I am ultimately heading with this series on category theory.

Last time I mentioned the origin of category theory as the formalism to clarify when two homology theories are equivalent. But category theory can be started from two other directions as well, and those alternative viewpoints help provide the intuition needed to navigate the abstractions of category theory. One thread of discussion starts with the idea of computability and the work of Alonzo Church and Alan Turing. Turing was Church's student and each started an essential line of research: lambda calculus and universal Turing machines.  Those later grew into one hand functional languages like Java script, and the other hand into object oriented languages like C++. What one can do with lambda calculus can be achieved with universal Turing machines, and the other way around. The essential idea of computer programming is to build complex structures out of simpler building blocks. Object oriented programming starts from the idea of packaging together actions and states. An object is a "black box" containing actions (functions performing computations) and information (the internal state of the object). Functional programming on the other hand lacks the concept of an internal state and you deal only with functions which take an input, crunch the numbers, and then produce an output. The typical example is FORTRAN: FORmula TRANslation (from higher level human understandable syntax into zeroes and ones understandable by a machine).

The second direction one can start category theory is intuitionistic logic and the foundation of set theory. The problem of naive set theory is that one can create paradoxes like Russel's paradox: the set of all sets which are not members of themselves. The solution Russel proposed was type theory. Types introduce structure to set theory preventing self-referential constructions. In computer programming, types are semantic rules which tell us how to understand various sequences of zeros and ones from computer memory as integers, boolean variables, etc.

In intuitionistic logic statements are not true by simply disproving their falsehood, but they are true by providing an explicit construction. Truth must be computed and the parallel with computer programming is obvious. There is a name for this relationship, the Curry-Howard isomorphism. The mathematical formalism needed to rigorously spell out this correspondence is category theory. At high level:
  • proofs are programs
  • propositions are types
More important is that we can attach logical and programming meaning to category theory constructions which helps dramatically reduce the difficulty of category theory to that of elementary linear algebra. 

There are two additional key points I want to make. First category theory ignores the structure of the objects: they can be sets, topological spaces, posets, even physical systems. As such uniqueness is relaxed in category theory and things are unique up to isomorphisms. Second, we are strengthening uniqueness by seeking universal propertiesThis gives category theory its abstract flavour: the generalization of standard mathematical concepts in category theory involve diagrams which must commute. The typical definition is something like: "if there is an "impostor" which claims to have the same properties as the concept being defined, then there exist a so and so isomorphism such that a certain diagram commutes which guarantees that the impostor is nothing but a restatement of the same concept up to isomorphism". Next time I will talk about the first key definition we need from category theory, that of a product, and by flipping the arrows that of a coproduct.    

Tuesday, February 28, 2017

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...

Sunday, February 19, 2017

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...

Sunday, February 12, 2017

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. 

Saturday, February 4, 2017

Trump, DeVos and the fleecing of America

Once upon a time there was a Sputnik circling the Earth and the fear it created spurred America to wake up and invest massively into education. Those days are long gone and now religious extremists (like late Jerry Falwell and his son Jerry Falwell Jr) wage war on science in United States. Sadly they are about to destroy the education system and the end result will be an Idiocracy society where we study only creationism and we water plans with sport drinks because they have electrolytes - what plants crave.

So what is president Trump's policy? He has only one policy: to continuously demonstrate he has the largest dick. 

From inauguration crowd size to popular vote size, it is all about how he is "yuuuge". Help him masturbate his ego in public and you get away with anything. One such person is Betsy DeVos. 

Betsy DeVos is a billionaire who made her money with Amway and she bought her way into the Trump administration by donating to the republican party something to the tune of 200 million dollars. On the recent confirmation hearings she said she is supports guns in schools to protect students from grizzly bears!!! Also she did not understand the difference between the value of something and the rate of increase of that value. Her intellectual level is that of a moronic imbecile who repeatedly failed to complete 3rd grade. Honestly, she deserves a prize for managing to beat Sarah Palin in stupidity: a really really hard thing to achieve.

So why does she want to lead the education department? Because the education budget is over 140 billion dollars. Cha-ching! By refusing to hold both public and private schools accountable to the same standards she opens to door to scams like her boss' "Trump University" And it is all paid for by us, the taxpayers. What? Did you believe those 200 million dollars were donated out of the goodness of her hearth?

Now maybe Amway is a legit business and she is not a nutcase. Do you know how Amway works? It is a pyramid scheme going under the name "multi-level marketing": you buy their soap and you sell it to say 5 of your friends and if you sign them up as Amway distributors you get a cut from their sale as well. Now you don't get rich by selling 3 dollars worth of soap in a month, but by signing up many more people and they in turn do the same. The end result is a pyramid of losers. Most of them end up broke and with a garage full of soap inventory. They loose money on books and brainwashing cult-like company seminars. So why is this not illegal? Because the Federal Trade Commission guys are crooks too: since there is an actual product flowing (which keeps the scam going) they merely get their cut of the scam by huge fines (which would have been impossible from traditional Ponzi schemes because in that case when the scheme crashes the money stops flowing).  Now Amway is not alone in MLM. Here is a nice video about the dangers of multi-level marketing:

Back to DeVos. She is promoted by Trump to buy republican support for keeping him in power, but she is a pure republican creation. Both republicans and Trump are willing to fleece America, their only difference is how they sell it to their base: republicans trick them by appealing to freedom, self-reliance, and independence, while Trump promises to make their dicks great again. Trump's ideology is nothing but the fascist utopia: "you are the best, screw everyone else" (like mexicans, muslims, and recently australians).

I don't want to end on a dark note so here is a nice video: 

because the best way to deal with Trump is by making fun of him. There are many more videos like this for various countries in the world. Have fun watching them all.

Sunday, January 29, 2017

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. 

Monday, January 16, 2017

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.

Monday, January 9, 2017

(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: 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:

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: 

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.